Carrier and Timing Synchronization of

BPSK via LDPC Code Feedback

Esteban Vallés, Richard Weseland John Villasenor

Electrical Engineering DepartmentUniv. California, Los Angeles

Christopher Jones Marvin Simon

Jet Propulsion LaboratoryPasadena, CA

2

IntroductionThis work addresses the problem of joint carrier-phase estimation and symbol timing recovery under low SNR conditions in a soft decision-directed LDPC-coded system.

Timing Recovery:Previous treatments in the literature focus on the use of output codewordsWe exploit information available from constraint node metrics of an LDPC code

Carrier-Phase Synchronization: Two basic trends:1. Classis Synchronization techniques: Costas Loop, PLL, squaring loops.2. Iterative Synchronization techniques

Modifying iterative detection/decoding algorithms and/or graph structure to include parameter estimationTurbo Synchronization: Pass messages between an independent phase estimation block and an essentially unmodified iterative decoder [Noels05]

We propose a soft decision-directed pilotless joint carrier-phase and symbol-timing recovery circuit with little modification to either the iterative decoder or the timing recovery block.

3

Joint Estimation Algorithm

Decoder has a carrier phase-synchronization block and a timing-recovery block that interact as follows:

1. Estimate carrier phaseThis can be done using a Costas Loop (CL) or an iterative Decision Directed Carrier Synchronization (DDCS) algorithmFor BPSK and QPSK constellation, BER/FER performance is similar using any of CL or DDCS since these constellations are not severely affected by carrier phase jitter.

2. Estimate Symbol Timing parametersFor symbol frequency and symbol time-delays use iterative timing recovery circuit that uses information from the constraint side of an LDPC graph to aid recovery processFor random walk effects, use a phase-lock loop (PLL)

3. Restart the iterative estimation process

4

Transmission Model

Parameters:di = BPSK symbols (Code length = n) hRRC(t) = Root-raised cosine (RRC) pulse T = Symbol periodτ(t) = Timing perturbationθc = Carrier Phase

Analog transmitted waveform s(t)comprised of a superposition of RRC pulses

( ) ( ) ( )1

0

( ) ( ) 2 sinn

i RRC c ci

r t d h t iT t P w t N tτ θ−

=

⎛ ⎞= − + + +⎜ ⎟⎝ ⎠∑ Received waveform

( )1

0( )

n

i RRCi

s t d h t iT−

=

= −∑

LDPCEncoder

im

AWGN

Const.Mapper

ixhRRC(t)

id+

Symbol &Carrier

Perturbations

( )2 sinccP w t θ+

( )r t

( )s t

5

Joint Symbol Timing and Carrier Phase Recovery

Block Diagram for DDCS

Carrier Freq.% Sat.

Constrants

Update SymbolEstimates

Update ChannelObservations

( )Rxy t

cw

Sampling &Interpolation

cX

CarrierPhaseEstimation

cθ

Sampling &Interpolation

sX

cZ

sZ

LDPCDecoder

Sym bolTim ing

Convert toBaseband

Convert toBaseband

6

BPSK Digital Circuit Implementation

θc

y1

Re

Im

[ ] [ ] sin( )[ ] [ ] cos( )

c

s

z k d k c Noisez k d k c Noise

θθ

= +

= +[ ] sin( )[ ] cos( )

c c

s c

u k A Noiseu k A Noise

θθ

= +

= +( ; ) ( ) sin( ) ( ; )

( ; ) ( ) cos( ) ( ; )c c s c c

s c s c c

x t PT m t Noise t

x t PT m t Noise t

θ θ θ

θ θ θ

= +

= +

( )( )2

22

2[ ] [ ] [ ] [ ] [ ] where 2

s

s c c s LLRsLLR

o

PTQ k z k w k z k w k E

N

σσ

= + =

( )ˆcosck cw θ=

LDPC LLR Inputs:

DetectedData

Symb.Timing( )sx t

Symb.Timing( )cx t

LDPCDecoder

NCO ACC

( )ˆsinsk cw θ=

ˆ[ ]ˆˆ[ ] [ ] n ky k d kA

= +

[ ]sz k

[ ]cz k [ ]cu k

[ ]su k

keQuadrant

1±

1±

cθ

cθ

90°

( )Rxy t

7

Carrier Phase Recovery: MotivationWhen designing communication systems engineers need to decide whether or not to suppress the transmitted carrier power

Total power = Carrier Power + Data Power Pt=Pc+PdCarrier Power: related to the accuracy of the carrier synch processData Power: related to the accuracy of the data detection process (in the presence of perfect carrier synchronization).

System design requires a proper trade off between these power requirements to minimize the average error probability of the system.

Suppressed-carrier circuits: Costas loop

Require a larger SNR to track the carrier with a given accuracy.Phase estimation has jitter

Decision Directed Carrier Synchronization:Parameter estimation improves with the iteration processSquaring loss tends to unity as iterations increase

8

Decision Directed Carrier-Synchronization (DDCS)

Use a soft estimate of the instantaneous data symbol (and thus of the instantaneous phase modulation) to reduce the amount of randomness (information) in the signal being processed in the carrier loop.

LDPC symbol estimates “wipe-off ” modulated symbols in a decision directed loop to enhance the carrier information such that a classic PLL can provide increasingly accurate (very low jitter) phase estimates over LDPC iterations.

Latency penalty: tracking improves with increased iterations

System complexity: No significant modifications to the current residual carrier recovery techniques used for BPSK/ QPSK modulation in NASA's deep-space network.

9

DelayΔ( ; )Rx cy t θ ( ; )Rx cy t θ−Δ

PLL( ; )cu t θ

cθ

ˆ( )ˆ( ) ( ) n ty t m tA

= −Δ +

DetectedData

-M ixer-Demodulator

-Decoder

BPSK Modulation:

Signal Input:

AWGN Noise:

IR Signal:

DDCS BPSK System Description

( ) ( ) ( ) ( )1; 2 sinRx c c cy ntt mP t tθ ω θ= + +

( ) ( ) ( )1 1 12 ( ) cos ( )sinc c c s c cn t N t w t N t w tθ θ= + − +⎡ ⎤⎣ ⎦

( )( ) k sk

m t d p t kT∞

=−∞

= −∑

( ) ( ); 2 sin [Noise terms]c c cu t P w tθ θ= + +

10

Carrier Detection Process

Assume N=5 transmitted BPSK symbolsSample transmitted waveform = [ 1, -1, 1, -1 ,1 ] Plot shows received waveform affected by symbol-wise noise

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1

0

1

t

Am

plitu

de

-80 -60 -40 -20 0 20 40 600

0.05

0.1

0.15

0.2

f

Am

plitu

de

1 -1 1 -1 1

11

Carrier Detection ProcessModulation is removed by multiplying the received waveform by soft-estimated symbols from the decoderSymbol Information randomness is reducedPlot shows “IR” waveform after the first iterationsAs iterations increase, soft-symbol estimation becomes more accurateFrequency spectrum has a distinctive tone that a PLL based circuit can now track.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1

0

1

t

Am

plitu

de

-80 -60 -40 -20 0 20 40 600

0.05

0.1

0.15

0.2

f

Am

plitu

de

Soft Symbol informationmay still beincorrect

Incorrect information has lowreliability

Unmodulatedfrequency spectrum begins to show the presence of a carrier tone.

12

PLL:

Loop SNR exhibits No squaring lossUtilizes carrier power for carrier estimation

Costas Loop:

SL is independent of the iteration process, for a given SNR.Suppressed carrier circuit

DDCS:

SL approaches unity as the number of LDPC iterations increaseSuppressed carrier circuit

Loop SNR

2

1

c

PLLo L

cPN Bφ

ρσ

= =

2

111

2. 1

c

CCo

LdL

t

NS

BP

Rφ

ρσ

−⎛ ⎞+⎜= ⎟

⎝ ⎠=

2

12

211 .c

DDCSDDCS Lo L

TPAN B

Sφ

ρ σσ

−⎛ ⎞+⎜ ⎟

⎝ ⎠= =

BL: Loop BandwidthPt: Total Power,SL IRCS: Squaring Loss,A: Estimated Decoder Amplitude

BL: Loop BandwidthPc: Carrier Power,No: Noise PSD

Pt: Total Power,SL IRCS: Squaring Loss,Rd: Input Data SNR

13

Averaging Costas Loop

The idea is to eliminate the phase jitter present in the estimationObserve that the mean of the error is in both cases very close to zero.

0 0.5 1 1.5 2

x 104

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Phase Error at EbNo=1.5dB

DDCSAveraging Costas

Instead of rotating the symbols by the “instantaneous”phase estimation, an average over a large number of symbols is used

14

Loop SNR: Larger Gains

Larger gainsPhase estimation process converges faster

DDCS Gains:Kp=8.92E-5KI=-8.75E-5

Costas Gains:Kp=8.85E-4KI=-8.75E-4

0 5 10 15 20 25 30 35 40 45 505

10

15

20

25

30

35

Loop

SN

R[d

B]

Iterations

0 5 10 15 20 25 30 35 40 45 505

10

15

20

25

30

35

Mea

n S

quar

e E

rror [

dB]

Iterations

IRCS 1.0dBIRCS 1.50dBIRCS 2.00dBCS 1.0dBCS 1.50dBCS 2.00dB

SteadyState

15

1st , 2nd Moments & Variance

Costas Variance =2.8E-2 LS ≈15 dBDDCS Variance = 6E-4 LS ≈32 dB

Loop SNR performance shows a difference of around 15dB for DDCS and Costas Loop

0 5 10 15 20 25 30 35 40 45 50-0.2

0

0.2

0.4

0.6

E[E

rror]

Iterations

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

E[E

rror2 ]

Iterations

DDCS 1.0dBDDCS 1.50dBDDCS 2.00dBCS 1.0dBCS 1.50dBCS 2.00dB

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

Var

Erro

r

Iterations

16

FER/BER

0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 2 . 21 0

- 6

1 0- 5

1 0- 4

1 0- 3

1 0- 2

1 0- 1

1 00

E b / N o

BE

R/F

ER

G e n i e [ 5 0 i t ]D D C S [ 5 0 i t ]C S T [ 1 0 L o o p - 5 0 L D P C ]C S T [ 5 0 L o o p - 5 0 L D P C ]

BER / FER results for 50 iterations assuming no symbol timing offsets

Results show that for small M-PSK constellations jitter effects do not affect average error symbol performance

17

BER /FER GAP from Genie Performance

0.8 1 1.2 1.4 1.6 1.8 2

0

0.05

0.1

FER

GA

P [d

B]

Eb/No

Genie [50it]DDCS [50it]CST [10Loop-50LDPC]CST [50Loop-50LDPC]

0.8 1 1.2 1.4 1.6 1.8 2

0

0.02

0.04

0.06

0.08

0.1

BE

R G

AP

[dB

]

Eb/No

BER / FER results for 50 iterations assuming no symbol timing offsets

18

Symbol-wise OffsetsThe traditional method used in the literature introduces an offset

to each of the conforming pulses of s(t) before any superposition is done

The major drawback of this method is that it introduces unnecessary ISI

Sample-wise Timing Offsets:Proposed models applies timing shifts to the superimposed signal

after superposition is done. This is done by having a time-varying timing perturbation function t(t) independent of the sample number.

( )1

0

( ) ( ) ( )n

i RRCi

r t a h t t iT N tτ−

=

= ⋅ + − +∑

( )1

0( ) ( )

n

i RRC ii

r t a h t iT N tτ−

=

= ⋅ + − +∑

Timing Offset

19

Timing Offset

Timing perturbations occur in the channel due to:

Relative motion between transmitter and receiverMismatch between clock crystal frequenciesReceiver-side timing estimation errors

Symbol-wise modeling introduces inter-symbol interference (ISI) in the transmitted waveform (even though an ISI free pulse shape is modulated e.g. RRC)as shown in the eye diagram

Symbol-wise perturbation model is not well suited for our channel model assumptions

-0.5 0 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

Am

plitu

de

Symbol-wise Perturbations

-0.5 0 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

Am

plitu

de

Sample-wise Perturbations

20

Timing Error Modalities

Constant Time Offset: All pulses affected by the same constant delay with respect totheir ideal sampling time

where k indexes the kth received sample.

Random Walk: Models an accumulation process. Timing error is equal to previous error plus a perturbation:

Frequency Offset: Produced by Mismatch between transmitter and receiver clock crystalsDoppler shift due to relative motion between transmitter and receiver

( )2[ ] [ 1] 0, dk kτ τ σ= − +N

[ ]k Dτ =

( )[ ] [ 1] %sk k T Freqτ τ= − +

21

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

0

2(b) Transmitted Waveform

Am

plitu

de

Time

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

0

5(a) Perturbation Profile

Tim

ing

Offs

et

Time

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

0

2(c) Received W aveform

Am

plitu

de

Time

Random Walk Timing Perturbation

Samples not uniformly spaced

Samples uniformly spaced

Positive Time Offset

Compression at the receiver

Timing offset which is tracked on a sample-by-sample basis

22

Constant Frequency Offset

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

0

2(b ) T ra n s m i t t e d W a ve fo rm

Am

plitu

de

T im e

-2 -1 0 1 2 3 4 5 6 7 80

5 0

1 0 0(a ) P e r t u rb a t io n P ro fi l e

Tim

ing

Offs

et

T im e

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

0

2(c ) R e c e ive d W a ve fo rm

Am

plitu

de

T im e

Frequency tracking is done in a Quasi-Static manner (ie. frequency offsets are tracked on a block-by-block basis). Positive frequency offset causes a time-compression on the received waveformSamples are still ISI-free

23

Symbol Timing Recovery Model

Loop 1Tracks large scale phase and frequency offsetInterpolator 1 corrects arbitrary frequency mismatch in LDPC blockInterpolator 2 corrects arbitrary phase offset in LDPC block

Loop 2Fine interpolation of frequency/phase compensated matched filter outputDynamic tracking within a block: random walk and residual loop 1 errorUse Müller-Mueller timing error detector (M&M TED) is used

r(t)Interpolator 1

Signal SamplingClock (1/Ts)

x[k](1/Ts) Matched

Filter

y[j](1/Ti)

z[i](1/T')

LDPC Decoder

ComputeFractional

Interval

μ[j](1/Ti)

η [j](1/Ti)

NCO

R esample

Overflows(1/Ti)

w[k](1/Ts)

v[i](1/Ti)

q[j](1/Ti)

01

Buffer

Loop 1: Time Delay & Frequency Offset Correction

Frequency Estimator

Time Delay Estimator

Interpolator 2

SatisfiedConstraints

(1/Nc)

p[h](1/Nv)

Timing ErrorDetector

DecodedSymbols

d[i](1/T')

Buffer

u[i](1/T')

LoopFilter

sel

s[i](1/Ti)

c[i](1/Ti)

Interpolator 3

Loop 2: Random Walk & Residue Error Correction

24

-1200 -800 -400 0 400 800 1200

Estimate [ppm]

Satis

fied

Con

stra

ints

-1600 1600

BestEstim ate800ppm

Window = 3200 ppmCenter = 0 ppmMin = -1600 ppmMax = 1600 ppmStep = 400 ppm

Search Ite ration 1

- Center next search at 800ppm- Reduce window and step size e.g. by a factor of 2- Perform l LDPC iterations (l<4) for each estimate

600 800

Estimate [ppm]

Satis

fied

Con

stra

ints

Search Ite ration 2

200 4000 16001200 14001000

BestEstim ate1000ppm

Real ppm = 940

Window = 1600 ppmCenter = 800 ppmMin = 0 ppmMax = 1600 ppmStep = 200 ppm

Estimation Error= 140 ppm

Estimation Error= 60 ppm

Real ppm = 940

Frequency Offset & Time Delay Search Example: Method A

Offers very accurate estimations but requires a large number of iterationsComplexity grows linearly with offset

25

-1000 -500 0 1000

Satis

fied

Con

stra

ints

1500

BestEstim ate750ppm

Window = 3500 ppmCenter = 0 ppmMin = -1750 ppmMax = 1750 ppmStep = 250 ppm

Se arch Ite ration 1

INTERPOLATE

Real ppm = 820

Ini tia l Estimation Error= 120 ppm×

-1500 500

Frequency Offset & Time Delay Search Example: Method BUse a smaller step but only one frequency searchProduce closer estimates that allow a curve fitting interpolation to reduce

the initial estimation error

Method ADiminishing returns after 3 search iterations and 3 LDPC iterationsEstimation accurate within 40 ppmInitial search window of ±1600ppm

3 search iter * 3 LDPC iterations * 9 estimates per search = 81 total LDPC iterations

Method BEstimation accurate within ≈60 ppmInitial search window of ±1750ppm

3 search iter * 1 LDPC iterations * 15 estimates per search = 45 total LDPC iterations

26

Tracking Random Walks

Tracked with a 1st order PLL-based circuit (loop 2)

Müller-Mueller timing error detector (M&M TED) is usedM&M TED updates its timing error estimate according to:

where s[i] is the symbol from the interpolator and d[i] is the decoded symbol from LDPC decoder

u[i] is noisy, hence attenuated with a 1st order loop filter:

where Kp is the proportional gain and u[i] is the output of the M&M TED

This circuit also corrects residual time delay and frequency offset

][]1[]1[][][ idisidisiu −−−=

][]1[][ iuKicic p ×+−=

27

Effects of Timing Errors (No Correction)

0 50 100 150 200

10-5

10-4

10-3

10-2

10-1

(1944,972) LDPC, Iterations = 20

Frequency Offset [ppm]

BE

R

1.0dB1.5dB2.0dB

0 0.1 0.2 0.3 0.4 0.5

10-5

10-4

10-3

10-2

10-1

(1944,972) LDPC, Iterations = 20

Random Walk σd/T [%]

BE

R1.0dB1.5dB2.0dB

SNR given in Eb/N0

28

Symbol Timing & Carrier Recovery

Fig.(a) FER 50 Iterations(solid) / FER 20 Iterations(dashed) performance.Legend format

[Carr.] indicates the presence of a =π/4 carrier phase offset,(Freq.) Symbol-frequency offset(TD) Symbol Time delay(Rw.) Random walk

Fig.(b) Shows the SNR gap with respect to the “genie-aided”performance for the same set of curves

29

SummaryCarrier synchronization circuit overcomes the performance loss due to a noisy signal reference at low SNRs, characteristic of suppressed carrier loops such as the Costas loop.Steady state operation is reached after around 15 joint DDCS/LDPC iterations.Two-stage pilotless symbol timing recovery model for tracking time delay, frequency offsets and random walks1st stage: LDPC constraint feedback to track large-scale time delays and symbol frequency offsets Windowed search method

Complexity grows linearly with offsetCan track any time delay frequency offset as long as it is within the initial search window

2nd stage: decoded LDPC symbols to track random walks and residual errors from the 1st stageSimulations results show within 0.4 dB of the ideal code performance for time delays and frequency offsets