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E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 1/38

Synchronization Techniques for Burst-Mode Continuous Phase Modulation

Ph.D. Dissertation DefenseEhsan Hosseini

Department of Electrical Engineering & Computer Science University of Kansas


Outline

• Chapter 1: Introduction

• Chapter 2: The Cramér-Rao Bound for Training Sequence Design for Burst-Mode CPM

• Chapter 3: Timing, Carrier and Frame Synchronization for Burst-Mode CPM

• Chapter 4: Applications to SOQPSK

• Chapter 5: Conclusions


External Contributions

• Chapter 1: Introduction

• Chapter 2: The Cramér-Rao Bound for Training Sequence Design for Burst-Mode CPM

• Chapter 3: Timing, Carrier and Frame Synchronization for Burst-Mode CPM

• Chapter 4: Applications to SOQPSK

• Chapter 5: Conclusions

• E. Hosseini and E. Perrins, "The Cramèr-Rao Bound for Training Sequence Design for Burst-Mode CPM," IEEE Transactions on Communications, vol. 61, no. 6, pp. 2396-2407, June 2013.

• E. Hosseini and E. Perrins, "Training Sequence Design for Data-Aided Synchronization of Burst-Mode CPM," in Proceedings of the IEEE Global Communications Conference (GLOBECOM'12), 2012..

• E. Hosseini and E. Perrins, "Maximum Likelihood Synchronization of Burst-Mode CPM," to appear in Proceedings of the IEEE Global Communications Conference (GLOBECOM'13), December 9-13, 2013

• E. Hosseini and E. Perrins, “Timing, Carrier, and Frame Synchronization of Burst-Mode CPM”, IEEE Transactions on Communications, Under Review.

• E. Hosseini and E. Perrins, "The Cramér-Rao Bound for Data-Aided Synchronization of SOQPSK," in Proceedings of the IEEE Military Communications Conference (MILCOM'12), 2012.

• E. Hosseini and E. Perrins, "On Burst-Mode Synchronization of SOQPSK," to appear in Proceedings of the IEEE Military Communications Conference (MILCOM'13), 2013.


Motivation: Aeronautical Telemetry

• integrated Network Enhanced Telemetry (iNET) project Packet-based telemetry in

contrast to streaming telemetry. Benefits: spectrum conservation,

multi-hop, enhanced mobility,.. A major challenge in the RF link

is detection and synchronizationof packets, i.e. bursts

Physical layer waveform: SOQPSK, a subset of CPM

• In this work: 1. Synchronization for general

CPM in burst-mode TX2. Apply the results to SOQPSK,

i.e. iNET standard

GS1

GS2

TA1

TA2

TA3

GS: Ground StationTA: Test Article


What is CPM?

• Continuous phase modulation:

• Frequency pulse:•• Duration of LTs

• Properties:• Constant envelope • Bandwidth efficient• Drawback: complex at Rx

i

sis

s iTtqhTEts )(2exp);( α h: modulation index

q(t): phase response

M-ary data symbols:i

dttdqtg )()(

tLTs

q(t)

21

g(t)

L=1: full-responseL>1: partial-response


Signal Transmission Model

• Burst-Mode Transmission:

• Burst Structure: • Preamble (training sequence): sequence of L0 known symbols• Unique Word (UW): burst identification• Payload: information bits

• Received baseband signal:

• Noise model: AWGN

Preamble UW PayloadGuardInterval

GuardInterval

L0 LUW Lpay

)()()( )2( twtsetr tfj d fd : Frequency offsetθ : Carrier phaseτ : time delay

Synchronization Parameters(Unknown)


Synchronization Problem

• Challenge: Acquisition time must be kept minimum in burst-mode communications to save bandwidth, thus:1. Feedforward approach for shorter acquisition time2. Data-aided (DA) algorithms to assist synchronization

• Problems:1. Preamble design: What is the best preamble?2. Timing & carrier recovery: Estimation of , ,3. Frame synchronization: Detection of preamble boundaries

Known Preamble Payload

DA Estimator Synchronizer


Comparison with Related Works

CPM-Frequency-Phase-Symbol Timing

-CPM-Burst-Mode Transmission

CPM-Frequency -Phase-Symbol Timing

Tavares (2007) & Shaw (2010)-PSKDebora (2002)-MSK-Only Symbol Timing

Zhao (2006)-CPM-Phase & Timing (No Freq.)Morelli (2007)-PSK

Choi (2002)-PSK-Continuous Transmission

Best Preamble

Timing & Carrier Recovery

Frame Synchronization

This Work Related Work


The Cramér-Rao Bound for Training Sequence Designfor Burst-Mode CPM


Optimum Training Sequence Design:Cramér-Rao Bound Method• Two approaches can be taken in order to derive the optimum

(best) training sequence for estimation of1. Minimize the estimation error variance:

2. Minimize the CRB, i.e. a theoretical lower bound on estimation error variance of any unbiased estimator:

• CRB is computed via

ααααα

|CRBminarg|CRB|ˆ opt2

iiii uuuuE

αuruIuIα ,;ln where)|(CRB2

,1, p

uuEu

jijiiii

Conditional CRB Fisher Info. Mat. (FIM) Likelihood Function

T,, dfu

MSE

ααα

|ˆminarg 2iiopt uuE


Overview of CRB Calculations

• Closed-form FIM for CPM and joint estimation

• It is shown the optimum training sequence must satisfy these two conditions at the same time1. 0

2. Maximize C

ChhBhAhBTThATT

NE

Ts

s 2220

20

220

30

2

0 828222

823/81)(

uI

0

0

11

11

0'

0'

L

ii

L

ii

B

iA

dutugugtRaTRRC g

L

i

L

iiisgig )()()( where)(2)0(

1

0

2

01

20 0


Proposed Optimum Training Sequence

• Proposed sequence based on our analysis

• It satisfies A’=B’=0,1. Optimum sequence for full-response CPM.2. Only 2 transitions => Near-optimum sequence for partial-response CPM

• We can reduce the approximations for partial-response by using /2 rather than L0 to reduce the edge-effect (EF)

L1/4 L1/2

L0

(M-1)

-(M-1)

00

11

11 0'(2) 0')1(

L

ii

L

ii BiA

1

0

1

01

20 0

)(2)0( Maximize )3(L

i

L

iiisgig aTRRC =0 for full-response CPM


Findings and Properties of the Optimum Sequence1. The same sequence is optimum for frequency, phase, and

timing, which is opposed to linear modulations.2. It follows the same pattern for all CPMs.3. The minimized frequency and phase CRBs are independent of

underlying CPM parameters.4. The minimized symbol timing CRB do depend on CPM

parameters such as frequency pulse.5. It decouples symbol timing estimation from phase and frequency

offset estimates.

*220

20

20

30

2

0

*

800022023/8

1)(Ch

TTTT

NE

Ts

s

uI

timing:phase:frequency:

3

2

1

uuu


More Findings

6. The proposed sequence is asymptotically optimum for partial-response CPMs

7. Computer search confirms the proposed sequence at small L0

8. We also computed CRB for random sequences and compared with the optimum one: The optimum sequence delivers significant gain for symbol timing

estimation of partial-response and/or M-ary CPMs. The frequency and phase estimation are less sensitive to the

selection of training sequence especially for long sequences.

1

1208

1~)(8~

|CRB|~CRB

000

*

Lsgg

sgsg

TRLRLTR

CTRC

αα

sequence Proposed:sequence alHypothetic :~

*αα


Timing, Carrier, and Frame Synchronization of Burst-Mode CPM


Timing and Carrier Recovery Framework

• Received Signal:• Decompose timing delay:

• For timing and carrier recovery, we assume integer delay is known

• Joint maximum likelihood (ML) estimator:

• fd and ε are embedded inside the above integral → It requires a 2-dimensional grid search

• Objective: decouple timing and frequency in the LLF

)()()( )2( twtsetr tfj d

ss TT

delay fractional :0.50.5-delayinteger :0

where

s

s

d

d

TT

T stfj

df

d dtTtstreftrf

0 )()(Re,,);(maxargˆ,ˆ,ˆ *2

,,


The Trick: Piecewise Linear Approximation

• Similar and consequent symbols cause the phase to increase/decreaseuniformly

• We use linear approximation of the phase in each segment• There is a lag for partial-response CPMs→ Tl

1

-1

0 4 8 12 16

0

t/Ts

Unw

rapp

ed P

hase

GMSK (BTs=0.3, M=2,h=1/2)

-2

2Original phaseApprox. phase

• Phase response of the optimum sequence


ν: normalized freq. offsetε: fractional delay

:baseband signal shifted by ε

Approximated Log-Likelihood Function (LLF)

• Discrete-time LLF:

• Approximated baseband signal:

ll

ll

ll

sl

sl

sl

TTtTTTTtTT

TTtT

TTTthMTTTthM

TTthMt

00

00

0

0

0*

4/34/34/

4/

/)()1(/)2/()1(

/)()1(,

α

1

0

**0

2),(L

isi iTtqht α

00

00

0

0

0

4/34/34/

4/

/1exp2//1exp

/1exp

NLnNLNLnNL

NLnn

LNnhMjLNnhMj

NnhMjns

]Re,,;1

0

*20

NL

n

nj nsnre r

ns

Linear phase approximation

Original phase

will be used in the above LLF

Time shift + sampling + modulation


Maximization of the LLF

• Our approximation decouples timing and frequency offsets:

• The maximization of the LLF becomes straightforward: Step 1: Frequency offset estimation:

Step 2: symbol timing estimation:

Step 3: carrier phase estimation:

2

11

1* Re),,;( hMjhMjj eee r

~)~( maxargˆ 21~

hM

)1(2ˆ)ˆ( argˆ

*21

)}ˆ()ˆ(arg{ˆ2

ˆ)1(1

ˆ)1( hMjhMj ee

Grid search

Functions of r


Implementation of Frequency Estimator

• The received r[n] is pre-processed

• λ1(ν) and λ2(ν) are computed using FFT operations:

• The precision can be increased via:1. Zero padding prior the FFTs → Kf

2. Interpolation of FFT results

ν

True maximumFFT

otherwise4/3

4/0

0][])1(exp[

][][ 00

0

01 NLnNLNLn

nrhLMjnr

nr

otherwise

4/34/0

][]2/)1(exp[][ 000

2

NLnNLnrhLMjnr

1

0

2/)1(11

0

][)(NL

n

njNhnMj eenr

1

0

2/)1(22

0

][)(NL

n

njNhnMj eenr


Joint ML Estimator Block Diagram


Estimator’s Performance:Frequency and Phase Offsets

0 1 2 3 4 5 6 7 8 9 1010-8

10-7

10-6

Es/N0 (dB)

Nor

mal

ized

Fre

quen

cy E

rror V

aria

nce

MSK1RC (M=2,h=1/2)GMSK2RC (M=4,h=1/4)CRB

0 1 2 3 4 5 6 7 8 9 1010-3

10-2

10-1

Es/N0 (dB)

Pha

se E

rror V

aria

nce

MSK1RC (M=2,h=1/2)GMSK2RC (M=4,h=1/4)CRB

• Similar performance for different CPM schemes with L0=64• Within 0.5 dB of CRB at low to medium SNRs


Estimator’s Performance:Timing Offset

• The error variance attains the CRB for various CPMs.• Exception: 1RC due to large deviations from our template signal.

0 1 2 3 4 5 6 7 8 9 1010-4

10-3

10-2

Es/N0 (dB)

Nor

mal

ized

Tim

ing

Erro

r Var

ianc

e

MSKCRB (MSK)1RC (M=2,h=1/2)CRB (1RC)GMSKCRB (GMSK)2RC (M=4,h=1/4)CRB (2RC)

L0 =64


Frame Synchronization:Overview• Prior to Timing and Carrier recovery we must determine preamble

boundaries, i.e. start-of-signal (SoS), in 2 steps:1. SoS detection algorithm: Detects the arrival of a new burst and

tentative location of the SoS

2. SoS estimation algorithm: Estimates the exact location of SoS within an observation window

PreambleIncoming Samples

Noise Only Preamble Random Data

Nw

δ Np

Nw : Observation window lengthNp : Preamble lengthδ : Estimation parameter


SoS Estimation Algorithm

• Apply ML estimation rules to the observation window

• Unwanted parameters, [ ν,θ,αd ] ,are averaged out

• Find δ for which the likelihood function is maximized

1 2)2(2

1

0

222 ][][1exp][1exp

)(1),,,;(

w

w

N

n

nj

nNd ensnrnrp

αr

1 1

1

1*

1**2 ][][)(][][][][2][)(;

w p w

p

pN

n

N

d

dN

Nnss

dN

n

dnrnrdRdnsnsdnrnrnrCp

r

1

1

1*

1**2 ][][)(][][][][2][)( maxargˆ

w w

p

pN

n

D

d

dN

Nnss

dN

ndnrnrdRdnsnsdnrnrnrC

ssimulation viaoptimized :0)()( qNC qw factor Complexity:1 pND


SoS Detection Algorithm

• We apply a simple likelihood ratio test (LRT)

• Using the same techniques as SoS estimator, the LRT becomes

• determines complexity• : test threshold; selected for a given probability of false alarm (Neyman-

Pearson criterion).• ROC: • The LRT is performed using a sliding window.

holdTest thres :preamble aligned-Fully :

preamble No :

samples ofvector :

1

0

HH

N ppr

:'1 pND 'D

1''0'' |)(Pr1 vs.|)(Pr HrLPHrLP DpDDDpDFA

′ ∗ ∗

1

0

′

1

1≷0

′

;;

≷


Frame Synchronization Performance

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-4

0.99

0.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

0.999

1

Probability of False Alarm (PFA)

Pro

babi

lity

of D

etec

tion

(PD

)

D'=2, L0=32

D'=3, L0=32

D'=4, L0=32

D'=2, L0=64

0 1 2 3 410

-5

10-4

10-3

10-2

10-1

100

Es/N0 (dB)

Pro

babi

lity

of F

alse

Loc

k (P

FL)

L0=64, D=63, q=1L0=64, D=8, q=0.5L0=64, D=4, q=0.4L0=64, D=2, q=0.3L0=32, D=31, q=1L0=32, D=8, q=0.5L0=32, D=4, q=0.4L0=32, D=2, q=0.3

SoS Detector Receiver Operating Characteristics (ROCs) SoS Estimator

• Probabilities are computed via simulations• Probability of false lock: Pr

Improvement

SNR = 1 dB


Putting Everything Together:Simulating a Burst-Mode CPM Receiver• Transmit bursts of optimum preamble (L0) + 4096 information bits• Apply AWGN + random frequency, phase and timing offsets• Receiver’s structure:

• Design parameters: N=2, Kf =2, Gaussian interpolator D=D’=4, Nw =2NL0

optimized for each scenario using simulations

SoSDetector

SoSEstimator

Joint MLEstimator MFs Viterbi

Algorithm

Frame Synchronization Timing & Carrier Recovery

CPM Demodulator+PLL

r[n] âi


Bit Error Rate (BER) Performance

GMSK (h=1/2, M=2) 2RC (h=1/4, M=4)

0 2 4 6 8 1010

-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

Bit

Erro

r Rat

e (B

ER

)

L0=32

L0=64

Perfect Synchronization

0 2 4 6 8 1010

-4

10-3

10-2

10-1

100

Eb/N0 (dB)

Bit

Erro

r Rat

e (B

ER

)

L0=32

L0=64

Perfect Synchronization

• A preamble of 64 bits performs within 0.1 dB of the ideal synchronization


Applications to SOQPSK(iNET)


What is SOQPSK?

• An OQSPK-like modulation, which has a CPM representation

• The precoder increases spectral efficiency of SOQPSK compared to a binary CPM by imposing: 1 cannot be followed by 1 and vice versa

• Partial-response SOQPSK-TG is adopted by the iNET standard

Precoder CPMModulator

)(ts 1,0,1n 1,1na

0 1 2 3 4 5 6 7 8-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Normalized Time (t/Ts)

Pha

se R

espo

nse

SOQPSK-MIL

SOQPSK-TG


iNET Synchronization Preamble

• Length of L0=128 bits• Periodic and consists of repeating a sequence of 16

bits 8 times as

• One period in terms of ternary symbols

7,,0for 0,0,1,0,1,1,0,10,1,0,1,0,1,0,1

12

2

ka

a

k

k

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-1

0

1

Time Index

Tern

ary

Sym

bols

( i )


Burst-Mode Synchronization of SOQSPK-TG:iNET Preamble• We apply our CPM techniques to SOQPSK by considering ternary

symbols• iNET preamble phase response

• SOQPSK-TG’s response is approximated by SOQPSK-MIL’s• Derivation of joint ML estimation algorithm is similar to the CPM’s

when h=1/2 and M=2

0 4 8 12 16 20 24 28 32

0

Normalized Time (t/Ts)

Unw

rapp

ed P

hase

(rad

)

SOQPSK-TGSOQPSK-MIL

2

3

4

Tl

iNET

Approximation


Estimation Error Variances for SOQPSK:Optimum vs. iNET preambles

0 1 2 3 4 5 6 7 8 9 1010-9

10-8

10-7

Es/N0 (dB)

Nor

mal

ized

Fre

quen

cy E

rror V

aria

nce

SOQPSK-MIL, OptimumCRB (SOQPSK-MIL, Optimum) SOQPSK-TG, OptimumCRB (SOQPSK-TG, Optimum)SOQPSK-TG, iNETCRB (SOQPSK-TG CRB, iNET)

0 1 2 3 4 5 6 7 8 9 1010-4

10-3

10-2

Es/N0 (dB)N

orm

aliz

ed T

imin

g E

rror V

aria

nce

SOQPSK-MIL, OptimumCRB (SOQPSK-MIL, Optimum) SOQPSK-TG, OptimumCRB (SOQPSK-TG, Optimum)SOQPSK-TG, iNETCRB (SOQPSK-TG, iNET)

• Performance degradation due to transitions in the iNET preamble• Phase error variance has the same behavior as the frequency’s


Overall Performance of SOQPSK:Optimum vs. iNET preambles

• iNET preamble performs very close to the optimum one when L0=128• The optimum preamble with L0 =64 is within 0.2 dB of the iNET’s• False locks for L0 =32 at low SNRs

0 1 2 3 4 5 6 7 8 9 1010

-5

10-4

10-3

10-2

10-1

100

Es/N0 (dB)

Bit

Erro

r Rat

e (B

ER

)

Optimum, L0=32

Optimum, L0=64

Optimum, L0=128

iNET, L0=128

Perfect Synchronizarion

Preamble dB loss @ BER=10-4

iNET 0.4

L0 =128 0.3

L0 = 64 0.6

L0 = 32 1


Conclusions

• Gave an overview of feedforwad synchronization methods for CPM

• Derived the optimum training sequence for CPM in AWGN Minimizes the CRBs for all three synchronization parameters Applicable to the entire CPM family including M-ary and partial response

• Developed a joint ML DA estimation algorithm for timing and carrier recovery Designed based on the optimized training sequence Performs within 0.5 dB of the CRB at SNRs as low as 0 dB for L0 = 64

• Simulated a burst-mode CPM receiver: only 0.1 dB SNR loss with the optimum preamble of L0=64

• Applied CPM results to SOQSPK for iNET standard• Final result: developed a complete feedforward DA

synchronization scheme for general CPM including: best training sequence, timing and carrier recovery, and frame synchronization


Areas of Future Study

• Application to frequency-selective channels: Training sequence design Timing and carrier recovery Frame synchronization

• Joint synchronization and channel estimation• Analysis of frame synchronization algorithm as it can

be applied to other modulations, Probability of false lock (SoS estimator) Probabilities of false alarm and detection (SoS detector)

• Hardware implementation and trade-offs between complexity and performance


Thank You!

Questions?

Please let it be OVER… Interesting

I wonder if this technique would work

for my problem What does that slide say?

Dunno, I’m playing Fruit Ninja


Derivation of CRB

• FIM, for CPM and AWGN, is computed via

• Closed-form FIM for CPM and joint estimation:

dtuu

tstsN

T

jiji ,,),,(Re2)(

0

0

*2

0,

αuαuuI

noise w/oPreamble Received

;2,, ααu tjtfj

s

s eeTEts d

Duration Preamble

00 sTLT

ChhBhAhBTThATT

NE

Ts

s 2220

20

220

30

2

0 828222

823/81)(

uI

PulseFrequency CPM :)(

1

0

1

00

1

00

1

00

0 00

00

00

tg

dtjTtgiTtgC

dtiTtgB

dtiTttgA

L

i

L

j

T

ssji

L

i

T

si

L

i

T

si


Training Sequence for Symbol Timing

• Symbol Timing CRB should be minimized,

• The CRB’s denominator should be maximized, or, simultaneously1. Maximize C2. Minimize the quadratic form: J is positive-definite It is

minimized for

• Assuming C is independent of (A,B):

• In practice,

1. Full-response CPM:

2. Partial-response CPM:

1

21

2221

1211222

28

288)|CRB(

hBhA

hBhACh

IIII

α

0 :form Quadratic T Jvv

0JvvT

0 BA0v0 subject to maxargopt BAC

αα

1

0

1

01

20 0

)(2)0(L

i

L

iiisgig aTRRC

.sufficient is 0const.0)( BACTR sg

.0 AND maximize 0)( 2

0 10

BATR L

i iisg


Proposing the training sequence

• A=B=0 is satisfied when:

• C is maximized when is maximized or sign transitions are minimized in the sequence.

• We propose the following binary sequence:

• It satisfies A’=B’=0,1. Optimum sequence for full-response CPM.2. Only 2 transitions => Near-optimum sequence for partial-response CPM.

0 0

1 111 0' AND 0'

L

i

L

iii BiA

2

0 10L

i ii

L0/4 L0/2 L0/4

1

-1


Generalization of the Proposed Sequence

• We can reduce the approximations for partial-response by using /2 rather than L0 to reduce the edge-effect (EF)

• M-ary CPM:

• Frequency and Phase CRBs:

Equalities hold, if λ1=λ2=0 or A=B=0 → Proposed sequence for symbol timing.

• Approximations:

onconstelati in the valuemaximum tos'Set 2iiC

233

32|CRB0

20

230

230

0

1

LABLACLACL

L

α

2

3

4434/

4/2

3|CRB 30

22

020

20

20

30

2

2

LABLL

BCL

BCLL

fd

α

Frequency Phase Timing

Full-response Exact Exact ExactPartial-response EF EF C and EF


Proposed Training Sequence and its Approximations• We propose the following sequence for general CPM and all

synchronization parameters:

• EF is reduced as the sequence length is increased• Approximations on C for partial-response CPM and symbol timing:

• The proposed sequence is asymptotically optimum

L1/4 L1/2

L0

(M-1)

-(M-1)

1

1208

1~)(8~

|CRB|~CRB

000

*

Lsgg

sgsg

TRLRLTR

CTRC

αα

sequence Proposed:sequence alHypothetic :~

*αα


PSD of the Optimum Sequence


Computer Search for the Optimum Sequence:Genetic Algorithm (GA)• The ratio of the minimum CRB found via GA to the proposed

sequence CRB:

• The GA is unable to find a better sequence than the proposed one even at short to moderate lengths

20 30 40 50 60 70 80 90 100 110 1200.98

1

1.02

1.04

1.06

1.08

1.1

Sequence Length

CR

BR

atio

1REC (h=1/2, M=2)GMSK (BT = 0.3)2RC (h=1/4, M=4)


Random vs. Optimum Sequences:Symbol Timing Estimation

True CRB vs. Opt. CRB for L=32Comparison of estimator performancefor “unknown and random” with“known and optimum” trainingsequences.

UCRB vs. Opt. CRBComparison of estimator performance for“known but randomly selected” with“known and optimum” training sequences.

-6 -4 -2 0 2 4 610

-3

10-2

10-1

Es/N0 (dB)

CR

B(

)

True CRB (1REC)Opt. CRB (1REC)True CRB (1RC)Opt. CRB (1RC)

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

L0U

CR

B( )

/CR

Bm

in(

) (dB

)

2RC (M=4, h=1/4)

GMSK (BT = 0.3)1REC (M=2, h=1/2)

1RC (M=2, h=1/2)


Random vs. Optimum Sequences:Frequency and Phase Estimation• UCRB vs. Optimum CRB

• Frequency and phase estimation are less sensitive to the selection of training sequence

10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

L0

UC

RB

(f d)/CR

Bm

in(f d) (

dB)

GMSK (BT=0.3)2RC (M=4,h=1/4)1REC (M=2,h=1/2)1RC (M=2,h=1/2)

10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

L0

UC

RB

( )/C

RB

min

( ) (

dB)

GMSK (BT=0.3)2RC (M=4,h=1/4)1REC (M=2,h=1/2)1RC (M=2,h=1/2)


Derivation of CRB

• Assumption: Joint estimation of u = [fd , θ, τ]T in AWGN,

• The FIM elements:

• Observation interval: T0 = L0Ts or L0 symbols• Computation of three partial derivatives:

• Note for CPM:

• Frequency pulse:

tweeTEtr tjfj

s

s d α,2)(

0

0

*

0,

,,Re2 T

jiji dt

uts

uts

NuuuI

uαuuuuu ,;,,,,,,2, tstjtstjststtsjfts

d

1,, * uu tsts Pulse Frequency / where; 1

0

0

ttqtgiTtgt L

isi

α


Derivation of CRBMore Simplifications• A, B, and C relate CRBs to frequency pulse and data symbols.

They are approximated as,

• Edge-effect: for partial-response CPM• Correlation function of g(t):

dutugugtRTjiRC

BduuugTiA

L

i

L

jgsgji

L

ii

L

i

LTsi

s

1

0

1

0

1

0

1

00

0 0

00

where

21 where

2

2/01 LLL

1

0

1

01

20 0

)(2)0(1for 0L

i

L

iiisgigsg aTRRCnnTR

Pulse Shape 2REC 3REC 2RC 3RC GMSK0.125 0.0833 0.1875 0.125 0.13270.0625 0.0556 0.0312 0.0589 0.0551

0 0.0278 0 0.0036 0.0036

)0(gR

)( sg TR

)2( sg TR ≈ 0


Exhaustive Computer Search for CPM

• Exact computer exhaustive search vs. approximated proposed sequence:

CPM Exhaustive Search Proposed Sequence ΔCRB

1REC 0

2REC 0.6%

3REC 0

1RC 0

2RC 0.7%

3RC 0

Length = 16, Binary CPM


CRB for Random Sequences

• Comparison of the optimum sequence and a random sequence• True CRB:

1. The likelihood function is averaged over α2. Only feasible for MSK-type CPMs, i.e. binary, h=1/2, and full-response

3. The above expectation is computed via simulations

• Unconditional CRB (UCRB):1. Conditional CRB is averaged over α.

2. A closed-form expression was found for symbol timing

1

0

1

0 00

2

0,

0 0

ˆ2tanhˆ2tanhˆˆ2 L

i

L

jj

si

s

l

j

k

islk a

NEa

NE

ua

uaE

NEuI

)(UCRB|CRBˆ|CRBˆ 2over Averaging2| αα αr

ααr EEE


CRB for Random Data Sequence:True CRB• NDA estimators with random (unknown) data

sequence:• Computation of true CRB, as a nuisance parameter.• Quite complex; may not be feasible for general CPM.

• Special case: binary, h=1/2 and full-response:1. PAM representation:

2. pseudo symbols: independent Gaussian RVs.3. Averaging LF over + partial derivatives→ FIM elements:

4. Monte-Carlo simulations → CRBs.

odd)(Im

even)(Re~ Receiver)(

20

22

02

0 ndtnTtctre

ndtnTtctreaiTtcats

Tss

tfjT

stfj

ni

sid

s

d

1

0

1

0 00

2

0,

0 0 ~2tanh~2tanh~~2 L

i

L

jj

si

s

l

j

k

islk a

NEa

NE

ua

uaE

NEuI

s'ias'ia


CRB for Random Data Sequence:Unconditional CRB (UCRB)• Averaging the conditional CRB w.r.t the training sequence results

in unconditional lower bound on the MSE:

• Symbol timing CRB can be expressed as:

• Where:

• Thus:

)(UCRB|CRBˆ|CRBˆ 2over Averaging2| αα αr

ααr EEE

Simplify

Numerical1/

)|(UCRB0 Qαα

α α Ts

s ENE

T


Computing Timing UCRB

• Define:• Approximation via Taylor series:

• Eigen-decomposition of Q:

• Expected value of Z

• Using the same approach and employing central limit theorem:

QααTZ

3

2

00 /1

/UCRB

ZEZE

NET

ZE

NET

s

s

s

s


UCRB:Computation of E{Z2}• Recall , the second moment of Z is written as

• Where and• requires computation of and . • Approximation:

• Xi`s are uncorrelated and hence independent.• Finally

0

1

2L

iiXZ

31,0~ remlimit theo Central

2MNXi


Maximization of LLF(Details 1/2)1. Discrete-time LLF:

2. Approximated CPM signal:

3. Approximated LLF

]Re,,;1

0

*20

NL

n

nj nsnre r

00

00

0

0

0

4/34/34/

4/

/1exp2//1exp

/1exp

NLnNLNLnNL

NLnn

LNnhMjLNnhMj

NnhMjns

1

4/3

)/()1(2

14/3

4/

)2//()1(214/

0

)/()1(2

0

0

0

0

0

00

][

][][Re),,;r(

NL

NLn

LNnhMjnj

NL

NLn

LNnhMjnjNL

n

NnhMjnjj

enre

enreenree


Maximization of LLF(Details 2/2)4. Rearrange LLF as

5. If , the LLF is maximized when phase is

6. The LLF becomes , which is maximized by maximizing

7. That is,

2

11

1 Re),,;( hMjhMjj eee r

)()(, 2)1(

1)1( hMjhMj ee

),(arg~

)~,~(

)~()~(Re2)~()~()~,~( *21

~1222

21

2 hMje

hM

)1(2)~()~(arg~

*21

~)~( maxargˆ 21~


Linear Phase Approximation:Other CPM Examples

0 4 8 12 16

0

1RC (M=2,h=1/2)

t/Ts

Unw

rapp

ed P

hase

0 4 8 12 16

0

t/Ts

2RC (M=4,h=1/4)

Unw

rapp

ed P

hase

2

-2

-2

2


FFT and Interpolation Precision

0 1 2 3 4 510-7

10-6

10-5

10-4

Es/N0 (dB)

Nor

mal

ized

Fre

quen

cy E

rror V

aria

nce

No Interpolation, Kf=1No Interpolation, Kf=2No Interpolation, Kf=4Parabolic, Kf=1Parabolic, Kf=2Parabolic, Kf=4Gaussian, Kf=1Gaussian, Kf=2Gaussian, Kf=4CRB

• Frequency MSE for different interpolators and zero padding factors.• Both interpolation and zero padding must be present based on simulations.

No zero padding

No interpolation

Our choice:Kf =2, Gaussian


SoS Estimation Algorithm(Details 1/3)1. Likelihood function of all unknown parameters

2. Drop constant factors

3. Approximate the exponential function using 2nd degree Taylor series

1 2)2(2

1

0

222 ][][1exp][1exp

)(1),,,;(

w

w

N

n

nj

nNd ensnrnrp

αr

1

)2(*22 ][][Re2expexp),,,;(

wN

n

vnjwd ensnrNp

αr

1 1)2)(2(*

4

1 1)2)(2(**

4

12*

2

][][][][Re1

][][][][Re1

][][Re21)(),,,;(

w w

w w

w

N

n

N

m

mnj

N

n

N

m

nmj

N

n

njd

emsnsmrnr

emsnsmrnr

ensnrCp

αr


SoS Estimation Algorithm(Details 2/3)4. Average the likelihood function w.r.t θ

5. Rearrange the above likelihood function

6. Take the expectation w.r.t data symbols

1 1)(2**

4 ][][][][Re1)(

),,,;(21),,;(

w wN

n

N

m

mnj

dd

emsnsmrnrC

drp

αr

1

1

1**2

12 ][][][][Re2|][|)(),,;(

w ww N

d

dN

n

djN

nd dnsnsdnrnrenrCp αr

1

1

1*

1**2

12 ][][)(][][][][Re2|][|)(),;(

p w

p

pw N

d

dN

Nnss

dN

n

djN

ndnrnrdRdnsnsdnrnrenrCp

r

Preamble Random Data


SoS Estimation Algorithm(Details 3/3)7. A frequency estimate can be derived from each term in the LLF

8. If we use each estimate in its corresponding term, the LLF becomes independent of frequency offset

9. Since the LLF is now only a function of δ, it can be used for estimation of SoS

1*

1** ][][)(][][][][arg

21ˆ

dN

Nn

dN

nssd

w

p

p

dnrnrdRdnsnsdnrnrd

1

1

1*

1**

12 ][][)(][][][][2|][|)();(

p w

p

pw N

d

dN

Nnss

dN

n

N

ndnrnrdRdnsnsdnrnrnrCp

r

1

1

1*

1**

12 ][][)(][][][][2|][|)(maxargˆ

p w

p

pw N

d

dN

Nnss

dN

n

N

ndnrnrdRdnsnsdnrnrnrC

qwNC )()(


Optimization of C(δ)

• PFL is computed via simulations• C(δ) is more important for larger values of D

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-3

10-2

10-1

100

q

Pro

babi

lity

of F

alse

Loc

k (P

FL)

D=2D=4D = 8D=63

qwNC )()( GMSK

Np= 64Es /N0=1 dB

synchronization techniques for burst-mode …ehsan hosseini department of electrical engineering...

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