advanced modulation/synchronization/coding schemes

Erasmus Summer School 20091

Advanced

Modulation/Synchronization/Coding

Schemes

Parameter Estimation and Synchronization in Digital Receivers

Dipl.Dipl.--IngIng. Dr. . Dr. techntechn. . WilfriedWilfried GappmairGappmair

Institute of Communication Networks and Satellite Communications

Graz University of Technology, Austria

2

Contents

� Introduction– Motivation and background

� Signal model– Modulation schemes

– Distortions

– Transmission parameters

– Pulse shaping and matched filter

� Synchronization in digital receivers– What does synchronization mean

– Basic receiver architecture

– Maximum-likelihood principle

– Estimation criteria

� Feedforward algorithms– Carrier phase estimation

– Carrier frequency estimation

– Symbol timing estimation

� Feedback algorithms– Digital recovery loops

– Carrier phase recovery

– Symbol timing recovery

� Research areas

3

Introduction

� Turbo codes

– Berrou et al. (ICC’93)

– Shannon bound (1948)

– Computational complexity

� Demodulator

– Synchronization with PLL as standard solution

o Continuous vs. burst mode

o Impact of phase noise (small/large bandwidths)

– Challenging operational conditions: very low SNR values, burst mode

– Major parameters to be estimated: carrier frequency/phase, symbol timing

4

Introduction

Basic receiver architecture

NCO

π/2

Decimation + MF

?

I

Q

Antialias filter IF ADC

Decimation + MF

FEC

5

Introduction

Performance of different FEC schemes (BPSK/AWGN)

0 2 4 6 8 10

1.×10−6

0.00001

0.0001

0.001

0.01

0.1

1

BPSK uncoded

ESA/NASA (Rc = ½)

Galileo Code (Rc = ¼)

Turbo Code (Rc = ¼)

Eb/N0 [dB]

BER

6

Signal model

� Modulation schemes

– PSK (spectral efficiency)

– QAM (nonlinear effects: TWT)

– APSK (DVB-S2 standard)

– CPM (MSK, GMSK)

� Distortions

– Additive white Gaussian noise (AWGN)

� Transmission parameters

– Carrier frequency offset: ∆f

– Carrier phase: θ

– Symbol period: T

– Symbol timing: τ

7

Signal model

R2

R1

π/4

π/6

Symbol constellation for 16-APSK

8

� Pulse shaping

– Nyquist criteria (no ISI)

– Root-raised cosines (RRCos): h(t)

– Matched filter: h*(–t)

– Roll-off factor: α

� MF input signal

– Parameter vector: v = (∆f, θ, τ)

– Additive white Gaussian noise (AWGN): w(t)

Signal model

)()()(),(),( )2( twkTthcetwtstrk

k

ftj +

−−=+= ∑+∆ τθπ

νν

9

Signal model

Impulse response h(t) of different RRCos filters

-3 -2 -1 0 1 2 3 4

0

0.25

0.5

0.75

1

1.25

1.5

t/T

α = 0.25 α = 0.50 α = 1.0

10

Signal model

Signal r(t,v) at MF input (α = 0.25)

2 4 6 8 10

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

t/T

Es/N0 = 10 dB

Es/N0 = 20 dB

Es/N0 → ∞

11

Signal model

� MF output signal

– Convolution with MF impulse response → RCos

– Additive non-white Gaussian noise: n(t)

)()()(),(),( )2(* tnkTtgcethtrtxk

k

ftj +

−−=−⊗= ∑+∆ τθπ

νν

)()()(

)2(1

)cos()(sinc)()()(

*

2

*

thtwtn

Tt

TtTtththtg

−⊗=

−=−⊗=

α

απ

12

Signal model

Impulse response g(t) of different RCos filters

-3 -2 -1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

t/T

α = 0.25 α = 0.50 α = 1.0

13

Signal model

Signal x(t,v) at MF output (α = 0.25)

2 4 6 8 10

-1.5

-1

-0.5

0

0.5

1

1.5

2

t/T

Es/N0 = 10 dB

Es/N0 = 20 dB

Es/N0 → ∞

14

Synchronization in digital receivers

� What does synchronization mean

– Synchronization in TDMA systems

– Synchronization as optimization problem

o Multi-dimensional

o Nonlinear

o Stochastic

– Object function

o Minimal Euclidean distance between received / replica signal

o Maximal correlation peak between received / replica signal

– Synchronization stages

o Acquisition (initialization)

o Tracking (controlling)

15


Euclidean distance

Parameter

global minimum

local minima

One-dimensional schematic of the synchronization process

16


� Basic receiver architecture– Demodulator

o IF converter + anti-alias filter

o AD conversion + mixer stage

o MF + decimation

– Synchronization

o Feedforward algorithms

� Used for acquisition (data-aided)

� Specified observation length

� No stability problems

o Feedback algorithms

� Used for tracking (decision-directed, non-data-aided)

� Probabilistic behavior of the settling time

� Stability problems (cycle slips, hang-ups)

17


NCO

π/2

Decimation + MF

Syncrhonization

(Acquisition + Tracking)

I

Q

Antialias-Filter IF ADC

Decimation + MF

Basic receiver architecture

18


� Maximum-likelihood principle (1)

– Replica signal with vector of trial parameters to be estimated

– Normal distribution of the probability function

– Observation length of L symbols

– Maximization procedure

)~(maxargˆ~

ννν

f=

−−= ∫

LT

w

dttstrC

Cf0

2

2

21 )~,(),(

2exp)~( ννν

σ

19



– Log-likelihood function

– Maximization of the correlation function

=Λ ∫

LT

dttstrf0

* )~,(),(Re~)~(ln)~( νννν

∑ −−= +∆

k

k

tfj kTthcets )~()~,( )~~

2( τθπν

−−=Λ ∑ ∫

+∆−

k

LT

tfj

k dtkTthtrec0

*)~~

2(* )~(),(Re)~( τθπνν

20



– Data-aided (DA)

– Decision-directed (DD)

– Non-data-aided (NDA)

=Λ

=Λ

=Λ

∑

∑

∑

k

k

k

kk

k

kk

yF

yc

yc

)]~([Re)~(

)~(ˆRe)~(

)~(Re)~(

NDA

*

DD

*

DA

νν

νν

νν

4444 34444 2143421

)~(:)~(

0

*

1«|~

|

)~~

2(

0

*)~~

2(

)~(),(

)~(),()~(

ττ

θπ

θπ

τ

τ

+=

∆

+∆−

+∆−

∫

∫

−−≈

−−=

kTxx

LT

Tf

Tfkj

LT

tfj

k

k

dtkTthtre

dtkTthtrey

ν

νν

21


� Estimator criteria

– Bias (offset)

– Jitter variance (CRLB)

– Consistency property

– Computational complexity

22

Feedforward algorithms

� Carrier phase estimation (1)

– Carrier phase constant (observation length: L symbols)

– No frequency error assumed

– Symbol timing established

– Modified CRLB

02

1MCRLB

NEL s

=

23



– Signal model

– Symbols ck known (DA)

– ML principle applicable

– DA unambiguity: |θ | < π

kk

j

k ncex += θ

=

= ∑∑

−

=

−1

0

*~

*

~ argRemaxargˆL

k

kk

k

j

kk xcexc θ

θθ

24



– Modulation scheme: MPSK

– Symbols ck unknown (NDA)

– Data modulation must be canceled

– Ad-hoc solution: power-law estimator

– NDA unambiguity: |θ | < π / M

= ∑

−

=

1

0

arg1ˆ

L

k

M

kxM

θ

25


θ

e-j(⋅)

1/M arg(⋅)

L

Phase shift

Power law

xk

NDA feedforward estimator for carrier phase

26


4 6 8 10 12 14 16

0.0001

0.0005

0.001

0.005

0.01

0.05

DA

NDA

MCRLB

Es/N0 [dB]

Jitter variance [rad2]

Jitter variance for QPSK (L = 32)

27


� Carrier frequency estimation (1)– Frequency offset due to Doppler shift and oscillator drifts

– Signal power affected at MF output

– Nyquist criterion violated: ISI problems

– Simplified signal model for analysis (|∆fT| « 1 )

– Symbol timing established

k

fTkj

kk necx += +∆ )2( θπ

28


∆fT

fT

1/2 1 -1/2 -1

signal power

Spectral conditions due to frequency errors

29


� Carrier frequency estimation (2)

– Maximum-likelihood (ML) solution

– Data-aided estimation

– Modified CRLB

=∆ ∑ +∆−

∆k

Tfkj

kkf

excf )~~

2(*

~,

~Remaxarg)ˆ,ˆ( θπ

θθ

0

22 )1(

1

2

3MCRLB

NELL s−=

π

30


� Carrier frequency estimation (3)

– ML too complex (searching)

– Application of DFT (FFT) algorithms: Rife-Boorstyn

– Autocorrelation principle: Luise-Reggiannini, Mengali-Morelli

– Main figures of merit

o Operational range

o Jitter variance

o Threshold effect

o Computational complexity

31


� Estimation of the symbol timing (1)

– Knowledge of carrier frequency/phase irrelevant

– Data symbols ck unknown (NDA)

– ML solution too complex → Oerder-Meyr algorithm

– Modified CRLB

0

2 )0(2

1MCRLB

NEgTL s&&−

=

222

312 8)31()0( ααπ −+=− gT &&

32

Feedback algorithms

� Tracking loops

– Carrier phase recovery

– Symbol timing recovery

– Main figures of merit

o Open-loop (detector) characteristic (S-curve)

o Jitter variance

� Linear range: ~ MCRLB

� Constant range: self-noise (ISI)

� Nonlinear range: threshold effect (DD, NDA)

– Linearization

o Loop filter: order of model

o Slope of S-curve in the stable equilibrium point: detector gain

33

Feedback algorithms

MCRLB

Es/N0 [dB]

log σ

v2

nonlinear

constant

linear

Jitter variance for feedback recovery loops

34

Feedback algorithms

Linearized tracker model

kv

Kd F(z) 1/(z – 1)

nk

vk

Running sum Loop filter Detector gain

∆vk

35

Feedback algorithms

� Digital recovery loops (1)

– First-order loops

o F(z) = KF

o Offset problems if input affected by a drift

o Loop gain K0 = KF Kd

o Transfer function H0(z)

0

01

1

)(

)()(

Kz

z

zv

zvzH

+−

−=

∆=

36

Feedback algorithms


– Second-order loops

o F(z) = a + b / (z – 1)

o No offset problems if input affected by a drift

o Transfer function H0(z)

dd bKzaKz

zzH

+−+−

−=

)1()1(

)1()(

2

2

0

37

Feedback algorithms


– Nonlinear effects: hang-ups, cycle slips

– One-sided equivalent noise bandwidth BLT

– Transfer function for loop noise: Hn(z) = 1 – H0(z)

– Jitter variance ~ BLT

∫=21

0

22 )(|)(| fTdeHTB fTj

nL

π

38

Feedback algorithms

� Carrier phase recovery (1)

– Modified CRLB: 2L→ 1 / BLT

– DA signal model

o Symbols ck known

o No carrier frequency offset

o Symbol timing established (no oversampling)

k

j

kk necx += θ

39

Feedback algorithms


– ML detector

o DA log-likelihood function

o Error signal uk

o Detector characteristic (S-curve)

=Λ ∑ −

k

j

kk exc θθ~

*

DA Re)~(

]Im[]Im[ *ˆ*

kk

j

kkk ycexcu == − θ

πθθθθ θ <=+=== ||,sin}])(Im[E{}0ˆ|E{)( *

DA k

j

kkk neccuS

40

Feedback algorithms


– DD detector: ck → decisions

– NDA detector: nonlinearity (power law)

– Unambiguity: |θ | < π / M

MSScc

ycu

kk

kkk

πθθθ <→→

=

||),()(:ˆ

]ˆIm[

DADD

*

MM

Mnec

MuS

yM

u

M

k

j

kk

M

kk

πθθθθ θ <=+===

=

||),sin(1

}])Im[(E{1

}0ˆ|E{)(

]Im[1

NDA

41

Feedback algorithms

e–j(⋅)

DD/NDA F(z)

kθ

xk

Σ

Decisions

uk

kc yk

Feedback loop for carrier phase recovery

42

Feedback algorithms

-3 -2 -1 1 2 3

-1

-0.5

0.5

1

DA

DD

NDA

θ [rad]

Detector characteristic for QPSK

43

Feedback algorithms

� Symbol timing recovery (1)

– Modified CRLB: 2L→ 1 / BLT

– Interpolation

o Asynchronous (equidistant) sampling

o Polynomial (Lagrange) interpolation

∑−

=+=

1

0

n

i

ikik xy λ

∏∏−

≠=

−

≠= −

−=

−

−=

1

0

1

0

ˆˆ n

ikk

n

ikk ki

ki

ki

k

tt

t ετλ

44

Feedback algorithms

xk

τ

λ3 λ2 λ1 λ0

z-1 z

-1 z

-1

yk

Cubic Lagrange interpolator (n = 4)

45

Feedback algorithms

� Symbol timing recovery (2)

– NDA detector

o ML solution too complex

o NDA and carrier-blind solution

o Gardner (GA) algorithm

o S-curve

])Re[( *

2/11 −−−= kkkk xxxu

)2sin()41(

)2sin()(

2πε

απ

παε

−=S

46

Feedback algorithms

kτ

KF GA Σ

Interpolator xk

uk

yk

Symbol timing recovery with first-order loops

47

Feedback algorithms

kτ

KF GA

e–j(⋅)

Σ

Interpolator

DD F(z)

kθ

xk

Σ

Decisions kc

Joint recovery of carrier phase and symbol timing

48

Research areas

� Satellite monitoring– Detection of modulation schemes

– Estimation of symbol period and roll-off factor

– SNR estimation

� Joint synchronization and decoding– Parameter estimation + data detection (FEC)

– Turbo synchronization

o Maximum-likelihood approach (complexity)

o Expectation-maximization solution (performance/complexity tradeoff )

o Per-survivor processing (low complexity)

� Fading channels– SNR estimation (ACM systems)

– Fading parameters

� PLL models– Nonlinear-stochastic differential equations

– Acquisition behavior

– Phase noise

49

References

1. Mengali, U., D’Andrea, A. N., Synchronization Techniques for Digital Receivers, Plenum Press:

New York, 1997.

2. Meyr, H., Moeneclaey, M., Fechtel, S. A., Digital Communication Receivers, Wiley: New York, 1998.

3. Kay, S. M., Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall: Upper Saddle River, NJ, 1993.

4. Proakis, J. G., Digital Communications, McGraw-Hill: New York, 1989.

5. Berrou, C., Glavieux, A., Thitimajshima, P., “Near Shannon Limit Error-Correcting Coding and Decoding: Turbo Codes”, in Proc. IEEE Int. Conf. Comm. ‘93, Geneva, Switzerland, pp. 1064–1070, May 1993.

6. Viterbi, A. J., Viterbi, A. M., “Nonlinear Estimation of PSK-Modulated Carrier Phase with Application to Burst Digital Transmission”, IEEE Trans. Inf. Theory, vol. 29, pp. 543–551, July 1983.

7. Morelli, M., Mengali, U., “Feedforward Frequency Estimation for PSK: a Tutorial Review”, Euro. Trans. Telecomm., vol. 9, pp. 103 – 116, March/April 1998.

8. Oerder, M., Meyr, H., “Digital Filter and Square Timing Recovery”, IEEE Trans. Comm., vol. 36, pp. 605–612, May 1988.

9. Gardner, F. M., “A BPSK/QPSK Timing-Error Detector for Sampled Receivers”, IEEE Trans.

Comm., vol. 34, pp. 423–429, May 1986.

10. Gardner, F. M., “Interpolation in Digital Modems – Part I: Fundamentals”, IEEE Trans. Comm., vol. 41, pp. 501–507, March 1993.

advanced modulation/synchronization/coding schemes

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