analysis, design, and optimization of rf cmos polyphase filters · 2017-11-29 · for design of rf...

Analysis, Design, and Optimization of

RF CMOS Polyphase Filters

Vom Fachbereich Ingenieurwissenschaften der

Universität Duisburg-Essen zur Erlangung des akademischen Grades eines

Doktors der Ingenieurwissenschaften

genehmigte Dissertation

von

Zisan Zhang

aus

Guoyang, Anhui Province, China

Referent: Prof. Dr.-Ing. habil. Peter Jung

Korreferent: Prof. Dr.-Ing. Horst Fiedler

Tag der mündlichen Prüfung: 29. September 2005

To my wife, Hongjin and my son, Berkeley

Abstract

Polyphase filters (PPFs) are an efficient solution for high accuracy quadrature

generation in radio frequency (RF) CMOS design. Although there are some guidelines

for design of RF CMOS PPFs, they give too much freedom. With layout considerations,

optimization of RF CMOS PPFs cannot be reached by using analytical calculations

because of many constraints and tradeoffs in the design. Thus, in design of RF CMOS

PPFs, intensive trials and several design iterations are needed to reach given

specifications. In this situation, a computer-aided analysis and design optimization tool

for RF CMOS PPFs is of great help to the community of RF CMOS PPFs designers,

especially for industrial projects that are under time-to-market pressures. However, such

a dedicated tool has not been available until now, to the author’s best knowledge.

Focusing on computer-aided analysis and design optimization tools for RF CMOS

PPFs, this dissertation has the following scientific contributions:

•

•

•

•

PPFANA, a dedicated computer-aided software tool for analysis of the

influences from the nonideal effects, namely, process tolerance, components

mismatch on the quadrature accuracy of the RF CMOS PPFs, was developed

by the author and presented in this dissertation.

Physical design considerations for RF CMOS PPFs in system-on-a-chip (SoC)

solution were proposed by the author through the analysis on the influences

from parasitic effects in RF SoC.

Silicon success of an RF CMOS PPF designed by the author for a low-IF

Bluetooth receiver using Fraunhofer-IMS 0.6 µm CMOS process was achieved.

PPFOPTIMA, a computer-aided synthesis and optimization tool for RF CMOS

PPFs with dedicated worst-case simulation and Monte Carlo simulation

toolboxes for verification of the performance of RF CMOS PPFs, was

developed by the author and presented in this dissertation.

Zusammenfassung Polyphasenfilter (PPF) sind eine leistungsfähige Lösung für die hochgenaue

Quadraturerzeugung im Hochfrequenz (HF)-CMOS-Entwurf. Obgleich es einige

Richtlinien für den Entwurf von HF-CMOS-PPF gibt, erlauben diese zu viele Freiheiten.

Eine Optimierung der Schaltung und der Layout kann mit Rücksicht auf viele technisch

und wirtschaftlich vorgegebene Randbedingungen und der daraus resultierenden

Kompromisse mit analytischen Methoden alleine nicht erreichen werden. Es werden

aufwändige Versuche und mehrere Entwurfsverbesserungen benötigt, um einen

zufriedenstellenden Entwurf zu erreichen. Deshalb ist ein computerbasiertes Werkzeug

zur Analyse und Optimierung des Entwurfs von HF-CMOS-PPF besonders für den

industriellen Anwendungsbereich wünschenswert. Nach bestem Wissen des Autors gab

es bislang ein solches Werkzeug nicht. Im Hinblick auf computerbasierte Werkzeuge

zur Analyse und Optimierung des Entwurfs von HF-CMOS-PPF bringt die vorliegende

Dissertation folgende wissenschaftliche Beiträge:

• PPFANA, ein spezielles Entwurfswerkzeug für die Simulation der Einflüsse

von den nichtidealen Effekten wie beispielsweise von Prozesstoleranzen, von

Fehlanpassungen der Komponenten und der Quadraturgenauigkeit, wurde vom

Autor entwickelt und vorgestellt.

• Richtlinien für den Entwurf von HF-CMOS-PPF auf „System-on-a-Chip“-

Lösungen werden vom Autor unter Berücksichtigung parasitärer Effekte

erarbeitet und vorgeschlagen.

• Die erfolgreiche Realisierung eines HF-CMOS-PPF für einen Bluetooth-

Empfänger mit niedriger Zwischenfrequenz wurde mit dem Fraunhofer-IMS

0.6 µm CMOS Prozeß nachgewiesen.

• PPFOPTIMA, ein Synthese- und Optimierungswerkzeug für HF-CMOS-PPF

auf Basis von Worst-Case- und Monte-Carlo-Simulationen wurde vom Autor

entwickelt und vorgestellt.

Acknowledgements

Time really flies; four years have passed. Looking back to the time I arrived at

Duisburg, the first colleague I met was Dr. Dieter Greifendorf, who picked my wife and

me up at the railway station of Duisburg in the middle of the night. In the last few years,

he has been the most important person who has given me countless helps in both my

academic works and daily life. My life here in Germany could not have been so easy if

there had not been a warm-hearted friend like him. Hereby I would like to give my

special thanks to him.

This dissertation would not have been completed; if there had not been continuous

supervisions and encouragements from my Ph.D supervisor, Professor Dr.-Ing. habil.

Peter Jung. His brilliance and solid academic foundations have given me invaluable

helps in the development of this dissertation.

Dr. Rainer Kokozinski, an expert in analog circuit design, has given me many

comments and suggestions in my research work. I cannot imagine how many difficulties

I should have had met, if there had not been his timely helps.

My group leaders Dr. Stephan Kolnsberg and Dr. Jörg Stammen have given me

many helps and supports. And I have enjoyed working under their supervision very

much.

I can never forget Dr. Thomas Kaiser, who accepted my application and gave me

the opportunity to working in Fraunhofer IMS. And he has given me countless support

in my academic development during the last four years.

My colleagues in the department of Wireless Chips and Systems (WCS),

Fraunhofer-Institute of Microelectronics Circuits and Systems have given me many

warm-hearted helps during my stay in Germany. Hereby I would like to give them my

sincere thanks.

Zisan Zhang

October, 2005

Duisburg, Germany

I

Table of Contents

List of Figures………………………………………………………..…………………………….V

List of Tables………………………………………………………….…………………………..IX

List of Abbreviations……………………………………………..…………………………….X

List of Symbols…………………………………………………………………………………XII

Chapter 1 Introduction……………………………………………...………………………1

1.1 Wireless communications with single chip transceivers………..……………………1

1.2 Analog conversion and complex mixing……………...……………...………………… 2

1.3 State-of-the-art of wireless transceiver architectures ………….…..…………………5

1.3.1 Introduction …….……………….……...………….…….………….……….…………5

1.3.2 Conventional heterodyne receiver architecture …………………………………6

1.3.3 Direct-conversion receiver architecture………………………………..………….8

1.3.4 Wideband IF double-conversion receiver architecture……………………….11

1.3.5 Low-IF receiver architecture………………………………………………...…….13

1.3.6 Selection of transceiver architecture in this work……………………….…….14

1.4 Analysis of RF quadrature generation techniques…………...…………………...….16

1.4.1 Introduction………………………………………………………………………...….16

1.4.2 Haven’s technique………………………………………………………………...….16

1.4.3 Frequency division……………………………………………………………….…..17

1.4.4 RC-CR network………………………………………………………………………18

1.4.5 RC polyphase filter…………………………………………………………………..19

1.5 Limitations of previous works …………………...……………………………………...23

1.6 Objectives and contributions of this dissertation …………………..………………..25

1.7 Outline of this dissertation …………………………………………...…………………..26

II

Chapter 2 Analysis of Nonidealities in RF CMOS Polyphase Filters

…………………….……………………………..………….29

2.1 Introduction………………………………….……………………………………………….29

2.2 Vector analysis of polyphase filters…………………………..………………………...31

2.3 PPFANA, a software tool for RF CMOS PPFs…………………..…………………..38

2.3.1 Overview………..……….…..…………..……..………..………………..………….38

2.3.2 Explicit transfer function of RF CMOS PPFs…………..……...…………….38

2.3.3 Development of PPFANA………………………………………………………..40

2.4 Analysis of quadrature accuracy in ideal PPFs……………………………….……....42

2.5 Analysis of quadrature accuracy influenced by process tolerances……………....46

2.6 Analysis of quadrature accuracy influenced by component mismatch….……….51

2.7 Analysis of influences from parasitic effects…….….………………………………...54

2.7.1 Simplified RF models of resistors and capacitors…………..……………….54

2.7.2 Substrate noise coupling………………………………………………………….58

2.8 Summary………………………….…………………………………………………………..62

Chapter 3 Design of an RF CMOS Polyphase Filter………..………………64

3.1 Overview…………………………………………….…………………………64

3.2 Schematic design………………...………………………………………………………….65

3.2.1 Derivation of specifications…………………………………..………………….65

3.2.2 Design guidelines………………………….………………………………………66

3.2.3 Tolerance design………………………………..………………………………….68

3.2.4 Analytical calculations……………………...…………….………………………69

3.2.5 Schematic simulation……………………………….……………………………..72

3.3 Physical layout design………………………………………...……………...……………73

3.3.1 Layout techniques……………………………….…………………………………73

3.3.2 On-chip noise isolation techniques………….…………………………...…….76

III

3.3.3 Layout and post-layout simulation……………………………………………..77

3.3.4 Chip photo…………………………………………………………………………...79

3.4 Measurement……………………...…………………………………………………………80

3.4.1 Introductory remarks……………………………………………………...…….…80

3.4.2 Modeling of test pads and bonding wires……………………………………..80

3.4.3 Design of the test PCB……………………………………………………………81

3.4.4 Measurement setup………………………………………………………………...83

3.4.5 Measurement results……………………………………………………………….84

3.5 Analysis of simulation and measurement results…………………...………………..88

3.6 Summary…………………………………...…………………………………………………88

Chapter 4 Optimization of RF CMOS Polyphase Filters…………………90

4.1 Introduction………………..………………………………………………………………...90

4.2 Design considerations and design flow…………………………….…………………..91

4.3 Problem formulation for RF CMOS PPFs optimization……………..…………….96

4.4 Analysis of possible optimization methods…………….…………..………………..100

4.4.1 Limitations of conventional optimization methods……………………..…100

4.4.2 Simulated annealing……………………………………………………………...102

4.4.3 Genetic algorithms……………………………………………………………….103

4.5 MATLAB implementation of genetic algorithms…………………………………105

4.5.1 Overview…………………………………………………...………………………105

4.5.2 Chromosome representation…...…………...………………………....105

4.5.3 Genetic operation……………………...……………………………....107

4.5.4 Initialization, termination and fitness functions.....…………………....111

4.6 Implementation of PPFOPTIMA………………………………..…………………….114

4.6.1 Introduction……………………………...……………………………………...…114

IV

4.6.2 Construction of fitness functions…………………………………………...…116

4.6.3 Graphic user interface………………………………………………………...…116

4.6.4 Verification toolboxes……………………………………………...…...……….118

4.7 Experimental results…………………..…………………………………………….....…123

4.7.1 Overview…………………..………….………………………………………...….123

4.7.2 Experiments on RF CMOS PPFs for a Bluetooth receiver……………124

4.7.3 Automated synthesis and verification of RF CMOS PPFs

for different communication standards………………………………………135

4.8 Summary……………………………………..……………………………………………..146

Chapter 5 Summary and Outlook………………………………………………….149

5.1 Summary…..………………………………………..………………………………………..149

5.2 Outlook………………………………………….…………………………………………...151

References………………………………………………………………………………………...153

Appendix 1 Publications From the Author…………..…………………………..163

Appendix 2 Transfer Functions of Multi-Stage RF CMOS PPFs…….165

Appendix 3 Source Code of PPFANA ………………………………………...……167

Appendix 4 Source Code of PPFOPTIMA……………………………………… 222

V

List of Figures

Figure 1.1 Mixing a real signal with a real sinusoid. ………………….….……………….…3

Figure 1.2 Mixing a real signal with a complex exponential. ……….….……………….…4

Figure 1.3 A practical implementation of the complex mixer. …………...…………….… 5

Figure 1.4 A generic heterodyne receiver……………………………………...…………….…6

Figure 1.5 A generic homodyne receiver…………………………………….…..………….… 8

Figure 1.6 Self-mixing of the LO and/or of an interfere in homodyne receiver frontend

……………………………………………….9

Figure 1.7 A wideband double-conversion receiver……………………….….…………….12

Figure 1.8 A generic low-IF receiver architecture………………………….…….………....14

Figure 1.9 Schematic of Haven’s technique. ………………………..……….………………17

Figure 1.10 A frequency divider as a quadrature generator. ……………………………...18

Figure 1.11 RC-CR networks as a quadrature generator. …………………………………19

Figure 1.12 Polyphase filter as a quadrature generator…………………………………… 20

Figure 2.1 One phase of a generalized N-phase PPFN………..……………………….…….32

Figure 2.2 One-stage RC sequence asymmetric PPFN……………………………...……....33

Figure 2.3 Quadrature signal generation using four-phase network.

(a) Circuit diagram. (b) Decomposition of the two-phase

input sequence into two four-phase symmetric sequences………………..…35

Figure 2.4 Graphic user interface of PPFANA for 4-stage RF CMOS PPF……….……41

Figure 2.5 Gain mismatch in ideal multi-stage RF CMOS PPFs tailored to

the Bluetooth frequency range 2.4GHz-2.48GHz...………………….……....43

Figure 2.6 Phase difference in ideal multi-stage RF PPFs tailored to

the Bluetooth frequency range 2.4GHz-2.48GHz..………...................…...…44

Figure 2.7 Gain mismatch influenced by process tolerance in a 2-stage

VI

RF CMOS PPF (2.4GHz-2.48GHz) with regular design…………..…………48

Figure 2.8 Phase mismatch influenced by process tolerance in a 2-stage

RF CMOS PPF (2.4GHz-2.48GHz) with regular design….…………………48

Figure 2.9 Gain mismatch influenced by process tolerance in a 2-stage

RF CMOS PPF (2.4GHz-2.48GHz) with tolerance design. …………………50

Figure 2.10 Simplified RF model of a polysilicon resistor……….……….…………..……55

Figure 2.11 Simplified RF model of a capacitor……..……..…………………..…..………...56

Figure 2.12 Principle of substrate noise coupling

in system-on-a-chip……………………...……………………..……………..……58

Figure 2.13 Simplified model of substrate noise coupling model

in RF CMOS PPFs…………………………………………..……………………...59

Figure 3.1 Schematic of a 3-stage RF CMOS PPF

for a low-IF Bluetooth receiver…………………..………………………………..71

Figure 3.2 Simulation results of a 3-stage RF CMOS PPFs

obtained by using SpectreRF………………...…………………….…….....72

Figure 3.3 Simulation results of a 3-stage RF CMOS PPFs

obtained by using PPFANA……………………………………………………..…72

Figure 3.4 Layout of the four resistors in one-stage of RF CMOS PPF ……...………74

Figure 3.5 Layout of the four capacitors and wires

in one-stage of an RF CMOS PPF……….…..……………………….…………..75

Figure 3.6 Layout of a 3-stage RF CMOS PPF with test pads…………………....………78

Figure 3.7 Chip photo of an RF CMOS PPF

for a low-IF Bluetooth receiver……………………………………..…...………..79

VII

Figure 3.8 Model of test pads and bonding wires…………………………………….....…...81

Figure 3.9 Schematic of the printed circuit board (PCB)

for test of the RF CMOS PPF…………..……………………..……………....…...82

Figure 3.10 Photo of the fabricated test PCB…………………………….…….………...…...83

Figure 3.11 Diagram of the measurement setup for test of the RF CMOS PPF ….…....84

Figure 3.12 Time domain measurement results of the RF CMOS PPF ……….….….….85

Figure 3.13 Gain mismatch of the RF CMOS PPF measured in frequency domain…..86

Figure 3.14 Phase difference of the RF CMOS PPF

measured in frequency domain…………………………………………………….87

Figure 4.1 A design flow for specification-driven

automated synthesis of RF CMOS PPFs………..…………………………….…95

Figure 4.2 A three-dimensional plot of the amplitude mismatch

in a 2-stage RF CMOS PPF, expressed in the equation (4.20)…..…..……100

Figure 4.3 Mapping variables into chromosomes……………………..…...……………….106

Figure 4.4 Roulette wheel selection……………………………………….………………….109

Figure 4.5 Flowchart of the genetic algorithm implemented in PPFOPTIMA……....113

Figure 4.6 Framework of PPFOPTIMA…………………………………………….…….....115

Figure 4.7 Graphic user interface of PPFOPTIMA……..…………………………..……..117

Figure 4.8 Flowchart of Monte Carlo simulation option in PPFOPTIMA….….…….121

Figure 4.9 Flowchart of the novel worst-case simulation

methodology in PPFOPTIMA…………………………………………………..122

Figure 4.10 Synthesis result of a 2-stage RF CMOS PPF for a Bluetooth receiver….125

Figure 4.11 Worst-case simulation results of a 2-stage RF CMOS PPF for

a Bluetooth receiver…..………..………………………………………………….126

Figure 4.12 Monte Carlo simulation results of a 2-stage RF CMOS PPF

for a Bluetooth receiver……………………….……………………..…………...127

VIII

Figure 4.13 Synthesis results of a 3-stage RF CMOS PPF for a Bluetooth receiver...29

Figure 4.14 Worst-case simulation results of a 3-stage RF CMOS PPF

for a Bluetooth receiver…….....…………………………...………...…………...130


for a Bluetooth receiver…..…..………………….……………….……………....131

Figure 4.16 Synthesis results of a 4-stage RF CMOS PPF for a Bluetooth receiver...132

Figure 4.17 Worst-case simulation results of a 4-stage RF CMOS PPF

for a Bluetooth receiver…………….…….………………………………..……..133


for a Bluetooth receiver…….…………………………..……………………......134

Figure 4.19 Graphic user interfaces of PPFOPTIMA

for automated synthesis of RF CMOS PPFs. ……………………………….136

Figure 4.20 Synthesis results of a 3-stage RF CMOS PPF for a GSM receiver….....137

Figure 4.21 Worst-case simulation results of

a 3-stage RF CMOS PPF for a GSM receiver…………………………...….138

Figure 4.22 Monte Carlo simulation results of

a 3-stage RF CMOS PPF for a GSM receiver..………………………….....139

Figure 4.23 Synthesis results of a 2-stage RF CMOS PPF for a DECT receiver….…140


a 2-stage RF CMOS PPF for a DECT receiver………………………....…141


a 2-stage RF CMOS PPF for a DECT receiver……………………………..42

Figure 4.26 Synthesis results of a 3-stage-RF CMOS PPF for a WCDMA receiver..144


a 3-stage RF CMOS PPF for a WCDMA receiver……………………….145


a 3-stage RF CMOS PPF for a WCDMA receiver………………………..146

IX

List of Tables

Table 1.1 Advantages and disadvantages of different receiver architectures. …….……15

Table 2.1 Quadrature accuracy of ideal RF CMOS PPFs (2.4GHz-2.48GHz),

obtained by using PPFANA and SpectreRF……….…………………………….45

Table 2.2 Quadrature accuracy of RF CMOS PPFs (2.4GHz-2.48GHz)

with process tolerance, obtained by PPFANA and SpectreRF……….………50

Table 2.3 Quadrature accuracy influenced by component mismatch in

multi-stage RF CMOS PPFs (2.4GHz-2.48GHz). ……………..………………53

Table 3.1 Specification of RF CMOS PPF for a low-IF Bluetooth receiver……..…..…66

Table 3.2 Dimension of a 3-stage RF CMOS PPF

for a low-IF Bluetooth receiver………………………………….……………..….71

Table 3.3 Simulation results of a 3-stage RF CMOS PPFs obtained

by using PPFANA and SpectreRF……………………………..………………..…73

Table 3.4 Simulation results obtained by using Cadence

post-layout simulation and PPFANA……….…………………..……………..….78

Table 4.1 Main functions used in the optimization engine of PPFOPTIMA………….114

X

List of Abbreviations

ABBREVIATION FULL EXPRESSION

AM Amplitude mismatch

BALUN Balance unbalance converter

BiCMOS Bipolar CMOS

C Capacitor

CDMA Code division multiple access

CEC Compression and error correction

CM Component mismatch

CMOS Complementary metal oxide semiconductor

CPU Central processing unit

DC Direct current

DCR Direct-down conversion receiver

DECT Digitally enhanced cordless telephony

ESD Electro static discharge

FM Frequency modulation

GA Genetic algorithms

GaAs Gallium arsenide

GM Gain mismatch

GSM Global system for mobile communications

GUI Graphical user interface

IC Integrated circuit

IF Intermediate frequency

IMS Institut für Mikroelektronische Schaltungen

und Systeme

I/O Input / output

I/Q In phase/quadrature phase

IRF Image reject filter

IRM Image reject mixer

IRR Image reject ratio

ISM Industrial, scientific and medical

XI

LNA Low noise amplifier

LO Local oscillator

LPF Low pass filter

MC Monte Carlo

MIM Metal insulator metal

MLR Maximum length of resistors

NF Noise figure

PA Power amplifier

PC Personal computer

PCB Printed circuit board

PD Phase difference

PDF Probability density function

PM Phase mismatching

PPF Polyphase filter

PPFANA Polyphase filter analysis toolbox

PPFN Polyphase filter network

PPFOPTIMA Polyphase filter optimization toolbox

PS Postlayout simulation

PT Process tolerances

QAM Quadrature amplitude modulation

QM Quadrature modulation

QPSK Quadrature phase shift keying

RC Resistance capacitance

RF Radio frequency

SA Simulated annealing

SiGe Silicon germanium

SoC System on a chip

VCO Voltage controlled oscillator

WLAN

WCDMA

Wireless local area network

Wideband code division multiple access

XII

List of Symbols

IA Amplitude of the I path output signal in RF CMOS PPFs

QA Amplitude of the Q path output signal in RF CMOS PPFs

eRA Fitting parameter for the standard deviation of eRσ

shRA Fitting parameter for the standard deviation of shRσ

wA∆ Fitting parameter for the standard deviation of wσ ∆

CC Parasitic capacitance between back plate of the capacitor

and the substrate

iC Capacitor of the thi stage in the RF CMOS PPFs

CM,iC Capacitor of the thi stage with component mismatch

between I path and Q path in the RF CMOS PPFs

PT,iC Capacitor of the thi stage with process tolerance

in the RF CMOS PPFs

iC ,I Capacitor of the thi stage in the I path of the RF CMOS PPFs

paraC Parasitic capacitance

iC ,Q Capacitor of the thi stage in the Q path of

the RF CMOS PPFs

RC Parasitic capacitance between polysilicon resistor

and the substrate

sumC Sum of parasitic capacitance between RF CMOS PPFs

and the substrate

uaCC Parasitic capacitance between unit area capacitor

and the substrate

iCuaC Parasitic capacitance between the capacitors in the thi stage

and the substrate

iCuaR Parasitic capacitance between the resistors in the thi stage

and the substrate

XIII

unitC Unit capacitance of the capacitor

phased Phase difference between I path and Q path in PPFs

f Frequency of a signal

af Constant describing influences of area on mismatch

pf Constant describing influences of periphery on mismatch

df Constant describing influences of distance on mismatch

iF Fitness of the individual i

H(j )ω Transfer function of polyphase filters

IH (j )ω Imaginary part of the transfer function of polyphase filters

RH (j )ω Real part of the transfer function of polyphase filters

inI Input current in RF CMOS PPFs

in,Ik The thk input current in RF CMOS PPFs

out,Ik The thk output current in RF CMOS PPFs

outI Output current in RF CMOS PPFs

Resmax,L Maximum length of the resistor

RL Length of the resistor

CM Mismatch between the neighboring capacitors

CMCM Component mismatch of the capacitors

CMRM Component mismatch of the resistors

RM Mismatch between the neighboring resistors

CMgain,m Gain mismatch in RF CMOS PPFs

with component mismatch

idealgain,m Gain mismatch in the ideal RF CMOS PPFs

PTgain,m Gain mismatch in RF CMOS PPFs

with process tolerances

phasem Phase mismatch in RF CMOS PPFs

CMphase,m Phase mismatch in RF CMOS PPFs

with component mismatch

XIV

idealphase,m Phase mismatch in the ideal RF CMOS PPFs

PTphase,m Phase mismatch in RF CMOS PPFs

with process tolerances

unitN Number of unit capacitor

jP Probability of selection of the j population

RP Process tolerance of the resistors

Q Quality factor

eR End resistance of a resistor

iR Resistor of the thi stage in the RF CMOS PPFs

CM,iR Resistor of the thi stage with component mismatch

between I path and Q path in the RF CMOS PPFs

iR ,I Resistors of the thi stage in the I path of the RF CMOS PPFs

PT,iR Resistor of the thi stage with process tolerance

in the RF CMOS PPFs

IRR Image reject ratio

iR ,Q Resistors of the thi stage in the Q path of the RF CMOS PPFs

shR Sheet resistance of resistor

subR Resistivity of the substrate

UIRR Ultimately obtainable image reject ratio

iSC Area of the capacitor in the thi stage in RF CMOS PPFs

iSR Area of the resistor in the thi stage in RF CMOS PPFs

RS Area of a resistor

inV Input voltage in RF CMOS PPFs

, inkV The thk input voltage in RF CMOS PPFs

, outkV The thk output voltage in RF CMOS PPFs

I, inV Real part of the input signal in RF CMOS PPFs

I, midV Intermediate I path signal

XV

Q, midV Intermediate Q path signal

IoutV I path output signal in RF CMOS PPFs

outV Output voltage in RF CMOS PPFs

Q, inV Imaginary part of the input signal in RF CMOS PPFs

QoutV Q path output signal in RF CMOS PPFs

RW Width of a resistor

iw Weight of the thi item

Y Admittances

ω Angular frequency

IFω Angular frequency of the intermediate frequency signal

imageω Angular frequency of the image

LOω Angular frequency of the signal from the local oscillator

RFω Angular frequency of the radio frequency signal

signalω Angular frequency of the signal

aσ Standard deviation of mismatch caused by area variation

componentσ Standard deviation of component mismatch

dσ Standard deviation of mismatch caused by distance

pσ Standard deviation of mismatch caused by periphery

Reσ Standard deviation of the end resistance

shRσ Standard deviation of the sheet resistance

∆wσ Standard deviation of the width of a resistor

φ Phase of a signal

θ Relative phase difference in RF CMOS PPFs

A∆ Amplitude mismatch in volts

φ∆ Phase mismatch in degrees

Chapter 1 Introduction

1

Chapter 1

Introduction

1.1 Wireless Communications With Single Chip Transceivers

The wireless communications market has grown substantially during the last

decades; there has always been pressure to reduce both the size and the cost of wireless

products. However, nowadays, in most designs, the analog part of receivers use multiple

packages and/or multiple integrated circuits (ICs) which are implemented in gallium

arsenide (GaAs) or silicon germanium (SiGe) bipolar processes, while baseband

components are fabricated in low-cost complementary metal oxide semiconductor

(CMOS) processes [1]. Since analog and digital components are designed for different

processes, it is cumbersome to integrate RF, analog and baseband in a single chip and

cost cannot be easily lowered.

Since a couple of decades ago, there have been tremendous endeavors in

developing CMOS wireless single chip transceivers, i.e., integrated transmitter and

receiver. Some pioneers, just name a few, Abidi [6, 8, 9] at University of California at

Los Angeles, Steyaert [23, 32] at Katholiek Universiteit Leuven, and Lee [2, 3] at

Stanford University, have contributed by both theoretical studies and practical system

implementations. Since the middle of the 1990s, more efforts have been put in

developing CMOS single chip transceivers with applications to various standards, for

example, global system for mobile communications (GSM), digitally enhanced cordless

telephony (DECT), Bluetooth, wireless local area network (WLAN), etc.


2

Even though there are a lot of GaAs, SiGe, bipolar circuits available, it is still

quite a challenge to implement similar circuits in CMOS processes. Oriented for digital

circuit design, a standard CMOS process has poor analog, and especially radio

frequency (RF) performance because of its process tolerances, parasitic effects, and low

quality (Q) factor passive components [2]. Thus, insights into transceiver architectures

which are integrateble and capable of providing good CMOS transceiver performances

become essential.

1.2 Analog Conversion and Complex Mixing

In RF receivers, frequency downconversion is an essential operation. Typically,

frequency downconversion of a passband signal is performed by multiplying, i.e.,

mixing, the signal with a sinusoid such as )πcos( LOtf2 . In Figure 1.1 [24], case 1, LOf is

not equal to the center frequency of the desired signal. This setup is used in a

heterodyne receiver. Figure 1.1 case 2 shows the situation for LOf equals to the center

frequency of the desired signal. This principle is used in the homodyne receiver. In the

next section, we will discuss the architecture of heterodyne and homodyne receivers in

more details. The spectra of the inputs of the mixer and the corresponding spectrum of

the output of the mixer are shown in Figure 1.1 to help understanding the frequency

response.

For a real signal, the negative and positive Fourier components are complex

conjugates of each other. A multiplication in the time domain is equivalent to a

convolution in the frequency domain. Since the Fourier transform of a sinusoid contains

two equal-amplitude impulses symmetrically placed around zero on the frequency axis,

the spectrum of the mixer output signal is the superposition of the positive and negative


3

shifted versions of the spectrum of the input signal. From Figure 1.1, we can see that,

two frequency bands symmetric around the multiplying frequency are downconverted to

the same output band. The undesired input signal band, which will be superimposed on

the desired signal band after mixing, is called the ‘image band’. This is an important

issue in the design of a heterodyne receiver. It is necessary to suppress any signal in the

image band prior to the mixing operation. This is the task of the image reject (IR) filter

which usually precedes the mixer. In the special case where the multiplying frequency

is equal to the center frequency of the incoming signal, the image band is the same as

the signal band, and the image cannot be eliminated by filtering.

Figure 1.1 Mixing a real signal with a real sinusoid.

)(tx )(ty

)πcos( LOtf2

LO- f LOff

f

Amplitude

LO- f LOff

f

Amplitude

Case1 Case2

Spectra of the signals

in the image band

Spectra of the

desired signals


4

The image problem arises due to the fact that the frequency spectrum of a real

sinusoid contains impulses at both positive and negative frequencies. One way to avoid

this problem is to mix the signal with a complex exponential which can be expressed as:

LOj2LO LOcos(2 ) - jsin(2 )f te f t f tπ π π− = . (1.1)

Mixing a real signal with this negative frequency complex exponential gives a complex

signal whose spectrum is a shifted version of the real signal spectrum. Theoretically,

this eliminates the image problem associated with frequency shifting when mixing with

a real sinusoid. The realization of a multiplier for mixing a real signal with a complex

exponential requires two real multipliers as shown in Figure 1. 2.

Figure 1.2 Mixing a real signal with a complex exponential [24].

Now, consider the mixing process shown in the above figure in a general way.

Mixing two complex signals ( )x t and ( )z t generates:

r r i i r i i r( ) ( ). ( ) ( ( ). ( ) - ( ). ( )) ( ( ). ( ) ( ). ( ))y t x t z t x t z t x t z t j x t z t x t z t= = + + , (1.2)

)().(-)().(=)( iirrr tztxtztxty , (1.3)

)().(+)().(=)( riiri tztxtztxty . (1.4)

)(r tx

t)πf2( LOsin

)πcos( LOtf2

)(r ty

)(i ty

90°

0° )πcos( LOtf2


5

A practical realization of this complex mixer using four real mixers and two

adders is shown in Figure 1.3. This realization is used in the wideband double-

conversion receiver discussed in the next section.

Figure 1.3 A practical implementation of the complex mixer [24].

1.3 State-of-the-Art of Wireless Transceiver Architectures

1.3.1 Introduction

In this section, we briefly review state-of-the-art receiver topologies and discuss

their advantages and disadvantages. For a given receiver architecture, there exists a

corresponding transmitter architecture with essentially the same fundamental building

blocks. Therefore, we focus on receiver structures.

)πcos( LOtf2

)πcos( LOtf2

)(r tx

)(i tx

)πsin( LOtf2

)(r ty

)(i ty

Adder

Adder


6

1.3.2 Conventional Heterodyne Receiver Architecture

Most of today’s commercially available RF transceivers utilize some variant of

the conventional heterodyne architecture. In a heterodyne receiver, as shown in Figure

1.4, the RF front-end preselection filter serves to remove out-of band signal energy as

well as partially reject image band signals. After this prefiltering, the received signal is

amplified by a low-noise amplifier (LNA). The IR filter following the LNA further

attenuates the undesired signals at the image band frequencies. The desired signal at the

output of the IR filter is then downconverted from the carrier frequency to a fixed

intermediate frequency (IF) by mixing with the output of a local oscillator (LO).

Figure 1.4 A generic heterodyne receiver.

In conventional heterodyne receivers, high performance, low phase noise voltage-

controlled oscillators (VCOs) employed as LOs are realized with discrete components

such as high quality (Q) factor inductors, varactors and diodes [3]. At the output of the

2LO 90°

0°

Mixer

Mixer

1LO

MixerBandpass

IR FilterLNA

RF

Preselection

Bandpass (RF)

Filter

Bandpass

Filter

Low-pass

Filter

Low-pass

Filter

ADC

ADC

PSD

PGA I

Q


7

mixer, an IF filter, typically followed by an IF programmable-gain amplifier (PGA),

selects the desired channel and reduces the distortion and dynamic range requirements

of the subsequent receiver blocks [3]. The signal can be shifted to baseband and

demodulated or alternatively further downconverted to lower IFs, and then shifted to

baseband and demodulated. Since, at the carrier frequency the desired band and image

band are separated by twice the IF, it is desirable to choose a high IF to reduce the

requirements on the IR filter. In fact, if the IF is chosen high enough that the RF filter

can sufficiently attenuate the image band; it might be possible to directly connect the

output of the LNA to the mixer without including an IR filter [24]. On the other hand,

since channel selection in a heterodyne system is done at the IF, a low IF allows the

employment of higher quality channel-select filters. Therefore, the choice of the IF

depends on the trade-off between image rejection and channel selection.

Conventionally, all the filters used in the heterodyne system are high-Q discrete

component filters, such as surface acoustic wave (SAW) or ceramic filters. Compared to

other receiver architectures discussed below, the heterodyne receiver has superior

performance with respect to selectivity, a measure of a receiver’s ability to separate the

desired band around the carrier frequency from signals received at other frequencies,

and sensitivity, the minimal signal level at the receiver input for which there is

sufficient signal-to-noise ratio (SNR) at the receiver output. This is achieved with the

use of high-Q discrete components [3], which mainstream CMOS processes do not have.

Thus, the conventional heterodyne receiver is not well suited to be integrated onto a

CMOS single chip.


8

1.3.3 Direct-Conversion Receiver Architecture

Direct-conversion receiver (DCR), also known as homodyne receiver or zero-IF

receiver [84], is a natural approach to convert an RF signal directly down to baseband.

This architecture, shown in Figure 1.5, employs low-pass filtering in the baseband to

suppress nearby interferers and select the desired channel.

The homodyne architecture has several fundamental advantages over its

heterodyne counterpart [8]. The IF stages and the IR filters are removed, furthermore,

the absence of the bulky off-chip IR filters removes the requirement on the LNA to

drive a low impedance load. The functions of channel selection and subsequent

amplification at a nonzero IF are replaced by low-pass filtering and baseband

amplification, amenable to monolithic integration [8]. Despite this suitability for higher

levels of integration, a homodyne receiver has the following issues that either do not

exist or are not as serious in a heterodyne receiver.

Figure 1.5 A generic homodyne receiver [8].

ADC

D S P

LO

°90

°0

ADC

Bandpass

RF Filter

LNA

Mixer PGA Low-pass

Filter

Mixer PGALow-pass

Filter

I

Q

RF


9

DC-Offsets — Perhaps the most serious problem is that of direct current (DC)

offsets in the baseband section of the homodyne receiver [4]. These extraneous offset

voltages can corrupt the desired signal and/or saturate the following stages. They arise

due to the self-mixing phenomenon of the local oscillator or the in-band interferer, aside

from the usual element mismatch in the signal path circuitry.

Figure 1.6 Self-mixing of the LO and/or of an interfere in homodyne receiver frontend.

To better understand the origin of these offsets, consider the received signal path

shown in Figure 1.6. First, the isolation between the LO port and the inputs of the mixer

and the LNA is not perfect, and a finite amount of feedthrough exists from the LO port

to the other mixer input and to the input of the LNA. Known as LO leakage, this effect

arises from capacitive and substrate coupling and, if the LO signal is provided

externally, through bond-wire coupling [21]. The leakage signal appearing at the inputs

of the LNA and the mixer is now mixed with the original LO signal, thus producing a

DC component at the output of the mixer [21]. This LO self-mixing can be quite severe,

and a time-varying DC-offset occurs when the LO signal leaks out through the antenna,

and is radiated and reflected from nearby objects back to the receiver [21]. A similar

MixerLNA

mixing-Self LO

Filter

RF

IF


10

effect occurs if a large interferer in the passband of the RF filter leaks from the LNA

output to the mixer LO port and gets multiplied by itself [24].

From the above discussion, we can see that direct-conversion receivers require

appropriate methods to remove undesired DC offsets. A simple approach is to use

alternate current (AC) coupling in the downconverted signal path [21]. However, the

spectra of all the spectrally efficient modulation schemes currently used exhibit

significant energy at and around DC, this kind of signals are corrupted by AC coupling

filters [4]. A better method is the use of baseband analog and/or digital signal

processing (DSP) techniques for offset estimation and cancellation [3][24]. However,

these techniques add complexity and do not solve the problems associated with 1/f noise

at low frequencies in CMOS implementations, another significant issue.

I/Q Mismatch — For most currently used modulation schemes, a homodyne

receiver must incorporate quadrature downconversion to avoid loss of information. This

requires shifting either the RF signal or the LO output by 90°, generating an in-phase (I)

path and a quadrature-phase (Q) path. Since phase-shifting the RF signal generally

entails severe noise, power, gain trade-offs [2] and is especially difficult for the

wideband signals used in high data-rate systems, it is often desirable to shift the LO

output. In either case, the errors in the nominally 90° phase shift and mismatches

between the amplitudes of the I and Q signal paths corrupt the downconverted signal

constellation, thereby increasing the bit error ratio (BER).

To gain more insight into the effect of I/Q mismatch, and show the versatility and

convenience of using complex formulation, consider the practical case where the

quadrature LO generates the complex signal:

LO LO LO( ) cos( ) - j(1 )sin( )x t t A tω ω φ= + ∆ + ∆ (1.5)


11

Here A∆ and φ∆ represent LO gain and phase errors. One can rewrite the quadrature

LO output as:

LO LOj - jj - jLO

1 1( ) [1-(1 ) ] [1+(1 ) ]

2 2t tx t A e e A e eω ωφ φ∆ ∆= + ∆ + + ∆ (1.6)

Ideally, the complex LO output should contain only the negative frequency. However,

from the above expression, it is apparent that, due to gain and phase errors, there is a

positive frequency component with a magnitude of j|1 (1 ) | / 2A e φ∆− + ∆ . This component

causes interfering images and, if not compensated, can deteriorate the sensitivity of the

receiver.

Aside from the problems mentioned above, direct-conversion receivers are

sensitive to even order distortions. Also, since the downconverted spectrum is located

around zero frequency, the flicker (1/f) noise of the devices, a severe problem in CMOS

implementations [2], has a profound effect on the signal-to-noise ratio (SNR).

Furthermore, integrating the high-frequency low-phase-noise channel select frequency

synthesizer is difficult to achieve with low-Q VCOs available in integrated circuits [3].

1.3.4 Wideband IF Double-Conversion Receiver Architecture

Shown in Figure 1.7 is the wideband double-conversion receiver [23]. In this

receiver, after filtering and amplification, all potential RF channels are complex mixed

and downconverted to IF. A second complex mixing is done from IF to baseband, using

a tunable channel-select frequency-synthesizer. In this complex mixer, by properly

adding the outputs of the real multipliers in pairs, the image frequencies are canceled

while the desired channels add constructively. If the IF is chosen high enough,

additional image rejection may be obtained from the RF frontend filter [3].


12

Comparing the homodyne receiver and the wideband IF double-conversion

receiver, we can see that channel selection is performed in the baseband in both

architectures, allowing the possibility of a programmable integrated channel-select filter

for multi-standard receiver applications.

Figure 1.7 A wideband IF double-conversion receiver [23].

However, the wideband IF double-conversion receiver architecture has some

advantages over its homodyne counterpart [21]. Due to the fact that channel tuning is

performed using the IF LO, the RF LO can be implemented as a fixed-frequency

crystal-controlled oscillator [25]. Several techniques may be utilized to realize a low-

phase-noise fixed LO with low-Q on-chip components [3]. Also, since tuning is

performed with the IF LO operating at a lower frequency, the phase-noise performance

1LO 90°

0°

RF

)πsin( LO2tf2

)πcos( LO2tf2

)πcos( LO2tf2

ADC

ADC

D

S

P

Adder

Adder

Bandpass

Filter

LNA

Low-pass

Filter

Low-pass

Filter

Low-pass

Filter

Low-pass

Filter

PGA

PGA

Mixer

Mixer

I

Q

1st Mixing

Stage

2nd Mixing

Stage


13

of this oscillator can be significantly better than that of the tunable RF oscillator

employed in a homodyne receiver. Furthermore, since in the wideband IF system there

is no LO operating at the same frequency as the incoming RF carrier, the potential

problems associated with LO leakage and time-varying DC offsets are minimized [24].

Although in the wideband IF double-conversion receiver the second LO is at the same

frequency as the desired IF channel, the DC offset which results at the baseband from

self-mixing is relatively constant and may be cancelled using adaptive signal processing

methods [3].

The limitations of wideband IF double-conversion receivers are the following:

Since the first LO is fixed in frequency, all channels must pass through the IF stage. The

desired frequency channel is selected using the second LO. This has two problematic

implications: First, as a result of moving the channel selection to a lower frequency, the

IF synthesizer requires a VCO with the capability of tuning across a broader frequency

range than a small percentage of the nominal operating frequency. Second, by removing

the channel-select filter from the first mixing stage, strong adjacent channel interferers

are now a concern for the second mixer stage as well as the baseband blocks. This

implies a higher dynamic range requirement for these latter receiver stages [25]. Also,

I/Q phase and gain mismatch degrade the performance of the receiver [3].

1.3.5 Low-IF Receiver Architecture

The idea behind low-IF topologies is to combine the advantages of both

heterodyne and homodyne receivers [23]. Figure 1.8 shows a low-IF receiver

architecture [84], in which the IF is usually chosen at a low frequency, typically at a few

hundred KHz. Low-IF architectures are suitable for the integration since they provide


14

on-chip image rejection, and the channel selection can be done by using a low-Q filter

following the mixer. Unlike the zero-IF architecture, a low-IF receiver is not sensitive to

the parasitic DC offset, LO leakage, and even-order distortion [3]. Low-IF architectures

also provide flexibility for processing the signal in multiple ways. One of the drawbacks

of this architecture is its limited image rejection (~40 dB) due to the on-chip matching

between I and Q generators [84]. Without proper pre-filtering, the dynamic range and

resolution requirement on an analog-to-digital (A/D) converter can substantially

increase. Also, an increase in current consumption results when this topology is used for

wide-channel bandwidth application.

Figure 1.8 A generic low-IF receiver architecture [84].

1.3.6 Selection of Transceiver Architecture in This Work

In order to select a well suited transceiver architecture for this work on multi-

standard CMOS single-chip transceivers, based on the above analysis, the

aforementioned receiver architectures, namely, heterodyne, homodyne, wideband IF

90°

0°

Mixer

Mixer

Band-pass

Filter

RF

LNA

LO

90°

ADC

PSD

Phase ShifterPGA

PGA

Low-pass

Filter

Adder

I

Q


15

double-conversion, and low-IF, were compared by using following Table 1.1. In this

table, eight important aspects in the receiver design are ranked as ‘+’ and ‘++’, which

stands for ‘advantage’ and ‘outstanding advantage’, ‘－’ and ‘－－’, which stands for

‘disadvantage’ and ‘severe disadvantage’, or ‘NO’, which stands for ‘no need to

consider’. From Table 1.1, we can see that the low-IF receiver architecture has the most

advantages. Thus, the low-IF receiver architecture is chosen in this work.

Table 1.1 Advantages and disadvantages of different receiver architectures.

Architecture

Items

Heterodyne Homodyne Wideband Dual

Conversion Low-IF

Sensitivity ++ － ++ +

Image reject ++ NO + +

DC offset NO －－ NO NO

Flicker noise NO －－－－

IQ mismatch －－－－－－

Multi-standard

compatibility －－ ++ ++ ++

Integratebility －－ ++ ++ ++

Simplicity －－ ++ －－ +


16

1.4 Analysis of RF Quadrature Generation Techniques

1.4.1 Introduction

As discussed in section 1.2, quadrature conversion is dominant in modern

wireless transceiver design, because frequency modulation and phase modulation are

deployed in most digital communication systems. In the low-IF receiver architecture we

chose, quadrature signal generation is an important function block, because quadrature

accuracy, that is the accuracy of °90 phase-shifted signal, directly determines the image

reject ratio (IRR), an important specification determining the sensitivity of the receiver

[23].

Generally, there are four quadrature generation techniques; namely, Havens’

technique, Frequency division, RC-CR networks and RC polyphase filters (PPFs).

Although RC PPFs are cascaded RC-CR networks, there are significant differences

between them in CMOS implementation. Thus, we take RC PPFs as an independent

quadrature generation technique based on the ‘stagger-tuning’ techniques [55] used in

this circuitry.

1.4.2 Haven’s Technique

Figure 1.9 shows a schematic implementation of Haven’s technique [84]. This

schematic consists of a phase splitter (phase shifter), two first limiters, two adders, and

two further limiters. The phase splitter first splits the input signal inV , creating 2 paths,

namely, I, midV and Q, midV , with approximately °90 phase difference, then adds and

subtracts them, producing the output signals IoutV and QoutV .


17

In Haven’s technique, if I, midV and Q, midV have equal amplitudes, the angle

between IoutV and QoutV is equal to °90 . The limiting stages will equalize the amplitudes

of I, midV and Q, midV by phase shift circuit. Moreover, the adders’ outputs will have

different amplitude if the phase difference between I, midV and Q, midV is not exactly 90° .

Another amplitude mismatch in Haven’s technique is the capacitive coupling between

the inputs of adders [23]. This may have significant effect in the phase imbalance at the

input, if output impedance of the limiter is not small.

Figure 1.9 Schematic of Haven’s technique [84].

1.4.3 Frequency Division

As shown in Figure 1.10, the idea of frequency division is to use a master-slave

flip-flop to divide a signal by a factor of 2, if inV has an exact 50% duty cycle, then

there is an accurate °90 phase difference between IoutV and QoutV [23].

Generating quadrature signals using frequency division at radio frequency has the

following main disadvantages:

inV

I, midV

Q, midV

IoutV

QoutV

Limiter Limiter

Limiter Limiter

Adder

Adder

Splitter Phase

o90≈

+

+

+

-


18

• It consumes a substantial power to generate a reference signal whose frequency

is twice the operating frequency; for some CMOS processes with low cut-off

frequency, this is simply impossible. For example, in the Fraunhofer-Institut

für Mikroelektronische Schaltungen und Systeme (Fraunhofer-IMS) 0.6 µm

CMOS process, the cut-off frequency is only 4.5 GHz; it is impossible to use

frequency division for generating quadrature LO signal in the low-IF Bluetooth

receiver, because a 4.9 GHz reference signal is required.

• There is a considerable phase mismatch when the input duty cycle deviates

from 50%, further more, mismatch in the signal path through the latches also

contribute to the phase mismatch.

Figure 1.10 A frequency divider as a quadrature generator [84].

1.4.4 RC-CR Networks

An RC-CR (R-resistor, C-capacitor) network is a quadrature generation technique

that shifts phases of the two signal paths by 4

π+ and 4

π− respectively [23]. As shown

in Figure 1.11, this circuit consists of a low-pass filter and a high-pass RC filter in two

signal paths. In the RC-CR network, the phase differences between IoutV and QoutV is

Latch

Latch

inV QoutVIoutV


19

°90 for all frequencies, however, the amplitudes of the outputs are equal only at the

corner frequency of the RC filter, that is, ω equals to 1/( )RC . As the absolute values of

the RC network vary with temperature and process, the corner frequency also varies; it

is thus impossible to get a robust quadrature signal in CMOS technology by using RC-

CR networks. Moreover, this technique is only suitable for narrow band operation,

because a good gain matching between two quadrature outputs can only be achieved

with a very narrow band centered on the corner frequency of this RC-CR network.

Figure 1.11 RC-CR networks as a quadrature generator [84].

1.4.5 RC Polyphase Filter (PPF)

From the above discussions, we can see that traditional quadrature generation

techniques widely used in processes with high precision resistors and capacitors are not

well suited to CMOS implementation. Therefore, alternative circuitries, which are

IoutV

QoutV

RC

R

C

inV

Low-pass

Filter

High-pass

Filter


20

robust against component mismatch and process variation, are needed. At this point, we

would like to point out that process variation is often also termed as process tolerances.

Originally, polyphase filters (PPFs) were developed by Gingell [45, 46] in 1971

for generating quadrature signals with even amplitude in audio. In 1994, integrated

PPFs were rediscovered as an efficient RF quadrature generation technique in CMOS

technology by Steyaert [55]. This has introduced PPFs into a new regime, which we

name as RF CMOS PPFs.

Figure 1.12 Polyphase filter as a quadrature generator [55].

+IoutV

IoutV −

+QoutV

QoutV −

mR

mC+inV

inV −

1R

1C

nR

nC

mR

mC

1R

1C

nR

nC

mR

mC

1R

1C

nR

nC

mR

mC

1R

1C

nR

nC

stage 1 stage m stage n


21

The topology of PPF is shown in Figure 1.12, and it can be viewed as ‘n’ stages

of RC networks in cascaded connection [55]. Each stage is composed of four identical

resistors and four identical capacitors. The resistors and capacitors vary from stage to

stage. In Figure 1.12, +inV and inV − are the differential inputs, +IoutV and IoutV − are

differential outputs of the I path, +QoutV and QoutV − are the differential outputs of the Q

path.

The principle of PPFs is a so-called ‘stagger tuning’ technique [55], that is, by

using two or more cascaded stages of RC-CR networks, the amplitude of the quadrature

signal generated can be smoothed, and the bandwidth can be extended. As a quadrature

generator, PPFs have the following advantages over other traditional quadrature

generation techniques [56]:

• They are simple in implementation, because only resistors and capacitors are

used.

• Since a stagger-tuning technique is used in PPFs, they can overcome the

problems brought by CMOS process variation without any other complicated

tuning circuitry.

• They are well suited for wideband quadrature signal generation.

The above-mentioned features make PPFs well suited for the CMOS

implementation, and nowadays, PPFs are widely used as quadrature generators at radio

frequency (RF). In this work, we name them as RF CMOS PPFs for this specific

application.

Though PPFs are well suited to function as RF quadrature signal generator in a

CMOS process, they have the following disadvantages:


22

• Being passive filters, RF CMOS PPFs have insertion loss of 3 dB/stage.

Buffers, i.e. amplifiers, are usually needed to compensate the signal attenuation.

For high accuracy quadrature signal generation, multiple cascaded stages are

needed in the PPFs, and the buffers are power hungry.

• If PPFs are used in the signal path, thermal noise of the resistors deteriorates

the noise figure (NF) of the receiver.

The problems mentioned above can be overcome or reduced by the following

techniques:

• In order to reduce power consumption of quadrature generation in wireless

transceiver design, direct-connection between voltage-controlled oscillator

(VCO) and RF CMOS PPFs can be used. Borremans [60] successfully

demonstrated that it is viable to connect RF CMOS PPF directly to VCO, thus

power-hungry buffers can be skipped.

• If RF CMOS PPFs are used in the signal path, proper values of resistors can be

chosen to avoid deteriorating noise figure (NF) of the receiver. The values of

resistors in the first stage of a PPF should be determined by the output

impedance of the preceding component, for example, the LNA. And the values

of resistors in the last stage of the PPFs should be as close as possible to 50

Ohms. By using this configuration, influences of the resistors on NF of the

receivers can be controlled in an acceptable range [32].

From the above discussions, we find that RF CMOS PPFs have advantages over

other conventional quadrature generation techniques in simplicity, bandwidth and

robustness against CMOS process variation. Moreover, frequency division technique is


23

not applicable in our project, thus, PPF is chosen as the quadrature generator in this

dissertation.

1.5 Limitations of Previous Works

Originally developed for generating quadrature signals with even amplitude in

audio by Gingell [45, 46], PPFs have received a lot of attention in both theoretical

analysis and practical designs [45-54]. In his doctoral dissertation, Gingell [48] did

some simple analytical work for guiding the practical designs. As in discrete

components implementation, matching and absolute values of resistors and capacitors

are under control by properly selecting components, process variation is not an issue for

this application. And in low frequency applications of PPFs, parasitic effects are usually

not under consideration.

In 1994, integrated PPFs were rediscovered as an efficient RF quadrature

generation technique in CMOS technology by Steyaert [55]. Design of integrated RF

CMOS PPFs faces many challenges, which were not considered by the pioneers. In

1994 and 1995, Schmidt [53] and Tetsuo [54] analyzed the PPFs in a systematic way by

using vector analysis. Though vector analysis of PPFs provides insight into the

mathematic representation of PPFs, it is too abstract for the designer to get quantitative

results of the influences from process tolerance and components mismatch on the

quadrature accuracy of the RF CMOS PPFs.

The most comprehensive and important contribution so far in RF CMOS PPFs

design is from Behbahani [62]; he analyzed practical design aspects and presented some


24

design guidelines. For understanding the limitations of the design guidelines, let us

analyze the main factors that should be considered in the design of an RF CMOS PPF.

1. CMOS process variation can be as much as 20%. The bandwidth of RF CMOS

PPFs should be extended so that the band of interest can be covered even the

worst case of the process variation occurs.

2. Quadrature accuracy (QA) is determined by component matching between the I

path and the Q path. In order to get good component matching, the layout of

resistors and capacitors should have large areas.

3. Parasitic effects in the RF CMOS PPFs cause signal losses and noise coupling

problems. Specifically, parasitic capacitors cause noise coupling from both

substrate and neighboring components; parasitic resistors make signal loss

more severe than the theoretical value of 3 dB insertion losses in each stage of

passive RC filters. In order to reduce unwanted parasitic effects, the layout of

resistors and capacitors should be kept as small as possible.

4. Compared to active components, in CMOS RF transceivers, passive

components usually consume more chip area, which is a main factor of cost. It

is thus required to design chips as small as possible.

From the above discussions, we can see that the main specification of the RF

CMOS PPF, quadrature accuracy, is very layout dependent; and the requirements from

component matching is contradictory to that of minimization of noise coupling, signal

loss and chip area. Thus, a tradeoff, that is, an optimization, must be made in a specific

design. By simply following Behbahani’s guidelines [62], it takes intensive trials and

many design iterations to reach a good design. Furthermore, with layout considerations,


25

optimization of RF CMOS PPFs cannot be achieved by using analytical calculations,

because of many constraints and tradeoffs in the design.

In this situation, a computer-aided analysis and design optimization tool for RF

CMOS PPFs is of great help to the community of RF CMOS PPFs designers, especially

for industrial projects, which are under time-to-market pressures. However, until now,

there has not been such a dedicated tool available in the open literature, to the author’s

best knowledge.

1.6 Objectives and Contributions of This Dissertation

In order to solve the problems discussed in the previous section, the objective of

this dissertation is to contribute a computer-aided analysis and design optimization tool

for RF CMOS PPFs, which is in need for the community of RF CMOS PPFs designers.

Based on the whole design flow and the silicon success of an RF CMOS PPF for a

Bluetooth low-IF transceiver in the Fraunhofer-IMS 0.6 µm CMOS process, this

dissertation has the following important contributions:

1. PPFANA, a dedicated computer-aided software tool developed by the author

for analyzing the influences from the nonideal effects, namely, process

tolerances and component mismatch on the quadrature accuracy of the RF

CMOS PPFs, is presented.

2. Physical design considerations for RF CMOS PPFs in system-on-a-chip (SoC)

solution are proposed by the author through the analysis of the influences

from parasitic effects in SoC.


26

3. Silicon success of an RF CMOS PPF for a low-IF Bluetooth receiver using

the Fraunhofer-IMS 0.6 µm CMOS process is reported.

4. The methodology of optimization of RF CMOS PPFs with layout

considerations by using genetic algorithms (GAs) is illustrated.

5. PPFOPTIMA, a computer-aided synthesis and optimization tool for RF

CMOS PPFs developed by the author, is presented.

6. Dedicated worst-case simulation and Monte Carlo (MC) simulation toolboxes

for verification of the performance of RF CMOS PPFs are illustrated.

1.7 Outline of This Dissertation

Chapter 1 begins with a discussion on CMOS single chip wireless transceivers,

and then advantages and disadvantages of the state-of-the-art transceiver architectures

are analyzed. Based on the analysis, the low-IF receiver architecture is chosen for the

CMOS single-chip Bluetooth transceiver design. RF quadrature signal generation is an

important functional block in modern wireless transceiver design. Through the analysis

of quadrature generation techniques, PPF is chosen as the RF quadrature generator in

our low-IF Bluetooth receiver. Existing problems in analysis, design and optimization

of RF CMOS PPFs are analyzed, limitations of previous work in this topic are pointed

out, and then objectives and contributions of this dissertation are clarified.

In chapter 2, nonidealities in RF CMOS PPF design, namely, process tolerances

and component mismatch, are analyzed by using the transfer function of PPFs.

PPFANA, a computer-aided software tool for analysis of influences from process

tolerances and component mismatch on the quadrature accuracy (QA) of RF CMOS


27

PPFs developed by the author, is described. Based on the simplified RF models of

resistors and capacitors, parasitic effects and their influences on RF CMOS PPFs are

analyzed; techniques for minimizing unwanted noise coupling and signal losses are

proposed and evaluated.

Based on the analytical results from chapter 2, physical design considerations for

RF CMOS PPFs to be implemented in System-on-a-Chip (SoC) solutions are proposed

in chapter 3. The author designed an RF CMOS PPF for a low-IF Bluetooth receiver by

using the Fraunhofer-IMS 0.6 µm CMOS process. The designed chip has been

fabricated at the Fraunhofer-IMS in Duisburg. The author characterized the

performance of the fabricated RF CMOS PPF by measurements. The design, fabrication,

measurement procedure and measurement results are presented in chapter 3. Silicon

success of this design lays a solid foundation for this dissertation.

Quadrature accuracy (QA), the main specification in design of an RF CMOS PPF

is determined by component matching which requires large chip area; however, in order

to minimize unwanted noise coupling, signal loss and the cost, the chip area must be

minimized. Thus, a tradeoff among these design parameters, that is, an optimization,

must be made in the design. Chapter 4 is committed to the optimization of RF CMOS

PPFs. First, the optimization problem is formulated with design specifications and

constraints. Then, possible optimization algorithms like calculus-based optimization,

gradient descend, simulated annealing (SA) and genetic algorithms (GA) are analyzed

by the author. Based on the results of the analysis, GA is chosen as the optimization

engine because of its advantages in escaping from local optima. The author

implemented PPFOPTIMA, a GA based optimization tool for RF CMOS PPFs by using

MATLAB. In order to facilitate designers in design automation, this tool has been


28

further developed into a synthesis and verification environment for RF CMOS PPFs by

the author, and the detailed implementation of PPFOPTIMA is described in chapter 4.

Experiments on multistage RF CMOS PPFs for a given standard, and experiments on

automated synthesis of RF CMOS PPF for different standards, such as, global system

for mobile communications (GSM), digitally enhanced cordless telephony (DECT) and

wide-band code division multiple access (WCDMA) demonstrate that PPFOPTIMA is

effective and efficient in synthesis, optimization, and verification of RF CMOS PPFs.

In chapter 5, contributions and conclusions from this dissertation are summarized,

and directions for possible future work are pointed out.

Chapter 2 Analysis of Nonidealities in RF CMOS PPFs

29

Chapter 2

Analysis of Nonidealities in RF CMOS PPFs

2.1 Introduction

Polyphase filters (PPFs) were invented by Gingell [45, 46] in 1971, for generation

of quadrature signals with even amplitude in audio. At that time, PPFs were

implemented by using discrete components. In his doctoral dissertation, Gingell [48] did

some simple analytical work for guiding the practical design. In 1994 and 1995,

Schmidt [53] and Tetsuo [54] analyzed the PPFN in a systematic way by using vector

analysis.

In 1994, integrated PPFs were rediscovered as an efficient RF quadrature

generation technique in CMOS technology by Stayaert [55], introducing PPFs into a

new regime, which we name as RF CMOS PPFs. The design of integrated RF CMOS

PPFs faces many challenges, which were not considered by the pioneers. In 2000, Galal

[58, 59] analyzed the sensitivity of the RF CMOS PPFs in RF integrated transceivers. In

2001, Behbahani [62] analyzed the application of RF CMOS PPFs in image rejection.

However, results from the open literature on the analysis of PPFs listed above

have the following limitations:


30

1. Gingell worked on audio frequency single sideband modulation [45-49]. At

such low operating frequencies, the influences from parasitic resistors and

capacitors are negligible. Thus, they were not considered in his publications.

2. In discrete components implementation of PPFs, components can be selected

with any given specifications. Thus, component mismatch was not analyzed in

[50-54].

3. References [55-61] provide useful information about the analysis and the

design of RF CMOS PPFs. However, until today, nonidealities, such as,

process tolerances, component mismatch and parasitic effects, which are

critical in RF CMOS PPF design and optimization, were not analyzed in a

systematic way.

4. Though vector analysis of PPFs [53, 54, 62] provides insight into the

mathematic representation of PPFs, it is too abstract for the designer to get

quantitative results about the influences from process tolerance and

components mismatch on the quadrature accuracy (QA) of RF CMOS PPFs. A

dedicated computer-aided software tool for solving this problem is required by

the design community of RF CMOS PPFs. However, this kind of computer-

aided tool has not been available to the author’s best knowledge.

In order to solve the shortcomings mentioned above, based on the explicit

transfer functions of PPFs, PPFANA, a dedicated computer-aided software tool for the


31

analysis and the simulation of influences from process tolerances and component

mismatch on the quadrature accuracy of the RF CMOS PPFs, has been developed by the

author, PPFANA will be presented in this chapter. By using simplified RF models of

resistors and capacitors, parasitic effects in RF CMOS PPFs are analyzed. Furthermore,

techniques for minimizing unwanted noise coupling and signal loss are proposed.

2.2 Vector Analysis of Polyphase Filters (PPFs)

A polyphase signal is a set of two or more vectors having the same frequency but

different phases [59]. If its vectors have the same magnitude and are equally spaced in

phase, it is said to be symmetric. Consequently, a symmetric four-phase signal consists

of two vectors of equal magnitudes having the same frequency and being separated in

phase by 90 degrees.

In general, polyphase filter networks (PPFNs) are networks having N input

terminals and N output terminals [54]. These networks are physically symmetric in a

way that the paths from each input to the corresponding output are symmetric. A PPFN

is driven by a polyphase signal consisting of a number of vectors that correspond to the

number of inputs of the PPFN. Although PPFNs are physically symmetric, they exhibit

asymmetric responses to inputs of opposite polarities [54]. This sequence-

discrimination property is the reason behind the name ‘sequence asymmetric’ [54]. The


32

response of a four-phase PPFN driven by a symmetric four phases input signal will be

analyzed now. Since the four phases are symmetric, the chain matrix of a single phase

represents the chain matrix of the network. The response of the same network to

asymmetric inputs can be derived by resolving the input into a summation of symmetric

inputs. Applying the rules of linear superposition, the output of the PPFN is then

considered as the sum of the outputs cascaded by each symmetric input alone.

Figure 2.1 One phase of a generalized N-phase PPFN.

Figure 2.1 shows the structure of one phase of a generalized N-phase PPFN,

where admittance 1Y is connected between the input and the corresponding output, and

admittance 2Y is skewed between the input and output of adjacent phases. Since the

input is symmetric and the PPFN is physically symmetric, we expect the output to be

also symmetric. The chain matrix of a single phase can be written as [58]:

,in ,out1 2

j,in 1 2 1 2 ,out1 2

, 11

2 (1 cos ), k k

k k

V VY Y

I YY Y Y IY e Yθ θ +

= − ++ (2.1)

j,inke Vθ

in,kV out,kV

- j,outke Vθ

in,kI out,kI

2Y

2Y

1Y


33

Where θ represents the relative phase difference between 1V and the neighboring inputs,

which in turn determines the polarity of the inputs. If θ <0, in,kV will be leading j,inke Vθ

in phase, which causes the inputs to be positive. On the other hand, if θ > 0, in,kV will

be lagging, thus causing the inputs to be negative.

Resistance–capacitance (RC) PPFNs are a special case of sequence asymmetric

PPFNs in which 1 is equal to 1/Y R and 2 is equal to jY Cω . The structure of an RC

four-phase PPFN is shown in Figure 2.2. In the following figure, ,inkV , ,outkV , k=1,2,3,4,

are the four inputs and the four outputs of the RC PPFN, respectively.

Figure 2.2 One-stage of an RC sequence asymmetric PPFN.

1,inVR

C

R

C

R

C

R

C

1,outV

2,inV

3,inV

4,inV

2,outV

3,outV

4,outV


34

In four-phase PPFNs such as shown in Figure 2.2, the value of θ is either 2/π

or 2/π− according to the polarity of the inputs. For such PPFNs, the chain matrix for

positive inputs can be written as [59]:

,in ,out

,in ,out

1 j ,1

2 j ,1 j1k k

k k

V VRC R

I C RC IRC

ωω ωω

+ = ++ , (2.2)

while for negative inputs, we find [59]:

,in ,out

,in ,out

1 j , 1

2 j ,1 j1k k

k k

V VRC R

I C RC IRC

ωω ωω

+ = +− . (2.3)

The open-circuit voltage transfer function for the negative part of the inputs is

found for out,kI equals to zero [59]:

,out

,in

1-H( )

1 jk

k

V RC

V RC

ωωω

= =+

, (2.4)

while for the positive part of the inputs, we yield [59]:

,out

,in

1H( )

1 jk

k

V RC

V RC

ωωω

+− = =+

(2.5)

Sequence-asymmetric PPFNs can be considered as a generalized structure of the

classical RC–CR network used for quadrature signal generation. Consider the M-stage

four-phase PPFN shown in Figure 2.3(a), having a positive part of the outputs transfer

function H( )ω and a negative part of the outputs transfer function H( )ω− , where

H( ) H( )ω ω≠ − . The PPFN is driven by only two phases representing a differential input

signal, while the other two inputs are connected as shown in Figure 2.3 (a). Figure 2.3(b)


35

shows the phasor representation of the input sequence which consists of two vectors

separated in phase by 180 degrees.

(a)

Inputs Negative part of the outputs Positive part of the outputs

(b)

Figure 2.3 Quadrature signal generation using four-phase PPFNs

(a) Circuit diagram.

(b) Decomposition of the two-phase inputs into two parts of four-phase outputs.

To be able to understand how quadrature phases are generated from a single input,

the inputs can be represented as the sum of two parts of four-phase outputs of different

3

1

3

2 4 1

3

4 2 = +

/ 4V

j / 4V−j / 4V

/ 4V−

/ 4V

j / 4V−j / 4V

/ 4V−

1 2/V

2/V−

out1,V

out2,V

out3,V

out4,V

in1,V

in2,V

in3,V

in4,V

PPFN

H( )ωV


36

polarities. Using linear superposition, the PPFN output can be expressed as the sum of

the outputs, separately. It should be noted that the voltage transfer function for any

phase in the positive part is H( )ω , while that for any phase in the negative part is

H( )ω− . Thus, the output phases can be written as:

1,out ( ) ((H( ) H(- ))2

VV ω ω ω= + , (2.6)

2,out ( ) - j ((H( ) - H(- ))2

VV ω ω ω= , (2.7)

3,out ( ) - ((H( ) H(- ))2

VV ω ω ω= + , (2.8)

4,out ( ) j ((H( ) - H(- ))2

VV ω ω ω= . (2.9)

If we consider the in-phase component to be 1,out 3,out( ) ( )V Vω ω− , and the

quadrature-phase component 2,out 4,out( ) - ( )V Vω ω , then we find:

I ( ) (H( ) H(- ))V Vω ω ω= + , (2.10)

Q ( ) - j (H( ) - H(- ))V Vω ω ω= , (2.11)

The ratio of the quadrature-phase to the in-phase outputs of an RC single stage

four-phase PPFN can be derived by substituting for H( )ω and H( )ω− . With (2.4) and

(2.5), we yield from (2.10) and (2.11),

Q

I

( j )j

( j )

VRC

V

ωω

ω= (2.12)

The disadvantage of the described PPFN is its limited frequency range within

which good gain matching can be achieved between quadrature outputs. The two

outputs, although in quadrature, have equal amplitudes only at a frequency equals to

RC1/ , which limits the use of this PPFN to narrow-band applications. Furthermore, the


37

gain and phase matching of the quadrature outputs is highly sensitive to the RC

mismatch.

To achieve broadband quadrature with better gain matching and reduced

sensitivity to components mismatch, a two-stage PPFN could be employed [65]. The

two outputs have equal amplitudes at two frequencies corresponding to the resonant

frequencies of each stage. The voltage transfer ratio can be derived by multiplying the

chain matrix of the two stages and evaluating the open-circuit voltage transfer function.

From (2.10) and (2.11), we yield [65]:

Q 1 1 2 22

I 1 1 2 2

( j ) ( )

( j ) 1 ( )

V j R C R C

V R R C C

ω ωω ω

+=+

(2.13)

This creates a wide-band quadrature generator defined by the resonant frequencies

112/1 CRπ and 222/1 CRπ . For any input signal with frequencies lying in this band, the

quadrature outputs are matched well in terms of gain and phase. The maximum gain

error occurs at frequency maxω , corresponding to the geometric mean of the two

resonant frequencies. The maximum gain error at this frequency is given

Q 1 1 2 2max

I 1 2 2 2

( j )

( j ) 2 ( )

V R C R C

V R R C C

ωω

ω+= , (2.14)

where max

1 2 2 2

1 is equal to

( )R R C Cω .


38

2.3 PPFANA, a Software Tool for RF CMOS PPFs

2.3.1 Overview

As discussed in the first chapter, vector analysis is very helpful in understanding

the mathematical representation of PPFs; however, it is too abstract for the designer to

get quantitative calculation on the quadrature accuracy (QA) of the PPFs. Furthermore,

in design and optimization of RF CMOS PPFs, it is of great importance to analyze and

simulate QA influenced by process tolerances and component mismatch. Unfortunately,

there is not any efficient simulation tool openly available. Thus, PPFANA, a software

analysis tool for solving the problems mentioned above has been developed in this work.

The core of PPFANA is made up from explicit transfer functions, which will be derived

in the following subsection.

2.3.2 Explicit Transfer Function of RF CMOS PPFs

Explicit transfer functions of RF CMOS PPFs are obtained by multiplying chain

matrices of each cascaded stages, then separating the real and imaginary parts. The

relation between input and output can thus be expressed by values of resistors and

capacitors and the operating frequency, explicitly.

In CMOS wireless transceiver RF front-end design; 2-stage, 3-stage, and 4-stage

RF CMOS PPFs are most widely used [68]. RF CMOS PPFs with more than four


39

cascaded stages are rarely used because of their severe insertion loss. For passive filters,

there is a theoretical insertion loss of 3dB/stage. We therefore derive the explicit

transfer function of a 4-stge RF CMOS PPF. Those of 2-stage and 3-stage RF CMOS

PPFs can be derived by the same method used here. Details of the derivation are given

in appendix 2 of this dissertation.

The transfer function of 4-stage RF CMOS PPFs is set out from

in 1 1 1

in 1 1 11 1

2 2 2

2 2 22 2

3 3 3

3 3 33 3

4 4

44 4

V 1+j R C R1=

I 2j C 1+j R C1- R C

1+j R C R1

2j C 1+j R C1- R C

1+j R C R1

2j C 1+j R C1- R C

1+j R C1

2j C1- R C

ωω ωωω

ω ωωω

ω ωωω

ωω

4 out

4 4 out

R V.

1+j R C Iω

(2.15)

The complex transfer function of the 4-stage RF CMOS PPF is given by

N(j )

H(j )=D(j )

ωωω

, (2.16)

1 1 2 2 3 3 4 4N(j )=(1- R C )(1- R C )(1- R C )(1- R C )ω ω ω ω ω , (2.17)

R ID(j )=D ( )+jD ( )ω ω ω , (2.18)

2R 1 2 1 2 1 3 1 3 2 3 2 3

1 4 1 4 2 4 2 4 3 4 3 4

1 3 1 4 2 3 2 4 1 2 2 4

1 2 1 4 1 3 3 4 2 3 3 4

1 3 2 3 1 2 2 3

D ( )=1- (R R C C +R R C C +R R C C

+R R C C +R R C C +R R C C

+2R R C C +2R R C C +2R R C C


+2R R C C +2R R C C

ω ω

1 4 2 4

1 4 3 4 2 4 3 4 1 2 1 3

41 3 2 4 1 2 3 4 1 2 3 4

+2R R C C


+4R R C C )+ (R R R R C C C C ),ω

(2.19)


40

I 1 1 2 2 3 3 4 4 3 4 1 2 1 3

3 2 3 2 4 1 4 1 2 3 1 2 3

1 2 4 1 2 4 1 3 4 1 3 4 2 3 4 2 3 4

1 2 3 1 2 4 1 2 3 2 3 4 1 2 3 1 3

D ( )= (R C +R C +R C +R C +2R C +2R C +2R C

+2R C +2R C +2R C )- (R R R C C C

+R R R C C C +R R R C C C +R R R C C C

+2R R R C C C +2R R R C C C +2R R R C C C

ω ωω

4

1 3 4 2 3 4 1 2 4 2 3 4 1 2 4 1 3 4 +2R R R C C C +2R R R C C C +2R R R C C C ).

(2.20)

Equation (2.15) can also be expressed as:

1 2H(j )=H (j )+jH (j )ω ω ω , (2.21)

2 4

11+a +b

H (j )=D(j )

ω ωωω

, (2.22)

1 2 1 2 1 3 1 3 2 3 2 3

1 4 1 4 2 4 2 4 3 4 3 4

a=R R C C +R R C C +R R C C

+R R C C +R R C C +R R C C , (2.23)

1 2 3 4 1 2 3 4b=R R R R C C C C , (2.24)

3

2cj +dj

H (j )=D(j )

ω ωωω

, (2.25)

1 1 2 2 3 3 4 4c=R C +R C +R C +R C , (2.26)

1 2 3 1 2 3 1 2 4 1 2 4

1 3 4 1 3 4 2 3 4 2 3 4

d=R R R C C C +R R R C C C

+R R R C C C +R R R C C C. (2.27)

2.3.3 Development of PPFANA

With the explicit transfer functions we derived in subsection 2.3.2, the quadrature

accuracy of an RF CMOS PPF with given values of resistors, capacitors and operating

frequency can be calculated by using MATLAB. In order to make it easy to use,

graphical user interfaces (GUIs) have been developed for PPFANA. The GUI of

PPFANA for 4-stage RF CMOS PPFs is shown in Figure 2.4. The GUIs for the 2-stage

and 3-stage RF CMOS PPFs are similar to this one.


41

Figure 2.4 Graphical user interface of PPFANA for 4-stage RF CMOS PPF.

In PPFANA, quadrature accuracy (QA), is measured by gain mismatch (GM) and

phase mismatch (PM), the smaller GM and PM are, the better QA is. From Figure 2.4,

we can see PPFANA has following features:

• QA influenced by process tolerance (PT) and component mismatch (CM) can

be analyzed independently.


42

• QA influenced by PT and CM can be analyzed at the same time.

• Worst-case QA can be analyzed.

• Default values for an RF CMOS PPF designed for a low-IF Bluetooth receiver

in Fraunhofer-IMS 0.6 µm CMOS technology are given as an example for new

users.

In the following sections, simulation results for QA influenced by PT and CM will

be presented. The simulation results will be compared with those obtained with the

commercial RF circuit simulator-Cadence SpectreRF to verify the effectiveness of

PPFANA.

2.4 Analysis of Quadrature Accuracy in Ideal PPFs

In this section, the quadrature accuracy (QA) of ideal PPFs is investigated by using

PPFANA. In the transfer function derived in the last section, 1H ( j )ω , 2H ( j )ω are

complex signal of the I, Q path respectively. From the definition of complex numbers,

we can derive the module and the complex angle. Gain mismatch, gainm , and phase

mismatch, φ∆ , can be expressed by

gain 10 1 10 2| 20 log | H ( j ) | 20log | H ( j ) ||m ω ω= − , (2.28)

phase 1 2|| H ( ) - H ( ) | -90 |m j jω ω= ∠ ∠ ° . (2.29)


43

Notice that in Cadence simulations, it is more convenient to get phase difference, phased ,

than phase mismatch, φ∆ , between I path and Q path by using AC simulation. For the

purpose of comparison, we define a parameter, phase difference, phased as

phase 1 2| H ( j ) - H ( j ) |d ω ω= ∠ ∠ . (2.30)

Please note that in all of the equations related to gainm , φ∆ , phased used in this dissertation,

the angular frequency ω is used for the convenience of expression of complex. Whereas

in all of the figures related to gainm , phasem , phased , frequency f is used for the purpose of

comparison with Cadence simulation results. It is well known that ω equals to 2πf .

Figure 2.5 Gain mismatch in ideal multi-stage RF CMOS PPFs tailored to

the Bluetooth frequency range 2.4 GH to 2.48 GHz.


44

Three ideal multi-stage (2-stage, 3-stage and 4-stage) RF CMOS PPFs for the

frequency band of Bluetooth, i.e., 2.4 GHz to 2.48 GHz, were designed by the author

and they were simulated by using PPFANA. Simulation results for the gain mismatch

and phase difference in these ideal RF CMOS PPFs are shown in Figure 2.5 and Figure

2.6, respectively.

Figure 2.6 Phase difference in ideal multi-stage RF CMOS PPFs tailored to

the Bluetooth frequency range 2.4 GH to 2.48 GHz.

From above figures, we can see the following:

• For ideal RF CMOS PPFs, at the desired frequency, gain mismatch is almost

zero, and phase difference is almost °90 .

• With more cascaded stages, the useful bandwidth with less gain mismatch and

phase mismatch can be extended.


45

In order to compare simulation results, these ideal RF CMOS PPFs were

simulated with Cadence SpectreRF as well. Simulation results from Cadence SpectreRF

and from PPFANA are listed in the following Table 2.1.

gainm /dB φ∆ /degrees Ideal RF

CMOS PPFs PPFANA SpectreRF PPFANA SpectreRF

2-stage 310− 310− 310− 310−

3-stage 310−< 310−< 310−< 310−<

4-stage 310−< 310−< 310−< 310−<

Table 2.1 Quadrature accuracy of ideal RF CMOS PPFs (2.4 GHz to 2.48 GHz),

obtained by using PPFANA and SpectreRF.

From these simulation results listed in the table above, we can see the following:

• For ideal 2-stage RF CMOS PPFs, the gain mismatch is 310− dB and the phase

mismatch is 310− degrees.

• For ideal RF CMOS PPFs with more than 2 cascaded stages, the gain mismatch

is less than 310− dB, and the phase mismatch is less than 310− degrees.

Considering computational errors, these values can be taken as zero. Thus, we

can say that ideal RF CMOS PPFs with more than two cascaded stages are

almost perfect in quadrature generation.


46

• The simulation results of PPFANA are in accordance with those of Cadence

SpectreRF. This validates the effectiveness of PPFANA.

2.5 Analysis of Quadrature Accuracy Influenced by Process

Tolerance

In CMOS processes, the actual values of identically designed and fabricated

components can be different from wafer to wafer. The variation of the component

values in a given CMOS process is called process tolerances, which is also named

process variation. In this dissertation, these two terminologies are used without any

difference.

In high performance CMOS analog/RF circuit, tuning techniques are usually

needed to overcome process tolerances. Most tuning techniques use variable capacitors

or variable resistors and additional control circuits, making circuitries complicated. In

RF CMOS PPFs, a special stagger-tuning technique [55] is used. The idea of this

technique is to increase the bandwidth of the circuit, to guarantee a proper filter

operation despite process tolerances. By using this special and simple tuning technique,

conventional tuning techniques [55] are no longer needed in RF CMOS PPFs to

overcome the influence from process variations.


47

Let us generalize the equations of GM and PM in multi-stage PPFs and derive

expressions for GM and PM influenced by process tolerances (PT). GM and PM in RF

CMOS PPFs with ideal components are expressed as idealgain,m and φ∆ ,

gain,ideal 1 1 n nG( , , , , , )i im R C R C R C= L L (2.31)

ideal 1 1 n nF( , , , , , )i iR C R C R Cφ∆ = L L (2.32)

Then we get GM and PM in RF CMOS PPFs with process tolerance PTgain,m and PTφ∆

according to

gain,PT PT1 PT1 PT PT PTn PTnG( , , , , , )i im R C R C R C= L L (2.33)

PT PT1 PT1 PT PT PTn PTnF( , , , , , )i iR C R C R Cφ∆ = L L (2.34)

where G , F are gain and phase expressions of RF CMOS PPFs’ transfer functions,

iR , iC , 1i n= L , are the nominal values of the resistors and capacitors, and iRPT , iCPT ,

1i n= L , are the resistors and capacitors with process tolerance.

In this work, gain mismatch and phase mismatch influenced by process tolerances

in multi-stage RF CMOS PPFs have been simulated by using PPFANA. These RF

CMOS PPFs are designed for the frequency band of Bluetooth, ranging from 2.4 GHz

to 2.48 GHz, using the Fraunhofer-IMS 0.6 µm CMOS technology. In these simulations,

the resistors are polysilicon resistors, in which process tolerance is %16± , and the

capacitors are poly-N+ capacitors, in which process tolerance is %14± . Gain mismatch

and phase mismatch are the worst value in the frequency band of interest, i.e. 2.4 GHz

to 2.48 GHz.


48

Figure 2.7 Gain mismatch influenced by process tolerance

in a 2-stage RF CMOS PPF (2.4 GHz to 2.48 GHz) with regular design.

Figure 2.8 Phase mismatch influenced by process tolerance

in a 2-stage RF CMOS PPF (2.4 GHz to 2.48 GHz) with regular design.


49

Figure 2.7 and Figure 2.8 show the gain mismatch and the phase mismatch

influenced by process tolerance in a 2-stage RF CMOS PPFs with regular design, that is,

it is designed with lower corner frequency at 2.4 GHz and higher corner frequency at

2.48 GHz without any technique to overcome the influences from process tolerance.

From these figures, we can see the following:

• Process tolerances have significant influence on the gain mismatch of the RF

CMOS PPF, whereas, their influences on the phase mismatch of the RF CMOS

PPF is not that obvious.

• With regular design, RF CMOS PPF is not robust enough to overcome the

influence from process tolerance.

In order to get robust RF CMOS PPFs, the influence of process tolerances on the

quadrature accuracy should be overcome. As the ‘stagger tuning’ technique is used in

the polyphase filters, in practical design, we can extend the bandwidth to guarantee that

the signal frequency band can be covered by worst-case lower and upper corner

frequencies. This is the concept of tolerance design [102] in RF CMOS PPFs design.

The 2-stage RF CMOS PPF was redesigned with tolerance design, and the gain

mismatch influenced by process tolerances is shown in the following Figure 2.9. From

this figure, we can clearly see that, the gain mismatch is no longer sensitive to process

tolerance. Thus, tolerance design should be applied in design of RF CMOS PPFs.


50

Figure 2.9 Gain mismatch influenced by process tolerance

in a 2-stage RF CMOS PPF (2.4 GHz to 2.48 GHz) with tolerance design.

PTgain,m /dB PTφ∆ /degrees RF CMOS

PPFs PPFANA SpectreRF PPFANA SpectreRF

2-stage 0.220 0.224 0.500 0.509

3-stage 0.015 0.017 0.050 0.053

4-stage 0.002 0.002 310−< 310−<

Table 2.2 Quadrature accuracy of RF CMOS PPFs (2.4 GHz to 2.48 GHz)

with process tolerance, obtained by PPFANA and SpectreRF


51

In order to compare the quadrature accuracy of multi-stage RF CMOS PPFs

influenced by process tolerance and to compare simulation results obtained by using

PPFANA and those by using Cadence SpectreRF, three RF CMOS PPFs with two, three

and four cascaded stages, respectively were designed for the frequency band of

Bluetooth with tolerance design. Simulation results for the gain mismatch and for the

phase mismatch of these RF CMOS PPFs are shown in Table 2.2, from which we can

see the following:

• Quadrature accuracy of RF CMOS PPFs is influenced by process tolerances,

significantly.

• By using more cascaded stages, the quadrature accuracy of the RF CMOS PPFs

can be improved.

• The simulation results from PPFANA are in accordance with those from

Cadence SpectreRF.

2.6 Analysis of Quadrature Accuracy Influenced by

Component Mismatch

In CMOS processes, two neighboring components that are ideally supposed to be

identical can have slight differences. This phenomenon is called component mismatch

(CM) [81]. Ideally, in RF CMOS PPFs, components in the I path and in the Q path


52

should be identical; however, in practice, this is impossible to achieve because of

component mismatch.

Now, let us generalize the transfer functions of multi-stage PPFs and analyze both

the GM and the PM influenced by component mismatch (CM) between I path and Q

path, we yield

gain,ideal I I1 I1 In In Q Q1 Q1 Qn Qn| G ( , , , ) - G ( , , , ) |m R C R C R C R C= L L , (2.35)

ideal I I1 I1 In In Q Q1 Q1 Qn Qn|| F ( , , , ) - G ( , , , ) | -90 |R C R C R C R Cφ∆ = oL L . (2.36)

Furthermore, we get

gain,CM I ICM1 ICM1 ICMn ICMn Q QCM1 QCM1 QCMn QCMn| G ( , , , ) - G ( , , , ) |m R C R C R C R C= L L , (2.37)

CM I ICM1 ICM1 ICMn ICMn Q QCM1 QCM1 QCMn QCMn|| F ( , , , ) - F ( , , , ) | -90 |R C R C R C R Cφ∆ = oL L .

(2.38)

In (2.37) and (2.38) ICM I equals (1 CM)k kR R± , QCM Q equals (1 CM)k kR Rm , ICMkC is

I(1 CM) kC± , and QCM Q is (1 CM)k kC Cm , 1k n= L , n is the number of cascaded stages

in a given RF CMOS PPF. In the equations above, idealgain,m and idealφ∆ are the gain

mismatch and the phase mismatch of RF CMOS PPFs with ideal resistors and

capacitors, CMgain,m and CMφ∆ are the gain mismatch and the phase mismatch of RF

CMOS PPFs with component mismatch, IF , QF , IG , QG are I path and Q path gain and

phase expressions derived from multi-stage RF CMOS PPFs’ transfer functions,

kRI , kCI , kRQ , kCQ are nominal values of resistors and capacitors in I path and Q path,


53

kRICM , kCICM , kRQCM , kCQCM are values of resistors and capacitors in I path and Q path

with component mismatch.

In this work, quadrature accuracy influenced by component mismatch in multi-

stage RF CMOS PPFs has been simulated by using PPFANA. These RF CMOS PPFs

were designed for the frequency band of Bluetooth, using Fraunhofer-IMS 0.6 µm

CMOS technology. In the simulations, the resistors are polysilicon resistors, and the

capacitors are poly-N+ capacitors. In order to compare the quadrature accuracy

influenced by different levels of component mismatch, 1% and 2% component

mismatch between neighboring resistors, and neighboring capacitors were simulated.

Gain and phase mismatches are the worst value in the band of interest, i.e., 2.4 GHz to

2.48 GHz. The simulation results are shown in Table 2.2.

resistors and capacitors with 1%

component mismatch

resistors and capacitors with 2%

component mismatch

CMgain,m /dB CMφ∆ /degrees CMgain,m /dB CMφ∆ /degrees

RF CMOS

PPFs

PPFANA PPFANA PPFANA PPFANA

2-stage 0.210 0.520 0.230 0.530

3-stage 0.025 0.030 0.060 0.090

4-stage 0.015 0.020 0.020 0.050

Table 2.3 Quadrature accuracy influenced by component mismatch in

multi-stage RF CMOS PPFs (2.4 GHz to 2.48 GHz).


54

From the simulation results in Table 2.3, we can see the following:

• Component mismatch has significant influence on quadrature accuracy of RF

CMOS PPF.

• Stronger component mismatch leads to worse quadrature accuracy, that is,

more gain mismatch and phase mismatch.

Though component mismatch cannot be totally eliminated in practice, layout

techniques, such as, inter-digitized [77] and common-centroid [77] layout structures, are

helpful to reduce the component mismatch to some extend. These techniques will be

discussed in the Chapter 3.

2.7 Analysis of Influences from Parasitic Effects

2.7.1 Simplified RF Models of Resistors and Capacitors

There are several types of resistors in CMOS processes; polysilicon resistors [81]

are the most favorable selection for RF applications for its area efficiency and

comparatively good robustness against parasitic influences. In compact models [75-80],

polysilicon resistors are usually modeled as distributed resistors and capacitors. As in

our application, the physical dimension of the resistors are much smaller than the

wavelength of the Bluetooth RF signal, for the purpose of analysis, we simplify the

distributed model into a simplified model as shown in Figure 2.8.


55

From Figure 2.10, we can see that polysilicon resistors with its parasitic

capacitance can form a low-pass filter. In RF CMOS PPFs, current flows through

resistors. If the corner frequency of the low pass filter formed is lower than that of the

signal frequency, the signal is then cut off. This phenomenon is unwanted and should be

avoided in RF CMOS PPF design.

Figure 2.10 Simplified RF model of a polysilicon resistor [81].

Now, let us investigate how the problem mentioned above can be avoided in the

design of RF CMOS PPFs. We have [81]

RCP 2para sh uap

sh uap

1 1 1

2 22f

LRC L R CLWR CW

π ππ= = = , (2.39)

RCPop ≤ff , (2.40)

op sh uap

1

2L

f R Cπ≤ , (2.41)

In

R

paraC

subR

paraC

subR

gnd gnd

Out


56

max,res

op sh uap

1

2L

f R Cπ= , (2.42)

where RCPf is the corner frequency of the filter formed by the polysilicon resistor and its

parasitic capacitor to the substrate, opf is the operating signal frequency, shR is the sheet

resistance of resistor, L is the length of the resistor, uapC is parasitic capacitance per

unit area to the substrate, resmax,L is the maximum length of resistors can be used.

From the derivations above, we can see that for a certain CMOS process and a

given operating frequency of the RF CMOS PPF, before physical layout, the maximum

length of the resistor should be calculated to avoid inappropriate physical design.

Figure 2.11 Simplified RF model of a capacitor [81].

1

p1R

p1C

subR

p1C

subR

gnd gnd

2

p2R

p2C p2C


57

In different CMOS processes, there are different kinds of capacitors, e.g. poly-

poly capacitors [81], poly-well capacitors [81], MOS capacitors [81], and metal-

insulator-metal (MIM) capacitors [81]. The first three kinds of capacitors are fairly

standard, whereas MIM capacitors are only available in advanced CMOS processes [81].

RF capacitors are modeled as distributed networks including parasitic resistors

and capacitors [81]. As in our application, the physical dimension of the capacitors are

much smaller than the wavelength of the Bluetooth RF signal, for the purpose of

analysis, we simplified the distributed model as the model shown in Figure 2.11. Now,

let us investigate the quality (Q) factors of the capacitor and make a comparison among

different capacitors mentioned above. The Q factor is given by [81]

ω

ωRC

Q1

)( = , (2.43)

sh unitR R N= , (2.44)

Insert (2.44) into (2.43), we get

sh unit

1( )Q

R N Cω

ω= . (2.45)

In the above equations, R is the parasitic resistance of the capacitor, C is the nominal

capacitance, ω is the operating frequency, shR is the sheet resistance of the plates, unitN

is the number of unit squares.

From (2.45) we can see that for a given capacitance C and a given operating

frequencyω , there are two ways to improve quality factor of the capacitor:

• Use capacitors with low sheet resistance plates.


58

• Use a low number of unit squares, e.g. unitN equals 1.

As metal plates have less sheet resistance, the quality factor of metal-insulator-

metal (MIM) capacitors is greater than that of poly-poly resistors, poly-well capacitors

and MOS capacitors [81]. Thus, if possible, the use of MIM capacitors is highly

recommended for design of RF CMOS PPFs. Furthermore, in the physical layout of

capacitors, square capacitors should be used to get a greater quality factor.

2.7.2 Substrate Noise Coupling

In CMOS single chip transceiver design, the whole transceiver system, including

RF, analog and digital circuits, is integrated into a chip [99]. This concept is call

system-on-a-chip (SoC) [99]. SoC design poses serious challenges different from

conventional multi-chip or multi-package design, because digital circuits are placed on

the same substrate with sensitive analog circuits [95].

Figure 2.12 Principle of substrate noise coupling in system-on-a-chip [99].

circuits Analog circuits Digital

Noise Noise

Noise Noise

CouplingNoise

CouplingNoise

SubstratetiveSemiconduc


59

The most critical issue in SoC design is substrate noise coupling [99], which

occurs as a result of the conductive nature of silicon and the lack of perfect isolation

between devices and the substrate [100]. Below frequencies of 5 GHz, the capacitive

noise coupling is the main contributor to the substrate noise coupling [87, 88]. In this

dissertation, the operating frequency is 2.45 GHz, which is well below 5 GHz. Thus, we

consider only the capacitive noise coupling.

In SoC design, digital circuits are usually noisy. Noise generated by digital

circuits is injected into the substrate [99]. Due to the semi-conductive nature of the

silicon substrate, the noise is then sensed by the sensitive analog circuit via capacitive

coupling [99]. The principle of substrate noise coupling in SoC is shown in Figure 2.12

[99].

Figure 2.13 Simplified model of substrate noise coupling in RF CMOS PPFs.

In

paraC

subR

gnd

Noise

paraC

subR

gnd

Out

Noise

PPF CMOS RF


60

Shown in Figure 2.13 is the simplified model of substrate noise coupling in RF

CMOS PPFs. In this model, simplified RF models of resistors and capacitors together

with noisy resistive substrate model are used.

Now, let us analyze the noise coupled to the unshielded RF CMOS PPFs by the

substrate noise coupling. The parasitic capacitance between the polysilicon resistor and

the substrate ( RC ), and the parasitic capacitance between the capacitor and the substrate

( CC ) are

uaRRuaRRRR == CSCWLC , (2.46)

uaCCuaCCCC == CSCWLC . (2.47)

Summing all capacitances, we get

∑1=

uaRRuaCCsum )×+(×4=n

iiiii CSCSC , (2.48)

where ni L1= , n is the number of cascaded stages, RC is the parasitic capacitance

between the polysilicon resistors and the substrate, RRR ,, SWL are length, width and

area of the polysilicon resistors, respectively, uaRC is the unit area parasitic capacitance

of the resistors, CC is the parasitic capacitance between the capacitor back plate and the

substrate, CCC ,, SWL are length, width and area of the capacitors, respectively, and uaCC

is the unit area parasitic capacitance of the capacitors.

In order to minimize noise coupling from the substrate, the parasitic capacitance

between RF CMOS PPF and the substrate, i.e., sumC , should be minimized. In (2.48),

for a certain CMOS process, uaRuaC,CC are known constants. Thus, what the designer


61

can do is to minimize the parasitic capacitance by minimizing the chip area. However,

in practice, the chip area can only be reduced to a certain range, so noise coupling still

exists [99], and in practice, it is impossible to get sufficiently low substrate noise

coupling by only minimizing the chip area. Thus, noise isolation techniques, namely, N-

Well and guard ring should be applied [99].

For different types of substrates, these noise isolation techniques have different

effects [93]. Categorized by resistivity in the substrates, there exist a high-resistivity

substrate and a low-resistivity substrate in CMOS processes [93]. A high resistivity

substrate is composed of a uniformly doped layer with a resistivity coefficient of 20

Ωcm to 50 Ωcm [93]. A low resistivity substrate consists of a thick, high resistivity

epitaxial layer (10~15 Ωcm) and a low resistivity bulk (1 mΩcm) . Low resistivity

substrates have been widely adopted for desirable latch-up suppression properties [93].

It has been found that at low and medium frequencies typically lower than 5 GHz,

substrates show a resistive behavior [93]. At higher frequencies, the transport patterns

are too complex to be accurately modeled by resistive or resistive-capacitive meshes

[98]. In this dissertation, the operating frequency is well below 5 GHz, thus we use

resistive network to model the substrate. In high-resistivity substrates, distance and

guard rings are effective attenuation techniques to reduce signal interaction [93]. In low

resistivity substrates, the current tends to flow through low impedance paths located

deep in the chip’s lower layers; as a result, guard rings are generally ineffective in


62

blocking substrate currents [94]. In low-resistivity substrates the use of a very low

impedance backplate contact is often preferred [99]. From the discussions above, we

can see that the choice of the isolation techniques should be based on the property of a

given CMOS process and the specific application.

2.8 Summary

In this chapter, based on vector analysis, explicit transfer functions of multi-stage

PPFs were derived. Non-idealities in RF CMOS PPFs design, namely, process tolerance,

component mismatch, and parasitic effects, were analyzed. Important contributions and

conclusions in this chapter are:

• PPFANA, a software tool developed by the author for the simulation of the

quadrature accuracy (QA) of RF CMOS PPFs, which are influenced by process

tolerance and component mismatch, is presented.

• As is shown that the process tolerance has significant influences on quadrature

accuracy. In order to make an RF CMOS PPF robust against process tolerance,

tolerance design is used.

• Component mismatch also has an influence on the quadrature accuracy (QA).

Solution for design: use layout technique to minimize component mismatch.


63

• Parasitic effects in RF CMOS PPFs have been analyzed by using simplified RF

models of resistors and capacitors. Physical design considerations for

minimizing and isolating substrate noise coupling have been proposed.

Chapter 3 Design of an RF CMOS Polyphase Filter

64

Chapter 3

Design of an RF CMOS Polyphase Filter

3.1 Overview

In order to verify the effectiveness of PPFANA, an RF CMOS polyphase filter

(PPF) for a low-IF Bluetooth receiver using Fraunhofer-IMS 0.6 µm CMOS process has

been designed, fabricated and measured. Influences from process tolerances and

component mismatch on the quadrature accuracy of the RF CMOS PPF designed have

been analyzed by using PPFANA. Schematic and post-layout simulation results

obtained by using Cadence SpectreRF have been compared with those obtained by

using PPFANA.

Silicon success of this design lays a solid foundation for this dissertation in the

following two aspects:

• Physical design considerations for RF CMOS PPFs in a system-on-a-chip (SoC)

solution have been proposed based on the design practice.

• Design heuristics from this practical design lay the basis for optimization of RF

CMOS PPFs.


65

3.2 Schematic Design

3.2.1 Derivation of Specifications

The specifications of RF CMOS PPFs can be derived from the wireless

communication standards and the selected transceiver architecture. In our design of a

Bluetooth receiver, a low-IF receiver architecture has been chosen because it

incorporates advantages over heterodyne and homodyne receivers, providing high

integratebility.

The target specification related to the RF CMOS PPFs in the receiver is the

desired image reject ratio (IRR), which can be obtained from system level simulation. In

our low-IF Bluetooth receiver, the IRR is required to be greater than or equal to 36 dB

in the frequency band of Bluetooth, that is, between 2.4 GHz and 2.48 GHz. The

relationship between the IRR, the gain mismatch A∆ and the phase mismatch φ∆ can

be expressed as follows [84]:

2 2

IR 10 2I Q

-20 log16( ) 4

AR

A A

φ ∆ ∆= + + (3.1)

In (3.1) IRR is the IRR, and IA and QA are the voltage gain of the I path and Q path

output signal.

Setting out from (3.1), the main specifications of the RF CMOS PPF, i.e. the gain

mismatch and phase mismatch have been derived. In order to compare these target

specifications and the simulation results obtained by using Cadence SpectreRF and

PPFANA, the gain mismatch is expressed as:


66

gain 10= -20logm ∆A (3.2)

Detailed specifications of this RF CMOS PPF are listed in the Table 3.1.

Frequency band / GHz gainm / dB φ∆ / degrees

RF CMOS PPF 2.4-2.48 ≤ 0.6 ≤ 5

Table 3.1 Specifications of RF CMOS PPF for a low-IF Bluetooth receiver.

3.2.2 Design Guidelines

In 2001, Behbahani [62] proposed the following guidelines for the design of RF

CMOS PPFs:

[Step1] Calculate the number of cascaded stages required in the PPF from the

targeting image reject ratio (IRR) and the fractional bandwidth over which this

rejection is required.

[Step2] As an initial guess, place the two lowest and highest poles at the

boundaries of the rejection band. Space the remaining poles equally on the

logarithmic frequency axis. The actual pole locations are fine-tuned by

simulations.

[Step3] Specify the matching between the resistors and between the capacitors

based on the desired IRR. This determines the physical area of the filter.


67

[Step4] Large resistors will lower the power consumption of the amplifier

driving the PPF input. Use the largest filter resistance, limited either by the

maximum noise at the output or by the cut-off frequency. In low-noise or low

frequency circuits, the first is the limiting factor, while in high-frequency PPFs

it is the second.

[Step5] To lower cascaded loss, taper down the resistance of the PPF stages

toward the input. The resulting impedance of the input stage of the PPF will

determine the drive requirements on the amplifier prior to the filter.

[Step6] Design the driving amplifier. If the cascade filter loss is still too large,

insert interstage amplifiers to preserve signal dynamic range within the PPF;

the collinear common-mode signal at each output shifts the zero crossings and

corrupts the final phase relationship among the outputs. Thus, the outputs must

be sensed using linear differential circuits that reject the common mode.

Though rather comprehensive, an important design consideration is missing in

these design guidelines, that is, process tolerance should be considered in selecting

lower and upper corner frequencies for the RF CMOS PPFs. The importance of

tolerance design in design of RF CMOS PPFs has already been analyzed in the previous

chapter 1; in this chapter we apply it to a practical design.


68

3.2.3 Tolerance Design

In the design of RF CMOS PPFs, to guarantee that the worst-case circuit

performance still satisfies the specifications, process tolerances should be considered by

using the concept of ‘tolerance design’ [102]. Specifically, by enlarging the bandwidth

of the PPF, the worst-case frequency range of the PPF can be guaranteed to be located

in a given operating frequency band. In the Fraunhofer-IMS 0.6 µm CMOS process,

sheet resistance of the polysilicon resistors used in this design varies from 36 Ω/square

to 50 Ω /square, while the nominal value is 43 Ω/square. The unit capacitance of the

poly_N+ capacitor in this process varies from 1.1 2fF/µm to 1.7 2fF/µm , whilst the

nominal value is 1.4 2fF/µm .

It is well known that the relationship between the corner frequency cornerf of an

RC filter and the values of resistors and capacitors is given by

corner =π1

f2 RC

. (3.3)

From (3.3), we get the expression of RC time constant as follows:

corner

=π

1RC

2 f, (3.4)

where R is the resistors value, C is the capacitor value, and cornerf is the corner

frequency of this RC filter. For convenience of calculation of the RC time constant with

tolerance design, we define the RC time constant variation ratio

RC

( )( )100%

R R C C

RCσ + ∆ + ∆= × (3.5)


69

In (3.5) R∆ and C∆ are the process variations of the resistors and capacitors,

respectively. Setting out from the Fraunhofer-IMS 0.6 µm CMOS process, we find RCσ

equal to 41.2%. In order to overcome the influence of RCσ , we need to decrease the

lower RC time constant, determined by higher corner frequency of the RC filter, and

increase the higher RC time constant, determined by lower corner frequency, by 41.2%

respectively. The lower corner frequency is set to 1.44 GHz, and the higher corner

frequency is set to 3.47 GHz with tolerance design of RF CMOS PPFs.

3.2.4 Analytical Calculations

In the schematic design, constraints in physical layout should also be considered

to avoid unreasonable circuit sizing. In the design of RF CMOS PPFs, the maximum

length of resistors, max,resL , allowed in a given CMOS process, and minimum

components area, required by a given standard deviation of the component mismatch,

need to be calculated.

• Calculation of maximum resistor length. As discussed in section 2.7.1, at

radio frequency, parasitic effects have significant influences on the circuit

performance. By using the simplified model of RF resistors, the maximum

length of the resistor, max,resL , allowed in a given CMOS process has been

derived, and it is expressed in equation (2.42). Using this equation, at 2.45 GHz,

the maximum length max,resL of the polysilicon resistor in the Fraunhofer-IMS

0.6 µm CMOS process is equal to 132 µm.


70

• Calculation of minimum component area. In the design of RF CMOS PPFs,

component matching is a critical factor to be considered. From Monte Carlo

simulations [73], it has been found that

UIR 10 Component> -20log (σ )R , (3.6)

where UIRR is the ultimately obtainable image reject ratio, componentσ is the

variance of the component mismatch. It is known from the experimental study

[73] that the variance of adjacent on-chip resistors and capacitors is

proportional to the inverse of their surface area. Measurement results in

Fraunhofer-IMS 0.6 µm CMOS process show that for two neighboring

polysilicon resistors with width of 20 µm and length of 250 µm, the maximum

mismatch is 1%, for two neighboring poly-N+ capacitors with 300 2µm , the

maximum mismatch is 0.5%.

In our design, the target specification of the image reject ratio (IRR) is 36 dB.

Calculated by using equation (3.6), a component mismatch of 1% is needed to satisfy

this target specification. Based on the guidelines and the hand calculation results, the

range of length, width of the resistor and the value of the capacitor were calculated as a

preparation for the physical layout. The schematic of the RF CMOS PPF is shown in

Figure 3.1.

This design was based on intensive trials and it took several iterations between

schematic design and layout to obtain proper tradeoffs among design parameters, i.e.,


71

quadrature accuracy, signal loss, and chip area. The circuit dimensions are shown in

the following Table 3.2.

Figure 3.1 Schematic of a 3-stage RF CMOS PPF for a low-IF Bluetooth receiver.

Stage Resistors Value Capacitors Value

1 R1 217.6 Ω C1 229.4 fF

2 R2 283.6 Ω C2 229.4 fF

3 R3 347.6 Ω C3 229.4 fF

Table 3.2 Dimensions of a 3-stage RF CMOS PPF for a low-IF Bluetooth receiver.

I_LO+

I_LO-

Q_LO+

Q_LO-

LO+

LO-

R0

C0

R1

C1

R1

C1

R1

C1

R1

C1

R0

C0

R0

C0

R0

C0

C2

R2

C2

R2

C2

R2

C2

R2

Stage 1 Stage 2 Stage 3


72

3.2.5 Schematic Simulation

Figure 3.2 Simulation results of a 3-stage RF CMOS PPF obtained by using SpectreRF

Figure 3.3 Simulation results of a 3-stage RF CMOS PPF obtained by using PPFANA.


73

gainm /dB phased /degrees

PPFANA 0.012 90.063

SpectreRF 0.010 90.061

Table 3.3 Simulation results of a 3-stage RF CMOS PPFs

obtained by using PPFANA and SpectreRF.

The simulation results obtained for the schematic by using Cadence SpectreRF

and PPFANA are shown in Figure 3.2 and Figure 3.3, respectively. Accurate results are

listed in Table 3.3. From figures and table above, we can see that the simulation results

match; the gain mismatch is approximately 0.01 dB and the phase difference is

approximately 90.06 degrees, measured between 2.4 GHz and 2.48 GHz, and ignoring

computational errors. As mentioned in the second chapter, the phase mismatch is phase

difference minus 90 degrees, thus phase mismatch in this RF CMOS PPF is

approximately 0.06 degrees. Compared with Table 3.1, we can see that the target

specifications are met in the schematic simulation.

3.3 Physical Layout Design

3.3.1 Layout Techniques

As discussed in the chapter 2, in the physical design, special attention must be

paid to the layout of the resistors and capacitors. The layout of the four resistors in one


74

stage of the RF CMOS PPF designed by the author is shown in Figure 3.4. To obtain

good matching, the following techniques are applied:

• Identical structures, including end contacts, are used for all the resistors.

• Resistors are placed very closely.

• All of the resistors have the same orientation.

• A common-centroid structure is used [77].

• Dummy resistors are used to minimize the boundary effects.

Figure 3.4 Layout of the four resistors in one stage of an RF CMOS PPF.

The layout of the four capacitors in one stage of the RF CMOS PPF designed by

the author is shown in Figure 3.5. In order to obtain good matching, the following

techniques are applied:

Resistor 1

Resistor 2

Resistor 3

Resistor 4

Dummy resistor

Dummy resistor

N-Well Guard rings


75

• Identical structures, including terminals, are used for all capacitors.

• All of the capacitors to be matched are placed very closely.

• Capacitors to be matched are placed with the same geometrical orientation.

• A common-centroid structure is used [77].

• Dummy capacitors are placed around the matched array to minimize the

boundary effects.

• Input and output wires are placed as symmetrically as possible.

Figure 3.5 Layout of the four capacitors and wires in one stage of an RF CMOS PPF.

Following the layout techniques mentioned above, component mismatch in RF

CMOS PPFs can be control well. The latest component mismatch model in [81] can be

Capacitor 1

Capacitor 2

Capacitor 3

Capacitor 4

Note: Surrouding capacitors are dummy capacitors

Wires

Guard ring


76

used to estimate the standard deviation of the mismatch in the layout. Thus a layout of

this design is taken as the layout pattern used in the chapter 4.

3.3.2 On-Chip Noise Isolation Techniques

As discussed in chapter 2, in the design of RF CMOS PPFs, special attentions

must be paid to on-chip noise isolation techniques to reduce unwanted substrate noise

coupling. On-chip shielding should be used to protect the sensitive RF CMOS PPFs

from noise generated by other circuits. The CMOS process used in this design, the

Fraunhofer-IMS 0.6 µm CMOS process, is a P-substrate, single N-well process. In our

application, N-well and a guard ring are used.

N-Wells: N-Well is an epitaxial layer fabricated on the P-substrate to

differentiate it from the substrate. In an N-well process, i.e., a process with P-substrate

and N-well, the N-well is originally used for the fabrication of P-transistors. With

proper biasing, the N-well can be used to provide isolation between a component and

the substrate.

As shown in Figure 3.4, resistors in this design are protected by N-wells, the

region with grids. These resistors are built in the shallow P-region within the buried N-

well. A depletion layer exists on both sides of the N-well and this provides dielectric

isolation between the active devices and the bulk region. Since in our application the N-

well is connected to the low-impedance positive supply, the N-well performs a Farady


77

shield around the resistors. Thus, resistors are protected against noise coupling by N-

well.

However, the N-well is not well suited to use for the capacitors in this design,

because the bottom plate of the poly-N+ is already in the N+ region. Despite this fact,

poly-N+ capacitors are the best capacitor in the sense of linearity in the Fraunhofer-IMS

0.6 µm CMOS process.

Guard Rings: The principle of guard ring, also named as guarding band, is to

use a narrow ring of P+ or N+ with stable and noise-free connections forming a local

clean zone to reduce noise coupling through the substrate. In the Fraunhofer-IMS 0.6

µm CMOS process, guard rings can be connected to either ground or Vdd. A grounded

guard ring uses a low-resistance P+ area to connect to ground. A guard ring that

connects to Vdd uses an N-well and N+ region on the substrate. Positive N+

connections attract electrons, and the grounded P+ connections attract holes. Thus, they

can provide a barrier against substrate noise coupling. As shown in Figure 3.4 and

Figure 3.5, a P+ guard ring and a N+ guard ring are used in the layout of the resistors, a

P+ guard ring is used in the layout of capacitors, and these guard rings are placed as

close as possible to the resistors and capacitor.

3.3.3 Layout and Post-Layout Simulation

In order to test this chip independently, test pads are added to the layout of the RF

CMOS PPF; shielded ground pads (Abschirm_GND) are used to avoid cross-coupling


78

between different paths and wiring in the layout is kept as symmetrical as possible to

minimize additional mismatch. The final layout of the RF CMOS PPF with test pads is

shown in Figure 3.10.

Figure 3.6 Layout of a 3-stage RF CMOS PPF with test pads.

gainm / dB φ∆ / degrees

Cadence post-

layout simulation 0.071 0.26

PPFANA 0.27 1.8

Table 3.4 Simulation results obtained by using Cadence post-layout simulation and PPFANA.

Stage 3 Stage 2 Stage 1 Decoupling capacitorTest pads


79

After the layout, post-layout simulation (PS) of the RF CMOS PPF with

coefficient extracted networks was done by using Cadence SpectreRF. In this extraction

method, all kinds of parasitic resistances and capacitances, such as, parasitics between

neighboring components, wires, and parasitics to the substrate, are extracted. In this

way, we can provide realistic simulation results. Unfortunately, by this simulation

method, the influences from process tolerances and component mismatching cannot be

easily simulated. In this situation, PPFANA, the software the author developed in this

dissertation shows its advantages over Cadence post-layout simulation. The simulation

results by using PPFANA and Cadence post-layout simulation are shown in Table 3.4

for the purpose of comparison.

3.3.4 Chip Photo

`

Figure 3.7 Chip photo of an RF CMOS PPF for a low-IF Bluetooth receiver.

Stage 3 Stage 2 Stage 1 Decoupling capacitorTest pads


80

A chip photo of the fabricated chip is shown in Figure 3.7. The squares in the top

of this photo are test pads; in the middle of the bottom part is the RF CMOS PPF

designed for a low-IF Bluetooth receiver.

3.4 Measurements

3.4.1 Introductory Remarks

As an RF CMOS PPF has three ports, namely one input port and two output ports,

and the wafer station in Fraunhofer-IMS can only measure components with two ports,

a printed circuit board (PCB) providing interfaces to the measurement equipments was

designed and manufactured. The fabricated chip was bonded to the test PCB to allow

measurements.

3.4.2 Modeling of Test Pads and Bonding Wires

As test pads and bonding wires have parasitic resistance, inductances and

capacitance, in order to simulate the influences from these parasitic components, a

model of test pads and bonding wires shown in Figure 3.8 was built by using the

Cadence package physical geometrical modeling tools. Both schematic simulation and

post-layout simulation of RF CMOS PPF with test pads and bonding wires show that

the parasitic components have no significant influences on the gain mismatch and the


81

phase mismatch of the RF CMOS PPF, because they are in a fully balanced

configuration and their layouts are symmetrical.

Figure 3.8 Model of test pads and bonding wires.

3.4.3 Design of the Test PCB

The functionality of the test PCB is to provide an interface to the RF

measurement equipment. Specifically, the test PCB includes two types of conversions,

i.e., impedance conversion and differential to single-end conversion. A schematic of the

test PCB is shown in the Figure 3.9. From this schematic, we can see that the

differential input of the RF CMOS PPF is accomplished by using a balance-unbalance

converter (BALUN), and the differential outputs of I path and Q path are converted by

BeforePad

C_Pad

gnd

∩∩∩∩ AfterBond

R_Pad

R_Bond L_BondR_ESD

=2.45GHzf

r=99 200m=r 800p=l

183f=c

174=r


82

BALUNs to single-ended outputs respectively. In order to get maximum output power

gain, resistors and capacitors with optimized values are connected to the outputs as

loads.

Figure 3.9 Schematic of the printed circuit board (PCB) for test of the RF CMOS PPF.

In Figure 3.10, the photo of the fabricated test PCB is shown. The metal

connectors are the SMA connectors. The black block in the center of the PCB is the

fabricated chip, which is bonded to the PCB.

SMA

in⊃⊃⊃⊃

⊂⊂⊂⊂

gnd

LO +

LO-

TOKO Balun617DB-1023

f_lo = 2.449GHzi_DC = 0

1:2

c = 240fr = 250gnd

⊃⊃⊃⊃

⊂⊂⊂⊂

gnd

I_LO +

I_LO-


2:1

SMA

in

gnd

r = 250 c = 240f

c = 240fr = 250gnd

⊃⊃⊃⊃

⊂⊂⊂⊂

gnd

Q_LO +

Q_LO-


2:1

SMA

in

gnd

r = 250 c = 240f


83

Figure 3.10 Photo of the fabricated test PCB.

3.4.4 Measurement Setup

In the measurement setup by the author, an Agilent E8267C PSG vector signal

generator and a Lecroy wavermaster digital oscilloscope were used to measure the

performance of the fabricated chip. A diagram of the measurement setup is shown in

Figure 3.11.

The Agilent E8267C PSG vector signal generator generates a 2.45 GHz

continuous sinusoidal signal, which is input to the test PCB via RF cable. Then the I (in-

phase) path and Q (quadrature-phase) path were connected to the Lecroy wavemaster

digital oscilloscope via an RF cable. A feature of this Lecroy wavemaster digital


84

oscilloscope is its embedded real time fast Fourier transform (FFT). With this digital

oscilloscope, time domain and frequency domain measurements of the two port outputs

can be conducted in real time.

Figure 3.11 Diagram of the measurement setup for test of the RF CMOS PPF.

3.4.5 Measurement Results

Two types of measurements, namely, time domain and frequency domain

measurements, on RF CMOS PPFs have been conducted in this work. Time domain

measurement results are shown in Figure 3.12. From the time domain signal, we can see

that the gain mismatch is approximately 0.17 dB, and the phase difference is

approximately 88.78 degrees, consequently, the phase mismatch is approximately 1.22

degrees.

RF CMOS PPFsTest PCB

In

I_out Q_out

Agilent E8267 PSGVector signal generator Out

P_in

Power Supply

P_out P_out

Lecroy WavemasterDigital oscilloscope

In1 In2P_in

Cable RF RF Cable

RF Cable


85

Figure 3.12 Time domain measurement results of the RF CMOS PPF.

As time domain measurement is observed at only that time interval, it has

significant measurement inaccuracy in the RF domain.


86

In order to get more accurate measurement result, frequency domain

measurement was conducted.

Figure 3.13 Gain mismatch of the RF CMOS PPF measured in frequency domain.


87

The frequency domain measurement results of the gain mismatch and the phase

mismatch in the RF CMOS PPF are shown in Figure 3.13 and Figure 3.14 respectively.

The measurement results are 0.45 dB gain mismatch and 3.2 degrees phase mismatch.

Figure 3.14 Phase difference of the RF CMOS PPF measured in frequency domain.


88

3.5 Analysis of Simulation and Measurement Results

By comparing schematic simulation results obtained by using PPFANA and

Cadence SpectreRF, which are shown in Table 3.3, we can see that they are in good

accordance with each other; the effectiveness of PPFANA is thus verified. From the

schematic and post-layout simulation of RF CMOS PPFs with test pads, we have found

that test pads do not have significant influences on the quadrature accuracy of RF

CMOS PPFs because they have fully balanced configuration and symmetrical layout,

whereas bonding wires, external components, i.e., load resistors and capacitors, and

BALUN contribute certain mismatch.

By comparing the post-layout simulation results of RF CMOS PPFs obtained by

using PPFANA and Cadence SpectreRF and the measurement results, we can see that

the simulation results obtained by using PPFANA are closer to the measurement results

because the influences from process tolerances and component mismatch are simulated.

In this situation, PPFANA provides a useful design aid for RF CMOS PPF design.

3.6 Summary

This chapter concentrates on a practical design of an RF CMOS PPF for a low-IF

Bluetooth receiver. Detailed design considerations, analytical calculations and layout

techniques for minimizing component mismatch were analyzed.


89

The fabricated chip was bonded to the test PCB. Measurement results show that

PPFANA developed by the author provides more realistic simulation results than the

commercially available Cadence tools. Further contributions of this chapter are the

following:

1. Silicon success of an RF CMOS PPF for a low-IF Bluetooth receiver was

presented.

2. Design heuristics from design practice reported here lay a solid foundation for

the optimization of RF CMOS PPFs.

3. The effectiveness of PPFANA was demonstrated.

Chapter 4 Optimization of RF CMOS Polyphase Filters

90

Chapter 4

Optimization of RF CMOS Polyphase Filters

4.1 Introduction

In the open literatures, there are some papers on analysis [55-60] and design[55-

68] of RF CMOS polyphase filters (PPFs). For instance, Behbahani [62] analyzed some

practical aspects in the design of RF CMOS PPFs and presented some design guidelines.

It is known that the main specification of the RF CMOS PPFs, quadrature accuracy, is

layout dependent. The requirements from component matching are contradictory to that

of minimization of noise coupling, signal loss and chip area [ZHA2]. Thus, a tradeoff

must be made in a specific design [ZHA2]. By simply following Behbahani’s guidelines

[62], it takes intensive trials and many design iterations to reach a good design [ZHA3].

Furthermore with layout considerations, optimization of RF CMOS PPFs cannot be

accomplished by using analytical calculations because of many constraints and tradeoffs

in the design [ZHA3]. In this situation, a computer-aided analysis and design

optimization tool for RF CMOS PPFs is of great help to the community of RF CMOS

PPFs designers, especially, for industrial projects, which are under time-to-market

pressures. Unfortunately, until now, there is not any such dedicated tool available

openly, to the author’s best knowledge. In this chapter, the author will present a novel

optimization tool for RF CMOS PPFs, based on MATLAB. The tool, which was


91

developed by the author, is termed PPFOPTIMA and it takes layout constraints

explicitly into considerations [ZHA3]. A genetic algorithm (GA) is proposed as the

optimization engine of PPFOPTIMA for its capability in avoiding local optima. In order

to enable users to carry out an automated design, PPFOPTIMA provides automated

synthesis and verification incorporating dedicated worst-case and Monte Carlo

simulation options. Experimental results illustrated in this chapter demonstrate that

PPFOPTIMA is effective and efficient in the synthesis, the optimization and the

verification of RF CMOS PPFs.

4.2 Design Considerations and Design Flow

As analyzed in chapter 2, CMOS process tolerances, component mismatch and

parasitic effects should be considered in the design of RF CMOS PPFs in order to

accomplish a robust design. Design considerations and heuristics from our design

practice in chapter 3 will be discussed in this section.

CMOS process tolerances should be considered by using the concept of

‘tolerance design’ [102]. Specifically, by enlarging the bandwidth of the PPF, the

operating frequency can be covered by upper and lower corners frequency of the RF

CMOS PPFs, no matter how they shift with process variations [102].


92

The most important specification for the design of an RF CMOS PPF is the

image reject ratio (IRR), which is related to the sensitivity of a receiver. From Monte

Carlo (MC) simulations, it has been found that [59]

UIR 10 Component-20log ( )R σ> , (4.1)

where UIRR is the ultimately obtainable IRR, and componentσ is the standard deviation of

the component mismatch. Typically, UIRR is in the range of 25 dB to 60 dB for RF

CMOS PPFs [ZHA3]. The standard deviation componentσ typically between 3% and 0.5%

[81]. It is known from the experimental study [73] that the variance of adjacent on-chip

resistors and capacitors and their area are reciprocally proportional, that is, neighboring

components with larger area have better matching than those with small area [73]. By

using the latest resistor and capacitor mismatch models [81] of (4.2) and (4.3), the

minimum chip area required by a target specification of the IRR can be calculated

[ZHA3].

The standard deviation componentσ of the component mismatch between two

neighboring resistors with identical layout can be calculated as follows [81]:

sh

2 2 2

2 2 2 2∆R/R R ∆W

sh e sh e( ) ( ) ( )eR

L L L

L R R L R R W Wσ σ σ σ

= + + ⋅ + ⋅ + + ∆ , (4.2)

with the standard deviation of the sheet resistance of the polysilicon resistors shR given

by

sh

sh

R

/Rσ =( )1 2

A

WL, (4.3)

with the standard deviation of the resistor width


93

21/∆W

∆WW

A=σ , (4.4)

and with the standard deviation of the end resistance coefficient eR defined as

ee RR A=σ , (4.5)

In (4.2) to (4.5), W and L are the resistor width and the resistor length, and W∆ is the

resistor width offset, shRA , WA∆ , and

eRA are fitting parameters obtained from massive

measurement results of shRσ , Wσ ∆ ,

eRσ , respectively.

For two neighboring capacitors with identical layout and capacitance C, the

variance 2/c Cσ ∆ of the component mismatch between them can be calculated as follows

[81]:

2d

2a

2p

2∆c/C σσσσ ++= , (4.6)

where pσ denotes the standard deviation of the periphery, given by

3/4p

p C

f=σ , (4.7)

with the standard deviation caused by area variations defined as

1/2a

a C

f=σ , (4.8)

and with the standard deviation caused by distance fluctuations

df .dd =σ , (4.9)

In (4.7) to (4.9), pf , af , df are constants describing the influences of periphery, area

and distance fluctuations. In the component mismatch model of capacitors, the

periphery component models the effect of edge roughness, and it is most significant for

small capacitors, which have a relatively large amount of edge capacitance. The area


94

component models the effect of short-range dielectric thickness variation, and it is most

significant for moderate size capacitors. The distance component models the effect of

global dielectric thickness variation across the wafer, and it becomes significant for

large capacitors or widely spaced capacitors [81].

Optimized for digital circuits, parasitic effects in CMOS processes are severe for

analog design. In the design of RF CMOS PPFs, there are two influences from parasitic

effects, namely [ZHA3]

• signal loss in addition to the theoretical 3 dB/stage caused by the lossy

substrate and

• capacitive noise coupling caused by parasitic capacitances.

For a given CMOS process, there are two ways to minimize the influences from

parasitic effects, one is to reduce the chip area as derived in Chapter 2 to decrease

capacitive noise coupling, the other is to use on-chip noise isolation techniques, namely

N-wells and guard rings, to protect noise sensitive analog circuits [ZHA2].

With considerations of parasitics, the maximum length of the resistor (MLR)

allowed in a CMOS process can be calculated by using the following equation [ZHA2]:

max,resop sh p

1=

πL

2 f R C (4.10)

where max,resL is the MLR, opf is the operating frequency of the RF CMOS PPF, shR is

the sheet resistance, and pC is parasitic capacitance per unit area between resistor and

the substrate. Before accomplishing physical layout, the MLR must be calculated to

avoid an inappropriate physical design.


95

Figure 4.1 A design flow for specification-driven automated synthesis of RF CMOS PPFs.

Specifications

Calculate minimum area of resistors and capacitors

Calculate values of resistors with

considerations of process tolerances

Increase the number of

cascaded stages by 1 Worst-case schematic simulation

Satisfy specifications?

Yes

No

RF CMOS PPF with 2 cascaded stages

Start

Calculate maximum lengths of resistors

Tradeoff between lengths and widths of resistors

Physical layout

Worst-case post-layout simulation

Satisfy specifications?

Yes

No

End


96

Based on aforementioned design considerations and the design heuristics, a

design flow for specification-driven automated synthesis of RF CMOS PPFs, as shown

in Figure 4.1, is proposed in this chapter. From Figure 4.1, we can see that all the design

considerations and constraints we have just discussed are included in this design flow.

Setting out from a given target specification, we start with a 2-stage RF CMOS PPF,

going through schematic design. If in the schematic simulation, the target specification

cannot be met, we move on to an RF CMOS PPF with three cascaded stages, and

deduce the compliance with the target specification. If we fail again, an RF CMOS PPF

with four cascaded stage is taken. After completing the schematic design, we carry out

the physical layout design and check whether the target specification can be met in the

post-layout simulation. If we fail, we go back to the schematic design using an RF

CMOS PPF with more cascaded stages, until the target specifications are met in the

post-layout simulation. Then we finish this design.

4.3 Problem Formulation for RF CMOS PPFs Optimization

In design of an RF CMOS PPF, the design objectives are

• to satisfy the target specifications required by a given wireless communication

standard and transceiver architecture,

• to minimize the chip area,

• to maximize the immunity to noise coupling, and


97

• to minimize additional signal losses.

Based on these design objectives, we can categorize the design considerations by

objective functions or constraints. Objective functions are the image reject ratio (IRR)

and the chip area. Here we use the relationship between the IRR IRR , the gain mismatch

gainm and the phase mismatch φ∆ as shown in the following equation (4.11) [84].

2 2gain

IR 10I Q

-20log ( )16( ) 4

mR

A A

φ∆= ++

, (4.11)

where the gain mismatch is defined by

gain I Qm A A= − , (4.12)

and where the phase mismatch is defined by

I Q|| | 90 |φ φ φ∆ = − − o . (4.13)

In (4.11) to (4.13), IA , QA , Iφ , Qφ are amplitudes and phases of the I path and the Q

path, respectively. All of these parameters can be calculated by using the transfer

functions of RF CMOS PPFs derived in chapter 3 of this dissertation.

In order to minimize the component mismatch, the layout pattern shown in Figure

3.4 and Figure 3.5 should be followed in the layout of resistors and capacitors. In this

layout pattern, the chip areas of an RF CMOS PPF with n cascaded stages is given by

[ZHA3]

∑=

+=n

iii SSS

1CR )166( , (4.14)

where the chip area of resistors in the ith stage is

R R Ri i iS L W= ⋅ , (4.15)


98

and where the chip area of the capacitor in the ith stage is

Cunit

= ii

CS

C , (4.16)

In (4.14) to (4.16), RiL , RiW are the length and width of resistors in the ith stage

respectively, iC is the capacitor value in the ith stage, unitC is the unit capacitance of the

capacitors, i=1,2,3,…n, n is the number of the cascaded stages in an RF CMOS PPF.

Though the chip areas defined in (4.14) is an objective to be minimized for the

reduction of substrate noise coupling, for a given target specification of the IRR and a

given CMOS process, there exist a minimum area of resistors and capacitors to avoid

bad component mismatch which cannot satisfy the specification of the IRR. Using the

equation (4.1), we find the corresponding standard deviation of resistors and capacitors

for a given target specification of IRR:

IR- / 20∆R/Rσ > 10 R , (4.17)

IR- /20∆C/Cσ > 10 R . (4.18)

In (4.17) and (4.18), IRR is the IRR and ∆R/Rσ , ∆C/Cσ are standard deviations of

matched resistors and capacitors, respectively. These figures can be calculated by using

the latest component mismatch models proposed by Cheng [81] as shown in (4.2) and

(4.6). Another constraint in practice is that the length of resistors should be less than the

MLR calculated by using (4.10).

In order to choose a suitable optimization algorithm, let us investigate the

relationship between the IRR and component values. For the purpose of simplification,

we take the amplitude mismatch between the I path and the Q path in a 2-stage RF


99

CMOS PPF for a Bluetooth receiver as an example. In the design of this 2-stage RF

CMOS PPFs, if we determine the values of the resistors in each stage, then the

capacitors values are automatically determined by the relationship between corner

frequency and RC time constant shown in (3.4). In this situation the amplitude

mismatch A∆ can be expressed as:

2

1 21/ 22 22

1

1 2 1 2 2 2

1ω ω

1 2ω ω ω ω ω

AR

R

ω

ω ω ω ω

+∆ =

− + + +

(4.19)

Where ω is the frequency of interest; 1ω and 2ω are the upper and the lower corner

frequency of this RF CMOS PPF, which are 2.4 GHz and 2.48 GHz respectively; 1R

and 2R are resistors values in the first stage and the second stage, which are in the range

of 50 Ohms to 500 Ohms. Taking these into account, we can rewrite (4.19) as

2

4 2

1

2 2 2

xz

x x x y

+=− + +

. (4.20)

Where z is A∆ ; x is the normalized frequency of interest, which is in the range of [0.5,

5]; and y is the ratio between 1R and 2R , which is in the range of [0.5, 5]. The three-

dimensional plot of A∆ is shown in Figure 4.2. From this plot, we can clearly see that it

has several maxima. This means that, the function in equation (4.20) has multiple local

optima. For RF CMOS PPFs with more cascaded stages, the relationship between

quadrature accuracy, component values, and frequency of interest is even more complex.

In this situation, an in-depth analysis of possible optimization methods is necessary.


100

Figure 4.2 A three-dimensional plot of the amplitude mismatch

in a 2-stage RF CMOS PPF, expressed in the equation (4.20).

4.4 Analysis of Possible Optimization Methodologies

4.4.1 Limitations of Conventional Optimization Methods

There are various optimization theories and techniques in engineering

mathematics [104]. Essentially, the targets of the optimization techniques are to search

for the maximum or the minimum values in a certain search space [104].

The easiest way to find the accurate maximum or minimum in a continuous

function is to use calculus-based search [104], in which the existence of the first order

and the second order derivatives are required [104]. However, this technique is not


101

suitable for solving the optimization problem in this dissertation for the following

reasons:

• Tendency to get trapped on local maxima. Even though a better solution may

exist, all moves from the local maxima seem to decrease the fitness of the

solution.

• Dependence on the existence of derivatives.

There is another optimization technique, gradient descent [104], which is also

referred to as hill-climbing [104]. This technique performs well on functions with only

one peak. However, for functions with multiple peaks, the first peak found is taken as

the maximum, no matter whether it is the highest peak or not [104]. Thus, hill-climbing

is not suitable for our optimization problem either.

For optimization problems with multiple peaks or multiple valleys, the direct way

is to use random search to find the optima. Unfortunately, for most of the real-life

optimization problems, this method is inefficient because of the vast amount of

computations and is therefore not selected by the author.

In order to save computational complexity, iterated climbing was proposed [104],

which is a combination of random search and gradient search. Once a peak has been

located, the hill-climbing is started again, but with another randomly chosen starting

point. However, since each random trial is performed in isolation, no overall idea of the

topology and photography of the search space is obtained. Furthermore, trials are

randomly allocated over the entire search space; many points in regions of low fitness


102

are evaluated as points in high fitness regions. Therefore, iterated climbing is not

considered as a proper candidate for the optimization of RF CMOS PPFs and is

therefore not selected by the author.

4.4.2 Simulated Annealing

Simulated annealing (SA) was invented by Kirkpatrick [109] in 1982 by

mimicking the process of cooling of a solid. Essentially, it is a modified version of hill-

climbing. The search process starts from a random point in the search space. Then, a

random move is made. In the case of climbing reaches a higher point by this move, the

move is accepted. Otherwise, it is accepted only with probability p(t) (where t is time).

The probability function p(t), essentially relates to the temperature. A value of p(t) close

to one refers to a high temperature. Decreasing p(t) means lowering the temperature. At

time t equal to zero, p(0) is approximately one. As time t passes, p(t) is decreased,

approaching zero at the end of the SA operation. This procedure is an analogy with the

cooling of a solid. Therefore, initially, any moves are accepted, but as p(t) decreases, the

probability of accepting a negative move is lowered. This optimization algorithm has

been successfully applied in analog design automation [109-112], especially in IC

layout routing. However simulated annealing deals with only one candidate at a time.

Like random search, it does not build an overall picture of the search space, and no

information from previous moves is used to guide the selection of new moves. This

leads to a low efficiency in solving complicated optimization problems. Therefore, SA


103

does not seem promising for the optimization of RF CMOS PPFs and is hence not

selected by the author.

4.4.3 Genetic Algorithms

A. Introductory Remarks

Genetic algorithms (GAs) are adaptive methods based on the genetic processes of

biological organisms. For GA basics, please refer to the famous textbook written by

Goldberg [113]. Here we focus on the analysis of the suitability of GAs to our

optimization problems. The following four key schemes, which characterize GAs,

create the benefit of GAs over conventional optimization techniques and the SA we had

mentioned in the previous subsections.

B. Direct Manipulation of Coding [118]

GAs manipulate decisions or control variable representations at a string level to

exploit similarities among high-performance strings [118]. Other methods usually deal

with functions and their control variables directly. GAs deal with parameters of finite

length, which are encoded using a finite alphabet, rather than directly manipulating the

parameters. This means that the search is constrained neither by the continuity of the

function under investigation, nor by the existence of a derivative function. Moreover, by

exploring similarities in codings, GAs can deal effectively with a broader class of

functions than can many other procedures.


104

C. Search from a Population, not a Single Point [118]

In GAs, the search sets out from a population of many individuals, rather than

starting from just one point. This parallelism means that the search will not become

trapped on a local maximum - especially if a measure of diversity-maintenance is

incorporated into the algorithm, even if one candidate may become trapped on a local

maximum, but the need to maintain diversity in the search population means that other

candidates will therefore avoid that particular area of the search space.

D. Search via Sampling, a Blind Search [118]

GAs remains general by exploiting information available in any search problem.

GAs process similarities in the coding with information and rank the structures

according to their survival capabilities in the current environment. By exploiting such

widely available information, GAs may be applied to virtually any problem.

E. Search Using Stochastic Operators, no Deterministic Rules [118]

The transition rules used by GAs are probabilistic, not deterministic. A

distinction, however, exists between the randomized operators of GAs and other

methods that are simple random walks. GAs use random choice to guide a highly

exploitative search.

From the analysis of GA, SA and conventional optimization techniques, we can

see that the GA is well suited for the solving of complicated optimization problems with


105

multiple objectives and, consequently, with multiple local optima. Since the

optimization of RF CMOS PPFs has multiple optimization objectives and multiple local

optima, conventional optimization algorithms are not recommendable. Therefore, the

GA was chosen for the optimization engine in PPFOPTIMA, the optimization tool for

RF CMOS PPFs developed by the author of this dissertation.

4.5 MATLAB Implementation of Genetic Algorithms

4.5.1 Overview

In this section, the MATLAB implementation of genetic algorithms is presented.

MATLAB is chosen as the programming platform for the following reasons:

• It provides many built-in auxiliary functions useful for function optimization.

• It is efficient for numerical computations.

4.5.2 Chromosome Representation

The process of GA begins with a set of potential solutions, termed chromosomes,

that are randomly generated or selected [113]. The entire set of those chromosomes

establishes a population [113]. For any GA, a chromosome represents an individual in

the population [113]. The representation scheme determines how the problem is

structured in the GA and how the genetic operators that are used [113]. Each

chromosome is made up of a sequence of genes, taken from a certain alphabet [113].


106

The sequence of genes makes the value of a particular chromosome. This value of a

chromosome is called genotypes [113]. The genotypes are uniquely mapped into the

decision variables, called phenotypic domain. The chromosomes evolve during several

iterations or generations.

The most commonly used representation in GAs is based on the binary alphabet

0, 1. In this case, each decision variable is encoded by a binary string. The binary

strings are concatenated to form a chromosome. For example, a problem with two

variables, X1 and X2, may be mapped onto the chromosome structure in the way shown

in Figure 4.3. X1 is encoded by 10 bits and X2 is represented by 15 bits. This encoding

may reflect the level of accuracy or range of the individual decision variables.

Examining the chromosome string in isolation yields no information about the problem

we are trying to solve. It is only with the decoding of the chromosome into its

phenotypic values that any meaning can be applied to the representation.

Figure 4.3 Mapping variables into chromosomes.

1 2

Chromosome comprising the values of two

concatenated decision variables X and X

1

Decision variable

X

1011010011 010111010100101

2

Decision variable

X


107

4.5.3 Genetic Operation


The genetic operation includes three basic operators, namely, reproduction,

crossover and mutation [113]. These three basic operators provide the basic search

mechanism of genetic algorithms (GAs) and they are used to create new solutions based

on existing solutions in the population [113].

B. Reproduction

Reproduction is the selection of individuals to produce successive generations

[113]. Reproduction plays an important role in a genetic algorithm. A probabilistic

selection is performed based upon the individual's fitness in such a way that the fitter

individuals have an increased chance of being selected. An individual in the population

can be selected more than once with all individuals in the population having a chance of

being selected to be reproduced into the next generation.

A common selection approach assigns a probability of selection, jP , to each

individual, j, based on its fitness value [113]. In general, j is an integer, which can vary

between 1 and N [113]. A series of N random numbers is generated and compared

against the cumulative probability, N

1j

j

p=∑ , of the population. The appropriate individual,

i, is selected and copied into the new population.


108

Various methods exist to assign probabilities to individuals, namely, roulette

wheel, linear ranking and geometric ranking. In the GA implemented in this work, the

roulette wheel mechanism was chosen to probabilistically select individuals based on

some measures of their performance.

The roulette wheel selection method, developed by Holland [113], was the first

selection method. The probability, jP , for each individual is defined by

N

1

jj

jj

FP

F=

=∑

(4.21)

Where jF is the fitness of individual j, and N is the population size.

As shown in Figure 4.4, the circumference of the roulette wheel is the sum of all

six individual’s fitness values; the sizes of the sectors stand for the fitness values of the

individuals. Individual 5, which has the largest sector, is the fittest individual, whereas

individuals 6 and 4 are the least fit and, correspondingly, have the smallest sectors

within the roulette wheel. To select an individual, a uniformly distributed random

number is generated in the predefined interval, and the individual whose sector spans

the random number is selected. This process is repeated until the desired number of

individuals has been selected. The roulette wheel selection is not only used in genetic

algorithms in conjunction with maximization, when combined with e.g. windowing and

scaling, it is well suited for minimization and negativity. Thus, roulette wheel selection

method has been chosen by the author.


109

Figure 4.4 Roulette wheel selections.

C. Crossover and Mutation

Crossover involves splitting two individuals and then combining one half of each

individual with the other pair [113]. Mutation involves altering one individual to

produce a single new solution [113]. If chromosomes are as binary types represented,

simple crossover and binary mutation can be adopted [113].

Let X and Y be two m-dimensional row vectors denoting individuals (parents)

from the population. Simple crossover generates a random sample r from a uniform

distribution from 1 to m and creates two new individuals according to equation (4.22)

and equation (4.23).

i

ii

x , if i rx

y , otherwise

<′ =

(4.22)

i

ii

y , if i ry

x , otherwise

<′ =

(4.23)

For example, consider the two pairs of binary strings


110

P1 = 1 0 0 1 0 1 1 0, (4.24)

P2 = 1 0 1 1 1 0 0 0. (4.25)

If an integer position, i, is selected uniformly and randomly, and the genetic

information exchanged between the individuals about this point, then two new offspring

strings are produced. The two offspring below are produced when the crossover point i

= 5 is selected,

O1 = 1 0 0 1 0 0 0 0, (4.26)

O2 = 1 0 1 1 1 1 1 0. (4.27)

This crossover operation is not necessarily performed on all strings in the

population. Instead, it is applied with a probability Px when the pairs are chosen for

breeding.

Binary mutation flips each bit in every individual in the population with

probability Pm according to following equation (4. 28):

mi

ii

U(0,1) P1 , if x

x , otherwise

x <−′ =

(4.28)

For example, mutating the fourth bit of O1 leads to the new string:

O1m = 1 0 0 0 0 0 0 0. (4.29)

Mutation is generally considered to be a background operator, which ensures that,

the probability of searching a particular subspace of the problem space is never zero.

This has the effect of tending to inhibit the possibility of converging to a local optimum,

rather than the global optimum.


111

4.5.4 Initialization, Termination and Fitness Functions

GAs simultaneously operate on a number of potential solutions, called a

population, consisting of some encoding of the parameter set [113]. The mode of

operation generates the required number of individuals using a random number

generator that uniformly distributes numbers in the desired range. However, since GAs

can iteratively improve existing solutions, i.e., solutions from other heuristics and/or

current practices, the initial population can be seeded with potentially good solutions,

with the remainder of the population being randomly generated solutions. One of the

variations is the extended random initialization procedure of Bramlette [113], whereby a

number of random initializations are tried for each individual and the one with the best

performance is chosen for the initial population. Typically, a population is composed of

between 30 and 100 individuals, which is about ten times the number of variables to be

optimized.

The GA moves from generation to generation, selecting parents and reproducing

offspring until a termination criterion is met. Because the GA is a stochastic search

method, it is difficult to formally specify convergence criteria. The most frequently used

stopping criterion is a specified maximum number of generations [118]. Another

termination strategy involves population convergence criteria. In general, GAs will

force much of the entire population to converge to a single solution. When the sum of

the deviations among individuals becomes smaller than some specified threshold, the

algorithm can be terminated. The algorithm can also be terminated due to a lack of


112

improvement in the best solution over a specified number of generations. Alternatively,

a target value for the evaluation measure can be established based on some arbitrarily

‘acceptable’ threshold. Several strategies can be used in conjunction with each other.

The objective function [118] is used to provide a measure of how individuals

have performed in the problem domain. In the case of a minimization problem, the

fittest individuals will have the lowest numerical value of the associated objective

function. This raw measure of fitness is usually only used as an intermediate stage in

determining the relative performance of individuals in a GA [118].

Another function, the fitness function, is normally used to transform the value of

the objective function f (x) into measure of relative fitness, thus:

F (x) = g (f (x)) (4.30)

where ( )g ⋅ transforms the value of the objective function to a non-negative number and

( )F ⋅ is the resulting relative fitness. This mapping is always necessary when the

objective function is to be minimized as the lower objective function values correspond

to fitter individuals.

A linear transformation which offsets the objective function is often used prior to

fitness assignment, such that,

F (x) = a f (x) + b, (4.31)

where a is a positive scaling factor. If we are minimizing, the offset b is used to ensure

that the resulting fitness values are non-negative.


113

Figure 4.5 Flowchart of the genetic algorithm implemented in PPFOPTIMA.

The main functions used in the optimization engine of PPFOPTIMA are listed in

the Table 4.1. These GA elements have been integrated into an operational GA. The

flowchart of this GA is shown in Figure 4.5.

Start

Create initial random population

Apply fitness measure to

individuals of the population

Termination

criteria satisfied?

Select genetic operations

Select one

individual

based on fitness

Select two

individuals

based on fitness

Select one

individual

based on fitness

Perform

reproduction Perform

crossover

Perform

mutation

Copy into new

population

Insert offsprings

into new

population

Insert mutation

into new

population

Select best

individuals

Design best

individual

for result

End

Create a new

population

No

Yes


114

Main functions Description

ga Basic function

roulette Selection function

simpleXover Crossover function

binaryMutation Mutation function.

initializega Initialization function

maxGenTerm Termination function

Table 4.1 Main functions used in the optimization engine of PPFOPTIMA.

4.6 Implementation of PPFOPTIMA

4.6.1 Introduction

In Figure 4.6, the framework of PPFOPTIMA is shown, from which we can see

that inputs needed from the user are specifications of the design, e.g. the operating

frequency, and process related information. Under these conditions, the GA used in the

optimization engine makes tradeoffs between resistors, capacitors and their

corresponding physical size. Values of the resistors and the capacitors are determined

by process tolerances and constraints from thermal noise. The physical sizes of the

resistors and the capacitors are determined by component mismatch properties and

constraints from noise coupling and insertion loss. The output of PPFOPTIMA

comprises the optimized circuit dimension for schematic and the physical layout, which


115

is convenient and efficient for designers under time-to-market pressures. Details of the

system implementation will be illustrated in the following subsections.

Figure 4.6 Framework of PPFOPTIMA.

noiseThermal noise

Coupled lossInsertion

capacitors and resistorsof valuesoff-Trade capacitors and resistors

of areas off-Trade

algorithms genetic usingby capacitors andresistors of areas and valuesoff-Trade

layout and schmaticfor sizingCircuit

accuracyQuadrature tolerance

Process mismatch Component sconstraint

Design

RF CMOS PPFsdesign specifications CMOS

process files

capacitors andresistors of

design Tolerance Maximum areaof resistors

and capacitors

Fewer stages,Minimum area valuesresistors

on sConstraintarea

Minimum


116

4.6.2 Construction of Fitness Functions

Unlike using conventional optimization methods, the first step of the GA is to

establish a fitness function F, which is the criterion in selecting the suitable population

for the next generation. In PPFOPTIMA, the fitness function is a sum of IRR, chip area

and the component mismatch of the resistors and the capacitors with different weights:

1 2 R C 3 IR( )F w S w M M w R= + + + (4.32)

The real numbers 1 2 3, and w w w are the weighting factors for chip area S, for the

component mismatch of resistors RM , and capacitors CM , and for the image reject ratio

(IRR) IRR , respectively. In our application, since S , RM and CM are in the order of

310− , 1w and 2w are chosen in the order of 410 to differentiate good population from

poor ones.

4.6.3 Graphical User Interface

In order to facilitate users input and visualize the outputs, a graphical user

interface (GUI) has been developed for PPFOPTIMA, which is depicted in Figure 4.7.

The blank area shown in Figure 4.7 will be filled by simulation results. The GUI has the

following features:

1. Explicit and regulated user input information: Operating frequency and

process information required are listed in the user input boxes. In order to

avoid possible errors caused by mis-operations, data types in input boxes are


117

regulated. For example, in the case of inputs that require numbers, strings will

not be accepted, and a popup window will notify the user about this.

Figure 4.7 Graphical user interface of PPFOPTIMA.


118

2. Visualized and classified outputs: The optimization results are classified and

displayed in different categorized windows.

3. Programmable weights for the fitness function: Weighting factors for

component mismatch, chip area and image reject ratio are taken as inputs.

4. Preset values for tutorial purpose: In order to support novice users, input

parameters for RF CMOS PPF designed for a Bluetooth receiver in Fraunhofer-

IMS 0.6 µm CMOS technology are provided in the GUI.

5. Integrated high-level verification toolboxes: In order to verify the optimization

results, high-level verification toolboxes, namely, Monte Carlo (MC)

simulation and worst-case simulation toolboxes, have been developed and

integrated into PPFOPTIMA.

4.6.4 Verification Toolboxes


In order to verify the performance of the synthesized circuit, dedicated

verification toolboxes, namely, Monte Carlo (MC) simulation and worst-case simulation

were developed as already mentioned in section 4.6.3. In our application, the Monte

Carlo (MC) simulation provides an estimate of the discrete probability density of gain

mismatch and phase mismatch, which are related to the required IRR, the main

specification of RF CMOS PPFs. The worst-case simulation gives us the possible worst-


119

case values for both the gain mismatch and the phase mismatch versus frequency,

respectively.

B. Monte Carlo Simulation

In a real design, there are a lot of processes happening in probabilistic ways, and

it is hard to model or describe them by using ordinary or differential equations.

Therefore, a statistical representation is a suitable way to characterize them. In statistical

simulations, sequences of random numbers with a certain probability distribution

function (PDF) are used to model the stochastic process. Based on this modeling

technique, responses of systems to stochastic inputs can be simulated. Usually, many

statistical simulations runs are conducted and averaged to reach good accuracy of the

simulation results. This is the method of Monte Carlo (MC) simulation, which has

following primary components:

• Probability distribution functions (PDF) - the statistical description of the

physical, or mathematical, system.

• Random number generator - a source of random numbers to mimic stochastic

factors in the physical system.

• Sampling rule - a prescription for sampling from random numbers with the

specified PDF.

• Scoring (or tallying) - the accumulated outcomes to meet overall tallies or

scores for the quantities of interest.


120

• Error estimation - an estimate of the statistical error (variance) as a function of

the number of trials and other quantities.

Process tolerances and component mismatch in integrated circuits are

consequences of stochastic processes within a certain range, and they are usually

available in CMOS process files derived by elaborate measurements. It is known that

both process tolerances and component mismatch have truncated Gaussian probability

distribution functions [102]. In our application, Monte Carlo (MC) simulation can be

applied to verify the statistical nature of the IRR with certain process tolerances and a

resultant component mismatch. In order to obtain comparatively stable and reliable

results, many simulation steps must be performed, and the results are taken as an

average over the number of Monte Carlo (MC) runs. A flowchart of the Monte Carlo

simulation toolbox implemented in PPFOPTIMA is shown in Figure 4.8.

The effect of the image reject ratio (IRR) used in this toolbox is represented by

the following equation:

IR gain phase ,1 ,1 , ,1

( , ) [ ( , , , )] /p

i i i n i ni

R F m m F R C R C p=

= = ∑ L , (4.33)

[ ] [ ], R1 1i j j RR R P M= ∗ ± ∗ ± , (4.34)

[ ] [ ], 1 1i j j C CC C P M= ∗ ± ∗ ± , (4.35)

where RP , CP , RM and CM are vectors of random numbers normalized by process

tolerances and component mismatch of a given CMOS process. In (4.33) 1i p= L , p is

the number of Monte Carlo (MC) runs. In (4.34) and (4.35), nj L1= , n is the number

of cascaded stages in an RF CMOS PPF.


121

Figure 4.8 Flowchart of the Monte Carlo simulation option in PPFOPTIMA

C．Worst-case Simulation

In circuit design, it is required that the worst-case circuit performance satisfies a

given target specifications. For most circuits, worst-case circuit performance occurs as

the combination of worst-case process tolerance, temperature and component mismatch.

Start

numbers random Generate

Normalize random numbers according to probability distrinution

functions of process tolerancesand component mismatch

Take random valuesof resitors and capacitors

to evaluate the IRR

Meet terminationcriteria?

End

Print out, and save output

Computer outputby averaging the statistical samples

Yes

No


122

Figure 4.9 Flowchart of the novel worst-case simulation methodology in PPFOPTIMA.

To obtain worst-case simulation results with widely used commercial tools like,

e.g. Cadence SpectreRF, the user need to carry out the simulations with variety of

models. For example, slow-to-fast, fast-to-slow, fast-to-fast, slow-to-slow and slow-gain

models must be considered. The various simulation results must then be compared in

order to find the worst-case performance of the circuit. This procedure is inconvenient

and expensive with respect to computation and simulation time.

Start

s toleranceprocesswithout PPFs CMOS RF of ratio

reject image worst Find

Shift corner frequencies, tomake worst image reject ratiofall into the operating band

capacitors and resistorsof areas the toaccording

mismatchcomponent Calculate

Add mismatch intoneighboring components

Calculate, save and printout simulation results

End


123

RF CMOS PPFs with varying numbers of cascaded stages have varying worst-

case boundaries. Through analysis, we have found that worst-case boundary of RF

CMOS PPFs does not necessarily to occur with the worst process variation. Instead, it

occurs in the case of a process variation where worst image reject ratio falls in the

operating frequency band. The verification toolboxes of PPFOPTIMA take this

observation into account, yielding a novel simulation methodology proposed by the

author of this dissertation. The flowchart of the novel worst-case simulation

methodology is shown in Figure 4.9.

4.7 Experimental Results

4.7.1 Overview

In order to verify effectiveness of PPFOPTIMA, two categories of experiments

have been conducted. The first category of experiments is to synthesize and verify

multi-stage RF CMOS PPFs for local oscillator (LO) quadrature generation in receivers

for a given communication standard. In these experiments, RF CMOS PPFs for LO

quadrature generation in a Bluetooth receiver are taken as examples. The second

category of experiments is the automated synthesis and verification of RF CMOS PPFs

in receivers for different communication standards, especially, for GSM, DECT, and

WCDMA receivers, by using PPFOPTIMA. The CMOS process used in all of these

experiments is Fraunhofer-IMS 0.6 µm CMOS process.


124

4.7.2 Experiments on RF CMOS PPFs for a Bluetooth Receiver


In this section, multi-stage RF CMOS PPFs following target specifications of a

low-IF Bluetooth receiver, which were synthesized and analyzed with PPFOPTIMA,

will be presented. These RF CMOS PPFs are designed for LO quadrature generation in

this low-IF Bluetooth receiver in which the LO frequency is 2.45 GHz. The

specification derived from the Bluetooth standard and the low-IF receiver architecture is

an image reject ratio (IRR) great than 30 dB, which can be translated into a gain

mismatch of less than 0.7 dB, and a phase mismatch of less than 5 degrees. Although

PPFOPTIMA is capable of synthesizing RF CMOS PPFs automatically, in these

experiments, RF CMOS PPFs have been synthesized and verified by using

PPFOPTIMA manually to verify the effectiveness of the tool.

The first step in using PPFOPTIMA is the dimensioning of the circuit by pressing

the button ‘Optimization by GA’ in the GUI. Synthesis results, which include values,

lengths, widths and layout areas of resistors and capacitors, are shown as ‘Optimization

Results’ in the GUI. After that worst-case simulations are used to check whether the

performance of the synthesized RF CMOS PPF satisfies the target specifications over

all of the process corners. Then, Monte Carlo simulations are used to check the

probability distribution of the gain mismatch and the phase difference, respectively.


125

B. Experiments on a 2-Stage RF CMOS PPF for a Bluetooth Receiver

Shown in Figure 4.10 is the synthesis result of a 2-stage RF CMOS PPF designed

for LO quadrature generation in a low-IF Bluetooth receiver.

Figure 4.10 Synthesis result of a 2-stage RF CMOS PPF for a Bluetooth receiver.

The worst-case simulation results of the RF CMOS PPF synthesized in Figure

4.10 are shown in Figure 4.11. The gain mismatch, gainm , is 0.81 dB and the phase


126

difference, phased , is 92 degrees, i.e., the phase mismatch, phasem , is 2 degrees. As the

gain mismatch cannot meet the target specification, RF CMOS PPFs with more

cascaded stages must be tried in the following subsections.

Figure 4.11 Worst-case simulation results of a 2-stage RF CMOS PPF for a Bluetooth receiver.

Upper plot: gain mismatch gainm versus frequency f.

Lower plot: phase difference phased versus frequency f.


127

As worst-case simulation only provides a boundary of the circuit performance, in

order to know the statistical distribution of the circuit performance, Monte Carlo (MC)

simulation runs using 2000 samples of RF CMOS PPFs have been conducted.

Figure 4.12 Monte Carlo simulation results of a 2-stage RF CMOS PPF for a Bluetooth receiver.

Upper plot: histogram of the gain mismatch gainm .

Lower plot: histogram of the phase difference phased .


128

The Monte Carlo simulation results are expressed as frequency of occurrence

histogram for different intervals of the gain mismatch and the phase difference, and they

are shown in Figure 4.12. In the MC simulations, after 200 MC runs, the simulation

results enter steady state. In 2000 samples of RF CMOS PPFs, 1880, that is, 94% of

them have a gain mismatch between 0.52 dB and 0.81 dB, and 1892, that is 94.6% of

them have a phase difference between 91.27 degrees and 91.97 degrees, that is, a phase

mismatch between 1.27 degrees and 1.97 degrees.

C. Experiments on a 3-Stage RF CMOS PPF for a Bluetooth Receiver

As the 2-stage RF CMOS PPF synthesized cannot satisfy the target specifications,

now let us check whether a 3-stage RF CMOS PPF can satisfy the specifications.

Synthesis results of a 3-stage RF CMOS PPF are shown in Figure 4.13. The

worst-case simulation results shown in Figure 4.14 reveal a gain mismatch of 0.083 dB,

and a phase difference of 90.15 degrees, i.e., a phase mismatch of 0.15 degrees. As the

target specifications of this design are met, a 3-stage RF CMOS PPF is chosen.

To verify the statistical distribution of gain mismatch and phase mismatch in this

design, MC simulation has been used. After 200 MC runs, the simulation results enter

steady state. The obtained results are shown in Figure 4.15. In 2000 samples of RF

CMOS PPFs, 90% of them have a gain mismatch of less than 0.047 dB and a phase

mismatch of less than 0.1 degrees.


129

Figure 4.13 Synthesis results of a 3-stage RF CMOS PPF for a Bluetooth receiver.

The author would like to mention that in Monte Carlo simulations gain mismatch

and phase mismatch could be worse than that from worst-case simulation, because in


130

the worst-case simulation, the worst image reject ratio (IRR), in which worst gain

mismatch and worst phase mismatch of the RF CMOS PPFs do not necessarily happen

at the same time.





131




D. Experiments on a 4-stage RF CMOS PPF for a Bluetooth Receiver

In this section, a 4-stage RF CMOS PPF for a Bluetooth receiver was synthesized.

The synthesis results are shown as ‘Optimization Results’ in Figure 4.16.


132

Figure 4.16 Synthesis results of a 4-stage RF CMOS PPF for a Bluetooth receiver.


133

The worst-case simulation results of the RF CMOS PPF are shown in Figure 4.17,

in which the gain mismatch is 0.028 dB, and the phase difference is 90.031 degrees, i.e.,

the phase mismatch is equal to 0.031 degrees.





134

Monte Carlo simulation results are shown in Figure 4.18, after 700 Monte Carlo

runs, the simulation result enter steady state. The MC simulation results show that in

2000 samples of RF CMOS PPFs, more than 90% of them have a gain mismatch less

than 0.005 dB and phase mismatch less than 0.01 degrees.





135

From the simulation results above, we can see that the performance of the 4-stage

RF CMOS PPF synthesized is more than sufficient for the target specifications in a low-

IF Bluetooth receiver. However, for passive filters like RF CMOS PPFs, they have a

nominal insertion loss of 3 dB/stage. In order to compensate the insertion loss, buffer

amplifiers are usually necessary for RF CMOS PPFs with many cascaded stages, thus,

causing higher power consumption. In the design of CMOS wireless receivers, low

power consumption is required. Thus, the strategy for topology selection in

PPFOPTIMA is to use as few cascaded stages as possible to satisfy the target

specifications.

4.7.3 Automated Synthesis and Verification of RF CMOS PPFs for

Different Communication Standards


In Figure 4.19 the GUI of PPFOPTIMA for automated synthesis of RF CMOS

PPFs is shown. The specification needed is image reject ratio (IRR) for CMOS wireless

receivers. With a given specification, a 2-stage RF CMOS PPF is chosen at first, and

optimized by PPFOPTIMA, and then the worst-case simulation is conducted to verify

whether the specifications can be met. If the specifications are met, a popup window

will notify the user that a 2-stage RF CMOS PPF is chosen and the optimized circuit

dimension is provided. If a 2-stage RF CMOS PPF cannot meet the specifications, the


136

number of stages of the RF CMOS PPFs is increased by one until the specification can

be met. The final results are an RF CMOS PPF with a selected number of stages and

optimized circuit dimensions.

Figure 4.19 Graphic user interfaces of PPFOPTIMA

for automated synthesis of RF CMOS PPFs.

In order to verify the effectiveness of PPFOPTIMA in automated synthesis and

verification, RF CMOS PPFs for LO quadrature generation in receivers of different

communication standards, for example, GSM, DECT and WCDMA were synthesized

and verified. The results will be presented in the remainder of this section 4.7.3.


137

B. Automated Synthesis and Verification of RF CMOS PPF for a GSM

Receiver

Figure 4.20 Synthesis results of a 3-stage-RF CMOS PPF for a GSM receiver.


138

In this experiment, an RF CMOS PPF targeting an image reject ratio (IRR) of at

least 40 dB for a low-IF GSM receiver was synthesized and verified. A 3-stage RF

CMOS PPF was chosen by PPFOPTIMA, the synthesis results are shown in Figure 4.20.

Figure 4.21 Worst-case simulation results of a 3-stage RF CMOS PPF for a GSM receiver.




139

The worst-case simulation results are shown in Figure 4.21, in which the IRR is

49.2 dB, the gain mismatch is 0.0512 dB and the phase difference is 90.1072 degrees,

i.e., the phase mismatch is 0.1072 degrees.

Figure 4.22 Monte Carlo simulation results of a 3-stage RF CMOS PPF for a GSM receiver.




140

In order to check the statistical distribution of gain mismatch and phase mismatch,

a Monte Carlo (MC) simulation in PPFOPTIMA was conducted. After 200 MC runs,

the results enter a steady state. The results of the MC simulation are shown in the Figure

4.22, from which we can see that in 2000 samples of RF CMOS PPFs, more than 90%

of them have a gain mismatch of less than 0.045 dB, and a phase mismatch of less than

0.1 degrees.

Figure 4.23 Synthesis results of a 2-stage-RF CMOS PPF for a DECT receiver.


141

C. Automated Synthesis and Verification of RF CMOS PPFs for a

DECT Receiver

In this section, an RF CMOS PPF targeting 28 dB IRR for a low-IF DECT

receiver has been synthesized and verified. A 2-stage RF CMOS PPF was chosen by

PPFOPTIMA, the synthesis results are shown in Figure 4.23.

Figure 4.24 Worst-case simulation results of a 2-stage RF CMOS PPF for a DECT receiver.




142

The worst-case simulation results of the RF CMOS PPF synthesized according to

Figure 4.23 are shown in Figure 4.24. The worst-case IRR is 30.7 dB, the gain

mismatch is 0.612 dB, and the phase difference is 91.51 degrees, i.e., the phase

mismatch is 1.51 degrees.

Figure 4.25 Monte Carlo simulation results of a 2-stage RF CMOS PPF for a DECT receiver.




143

To check the statistical distribution of gain and the phase mismatch, Monte Carlo

simulation in PPFOPTIMA was applied. After 200 MC runs, the simulation results

enter a steady state. The MC simulation results are shown in Figure 4.25. In 2000

samples of RF CMOS PPFs, more than 90% of them have the gain mismatch between

0.45 dB to 0.61 dB, and the phase mismatch between 0.9 degrees and 1.5 degrees.

D. Automated Synthesis and Verification of RF CMOS PPFs for a

WCDMA Receiver

In this section, an RF CMOS PPF for LO quadrature generation in a low-IF

WCDMA receiver was synthesized and verified. The LO frequency is set to 1.90 GHz,

and the targeting specification of RF CMOS PPF is 36 dB IRR. A 3-stage RF CMOS

PPF was chosen by PPFOPTIMA, the synthesis results are shown in Figure 4.26.

The worst-case simulation results are shown in Figure 4.27, in which the worst-

case IRR is 49.1 dB, the gain mismatch is 0.0541 dB, the phase difference is 90.1201

degrees, i.e., the phase mismatch is 0.1201 degrees.

To check the statistical distribution of the gain mismatch and the phase mismatch,

a Monte Carlo simulation in PPFOPTIMA has been conducted. The MC simulation

results shown in Figure 4.28 demonstrate that, in 2000 samples of RF CMOS PPFs,

more than 90% of them have the gain mismatch less than 0.04 dB, and the phase

mismatch less than 0.1 degrees.


144

Figure 4.26 Synthesis results of a 3-stage-RF CMOS PPF for a WCDMA receiver.


145

Figure 4.27 Worst-case simulation results of a 3-stage RF CMOS PPF for a WCDMA receiver.




146

Figure 4.28 Monte Carlo simulation results of a 3-stage RF CMOS PPF for a WCDMA receiver.



4.8 Summary

In this chapter, the optimization problem for RF CMOS PPFs has been

formulated. Possible optimization algorithms like calculus-based optimization, gradient


147

descend, simulated annealing (SA) and genetic algorithms (GA) have been analyzed.

GA has been selected for the optimization engine because of its advantages in escaping

from local optima.

PPFOPTIMA, a GA based optimization tool for RF CMOS PPFs has been

implemented by using MATLAB. In order to facilitate designers in design automation,

this tool has been developed further into a specification-driven automated synthesis and

verification environment incorporating dedicated worst-case and Monte Carlo

simulation.

In order to verify the effectiveness of PPFOPTIMA, two types of experiments

have been conducted. The first type of experiments consists of the synthesis and the

verification of multi-stage RF CMOS PPFs in a receiver for a given communication

standard. Here, a low-IF Bluetooth receiver has been taken as an example. The second

type of experiments consists of the synthesis and verification of the RF CMOS PPFs for

different communications standards, namely GSM, DECT, and WCDMA. The

experimental results demonstrate that PPFOPTIMA is effective in synthesis,

optimization and verification of RF CMOS PPFs.

Focusing on the optimization of RF CMOS PPFs, this chapter has the following

contributions:

1. The methodology of optimization of RF CMOS PPFs with layout

considerations by using genetic algorithm (GA) has been illustrated.


148

2. PPFOPTIMA, a computer-aided synthesis and optimization tool for RF

CMOS PPFs, has been presented.

3. Dedicated worst-case simulation and Monte Carlo simulation toolboxes for

verification of the performance of RF CMOS PPF have been developed.

Chapter 5 Summary and Outlook

149

Chapter 5

Summary and Outlook

5.1 Summary

RF CMOS polyphase filters (PPFs) is an efficient solution for high accuracy radio

frequency quadrature generation in CMOS technology. The main specification of the

RF CMOS PPF, namely the quadrature accuracy, is layout dependent; and the

requirements from component match are contradictory to that of minimization of noise

coupling, signal loss and chip area, thus a tradeoff must be made in a specific design.

By simply following guidelines proposed by Behbahani [62], it takes intensive trials and

many design iterations to reach a good design. Furthermore, with layout considerations,

the optimization of RF CMOS PPFs cannot be solved by using analytical calculations,

because of many constraints and tradeoffs in the design.

In this situation, a computer-aided analysis and design optimization tool for RF

CMOS PPFs is of great help to the community of RF CMOS PPFs designers, especially

for industrial projects, which are under time-to-market pressure. Unfortunately, until

now, there is no such dedicated tool available, to the author’s best knowledge.

Focused on analysis, design and optimization of RF CMOS PPFs, this dissertation

has following contributions:

1. PPFANA, a dedicated software tool for simulation of quadrature accuracy in

RF CMOS PPFs influenced by process tolerances and component mismatch,


150

has been developed. It provides an efficient computer-aided analysis for RF

CMOS PPFs.

2. Based on the simplified RF CMOS models of resistors, and capacitors,

parasitic effects in RF CMOS PPFs have been analyzed. Techniques of

minimizing those unwanted influences from parasitics have been proposed, and

physical design considerations for RF CMOS polyphase filter in system-on-a-

chip (SoC) solution have been analyzed.

3. Silicon success of an RF CMOS polyphase filter for a low-IF Bluetooth

receiver was achieved. Measurement results from this practical design further

verified the effectiveness of PPFANA. Design consideration and design

heuristics from this design practice lay a solid foundation for this dissertation.

4. Based on heuristics from the practical design of an RF CMOS PPF, a design

automation oriented design flow for RF CMOS PPFs has been proposed.

5. By analysis on existing optimization algorithms, the methodology of synthesis

and verification of RF CMOS PPFs with layout considerations by using genetic

algorithm (GA) has been proposed in this dissertation.

6. PPFOPTIMA, a GA based optimization tool for RF CMOS PPFs has been

implemented by using MATLAB. In order to enable the designers in design

automation, PPFOPTIMA has been developed further towards a specification-

driven automated synthesis and verification environment incorporating

dedicated worst-case and Monte Carlo simulation options.


151

5.2 Outlook

Compared to Electronic Design Automation (EDA) tools for digital circuit, those

for analog circuit, especially for RF circuit, are still in their infancy. Nowadays, RF

analog design is still a kind of ‘art’, which is done by ‘Analog Artists’. Although some

research groups, e.g. at the University of California at Los Angeles, and at the

Katholieke Universiteit Leuven, have contributed some design automation tools for

some RF components, there is still a far way to go in the research topic.

Optimization is a key step in design automation of analog and RF circuits. Though

RF circuits are comparatively simple in the sense of the number of components, taking

layout considerations into account, optimization of RF circuits is complicated because

of multiple design objectives and constraints. In this situation, genetic algorithm (GA)

has advantages over other conventional optimization techniques for its capability in

escaping from local optima. The successful application of GA in the optimization of RF

CMOS PPFs demonstrates that GA is effective and efficient in RF circuit optimization.

However, this optimization methodology has following drawbacks:

1. GA is able to escape from local optima, but it is not good at finding exact

optimum point.

2. GA is not intelligent enough to be trained, that is, for GA based optimization

tools, if modification is needed; change of fitness function and more work is

needed.

3. It is still heuristics based trials to find suitable weights in applications of GAs.


152

Compared to GA, an artificial neural network (ANN) may lead to improvements

because of its self-learning mechanisms. For RF circuit synthesis and optimization, an

ANN may find a wider application in the future.

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[119] Allstot, D. J.; Choi, K.; Park, J.: Parasitic-aware optimization of CMOS RF circuits.

Boston: Kluwer Academic Publishers, February 2003

XV. Ph.D Dissertations

[120] Kuhn, W. B.: Design of integrated, low power, radio receivers in BiCMOS technologies.

Ph.D dissertation in Electrical Engineering, Virginia Polytechnic Institute and State

University, 1995.

[121] Min, J.: Analysis and design of a frequency-hopped spread-spectrum transceiver for

wireless personal communications. Ph.D. dissertation in Electrical Engineering,

University of California, Los Angeles, 1996.

[122] Chang, J.: An integrated 900 MHz spread-spectrum wireless receiver in 1-µm CMOS and

a suspended inductor technique. Ph.D dissertation in Electrical Engineering, University

of California, Los Angeles, March 1998.

[123] Shaeffer, D.K.: The design and implementation of low-power CMOS radio receivers.

Ph.D dissertation in Electrical Engineering, Stanford University, 1998.

[124] Basedau, P. M.: Analysis and design of LC and crystal oscillator. Ph.D dissertation in

Electric Engineering, Swiss Federal Institute of Technology, Zurich, 1999.

[125] Delaurenti, M.: Design and optimization techniques of high–speed VLSI circuits. PhD

dissertation in Electrical Engineering, Politecnico di Torino, December 1999.

References

162

[126] Piazza, F.: Low power RF-receiver front-end ICs for mobile communications. Ph.D

dissertation in Electrical Engineering, Swiss Federal Institute of Technology, Zurich,

2000.

[127] Lin, L.: Design techniques for high performance integrated frequency synthesizers for

multi-standard wireless communication applications. Ph.D dissertation in Electrical

Engineering, University of California, Berkeley, 2000.

[128] Hitko, D. A.: Circuit design and technological limitations of silicon RFICs for wireless

applications. Ph.D dissertation in Electrical Engineering, Massachusetts Institute of

Technology, 2002

Appendix 1. Publications from the Author

163

Appendix 1

Publications from the Author

Part 1. Publications from the Author After 2001

ZHA1. Zhang, Z.S.; Christoffers, N.; Hosticka B.J.; Kaiser, T.; Kokozinski, R.; Lin, J.: A survey of

oscillator phase-noise: analysis and simulation. Proceedings of the Second Joint Symposium on

Opto- & Microelectronic Devices and Circuits (SODC’02), pp. 322-326, Stuttgart/Germany,

March 2002. (in English)

ZHA2. Zhang, Z.S.; Kolnsberg, S.; Kokozinski, R.: Design considerations for RF CMOS polyphase

filter in low-cost digital SoC solution. On-line Proceedings of The sixth workshop on Analog

Circuit: http://www.imtek.uni-freiburg.de/workshop2004/, Freiburg/Germany, March 2004. (in

English)

ZHA3. Zhang, Z.S.; Kolnsberg, S.; Kokozinski, R.: Synthesis of RF CMOS polyphase filters with

layout considerations. Chinese Journal of Semiconductors, vol.25 (2004), No.12, pp. 1612-1617.

(in English)

Part 2. Publications from the Author Before 2001

ZHA4. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Sun, M.: Design and experiments of PHV-I portable high-

frequency vibrator, Instrument Technology & Sensors, vol.36 (2000), No.10, pp. 14-18. (in

Chinese)

ZHA5. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Yu, S.B.: Development of portable high-frequency vibrator.

Automation in Petroleum Industry, vol.14 (2000), No.6, pp. 14-17. (in Chinese)

ZHA6. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Yu, S.B.: Design and implementation of portable high

frequency vibrator. Measurement & Control Technique, vol.19 (2000), No.10, pp. 10-53. (in

Chinese)

ZHA7. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Yu, S.B.: An industrial PC-based measurement & control

system in portable high-frequency vibrator. Computer Automated Measurement & Control, vol.8

(2000), No.4, pp. 10-13. (in Chinese)

ZHA8. Zhang, Z.S.; Lin, J.; Jiang, H.: A novel interface between non-standard keyboard and industrial

PC in portable measurement & control instruments. Measurement & Control Technique, vol.19

(2000), No.6, pp. 10-61. (in Chinese)

ZHA9. Zhuang, J.; Lin, J.; Wu D. J.; Zhang, B.J.; Zhang, Z.S.: Analysis on amplitude characteristics of

the coupling process between PHV and the earth. Journal of Changchun University of Science &

Technology, vol.29 (1999), No.2, pp. 184-188. (in Chinese)

Appendix 1. Publications from the Author

164

ZHA10. Yu, S.B.; Lin, J.; Zhang, Z.S.: The Study on the frequency characteristic of portable high-

frequency vibrator. Proceedings of The Fourth International Conference on Electronic

Measurement & Instruments (ICEMI’99), Harbin/China, pp. 191-195, 1999. (in English)

ZHA11. Zhang, Z.S.; Lin, J.; Yu, S.B.: The optimal signal design of linear sweep in PHVS system.

Petroleum Instruments, vol.12 (1998), No.3, pp. 1-5. (in Chinese)

ZHA12. Zhang, Z.S.: Serial ADC DSP101/102 directly interfaced with DSP and its applications.

Petroleum Instruments, vol.12 (1998), No.2, pp. 19-22. (in Chinese)

ZHA13. Zhang, Z.S.; Lin, J.; Fu X.J.; Sun C.T.: An analysis on sweep signal design of vibroseis

system. Petroleum Instruments, vol.11 (1997), No.6, pp. 10-14. (in Chinese)

ZHA14. Zhang, Z.S.: Realizing digital programmable filter by using universal filter UAF42.

Electronic Technology, vol.24 (1997), No.9, pp. 13-34. (in Chinese)

Appendix 2 Transfer Functions of Multi-stage RF CMOS PPFs

165

Appendix 2

Transfer Functions of Multi-Stage RF CMOS PPFs

A2.1 Explicit Transfer Function of 2-Stage RF CMOS PPFs

The transfer function of the 2-stage RF CMOS PPF can be obtained by multiplying the chain

matrices of each stage as follows:

in 1 1 1

in 1 1 11 1

2 2 2 out

2 2 2 out2 2

V 1+j R C R1=


1+j R C R V1 .

2j C 1+j R C I1-ωR C

ωω ωωω

ω ω

(A2.1)

The open circuit ( outI = 0) complex transfer function is:

1 1 2 2

21 1 2 2 1 2 1 2 1 2

(1- R C )(1- R C )H(j )=

1+j (R C +R C +2R C )- (R R C C )

ω ωωω ω

. (A2.2)

Equation (A2.1) can also be expressed as:

1 2H(j )=H (j )+jH (j )ω ω ω , (A2.3)

Where 2

1 2 1 21

21 1 2 2 1 2 1 2 1 2

1+ (R R C C )H (j )=

1+j (R C +R C +2R C )- (R R C C )

ωωω ω

, (A2.4)

1 1 2 22

21 1 2 2 1 2 1 2 1 2

j (R C +R C )H (j )=

1+j (R C +R C +2R C )- (R R C C )

ωωω ω

, (A2.5)

A2.2 Explicit Transfer Function of 3-Stage RF CMOS PPFs

The transfer function of the 3-stage RF CMOS PPF can be derived by multiplying

the chain matrices of each stage as follows:

in 1 1 1

in 1 1 11 1

2 2 2

2 2 22 2

3 3 3 out

3 3 3 out3 3

V 1+j R C R1=


1+j R C R1

2j C 1+j R C1-ωR C

1+j R C R V1 .

2j C 1+j R C I1- R C

ωω ωωω

ω ωω

ω ωω

(A2.6)

The complex transfer function is:

Appendix 2 Transfer Functions of Multi-stage RF CMOS PPFs

166

N(j )

H(j )=D(j )

ωωω

, (A2.7)

Where 1 1 2 2 3 3N(j )=(1- R C )(1- R C )(1- R C )ω ω ω ω , (A2.8)

R ID(j )=D ( )+jD ( )ω ω ω , (A2.9)

2

R 1 2 1 2 1 3 1 3 2 3 2 3

1 3 2 3 1 2 1 3 1 2 2 3

D ( )=1- (R R C C +R R C C +R R C C

+2R R C C +2R R C C +2R R C C ),

ω ω (A2.10)

I 1 1 2 2 3 3 1 2 1 3

32 3 1 2 3 1 2 3

D ( )= (R C +R C +R C +2R C +2R C

+2R C )- (R R R C C C ).

ω ωω

(A2.11)

Equation (A2.7) can also be expressed as:

1 2H(j )=H (j )+jH (j )ω ω ω (A2.12)

Where 2

1 2 1 2 1 3 1 3 2 3 2 31

1+ (R R C C +R R C C +R R C C )H (j )=

D(j )

ωωω

, (A2.13)

3

1 1 3 3 2 2 1 2 3 1 2 32

j (R C +R C +R C )+j (R R R C C C )H (j )=

D(j )

ω ωωω

. (A2.14)

Appendix 3 Source Code of PPFANA

167

Appendix 3

Source Code of PPFANA

A3.1 Source Code for Analysis of Gain Mismatch Influenced by Process Tolerances

A.3.1.1 Source Code for Analysis of Gain Mismatch Influenced by Process

Tolerances in 2-stage RF CMOS PPFs

global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global pt_r; global pt_c;

f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input

f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));

rt_1=findobj(gcf,'Tag','rt_1'); rt_1=str2double(get(rt_1,'string'));


ct_1=findobj(gcf,'Tag','ct_1'); ct_1=str2double(get(ct_1,'string'));


pt_r=findobj(gcf,'Tag','pt_r'); pt_r=str2double(get(pt_r,'string'));

if isempty(pt_r), pt_r=0.16; else pt_r= pt_r; end

pt_c=findobj(gcf,'Tag','pt_c'); pt_c=str2double(get(pt_c,'string'));

if isempty(pt_c), pt_c=0.21; else pt_c=pt_c; end

if isempty(f_1), f_1=1.44e9; else f_1=f_1; end


if isempty(rt_1), rt_1=223.5; else rt_1=rt_1; end


if isempty(ct_1), ct_1=205.2; else ct_1=ct_1; end


figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);

c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15;

r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);

a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./

(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1

*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RinCin=abs(h-n); plot(w,RinCin,'g -s')

c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);

a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


168


*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RdeCde=abs(h-n); plot(w,RdeCde,'b -*')

c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);

a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RinCde=abs(h-n); plot(w,RinCde,'r -^')

c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);

a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RdeCin=abs(h-n); plot(w,RdeCin,'k -d')

legend('RinCin','RdeCde','RinCde','RdeCin')

w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4);

c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);

a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RinCin=abs(h-n); plot(w,RinCin,'g -')


a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RdeCde=abs(h-n); plot(w,RdeCde,'b -.') % %



169

a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RinCde=abs(h-n); plot(w,RinCde,'r --') % %


a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=20.*log10(a); n=20.*log10(d); RdeCin=abs(h-n); plot(w,RdeCin,'k :')

xlabel('\itf / GHz','FontSize', 14)

ylabel('\itm-\itgain / dB','FontSize', 14)

title('Gain mismatch influenced by process tolerances','FontSize', 14)

A.3.1.2 Source Code for Analysis of Gain Mismatch Influenced by Process Tolerances in 3-stage RF CMOS PPFs

global f_1; global f_2; global rth_1; global rth_2; global rth_3; global cth_1; global cth_2; global cth_3;

global cm; global pt_r; global pt_c;



rth_1=findobj(gcf,'Tag','rth_1'); rth_1=str2double(get(rth_1,'string'));



cth_1=findobj(gcf,'Tag','cth_1'); cth_1=str2double(get(cth_1,'string'));






if isempty(pt_c), pt_c=0.21;else pt_c=pt_c; end



if isempty(rth_1), rth_1=263.9; else rth_1=rth_1; end




170

if isempty(cth_1), cth_1=173.8; else cth_1=cth_1; end



figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);

c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15;

c3_1=cth_1.*(1+pt_c).*1e-15; r3_1=rth_1.*(1+pt_r);

r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);

c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./

(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-

(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2

+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.

*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

b=abs((-

j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3

_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-


+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).

^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -s')

c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;

r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -*')

c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15; c3_1=cth_1.*(1+pt_c).*1e-15;



171


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'r -^')


r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'k -d')


w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4); c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15;

c3_1=cth_1.*(1+pt_c).*1e-15; r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));


172

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -.')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'r --')



173



(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'k :')

xlabel('\itf / GHz','FontSize', 14) ylabel('\itm-\itgain / dB','FontSize', 14)


A.3.1.3 Source Code for Analysis of Gain Mismatch Influenced by Process Tolerances in 4-stage RF CMOS PPFs

global f_1; global f_4; global rf_1; global rf_2; global rf_3; global rf_4; global cf_1; global cf_2;

global cf_3; global cf_4; global pt_r; global pt_c;



rf_1=findobj(gcf,'Tag','rf_1'); rf_1=str2double(get(rf_1,'string'));




cf_1=findobj(gcf,'Tag','cf_1'); cf_1=str2double(get(cf_1,'string'));










if isempty(rf_1), rf_1=289.9; else rf_1=rf_1; end

if isempty(rf_2), rf_2=329; else rf_2=rf_2; end


174



if isempty(cf_1), cf_1=158.2; else cf_1=cf_1; end




figure hold on

w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15;

c2=cf_2.*(1+pt_c).*1e-15; c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r);

r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);

d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4

*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3

+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4

+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1

*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)

+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)-

(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4

+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4

+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c

4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3


+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4

+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c

3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c

4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RinCin=abs(m-k); plot(w,RinCin,'g -s')

c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;

c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);



+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.



+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) -



+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));


175









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RdeCde=abs(m-k); plot(w,RdeCde,'b -*') %

c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15; c2=cf_2.*(1+pt_c).*1e-15;

c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));


4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3




+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) -

(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r

1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4+2*r1*r2*r4*c3

*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RdeCin=abs(m-k); plot(w,RdeCin,'r -^') %


c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);






3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*

c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


176


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RinCde=abs(m-k); plot(w,RinCde,'k -d')


w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4); c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15;

c2=cf_2.*(1+pt_c).*1e-15; c1=cf_1.*(1+pt_c).*1e-15;

r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);







c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RinCin=abs(m-k); plot(w,RinCin,'g -')


c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1-pt_r);

r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);





177






+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RdeCde=abs(m-k); plot(w,RdeCde,'b -.') %




*c3)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3







+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RdeCin=abs(m-k); plot(w,RdeCin,'r --') %



d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*

c4*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3


178







+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));


4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-

(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*

r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*

c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2

.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+

2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)-


1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4

+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

m=20.*log10(d); k=20.*log10(a); RinCde=abs(m-k); plot(w,RinCde,'k :')



A3.2 Source Code for Analysis of Phase Mismatch Influenced

by Process Tolerances

A.3.2.1 Source Code for Analysis of Phase Mismatch Influenced by

Process Tolerances in 2-stage RF CMOS PPFs

global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global pt_r; global pt_c;












179







figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2); c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15;

r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);

a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.

*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*

w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

h=atan(a); n=atan(d); RinCin=(h-n).*180./pi+90; plot(w,RinCin,'g -s')



*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));


w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

h=atan(a); n=atan(d); RdeCde=(h-n).*180./pi+90; plot(w,RdeCde,'b -*')



*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));


w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

h=atan(a); n=atan(d); RinCde=(h-n).*180./pi+90; plot(w,RinCde,'r -^')



*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

h=atan(a); n=atan(d); RdeCin=(h-n).*180./pi+90; plot(w,RdeCin,'k -d')


w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4);

c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));


w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

h=atan(a); n=atan(d); RinCin=(h-n).*180./pi+90; plot(w,RinCin,'g -')


180



*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));


w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

h=atan(a); n=atan(d); RdeCde=(h-n).*180./pi+90; plot(w,RdeCde,'b -.') % %



*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));


w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

h=atan(a); n=atan(d); RinCde=(h-n).*180./pi+90; plot(w,RinCde,'r --') % %



*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));


w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

h=atan(a); n=atan(d); RdeCin=(h-n).*180./pi+90; plot(w,RdeCin,'k :')

xlabel('\itf / GHz','FontSize', 14) ylabel('\itd-\rmphase / degrees','FontSize', 14)

title('Phase difference influenced by process tolerances','FontSize', 14)

A.3.2.2 Source Code for Analysis of Phase Mismatch Influenced by Process Tolerance in 3-stage RF CMOS PPFs

global f_1; global f_2; global rth_1; global rth_2; global rth_3; global cth_1; global cth_2;

global cth_3; global cm; global pt_r; global pt_c;

















181






figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);



h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./

(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-

(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3

_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.

*r3_3.*c3_1.*c3_2.*c3_3));

t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c

3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -s')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));


3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -*')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)


182

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-

(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2.


*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90; plot(w,RdeCin,'r -d')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'k -d')


w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4); c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15;

c3_1=cth_1.*(1+pt_c).*1e-15; r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)


183

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));


3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -.')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

t=abs(( j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c

3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90; plot(w,RdeCin,'r --')




(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



184


*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));


3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'k :')



A.3.2.3 Source Code for Analysis of Phase Mismatch Influenced by Process Tolerance in 4-stage RF CMOS PPFs

global f_1; global f_4; global rf_1; global rf_2; global rf_3; global rf_4; global cf_1;

global cf_2; global cf_3; global cf_4; global cm; global pt_r; global pt_c;


























185

figure hold on

w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*(1+pt_c).*1e-15;


r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);

c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4








+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c






4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RinCin=(n-h).*180./pi+90; plot(w,RinCin,'g -s')











+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









186


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RdeCde=(n-h).*180./pi+90; plot(w,RdeCde,'b -*') %






+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4.+2*r1





+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RdeCin=(n-h).*180./pi+90; plot(w,RdeCin,'r -^') %











+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));






3*c4)


187

+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)

-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RinCde=(n-h).*180./pi+90; plot(w,RinCde,'k -d')


w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4); c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15;

c2=cf_2.*(1+pt_c).*1e-15; c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r);

r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RinCin=(n-h).*180./pi+90; plot(w,RinCin,'g -')




*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))/(1-




.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2

+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r

3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c

4+2*r1*r4*r3*c3*c2*c4+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));




188




+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)-


1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4+2*r1*r2*r4*c3

*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RdeCde=(n-h).*180./pi+90; plot(w,RdeCde,'b -.') %











+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));






+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)

-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RdeCin=(n-h).*180./pi+90; plot(w,RdeCin,'r --') %











+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));


189









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RinCde=(n-h).*180./pi+90; plot(w,RinCde,'k :')



A3.3 Source Code for Analysis of Gain Mismatch Influenced

by Component Mismatch

A.3.3.1 Source Code for Analysis of Gain Mismatch Influenced by Component Mismatch in 2-stage RF CMOS PPFs

global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global cm; global pt;



rt_1=findobj(gcf,'Tag','rt_1'); rt_1=str2double(get(rt_1,'string')); rt_2=findobj(gcf,'Tag','rt_2');

rt_2=str2double(get(rt_2,'string')); ct_1=findobj(gcf,'Tag','ct_1'); ct_1=str2double(get(ct_1,'string'));


cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string'));

if isempty(cm), cm=0.01;else cm=cm; end

if isempty(f_1), f_1=1.44e9; else f_1=f_1;end

if isempty(f_2), f_2=3.47e9; else f_2=f_2;end

if isempty(rt_1), rt_1=223.5; else rt_1=rt_1;end

if isempty(rt_2), rt_2=265.6; else rt_2=rt_2;end



figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2); c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15;

r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.

*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15;


190

r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);

c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w

)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b); k=20.*log10(c); RinCin=abs(m-k); plot(w,RinCin,'g -s')

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);


)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b); k=20.*log10(c); RdeCde=abs(m-k); plot(w,RdeCde,'b -*')

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1+cm);r2_2=rt_2.*(1+cm);


)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b); k=20.*log10(c); RinCde=abs(m-k); plot(w,RinCde,'r - ^')

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);


)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b); k=20.*log10(c); RdeCin=abs(m-k); plot(w,RdeCin,'k -d')


w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2); c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);


)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b); k=20.*log10(c); RinCin=abs(m-k); plot(w,RinCin,'g -')

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));



)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b); k=20.*log10(c); RdeCde=abs(m-k); plot(w,RdeCde,'b -.') % %

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;


191


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);

c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.

*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b);k=20.*log10(c); RinCde=abs(m-k); plot(w,RinCde,'r --')% %

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));



)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));

m=20.*log10(b); k=20.*log10(c); RdeCin=abs(m-k); plot(w,RdeCin,'k :')


title('Gain mismatch influenced by component mismatch','FontSize', 14)


global f_1; global f_2; global rth_1; global rth_2; global rth_3; global cth_1;

global cth_2; global cth_3; global cm;










if isempty(cm), cm=0.01; else cm=cm; end









figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2); c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15;


192

c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;

r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);

b=abs((*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3

_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -s') c3_3=cth_3.*1e-15;

c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;

r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -*') c3_3=cth_3.*1e-15;



(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));


193



b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'r -^') c3_3=cth_3.*1e-15;



(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));



b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'k -d')


w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2); c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;

r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)


194

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -')

c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -.') %



(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'r --') %



195


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'k :') %




global f_1; global f_4; global rf_1; global rf_2; global rf_3; global rf_4; global cf_1; global cf_2;

global cf_3; global cf_4; global cm;



















196






figure hold on

w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15;

c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;

a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4






(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*

r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4

+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1+cm).*1e-15; c3=cf_3.*(1+cm).*1e-15; c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;

r1=rf_1.*(1+cm); r2=rf_2.*(1+cm); r3=rf_3.*(1+cm); r4=rf_4.*(1+cm);

d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c




*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4).




+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RinCin=abs(l-m); plot(w,RinCin,'g -s')

c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1;

r2=rf_2; r3=rf_3; r4=rf_4;









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1-cm).*1e-15; c3=cf_3.*(1-cm).*1e-15; c2=cf_2.*(1-cm).*1e-15; c1=cf_1.*(1-cm).*1e-15;


197

r1=rf_1.*(1-cm); r2=rf_2.*(1-cm); r3=rf_3.*(1-cm); r4=rf_4.*(1-cm);









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RdeCde=abs(l-m); plot(w,RdeCde,'b -*') % %

c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));




4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-





2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)-



+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RinCde=abs(l-m); plot(w,RinCde,'r -^') % %









198



+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));








+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)-



+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RdeCin=abs(l-m); plot(w,RdeCin,'k -d')


w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4); c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15;






*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4).




+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));




4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-





2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r

4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2

*r1*r4*r3*c3*c2*c4 +2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RinCin=abs(l-m); plot(w,RinCin,'g -')



199









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));









(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RdeCde=abs(l-m); plot(w,RdeCde,'b -.') % %







+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)

-(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));












200

+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RinCde=abs(l-m); plot(w,RinCde,'r --') % %

c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1;

r2=rf_2; r3=rf_3; r4=rf_4;







c4)j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1+cm).*1e-15; c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;



4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))/(1-





2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)-

j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

l=20.*log10(a); m=20.*log10(d); RdeCin=abs(l-m); plot(w,RdeCin,'k :')%



A3.4 Source Code for Analysis of Phase Mismatch Influenced by Component Mismatch

A.3.4.1 Source Code for Analysis of Phase Mismatch Influenced by Component Mismatch in 2-stage RF CMOS PPFs

global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global cm; global pt;

f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string'));%get the value from GUI input






201










figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1;r2_2=rt_2;


*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));


c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); RinCin=(m-k).*180./pi+90;

plot(w,RinCin,'g -s') c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));


c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); RdeCde=(m-k).*180./pi+90; plot(w,RdeCde,'b -*')

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));


c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); RinCde=(m-k).*180./pi+90; plot(w,RinCde,'r -^')

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));


202


c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); RdeCin=(m-k).*180./pi+90; plot(w,RdeCin,'k -d')


w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2);

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));


c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); RinCin=(m-k).*180./pi+90; plot(w,RinCin,'g -')

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));


c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); RdeCde=(m-k).*180./pi+90; plot(w,RdeCde,'b -.') % %

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));


c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); RinCde=(m-k).*180./pi+90; plot(w,RinCde,'r --') % %

c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

c2_2=ct_2.*(1+cm).*1e-15;c2_1=ct_1.*(1+cm).*1e-15;r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);

c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


203


*c2_2)));

m=atan(b); k=atan(c); RdeCin=(m-k).*180./pi+90; plot(w,RdeCin,'k :')


title('Phase difference influenced by component mismatch','FontSize', 14)


global f_1; global f_2; global rth_1; global rth_2; global rth_3;

global cth_1; global cth_2; global cth_3; global cm;



















figure hold on

w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);

c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;

r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));




204

t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -s')

c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1;

r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -*')

c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;

r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



205


*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'r -^')

c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1;

r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90;

plot(w,RdeCin,'k -d') legend('RinCin','RdeCde','RinCde','RdeCin')

w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2); c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15;

c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -')



206


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_3.*c3_1.*c3_2.*c3_3));



t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -.') %

c3_3=cth_3.*1e-15;c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;

r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));



t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'r --') %



(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));


207



t=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90; plot(w,RdeCin,'k :') %




global f_1;global f_4;global rf_1;global rf_2;global rf_3;global rf_4;

global cf_1;global cf_2; global cf_3;global cf_4;global cm;























figure hold on


208

w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15;







+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)

-(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4


+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));











+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RinCin=(n-h).*180./pi+90; plot(w,RinCin,'g -s')

c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15;

r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;







c4)-



+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;c1=cf_1.*(1-cm).*1e-15;

r1=rf_1.*(1-cm);r2=rf_2.*(1-cm);r3=rf_3.*(1-cm);r4=rf_4.*(1-cm);






209





+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RdeCde=(n-h).*180./pi+90; plot(w,RdeCde,'b -*') % %

c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15;

r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;

c1=cf_1.*(1-cm).*1e-15;r1=rf_1.*(1+cm);r2=rf_2.*(1+cm);r3=rf_3.*(1+cm);r4=rf_4.*(1+cm);









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c);n=atan(b);RinCde=(n-h).*180./pi+90;plot(w,RinCde,'r -^')% %

c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;r1=rf_1;r2=rf_2; r3=rf_3;r4=rf_4;









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1+cm).*1e-15;c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15;c1=cf_1.*(1+cm).*1e-15;



210









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c);n=atan(b);RdeCin=(n-h).*180./pi+90;plot(w,RdeCin,'k -d')


w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4);

c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;

r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1+cm).*1e-15;c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15;c1=cf_1.*(1+cm).*1e-15;

r1=rf_1.*(1+cm);r2=rf_2.*(1+cm);r3=rf_3.*(1+cm);r4=rf_4.*(1+cm);


4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-








h=atan(c);n=atan(b);RinCin=(n-h).*180./pi+90;plot(w,RinCin,'g -')

c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;

r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;






211





+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;

c1=cf_1.*(1-cm).*1e-15;r1=rf_1.*(1-cm);r2=rf_2.*(1-cm);r3=rf_3.*(1-cm);r4=rf_4.*(1-cm);









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c);n=atan(b);RdeCde=(n-h).*180./pi+90;plot(w,RdeCde,'b -.') % %

c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;c1=cf_1.*(1-cm).*1e-15;

r1=rf_1.*(1+cm);r2=rf_2.*(1+cm);r3=rf_3.*(1+cm);r4=rf_4.*(1+cm);









+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

h=atan(c); n=atan(b); RinCde=(n-h).*180./pi+90; plot(w,RinCde,'r --') % %

c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;


212



+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c..






+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));

c4=cf_4.*(1+cm).*1e-15;c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;



4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-








h=atan(c);n=atan(b);RdeCin=(n-h).*180./pi+90;plot(w,RdeCin,'k :')%



A3.5 Source Code for Analysis of Gain Mismatch in the Worst-Case Simulation

A.3.5.1 Source Code for Analysis of Gain Mismatch in the Worst-Case Simulation of 2-stage RF CMOS PPFs

global f_1;global f_2;global rt_1;global rt_2;global ct_1;global ct_2;global cm; global pt;












213





h = waitbar(0,'Simulation is processing, please wait'); for x=1:100, waitbar(x/100,h), end close(h)

rand('seed',0); % Crossover Operators xFns = 'simpleXover'; xOpts = [.4];

% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];

% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations

% Selection Function electFn = 'roulette'; selectOps = [];

% Evaluation Function evalFn = 'ppfana2stage'; evalOps = []; %type gaMichEval3stage;

% Bounds on the variables b_1=f_1; b_2=f_2; bounds = [b_1 b_2];

% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];

% Generate an intialize population of size 20

startPop = initializega(20,bounds,'ppfana2stage',[],[1e-6 0]);

% Lets run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,

termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);

% x is the best solution found %x; % endPop is the ending population %endPop;

% trace is a trace of the best value and average value of generations %trace;

% Lets increase the population size by running the defaults rand('seed',0); termOps=[100];

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,

selectFn,selectOps); % x is the best solution found x; fmax=x; f=f_1+(f_2-f_1)./2; ptt=fmax./f;

pt_two=1-sqrt(1./ptt); figure

w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3); c2_2=ct_2.*1e-15+ct_2.*1e-5.*pt_two;

c2_1=ct_1.*1e-15+ct_1.*1e-15.*pt_two; r2_1=rt_1+rt_1.*pt_two; r2_2=rt_2+rt_2.*pt_two;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

c2_2=ct_2.*1e-15+ct_2.*1e-15.*(1+cm).*pt_two; c2_1=ct_1.*1e-15+ct_1.*1e-15.*(1+cm).*pt_two;

r2_1=rt_1+rt_1.*(1+cm).*pt_two; r2_2=rt_2+rt_2.*(1+cm).*pt_two;

c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=20.*log10(b); k=20.*log10(c); wcga=abs(m-k); plot(w,wcga,'b -')


title(' Worst case gain mismatch of 2-stage RF CMOS PPFs','FontSize', 14)


global f_1; global f_2; global rth_1; global rth_2; global rth_3;

global cth_1; global cth_2; global cth_3; global cm;


214


f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string')); rth_1=findobj(gcf,'Tag','rth_1');

rth_1=str2double(get(rth_1,'string')); rth_2=findobj(gcf,'Tag','rth_2');

rth_2=str2double(get(rth_2,'string')); rth_3=findobj(gcf,'Tag','rth_3');

rth_3=str2double(get(rth_3,'string')); cth_1=findobj(gcf,'Tag','cth_1');

cth_1=str2double(get(cth_1,'string')); cth_2=findobj(gcf,'Tag','cth_2');

cth_2=str2double(get(cth_2,'string')); cth_3=findobj(gcf,'Tag','cth_3');

cth_3=str2double(get(cth_3,'string')); cm=findobj(gcf,'Tag','cm');

cm=str2double(get(cm,'string')); if isempty(cm), cm=0.01; else cm=cm; end









h = waitbar(0,'Simulation is processing, please wait');

for x=1:100, waitbar(x/100,h), end close(h)




% Selection Function selectFn = 'roulette'; selectOps = [];

% Evaluation Function evalFn = 'ppfana3stage'; evalOps = [];

%type gaMichEval3stage;

% Bounds on the variables

% b_1=f_1./(1-2.*pt); % b_2=f_2./(1+2.*pt); b_1=f_1; b_2=f_2; bounds = [b_1 b_2];


% Generate an intialize population of size 20 startPop = initializega(20,bounds,'ppfana3stage',[],[1e-6 0]);



% x is the best solution found x;% endPop is the ending population%endPop;

% trace is a trace of the best value and average value of generations trace;

% Lets increase the population size by running the defaults

rand('seed',0); ermOps=[100]; x endPop bestPop

trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps, selectFn,selectOps);

% x is the best solution found

x; max=x; f=f_1+(f_2-f_1)./2;ptt=fmax./f;

pt_three=1-sqrt(1./ptt);figure

w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3);c3_3=cth_3.*(1+pt_three).*1e-15;

c3_2=cth_2.*(1+pt_three).*1e-15;c3_1=cth_1.*(1+pt_three).*1e-15;


215

r3_1=rth_1.*(1+pt_three);r3_2=rth_2.*(1+pt_three);r3_3=rth_3.*(1+pt_three);


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

c3_3=cth_3.*(1+pt_three).*(1+cm).*1e-15;c3_2=cth_2.*(1+pt_three).*(1+cm).*1e-15;

c3_1=cth_1.*(1+pt_three).*(1+cm).*1e-15;r3_1=rth_1.*(1+pt_three).*(1+cm);

r3_2=rth_2.*(1+pt_three).*(1+cm);r3_3=rth_3.*(1+pt_three).*(1+cm);

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=20.*log10(c); k=20.*log10(b); wcga3=abs(m-k); plot(w,wcga3,'r-')


title(' Worst case gain mismatch of 3-stage RF CMOS PPFs','FontSize', 14)


global f_1;global f_4;global rf_1;global rf_2;global rf_3;global rf_4;

global cf_1;global cf_2;global cf_3;global cf_4;global cm;

















216










rand('seed',0);

% Crossover Operators xFns = 'simpleXover'; xOpts = [.4];





%type gaMichEval3stage; % Bounds on the variables






% x is the best solution found %x; % endPop is the ending population %endPop; % trace is a trace of the

best value and average value of generations %trace;

% Lets increase the population size by running the defaults rand('seed',0);

termOps=[100]; [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,

selectFn,selectOps);

% x is the best solution found x; fmax=x; f=f_1+(f_4-f_1)./2; ptt=fmax./f; pt_four=1-sqrt(1./ptt);

w=f_1.*(1-0.3):f_1./1e3:f_4.*(1+0.3);

c1=cf_1.*1e-15; c2=cf_2.*1e-15; c3=cf_3.*1e-15; c4=cf_4.*1e-15; dr_1=pt_four.*rf_1;

dr_2=pt_four.*rf_2; dr_3=pt_four.*rf_3; dr_4=pt_four.*rf_4; dc_1=pt_four.*c1; dc_2=pt_four.*c2;

dc_3=pt_four.*c3; dc_4=pt_four.*c4; dr_11=cm*rf_1; dr_22=cm*rf_2; dr_33=cm*rf_3;

dr_44=cm*rf_4; dc_11=cm*c1; dc_22=cm*c2; dc_33=cm*c3; dc_44=cm*c4;

r1=rf_1+dr_1+dr_11; r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33; r4=rf_4+dr_4+dr_44;

c41=c1+dc_1+dc_11; c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;

d=F_amplitudew_in(w,r1,c41,r2,c42,r3,c43,r4,c44);

n=20*log10(d); r1=rf_1+dr_1+dr_11; r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33; r4=rf_4+dr_4+dr_44;

c41=c1+dc_1+dc_11; c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;

b=F_amplitudew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44);

m=20*log10(b); gmis=abs(m-n); figure plot(w,gmis,'m-') xlabel('\itf / GHz','FontSize', 14)

ylabel('\itm-\itgain / dB','FontSize', 14) title(' Worst case gain mismatch of 4-stage RF CMOS

PPFs','FontSize', 14)


217

A3.6 Source Code for Analysis of Phase Mismatch in the Worst-Case Simulation

A.3.6.1 Source Code for Analysis of Phase Mismatch in the Worst-Case simulation of 2-stage RF CMOS PPFs

global f_1; global f_2; global rt_1; global rt_2;

global ct_1; global ct_2;global cm; global pt;
















for x=1:100, waitbar(x/100,h), end close(h) rand('seed',0);






%type gaMichEval3stage;

% Bounds on the variables

b_1=f_1; b_2=f_2; bounds = [b_1 b_2];





% x is the best solution found %x; % endPop is the ending population

%endPop; % trace is a trace of the best value and average value of generations

%trace;



218

rand('seed',0); termOps=[100]; [x endPop bestPop

trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,


% x is the best solution found x; fmax=x; f=f_1+(f_2-f_1)./2; ptt=fmax./f;

pt_two=1-sqrt(1./ptt); figure

w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3);

c2_2=ct_2.*1e-15+ct_2.*1e-15.*pt_two;c2_1=ct_1.*1e-15+ct_1.*1e-15.*pt_two;

r2_1=rt_1+rt_1.*pt_two;r2_2=rt_2+rt_2.*pt_two;

b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./


*c2_2)));

c2_2=ct_2.*1e-15+ct_2.*1e-15.*(1+cm).*pt_two; c2_1=ct_1.*1e-15+ct_1.*1e-15.*(1+cm).*pt_two;

r2_1=rt_1+rt_1.*(1+cm).*pt_two; r2_2=rt_2+rt_2.*(1+cm).*pt_two;

c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./


*c2_2)));

m=atan(b); k=atan(c); wcga=(m-k).*180./pi+90; plot(w,wcga,'r -')


title(' Worst case phase mismatch of 2-stage RF CMOS PPFs','FontSize', 14)

A.3.6.2 Source Code for Analysis of Phase Mismatch in the Worst- Case

simulation of 2-stage RF CMOS PPFs

global f_1;global f_2;global rth_1;global rth_2;global rth_3;

global cth_1;global cth_2;global cth_3;global cm;










if isempty(cm), cm=0.01;else cm=cm;end

if isempty(f_1), f_1=1.44e9;else f_1=f_1;end


if isempty(rth_1), rth_1=263.9;else rth_1=rth_1;end



if isempty(cth_1), cth_1=173.8;else cth_1=cth_1;end


219





rand('seed',0);












% x is the best solution found%x; % endPop is the ending population

%endPop; % trace is a trace of the best value and average value of generations

%trace; % Lets increase the population size by running the defaults

rand('seed',0);

termOps=[100];



% x is the best solution found x; fmax=x; f=f_1+(f_2-f_1)./2; ptt=fmax./f; pt_three=1-sqrt(1./ptt);

figure

w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3);

c3_3=cth_3.*(1+pt_three).*1e-15; c3_2=cth_2.*(1+pt_three).*1e-15;

c3_1=cth_1.*(1+pt_three).*1e-15; r3_1=rth_1.*(1+pt_three);

r3_2=rth_2.*(1+pt_three); r3_3=rth_3.*(1+pt_three);


(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

c3_3=cth_3.*(1+pt_three).*(1+cm).*1e-15; c3_2=cth_2.*(1+pt_three).*(1+cm).*1e-15;

c3_1=cth_1.*(1+pt_three).*(1+cm).*1e-15;r3_1=rth_1.*(1+pt_three).*(1+cm);

r3_2=rth_2.*(1+pt_three).*(1+cm);r3_3=rth_3.*(1+pt_three).*(1+cm);

b=abs((-


_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)


220

+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)-



*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));

m=atan(c); k=atan(b); wcph3=(m-k).*180./pi+90; plot(w,wcph3,'b-')



A.3.6.3 Source Code for Analysis of Phase Mismatch in the Worst-Case simulation of 4-stage RF CMOS PPFs

global f_1;global f_4;global rf_1;global rf_2;global rf_3; global rf_4;

global cf_1;global cf_2;global cf_3;global cf_4;global cm;















if isempty(rf_1), rf_1=289.9;else rf_1=rf_1;end

if isempty(rf_2), rf_2=329;else rf_2=rf_2;end

if isempty(rf_3), rf_3=377.4;else rf_3=rf_3;end

if isempty(rf_4), rf_4=390; rf_4=rf_4;end

if isempty(cf_1), cf_1=158.2;else cf_1=cf_1;end






rand('seed',0);

% Crossover Operators xFns = 'simpleXover';xOpts = [.4];




221




% b_1=f_1./(1-2.*pt); % b_2=f_2./(1+2.*pt);

b_1=f_1; b_2=f_4; bounds = [b_1 b_2];



% Lets run the GA

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,





rand('seed',0); termOps=[100];



% x is the best solution found x; fmax=x; f=f_1+(f_4-f_1)./2; ptt=fmax./f;

pt_four=1-sqrt(1./ptt); figure w=f_1.*(1-0.3):f_1./1e3:f_4.*(1+0.3); c1=cf_1.*1e-15; c2=cf_2.*1e-15;

c3=cf_3.*1e-15; c4=cf_4.*1e-15; dr_1=pt_four.*rf_1; dr_2=pt_four.*rf_2; dr_3=pt_four.*rf_3;

dr_4=pt_four.*rf_4; dc_1=pt_four.*c1; dc_2=pt_four.*c2; dc_3=pt_four.*c3; dc_4=pt_four.*c4;

dr_11=cm*rf_1; dr_22=cm*rf_2; dr_33=cm*rf_3; dr_44=cm*rf_4; dc_11=cm*c1; dc_22=cm*c2;

dc_33=cm*c3; dc_44=cm*c4; r1=rf_1+dr_1+dr_11; r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33;

r4=rf_4+dr_4+dr_44; c41=c1+dc_1+dc_11; c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33;

c44=c4+dc_4+dc_44; a=F_phasew_in(w,r1,c41,r2,c42,r3,c43,r4,c44); r1=rf_1+dr_1+dr_11;

r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33; r4=rf_4+dr_4+dr_44; c41=c1+dc_1+dc_11;

c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;

d=F_phasew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44); wcph4=d-a+90;

plot(w,wcph4,'r-') xlabel('\itf / GHz','FontSize', 14) ylabel('\itm-\itgain / dB','FontSize', 14)


Appendix 4 Source Code of PPFOPTIMA

222

Appendix 4

Source Code of PPFOPTIMA

A4.1 Source Code for an Implementation of Genetic Algorithm

function [x,endPop,bPop,traceInfo] = ga(bounds,evalFN,evalOps,startPop,opts,...

termFN, termOps,selectFN,selectOps,xOverFNs,xOverOps,mutFNs,mutOps)

% GA runs a genetic algorithm % Output Arguments: % x - the best solution found during the course of

the run % endPop - the final population % Input Arguments: % bounds - a matrix of upper and lower

ounds on the variables % evalFN - the fitness function % evalOps - options to pass to the evaluation

function ([NULL]) % startPop - a matrix of solutions that can be initialized % from initialize.m

% termFN - name of the .m termination function (['maxGenTerm']) % termOps - options string to be

passed to the termination function % selectFN - name of the .m selection function

% selectOpts - roulette selection % xOverFNS – simpleXover % xOverOps - A matrix of options to pass

to Xover.m files with the % first column being the number of that xOver to perform % similiarly for

mutation % mutFNs – binaryMutation % mutOps - A matrix of options to pass to Xover.m files with the

% first column being the number of that xOver to perform % similiarly for mutation

n=nargin; if n<2 n==6 n==10 n==12 disp('Insufficient arguements') end

if n<3 %Default evalation opts. evalOps=[]; end if n<5 opts = [1e-6 1 0]; end

if isempty(opts) opts = [1e-6 1 0]; end

if any(evalFN<48) %Not using a .m file

e1str=['x=b2f(endPop(j,:),bounds,bits); endPop(j,xZomeLength)=',... evalFN ';']; else %Are using a .m

file e1str=['x=b2f(endPop(j,:),bounds,bits);[x v]=' evalFN ... '(x,[gen evalOps]);

ndPop(j,:)=[f2b(x,bounds,bits) v];']; end

if n<6 %Default termination information termOps=[100]; termFN='maxGenTerm'; end

if n<12 %Default muatation information mutFNs=['binaryMutation']; mutOps=[0.05]; end

if n<10 %Default crossover information xOverFNs=['simpleXover']; xOverOps=[0.6]; end

if n<9 %Default select opts only i.e. roullete wheel. selectFN=['roulette']; selectOps=[]; end

if n<6 %Default termination information termOps=[100]; termFN='maxGenTerm'; end

if n<4 %No starting population passed given startPop=[]; end if isempty(startPop) %Generate a

population at random %startPop=zeros(80,size(bounds,1)+1);

startPop=initializega(80,bounds,evalFN,evalOps,opts(1:2)); end

if opts(2)==0 %binary bits=calcbits(bounds,opts(1)); end

xOverFNs=parse(xOverFNs); mutFNs=parse(mutFNs); xZomeLength = size(startPop,2); %Length of the

xzome=numVars+fitness numVar = xZomeLength-1; %Number of variables popSize = size(startPop,1);

%Number of individuals in the pop endPop = zeros(popSize,xZomeLength); %A secondary population

atrix c1 = zeros(1,xZomeLength); %An individual c2 = zeros(1,xZomeLength); %An individual

numXOvers = size(xOverFNs,1); %Number of Crossover operators numMuts = size(mutFNs,1);

%Number of Mutation operators epsilon = opts(1); %Threshold for two fittness to differ oval =


223

max(startPop(:,xZomeLength)); %Best value in start pop FoundIn = 1; %Number of times best has

changed done = 0; %Done with simulated evolution gen = 1; %Current Generation Number collectTrace

= (nargout>3); %Should we collect info every gen floatGA = opts(2)==1; %Probabilistic application of

ops display = opts(3); %Display progress while(~done) %Elitist Model [bval,bindx] =

max(startPop(:,xZomeLength)); %Best of current pop best = startPop(bindx,:); if ( (abs(bval -

oval)>epsilon) (gen==1)) %If we have a new best sol if display fprintf(1,'\n%d %f\n',gen,bval); %Update

the display end bPop(bFoundIn,:)=[gen b2f(startPop(bindx,1:numVar),bounds,bits)...

startPop(bindx,xZomeLength)]; bFoundIn=bFoundIn+1; %Update number of changes

oval=bval; %Update the best val else if display fprintf(1,'%d ',gen); %Otherwise just update num gen

end end endPop = feval(selectFN,startPop,[gen selectOps]); %Select for i=1:numMuts,

for j=1:mutOps(i,1), a = round(rand*(popSize-1)+1);

c1 = feval(deblank(mutFNs(i,:)),endPop(a,:),bounds,[gen mutOps(i,:)]);

if c1(1:numVar)==endPop(a,(1:numVar)) c1(xZomeLength)=endPop(a,xZomeLength);

else %[c1(xZomeLength) c1] = feval(evalFN,c1,[gen evalOps]); eval(e1str); end endPop(a,:)=c1; end

end else %We are running a probabilistic model of genetic operators for i=1:numXOvers,

xN=deblank(xOverFNs(i,:)); %Get the name of crossover function

cp=find(rand(popSize,1)<xOverOps(i,1)==1); if rem(size(cp,1),2) cp=cp(1:(size(cp,1)-1)); end

cp=reshape(cp,size(cp,1)/2,2); for j=1:size(cp,1) a=cp(j,1); b=cp(j,2); [endPop(a,:) endPop(b,:)] =

feval(xN,endPop(a,:),endPop(b,:),... bounds,[gen xOverOps(i,:)]); end end

for i=1:numMuts mN=deblank(mutFNs(i,:)); for j=1:popSize endPop(j,:) =

eval(mN,endPop(j,:),bounds,[gen mutOps(i,:)]); eval(e1str); end end end gen=gen+1;

done=feval(termFN,[gen termOps],bPop,endPop); %See if the ga is done startPop=endPop; %Swap the

populations [bval,bindx] = min(startPop(:,xZomeLength)); %Keep the best solution

startPop(bindx,:) = best; %replace it with the worst end [bval,bindx] = max(startPop(:,xZomeLength));

if display fprintf(1,'\n%d %f\n',gen,bval); end x=startPop(bindx,:); if opts(2)==0 %binary

x=b2f(x,bounds,bits); bPop(bFoundIn,:)=[gen b2f(startPop(bindx,1:numVar),bounds,bits)...

startPop(bindx,xZomeLength)]; else bPop(bFoundIn,:)=[gen startPop(bindx,:)];

end

A4.2 Source Code of Monte Carlo Simulation

A4.2.1 Source Code for Monte Carlo Simulation of 2-stage RF CMOS PPFs

% load optimization results load twostageppfopt.mat r_1 r_2 c_1 c_2;

global f; w=f; global r_1; global r_2; global c_1;

global c_2; global pt_r; global pt_c; global cm; global m2_mu;

% visiable waitbar h = waitbar(0,'Simulation is processing, please ait...');

for x=1:100, % computation here % waitbar(x/100,h) end close(h) % get the value of resistor process

tolerance from input pt_r=findobj(gcf,'Tag','pt_r'); pt_r=str2double(get(pt_r,'string'));



224

% get the value of capacitor process tolerance from input

pt_c=findobj(gcf,'Tag','pt_c'); pt_c=str2double(get(pt_c,'string')); if isempty(pt_c), pt_c=0.21; else

pt_c=pt_c; end % get the value of maximum component mismatch from input

cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string')); if isempty(cm), cm=0.01; else cm=cm; end

% get the number of runs of MC from input m2_mu=findobj(gcf,'Tag','m2_mu');

m2_mu=str2double(get(m2_mu,'string')); if isempty(m2_mu), m2_mu=200; else m2_mu=m2_mu; end

% caculate maximum R & C variant dc_1=c_1.*pt_c.*1e-15; dc_2=c_2.*pt_c.*1e-15; dr_1=r_1.*pt_r;

dr_2=r_2.*pt_r; % generate random numbers with truncated Gaussian distribution amm_2=0;

pmm_2=0; for i=1:m2_mu; i=i+1; N=2000; % size of random numbers M=1; mu=0; % mean value

sig=0.3; % standard deviation xlo=-1; % low truncation point xhi=1; % high truncation point

s=truncnormrnd(N,M,mu,sig,xlo,xhi); N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;

t=truncnormrnd(N,M,mu,sig,xlo,xhi); % MC simulations of gain mismatch and phase difference

r2_1=r_1+dr_1.*s; r2_2=r_2+dr_2.*s; c2_1=c_1.*1e-15+dc_1.*s; c2_2=c_2.*1e-15+dc_2.*s;

d=T_amplitudept_in(w,r2_1,c2_1,r2_2,c2_2); m=T_phasept_in(w,r2_1,c2_1,r2_2,c2_2);

r2_1=(r_1+dr_1.*s).*(1+cm.*t); r2_2=(r_2+dr_2.*s).*(1+cm.*t); c2_1=(c_1.*1e-15+dc_1.*s).*(1+cm.*t);

c2_2=(c_2.*1e-15+dc_2.*s).*(1+cm.*t); b=T_amplitudept_qu(w,r2_1,c2_1,r2_2,c2_2);

n=T_phasept_qu(w,r2_1,c2_1,r2_2,c2_2); amm_1=abs(20.*log10(b)-20.*log10(d));

t1=sort(amm_1); amm_2=amm_2+t1; pmm_1=n-m+90; t2=sort(pmm_1); pmm_2=pmm_2+t2;

end amm=amm_2./m2_mu; pmm=pmm_2./m2_mu; % caculate standard deviation kerse_1=std(amm)

kerse_1=round(kerse_1.*1000)./1000; st1=findobj(gcf,'Tag','kerse_1'); set(st1,'String',[num2str(kerse_1)])

kerse_2=std(pmm-90) kerse_2=round(kerse_2.*1000)./1000; stm=findobj(gcf,'Tag','kerse_2');

set(stm,'String',[num2str(kerse_2)]) figure subplot(2,1,1);

% plot gain mismatch histogram of gain mismatch

hist(amm) xlabel('\itm-\rmgain / dB','FontSize', 14)

title('Monte Carlo Simulation for 2-stage RF CMOS PPF','FontSize', 14)

ylabel('Number of Samples','FontSize', 14)

subplot(2,1,2); % plot phase difference histogram of gain mismatch hist(pmm)

xlabel('\itd-\rmphase / degree','FontSize', 14)



load threestageppfopt.mat r1 r2 r3 c1 c2 c3;

global f; w=f; global r1; global r2; global r3; global c1;global c2; global c3; global pt_r; global pt_c;

global cm; global m3_mu;

h = waitbar(0,'Simulation is processing, please wait...');

for x=1:100, % computation here % waitbar(x/100,h) end close(h)



dr_1=r1.*pt_r; dr_2=r2.*pt_r; dr_3=r3.*pt_r; pt_c=findobj(gcf,'Tag','pt_c');


225

pt_c=str2double(get(pt_c,'string'));


dc_1=c1.*pt_c.*1e-15;dc_2=c2.*pt_c.*1e-15; dc_3=c3.*pt_c.*1e-15


if isempty(cm), cm=0.01; else cm=cm; end;

m3_mu=findobj(gcf,'Tag','m3_mu');

m3_mu=str2double(get(m3_mu,'string'));

if isempty(m3_mu), m3_mu=200; else m3_mu=m3_mu; end

amm_2=0; pmm_2=0;

for i=1:m3_mu; i=i+1; N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;

s=truncnormrnd(N,M,mu,sig,xlo,xhi); N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;

t=truncnormrnd(N,M,mu,sig,xlo,xhi);

r3_1=r1+dr_1.*s; r3_2=r2+dr_2.*s; r3_3=r3+dr_3.*s; c3_1=(c1.*1e-15)+dc_1.*s;

c3_2=(c2.*1e-15)+dc_2.*s; c3_3=(c3.*1e-15)+dc_3.*s;

d=Th_amplitudept_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

m=Th_phasept_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

r3_1=(r1+dr_1.*s).*(1+cm.*t); r3_2=(r2+dr_2.*s).*(1+cm.*t);

r3_3=(r3+dr_3.*s).*(1+cm.*t); c3_1=((c1.*1e-15)+dc_1.*s).*(1+cm.*t);

c3_2=((c2.*1e-15)+dc_2.*s).*(1+cm.*t); c3_3=((c3.*1e-15)+dc_3.*s).*(1+cm.*t);

b=Th_amplitudept_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

n=Th_phasept_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

amm_1=abs(20.*log10(d)-20.*log10(b)); t1=sort(amm_1); amm_2=amm_2+t1;

pmm_1=(m-n)+90; t2=sort(pmm_1); pmm_2=pmm_2+t2; end

amm=amm_2./m3_mu; pmm=pmm_2./m3_mu; figure subplot(2,1,1);

hist(amm) xlabel('\itm-\rmgain / dB','FontSize', 14)

title('Monte Carlo Simulation of 3-stage RF CMOS PPF','FontSize', 14)

ylabel('Number of Samples','FontSize', 14) subplot(2,1,2); hist(pmm)

xlabel('\itd-\rmphase / degrees','FontSize', 14) ylabel('Number of Samples','FontSize', 14)


load fourstageoptrc.mat;

global f; w=f; global r4_1; global r4_2; global r4_3; global r4_4; global c4_1;

global c4_2; global c4_3; global c4_4; global pt_r; global pt_c; global cm; global m4_mu;

r_1=r4_1; r_2=r4_2; r_3=r4_3; r_4=r4_4; c_1=c4_1*1e-15;

c_2=c4_2*1e-15; c_3=c4_3*1e-15; c_4=c4_4*1e-15;

h = waitbar(0,'Simulation is processing, please wait...');

for x=1:100, % computation here % waitbar(x/100,h) end close(h)




226


if isempty(pt_c), pt_c=0.21; else pt_c=pt_c; end cm=findobj(gcf,'Tag','cm');

cm=str2double(get(cm,'string'));


m4_mu=findobj(gcf,'Tag','m4_mu'); m4_mu=str2double(get(m4_mu,'string'));

if isempty(m4_mu), m4_mu=200; else m4_mu=m4_mu; end

amm_2=0; pmm_2=0; for i=1:m4_mu; i=i+1; N=2000; M=1;mu=0; sig=0.3; xlo=-1; xhi=1;

t=truncnormrnd(N,M,mu,sig,xlo,xhi); N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;

s=truncnormrnd(N,M,mu,sig,xlo,xhi); dr4_1=r_1.*pt_r; dr4_2=r_2.*pt_r; dr4_3=r_3.*pt_r;

dr4_4=r_4.*pt_r; dc_1=c_1.*pt_c; dc_2=c_2.*pt_c; dc_3=c_3.*pt_c; dc_4=c_4.*pt_c;

c1=c_1+dc_1.*t; c2=c_2+dc_2.*t; c3=c_3+dc_3.*t; c4=c_4+dc_4.*t; r4_1=r_1+dr4_1.*t;

r4_2=r_2+dr4_2.*t; r4_3=r_3+dr4_3.*t; r4_4=r_4+dr4_4.*t;

d=F_amplitudept_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

m=F_phasept_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

c1=(c_1+dc_1.*t).*(1+cm.*s); c2=(c_2+dc_2.*t).*(1+cm.*s);

c3=(c_3+dc_3.*t).*(1+cm.*s); c4=(c_4+dc_4.*t).*(1+cm.*s);

r4_1=(r_1+dr4_1.*t).*(1+cm.*s); r4_2=(r_2+dr4_2.*t).*(1+cm.*s);

r4_3=(r_3+dr4_3.*t).*(1+cm.*s); r4_4=(r_4+dr4_4.*t).*(1+cm.*s);

b=F_amplitudept_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

n=F_phasept_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

amm_1=abs(20.*log10(d)-20.*log10(b)); t1=sort(amm_1); amm_2=amm_2+t1; pmm_1=(m-n)+90;

t2=sort(pmm_1); pmm_2=pmm_2+t2; end amm=amm_2./m4_mu; pmm=pmm_2./m4_mu; figure

subplot(2,1,1); hist(amm)

xlabel('\itm-\rmgain / dB','FontSize', 14)

title('Monte Carlo Simulation of 3-stage RF CMOS PPF','FontSize', 14)


subplot(2,1,2); hist(pmm)

xlabel('\itd-\rmphase / degrees','FontSize', 14)


A4.3 Source Code of the Worst-Case Simulation

A4.3.1 Source Code for the Worst-Case Simulation of 2-stage RF CMOS PPFs

% load optimization results load twostageppfopt.mat r_1 r_2 c_1 c_2;

global f_1; global f_2;global f; global r_1; global r_2;

global c_1; global c_2; global cm; global irrt;

% get the value of maximum component mismatch from input


if isempty(cm), cm=0.01; else cm=cm; end % visiable waitbar


227

h = waitbar(0,'Simulation is processing, please wait...'); for x=1:100, % computation here %

waitbar(x/100,h), end close(h) % search worst IRR corresponding frequency point with R&C nominal

values rand('seed',0); % Crossover Operators xFns = 'simpleXover'; xOpts = [.4]; % Mutation Operators

mFns = 'binaryMutation'; mOpts = [0.005]; % Termination Operators termFns = 'maxGenTerm';

termOps = [100]; % 200 Generations % Selection Function selectFn = 'roulette'; selectOps = [];

% Evaluation Function evalFn = 'gaMichEval2stage2'; evalOps = []; b_1=f_1; b_2=f_2; bounds = [b_1

b_2]; gaOpts=[1e-6 0 1]; % Generate an intialize population of size 20

startPop = initializega(20,bounds,'gaMichEval2stage2',[],[1e-6 0]); % lets run the GA

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,...


rand('seed',0); termOps=[100];

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,...


% x is the best solution found x; fmax=x; w=fmax; % caculate worst case IRR

r2_1=r_1; r2_2=r_2; c2_1=c_1*1e-15; c2_2=c_2*1e-15;

d=T_amplitude_qu(w,r2_1,c2_1,r2_2,c2_2);

k=T_phase_qu(w,r2_1,c2_1,r2_2,c2_2);

n_1=(d); p1_1=(k); b=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2); h=T_phase_in(w,r2_1,c2_1,r2_2,c2_2);

p2_1=(h); m_1=(b); a_delta_1=abs(m_1-n_1); p_delta_1=abs((p2_1-p1_1).*pi./180);

r2_1=r_1; r2_2=r_2; c2_1=c_1*1e-15; c2_2=c_2*1e-15; d=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2);

k=T_phase_in(w,r2_1,c2_1,r2_2,c2_2); n_2=(d); p1_2=(k); dr_11=cm*r_1; dr_22=cm*r_2;

dc_11=cm*c_1*1e-15; dc_22=cm*c_2*1e-15; r2_1=r_1+dr_11; r2_2=r_2+dr_22; c2_1=c_1*1e-

15+dc_11; c2_2=c_2*1e-15+dc_22; b=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2);

h=T_phase_in(w,r2_1,c2_1,r2_2,c2_2); p2_2=(h); m_2=(b); a_delta_2=abs(m_2-n_2);

a=sqrt(m_2.^2+n_2.^2); p_delta_2=abs((p2_2-p1_2).*pi./180); a_delta=(a_delta_1+a_delta_2);

p_delta=(p_delta_1+p_delta_2); irrt=ImageRR(a_delta,a,p_delta); irrw=round(irrt.*10)./10;

hh=findobj(gcf,'Tag','irrw'); set(hh,'String',[num2str(irrw)]) % load IRR of design specification

load designspeci.mat irrspec; % compare with worst case IRR if irrw<irrspec,

warndlg('Two-stage PPF IRR can not satisfy the design specification, please use higher

stage PPF.','Warning') end

% caculate corresponding process tolerance of R&C ptt=fmax./f; pt_two=1-sqrt(1./ptt); % caculate R & C

variant dr_1=r_1*pt_two; dr_2=r_2*pt_two; dc_1=c_1*pt_two*1e-15; dc_2=c_2*pt_two*1e-15;

% caculate gain mismatch and phase difference r2_1=r_1+dr_1; r2_2=r_2+dr_2; c2_1=c_1*1e-15+dc_1;

c2_2=c_2*1e-15+dc_2; w=f_1*(1-0.2):f_1./1e4:f_2*(1+0.2); d=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2);

k=T_phase_in(w,r2_1,c2_1,r2_2,c2_2); n=20*log10(d); p1=(k); dr_11=cm*r_1;dr_22=cm*r_2;

dc_11=cm*c_1*1e-15;dc_22=cm*c_2*1e-15; r2_1=r_1+dr_1+dr_11; r2_2=r_2+dr_2+dr_22;

c2_1=c_1*1e-15+dc_1+dc_11; c2_2=c_2*1e-15+dc_2+dc_22;

b=T_amplitude_qu(w,r2_1,c2_1,r2_2,c2_2); h=T_phase_qu(w,r2_1,c2_1,r2_2,c2_2);

p2=(h); m=20*log10(b); gmis=abs(m-n); figure % polt gain mismatch plot(w,gmis,'m-')

xlabel('\itf / Hz','FontSize', 14) ylabel('\itm-\rmgain / dB','FontSize', 14) title('Gain Mismatch in 2-

Stage RF CMOS PPF','FontSize', 14) figure pmis=-p1+p2+90; % plot phase difference


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plot(w,pmis,'m-') xlabel('\itf / Hz','FontSize', 14) ylabel('\itd-\rmphase / degrees','FontSize', 14)

title('Phase Difference in 2-Stage RF CMOS PPF','FontSize', 14)

A4.3.2 Source Code for the Worst-Case simulation of 3-stage RF CMOS PPFs

load threestageppfopt.mat r1 r2 r3 c1 c2 c3 we;

global f_1; global f_2; global f; global r1; global r2; global r3; global c1;

global c2; global c3; global cm; global irrth;




waitbar(x/100,h) end close(h)





% Evaluation Function evalFn = 'gaMichEval3stage2'; evalOps = []; bounds = [f_1 f_2];



startPop = initializega(20,bounds,'gaMichEval3stage2',[],[1e-6 0]);

% Lets run the GA





% Lets increase the population size by running the defaults rand('seed',0);

termOps=[100]; [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,...


% x is the best solution found x; fmax=x; w=fmax;

r3_1=r1; r3_2=r2; r3_3=r3; c3_1=c1*1e-15; c3_2=c2*1e-15; c3_3=c3*1e-15;

d=Th_amplitude_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

k=Th_phase_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

n=(d); p1=(k); b=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

h=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

p2=(h); m=(b); a_delta_1=abs(m-n);

p_delta_1=abs((p2-p1).*pi./180);

a=n; r3_1=r1; r3_2=r2; r3_3=r3; c3_1=c1*1e-15; c3_2=c2*1e-15; c3_3=c3*1e-15;

d=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

k=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3); n=(d); p1=(k);

dr_11=cm*r1; dr_22=cm*r2; dr_33=cm*r3; dc_11=cm*c1*1e-15;


229

dc_22=cm*c2*1e-15; dc_33=cm*c3*1e-15; r3_1=r1+dr_11;

r3_2=r2+dr_22; r3_3=r3+dr_33; c3_1=c1*1e-15+dc_11; c3_2=c2*1e-15+dc_22;

c3_3=c3*1e-15+dc_33; b=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

h=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3); p2=(h); m=(b); a_delta_2=abs(m-n);

% a=sqrt(m.^2+n.^2); p_delta_2=abs((p2-p1).*pi./180); p_delta=p_delta_1+p_delta_2;

a_delta=a_delta_1+a_delta_2; irrth=ImageRR(a_delta,a,p_delta); irrw=round(irrth.*10)./10;

hh=findobj(gcf,'Tag','irrw'); set(hh,'String',[num2str(irrw)]) %save threestageppfirr.mat irrth;

load designspeci.mat irrspec; if irrw<irrspec, warndlg('Three-stage PPF IRR can not satisfy the design

specification, please use higher stage PPF.','Warning') end ptt=fmax./f;

pt_three=1-sqrt(1./ptt); a=pt_three; w=f_1*(1-0.3):f_1./1e4:f_2*(1+0.5);

dr_1=r1*pt_three; dr_2=r2*pt_three; dr_3=r3*pt_three; dc_1=c1*pt_three*1e-15;

dc_2=c2*pt_three*1e-15; dc_3=c3*pt_three*1e-15; r3_1=r1+dr_1;

r3_2=r2+dr_2; r3_3=r3+dr_3; c3_1=c1*1e-15+dc_1; c3_2=c2*1e-15+dc_2;

c3_3=c3*1e-15+dc_3; d=Th_amplitude_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

k=Th_phase_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3); n=20*log10(d);

p1=(k); dr_11=cm*r1; dr_22=cm*r2; dr_33=cm*r3; dc_11=cm*c1*1e-15;

dc_22=cm*c2*1e-15; dc_33=cm*c3*1e-15; r3_1=r1+dr_1+dr_11; r3_2=r2+dr_2+dr_22;

r3_3=r3+dr_3+dr_33; c3_1=c1*1e-15+dc_1+dc_11; c3_2=c2*1e-15+dc_2+dc_22;

c3_3=c3*1e-15+dc_3+dc_33; b=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

h=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);

p2=(h); m=20*log10(b); % gmis=we.*abs(m-n); gmis=abs(m-n);

figure subplot(2,1,1); plot(w,gmis,'m-')

xlabel('\itf / GHz','FontSize', 14) ylabel('\itm-\rmgain / dB','FontSize', 14)

title('Gain Mismatch in 3-Stage RF CMOS PPF','FontSize', 14)

subplot(2,1,2); pmis=-p1+p2+90; plot(w,pmis,'m-')


title('Phase Mismatch in 3-Stage RF CMOS PPF','FontSize', 14)

A4.3.3 Source Code for the Worst-Case simulation of 4-stage RF CMOS PPFs

load fourstageoptrc.mat;

global f_1; global f_4; global f; global r4_1; global r4_2; global r4_3; global r4_4;

global c4_1; global c4_2; global c4_3; global c4_4; global cm; global irrf;


waitbar(x/100,h) end close(h)





f=findobj(gcf,'Tag','f'); f=str2double(get(f,'string')); rand('seed',0);


230





% Evaluation Function evalFn = 'gaMichEval4stage2'; evalOps = []; bounds = [f_1 f_4];



startPop = initializega(20,bounds,'gaMichEval4stage2',[],[1e-6 0]);

% Lets run the GA






rand('seed',0); termOps=[100]; [x endPop bestPop

trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,...


% x is the best solution found x; fmax=x; w=fmax;

c1=c4_1*1e-15; c2=c4_2*1e-15; c3=c4_3*1e-15; c4=c4_4*1e-15;

r4_1=r4_1; r4_2=r4_2; r4_3=r4_3; r4_4=r4_4; d=F_amplitude_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

k=F_phase_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,4); n=(d); p1=(k);

b=F_amplitude_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

h=F_phase_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4); p2=(h); m=(b); a_delta_1=abs(m-n);

p_delta_1=abs((p2-p1).*pi./180); c1=c4_1*1e-15;c2=c4_2*1e-15; c3=c4_3*1e-15; c4=c4_4*1e-15;

r4_1=r4_1; r4_2=r4_2; r4_3=r4_3; r4_4=r4_4;

d=F_amplitude_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

k=F_phase_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

n=(d); p1=(k); dr_11=cm*r4_1; dr_22=cm*r4_2; dr_33=cm*r4_3; dr_44=cm*r4_4; dc_11=cm*c1;

dc_22=cm*c2; dc_33=cm*c3; dc_44=cm*c4; r4_1=r4_1+dr_11; r4_2=r4_2+dr_22; r4_3=r4_3+dr_33;

r4_4=r4_4+dr_44; c1=c1+dc_11; c2=c2+dc_22; c3=c3+dc_33; c4=c4+dc_44;

b=F_amplitude_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

h=F_phase_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);

p2=(h); m=(b); a_delta_2=abs(m-n); p_delta_2=abs((p2-p1).*pi./180);

a=sqrt(m.^2+n.^2); a_delta=a_delta_1+a_delta_2; p_delta=p_delta_1+p_delta_2;

irrf=ImageRR(a_delta,a,p_delta); irrw=round(irrf.*10)./10; hh=findobj(gcf,'Tag','irrw');

set(hh,'String',[num2str(irrw)]) %save fourstageppfirr.mat irrf;

load designspeci.mat irrspec; if irrw<irrspec,

warndlg('Four-stage PPF IRR can not satisfy the design specification, this tool box can not help you any

more, please use Double Quadrature Transceiver architecture.','Warning') end

ptt=fmax./f; pt_four=1-sqrt(1./ptt); a=pt_four; load fourstageoptrc.mat;

w=f_1.*(1-0.30):f_1./1e4:f_4.*(1+0.30); c1=c4_1.*1e-15; c2=c4_2.*1e-15; c3=c4_3.*1e-15;


231

c4=c4_4.*1e-15; dr_1=pt_four.*r4_1; dr_2=pt_four.*r4_2; dr_3=pt_four.*r4_3; dr_4=pt_four.*r4_4;

dc_1=pt_four.*c1; dc_2=pt_four.*c2; dc_3=pt_four.*c3; dc_4=pt_four.*c4; dr_11=cm*r4_1;

dr_22=cm*r4_2; dr_33=cm*r4_3; dr_44=cm*r4_4; dc_11=cm*c1; dc_22=cm*c2; dc_33=cm*c3;

dc_44=cm*c4; r1=r4_1+dr_1; r2=r4_2+dr_2; r3=r4_3+dr_3; r4=r4_4+dr_4; c41=c1+dc_1;

c42=c2+dc_2; c43=c3+dc_3; c44=c4+dc_4; d=F_amplitudew_in(w,r1,c41,r2,c42,r3,c43,r4,c44);

k=F_phasew_in(w,r1,c41,r2,c42,r3,c43,r4,c44);n=20*log10(d); p1=(k); r1=r4_1+dr_1+dr_11;

r2=r4_2+dr_2+dr_22; r3=r4_3+dr_3+dr_33; r4=r4_4+dr_4+dr_44; c41=c1+dc_1+dc_11;

c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;

b=F_amplitudew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44);

h=F_phasew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44); p2=(h);

m=20*log10(b); gmis=abs(m-n); figure subplot(2,1,1); plot(w,gmis,'m-')


ylabel('\itm-\rmgain / dB','FontSize', 14)

title('Gain Mismatch in 4-Stage RF CMOS PPF','FontSize', 14)

subplot(2,1,2); pmis=-p1+p2+90; plot(w,pmis,'m-')


ylabel('\itd-\rmphase / degrees','FontSize', 14)

title('Phase Mismatch in 4-Stage RF CMOS PPF','FontSize', 14)

analysis, design, and optimization of rf cmos polyphase filters · 2017-11-29 · for design of rf...

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