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DSP Based Corrections of Analog Components in

Digital Receivers

DSP Based Corrections of DSP Based Corrections of Analog Components in Analog Components in

Digital ReceiversDigital Receivers

fred harrisfred harris

IEEE Communications, Signal Processing, and Vehicular Technology Chapters

Coastal Los Angeles Section

24-April 2008

It’s all done with Computer Chips

We each own a BillionTransistors

We have an amazing wealth of resources at our disposal! Just How big is a Billion? A stack of a billion dollar notes would be

76.2 kilometers High. A billion seconds ago was 32.5 years ago.

We each own a Billion Transistors

For Comparison, the Eiffel Tower Contains 18,084 PartsIt is Fastened together by 2.5 Million Rivets

The world manufactures more transistors than it grows grains of rice.

We each own a Billion Transistors

How big is a billion grains of rice?• 8mm x 2mm x 2mm (Long Grain)• 1-billion grains of rice• 8 Meters x 2 Meters x 2 Meters• Or 32 Cubic Meters• Or a cube 3.2 Meters on a side• It weighs 24,000 kgr (26.6 tons)• Market price, $1,000/ton; $26,600 • CLS-350 Mercedes Benz weighs 2,200 kgr

(gross wt)

Digital Translator

Analog IF

I-QBalance

DCCancel

AnalogSynthesizer

DDS DDS

Sampling Clock

2 2 2

ADC DAC

4-to-1 Filter

1-to-8 Filter

LineCancel

VariableBW Filter

Equalizersin(x)(x)

Comp

DC CancelI-Q Balance

Digital TranslationLine Cancel

Variable BW FilterEqualizer

DAC sin(x)/x Compensation

From Where Did DC Come?

• Self Mixing in Analog Down Converter• ADC Injects DC• 2’s Complement Arithmetic Injects DC

Self Mixing in Down Converter

LNA

FrequencySynthesizer

Low Pass Filter

Low Pass Filter

ParasiticCoupling

f f f

DC Offset: Self MixingDesired Component: MixingSpectral Image: I-Q Imbalance

DC From A-to-D Converter

x x

x- x-

x x

X(n) X(n)

xq xq

xq xq

xq(n) Xq(n)ADC ADC

Rounding Quantizer Truncating Quantizer

CLK CLK

Truncating QuantizerHighest Allowable Output Value

Not Greater than Measured Value

Nq

(n+1)q

(N-1)q

Sampled Signal Values

Sampled Signal Values

Quantized Sample ValuesQuantized Sample Values

Quantization Errors

Quantization Errors

Amplitude

Sample Time

2’s Complement Number Representation

0 +Nmax-Nmin

b-bitsb-bits

(b-2)-bits(b-2)-bits ErrorError

Positive numbers:Measure displacement from reference. Reference = 0

Negative numbers:Measure displacement from reference. Reference = -Nmin

QuantizedNumber Line

Errors Due to finite Arithmetic

x1 x

x2 hQ

hQh

qA qMqC

s = x + x +q1 2 A p = x h +qQ M.

Quantize Addition Quantize Coefficient Quantize Multiplication

Finite Arithmetic in Radix-2 FFT Algorithm

+

+

+

-

SCL

(2 , 20 -1)e

-j 2πN k

N

N N

NN

N

NN N

2 22

2

22 2

-Pnt

-Pnt

DFT

DFT

k1

n1

n2

k

k+k2

Time Time Freq Freq

n

Radix-2 FFT Signal Flow Diagram

000

011

110

001

100

111

010

101

-

-- -

-

- -

-

- - -

-

w0

w0

w0

w2

w2

w2

w3

w1

Algorithm Noise Due to Finite Arithmetic, Coefficient Noise

Algorithm Noise due to Finite Arithmetic, Scaling Noise

DC Canceller, DC Notch Filter

α

z-1-

x(z)

f(z)

y(z)

)1(1)(α−−

−=

ZZZH

+

1-α

Spectral Response

DC Canceller with Embedded Sigma-Delta

Converter

Z

Z

-1

-1

-

- μ

R1

R2

x(n)y (n)2

Qq_loop

q_out

Suppress Bit Growth withSigma-Delta Converter inFeedback Path

DC Canceller Time Series

Spectra Input and Output of Canceller

Tunable Notch, Spin the Delay Line

α

z-1-

x(z)

f(z)

y(z)φje

φ

φ

α j

j

eZeZZH

)1()(

−−−

=

+

1-α

φ

Spectral Response

Tuning With LP-to-BP Transformation

2α

z-1-

x(z)

f(z)

y(z)

c

cZcZ−

−−

1

)cos(:)21()1(2

12)( 2

2

φαα

=−+−−

+−= c

ZcZcZZZH

++

1-α

1-α

φ

Implementing LP-to-BP Transformation

-

-z z-1 -1

c=cos( )θ mu

x y

b1 b1

Z Z

Z ZZ

-1 -1

-1

-1

-1 -1-1

- -

x(z) x(z)

y(z) y(z)Y(z) zY(z) z

1-cZZ-c

Spectral Response

Self Tuning:Reference Canceling

y(n)x(n)

r(n)

Z-1

w(n) w(n+1)

μ

**

y(n)x(n)

ej nθ e

j nθ

Z-1

w(n) w(n+1)

μ

**

Filters have Same Transfer Function

y(n)x(n)

ej nθ e

j nθ

Z-1

w(n) w(n+1)

μ

**

α

z-1-

x(z)

f(z)

y(z)φje

I-Q Imbalance

Ideal I-Q Mixing

Shape

Shape

Match

Match

CHANNEL

cos( t)ω0 cos( t)ω0

-sin( t)ω0 -sin( t)ω0

n(t)

I(t) I(t)

Q(t)Q(t)

^

^

Real Sinusoids, Time and Frequency

Complex Sinusoids-I

Complex Sinusoids-II

Imbalance: New Spectral Terms

f f

f f

f f

RealReal

RealReal

RealReal

ImagImag

ImagImag

ImagImag

A

AA

A

A

A

A

A

A

A

A

A

A

A

A

2

22

2

2

2

2

2

2

2

2

2

2

_

__

_

_

_

_

_

_

_

_

_

_ α

α

(1+ )ε

(1+ )ε

α

α

ε

-

f0

f0

f0

f0

f0

-f0

-f0

-f0 -f0

-f0

-f0

f0

A cos(2 f t)π 0

A cos(2 f t)π 0

(1+ ) A sin(2 f t+ε π α)0

A sin(2 f t)π 0

A exp(j 2 f t)π 0

X={ }

X=F{ }

i Y={ i }i Y=F{i }

X+i Y={ }

X+i Y=F{sum}

Arbitrary Signals,Time and Frequency

Complex Baseband & Complex Band-Centered

Complex Baseband & Real Band-Centered

Complex Down Conversion

Gain and Phase Imbalance inI-Q Mixers

Balanced Mixers

Gain Imbalance

Phase Imbalance

Gain and Phase Imbalance

Filter Imbalance

Effect of Gain Imbalance

Effect of Phase Imbalance

Description and Model of Baseband Down Conversion

Effect of I-Q Imbalance on 16-QAM Constellation

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5I-Q SAMPLES OF 16-QAM SIGNAL W ITH I-Q GAIN AND PHASE MISMATCH

Input Signal Conditioning and Correction

Analog I/QDown Convert

AnalogLow Pass Filters

A-to-DConverter

DCCancel

PhaseBalance

GainBalance

FixedEqualizer

Synthesizer

Analog Signals Digital Signals

DC-Cancel and I-Q Imbalance Correction

I I’

Q Q’

α1+ε

DCI

DCQ

II’

QQ’

α

1-ε

- ^

^

^

^ αestimate

εestimate

DCCancel

DCCancel

I-Q Imbalance Correction Algorithms

Gain and Phase Parameter Trajectories

0 1000 2000 3000 4000 5000 6000 7000 80000

0.1

0.2

0.3

0.4

0.5PHASE PARAMETER

AM

PLI

TUD

E

0 1000 2000 3000 4000 5000 6000 7000 80000.8

0.85

0.9

0.95

1

1.05GAIN PARAMETER

TIME (in SAMPLES)

AM

PLI

TUD

E

Constellation after Correcting Gain and Phase Imbalance

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5CONSTELLATION OF I-Q SAMPLES AFTER PHASE AND GAIN ADAPTION

• That was the Easy part

• The Easy part is over

• Consider a channelizer!

Analog Block Conversion

Digital Second Conversion

Description and Model of Block Down Conversion

Constellations of channel +k and -k

Crosstalk Between Channels k and –kDue to gain and Phase Imbalance

Constellation after Gradient Descent Correction of Gain and Phase Imbalance

Crosstalk Between Channels k and Empty Channel–k

Constellation after Gradient Descent Correction of Gain and Phase Imbalance

Estimating Correcting Gain Terms

The Plot Thickens

Analog Block Down ConversionWith a Frequency Offset

Removed by Digital Down Conversion

Analog Block Conversion with Frequency Offset

Digital Down Conversion and Removal of Frequency Offset

Description of Model and Block Down Conversion With frequency Offset

Estimating Correcting Gain Terms with Frequency Offset

Constellations with I-Q Imbalance with and without Frequency Offset

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Constellations, With I-Q Imbalance

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Constellations, With I-Q Imbalance and Frequency Offset

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Constellation Trajectories During Adaptive Convergence

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Constellations, With I-Q Imbalance and Frequency Offset

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Constellations, Compensated for I-Q Imbalance

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Processing Task

Variable BWVariable Length FIR Filter

Narrower Bandwidth Filters Have Narrower Transition Bandwidths

Hence Are Longer Filters

Spectrum and Group DelayIIR Filter and Equalizer

Spectrum and Group Delay2- BW IIR Filters and Equalizers

Pole-Zero Diagrams for IIR Filter and Equalizer

Filter Coefficient Variation with Filter Bandwidth

21

21 2

1( ) Z b ZH ZZ a Z a

+ +=

+ +

Equalizer Coefficient Variation with Filter Bandwidth

4 22 1

4 21 2

1( ) a Z a ZE ZZ a Z a

+ +=

+ +

Recursive Filter for Variable Bandwidth with Companion All-

Pass Equalizer

a (bw)n

d (bw)n

c (bw)n

b (bw)n

Recursive Filter Recursive Equalizerx(n) y(n)

bwCoefficient Polynomials

Compare FIR Filter and Equalized IIR Filter

Narrower Bandwidth-> Longer Filter, Longer Filter -> More Taps

True for Fixed Sample Rate

Narrower Bandwidth-> Longer Filter, Longer Filter -> Same Taps

at Lower Sample Rate

Continuously Variable Bandwidth FIR Filter

with Fixed Number of Taps

Fixed BWFixed Length FIR Filter

P-to-Q ArbitraryInterpolator

Q-to-P ArbitraryInterpolator

Spectra at Input and Output of the Three Processing blocks of the Variable

Bandwidth Filter

Impulse Response of Variable BW FIR Filter

Power Amplifier Linearization

AnalogSynthesizer

DDS

2

2

DAC

ADC

1-to-8 Filter

LPFilter

IFFilter

IFFilter

Pre-Distort

NLModel

GainCorrect Table

sin(x)(x)

PA

Comp

PA LinearizationPeak-to-Average Ratio Control

Non Linear Amplifier and Precompensating Gain

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3Nonlinear Transfer Function of Amplifier

1-dB Compression Point

Input Magnitude

Out

put M

agni

tude

0 0.5 1 1.5 2 2.5 30.95

1

1.05

1.1

1.15

1.2

1.25

1.3Compensating Gain of Transfer Function

Input Magnitude

Com

pens

atin

g G

ain

Transition Diagram Input and Output of Amplifier and input and output of Precompensator

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Transition Diagram: Input and Compressed Amplifier

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Transition Diagram: Input and Precompensated

Spectra: Input and Output of Amplifier and Output of Precompensator and Precompensated Amplifier

To Clip or Not to Clip; That is the Question!

s (t)1

s (t)=3 s (t)-s (t)2 1

+LCLIP

-LCLIP

s (t)2

-LCLIP

-LCLIP -LCLIP

LCLIPLCLIP

LCLIP

s2 s3

s1 s1

Band LimitedSubtractive Clipping

L L

-L -L

S (n)1

S (n)3

a1

a1

a2

a2

C(k )1

k1 k1

k1

k2

k2 k2

C(k )2

-

Band-Limited Clipping

LCLIP

s (t)2

s (t)3

s (t)1CLIP-1

-LCLIP

s (t)2

s (t)3

s (t)1

CLIP-2

LCLIP

s (t)2

s (t)3s (t)4

s (t)1

CLIP-2 Filter

Delay

Spectra: Input Signal, Clipping Component and Clipped Signal

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5Magnitude of Input Signal

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4Magnitude of Input Signal Exceeding Top

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5Magnitude of Input Signal - Level Exceeding Top

Spectra: Input Signal, Band Limited Clipping Component and Clipped Signal

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5Magnitude of Input Signal

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4Magnitude of Input Signal - Level Exceeding Top and Filtered Version

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5Magnitude of Input Signal - Filtered Level Exceeding Top

Spectra: Input, Clip Component, Band Limited Clip, and Band Limited Clip

-4 -2 0 2 4-80

-60

-40

-20

0

Input Spectra to PAR Control

Normalized Spectra

20*lo

g 10(m

ag),

(dB

)

-4 -2 0 2 4-80

-60

-40

-20

0

Spectra; Signal Exceeding Top

Normalized Spectra

20*lo

g 10(m

ag),

(dB

)

-4 -2 0 2 4-80

-60

-40

-20

0

Spectra; Bandlimited Version Exceeding Top

Normalized Spectra

20*lo

g 10(m

ag),

(dB

)

-4 -2 0 2 4-80

-60

-40

-20

0

Spectra; Signal with Bandlimiting Top

Normalized Spectra

20*lo

g 10(m

ag),

(dB

)

Professor harris, may I be excused? My brain is full.

SOFTWAREDEFINED

RADIOMAN

Is Open For Questions