1

Iterative Equalization and Decoding

John G. Proakisproakis@neu.edu

COMSOC Distinguished Lecture Tour

2

Conventional Equalization

Equalizer DecoderSoft OutputFrom Receiver

Filter

Hard Output

Possible Equalizer Types:

•Linear Equalizer

•Decision Feedback Equalizer (DFE)

•Maximum A posteriori Probability (MAP) Equalizer

•Soft-output Viterbi (MLSE) Equalizer

Possible Decoder Types:

•Maximum A posteriori Probability (MAP) Decoder

•Viterbi (MLSE) Decoder

3

Turbo Principle / Turbo Coding

Turbo Encoder:

• Parallel concatenated recursive systematic convolutional encoders

• Encoders separated by an interleaver

d(n)

b(n)

D D+

cs(n)

cp1(n)

D D

cp2(n)

d`(n)

Puncturer and P

/S converter

Encoder 1

Encoder 2

+

+

+

Turbo Decoder:

• Two Soft-Input Soft-Output (SISO) decoders separated by interleavers

• SISO modules can be

• SOVA

• MAP

• Extrinsic information passed between modules

SISODecoder 1

SISODecoder 2

xp1

xs

xp2

Le

21

Le

12

Ld

4

Serially Concatenated Systems

Serially Concatenated Coding:

• Serial concatenated (recursive) convolutional encoders

• Encoders separated by an interleaver

Encoder 1d(n)

c(n) c’(n) b(n)

Encoder 2

Coded Transmission over Multipath Channels:

• (recursive) convolutional encoder

• Interleaved bits mapped to symbols

• Symbols passed through a multipath channel

Encoderd(n)

c(n) c’(n) Symbol

Mapper

Multipath

Channel

x(n)

r(n)

5

Channel Model / Precoding

Multipath Channel Model:• Received signal:

•Rate 1/1 convolutional code

D

Xh1(n)

D

Xh2(n)

D

XhL-1(n)

Xh0(n)

+

+w(n)

x(n)

r(n)

)()()()(0

nwlnxnhnrL

l

l

Precoded System:• Iteration gain only possible with recursive inner code (channel) [1], [2]• Recursive rate 1/1 precoder is employed before transmission• Most common precoder: Differential encoder

D

Xh1(n)

D

Xh2(n)

D

XhL-1(n)

Xh0(n)

+

+w(n)

x(n)

r(n)

y(n)

y(n-1)

6

Iterative Equalization and Decoding (Turbo Equalizer)

ConvolutionalEncoder Symbol

MapperMultipathChannel

dn cn c’n xn rn

MAPDecoder+

MAPEqualizer

+ChannelEstimator

r

LD

e(c’) LD

e(c) LD(c)

LD(d)

LE

e(c)LE

e(c’)LE(c’)

-

-

• Data bits are convolutionally encoded and interleaved• M-ary PSK modulated signals transmitted through a multipath channel, which is treated as an encoder

• Received signals are jointly equalized and decoded in a turbo structure• First proposed by Douillard, et.al [3], where SOVA modules are employed• Bauch extended the idea by employing MAP modules [4]

7

Time-invariant Test Channel

0.688

0.460

0.227

0.460

0.227

Impulse Response Frequency Response

t

• Proakis C channel [5]

8

Low Complexity Alternative Equalizers: DFE and MLSE

• Performance of DFE and MLSE over the Proakis C channel [5]

Bit error rate performance [5]

9

Iterative Equalization and Decoding Performance

• Iterative equalization and decoding with MAP modules• Recursive systematic convolutional encoder with R=1/2, K=5, • Time-invariant 5 tap channel with a spectral null (Proakis C [5])• Equalizer has perfect knowledge of the channel • Block length 4096

Bit error rate performance of turbo equalizer [4]

10

Hard Iterative DFE and MAP Decoding

Forward

Filter

Feedback

Filter

Symbol

Detector

+

MAP

DecoderInput from

receiver filter

DFE withhard input feedback

kI

kI~

Hard encodedsymbols

OutputData

• During the first pass, symbol detector output is passed to the feedback filter• After the first pass, hard encoded symbol output of the decoder is used in the feedback filter

11

Performance of Hard Iterative DFE and MAP decoder

• BPSK modulation• R=1/2, K=7 convolutional coding• Block length 2048• Channel Proakis C

12 12.5 13 13.5 14 14.5 15 15.510

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

PB

ER

Performance of Hard Iterative DFE(K1=K2=4) in Coded BPSK System

First iteration K=7 Second iteration K=7Third iteration K=7 Fourth iteration K=7

12

Soft Iterative DFE and MAP Decoding

Forward

Filter

Feedback

Filter

Decision

Device

+

MAP

Decoder

DFE withhard input feedback

kI

Soft encodedsymbols

OutputData

Soft decisions of the decoder is combined with the soft outputs of the DFE:

2

2

+

kI

Soft APPfrom lastiteration

Hard detectedsymbols

)ˆ( kIL

13

Histogram of DFE Output

-4 -3 -2 -1 0 1 2 3 40

100

200

300

400

500

600

700

E qua lizer Output va lues

Dis

trib

uta

tion

of E

qu

ali

zer

Ou

tpu

t Va

lue

s

F i rs t Itera tion, S NR=12, K =3

-5 0 50

200

400

600

800

1000

1200

1400

1600

1800

2000

E qua lizer Output va lues

Dis

trib

uta

tion

of

Eq

ua

lize

r O

utp

ut V

alu

es

F i rs t Itera tion, S NR=20, K =3

The histogram of equalizer estimated output for SNR = 12 dB

The histogram of equalizer estimated output for SNR = 20 dB

14

Modified Soft Iterative DFE and MAP Decoding

Forward

Filter

Feedback

Filter

Decision

Device

+

MAP

Decoder

DFE withhard input feedback

kI

Soft encodedsymbols

OutputData

+

-

Only extrinsic information is passed to the DFE from the decoder

Re

Im

22

2

22

2

Variance Estimator

Variance Estimator

+ +kI

Soft APPfrom lastiteration

Hard detectedsymbols

Conversion

to LLR

15

Performance of Soft Iterative DFE and MAP Decoder

• Recursive systematic convolutional encoder with R=1/2, K=5, • Time-invariant 5 tap channel with a spectral null (Proakis C [5])• RLS updates at the DFE• Block length 4096• BPSK modulation

16

Performance of Soft Iterative DFE and MAP Decoder

• Recursive systematic convolutional encoder with R=1/2, K=5, • Time-invariant 5 tap channel with a spectral null (Proakis C [5])• RLS updates at the DFE• Block length 4096• QPSK modulation

17

Iterative Linear MMSE Equalization and Decoding

SISO Linear MMSE Equalizer [6]:

• Known channel:

• Received signal:

• Likelihood ratio for MMSE estimator output, :

• Channel matrix:

• MMSE estimator output:

where,

211 ,,1,, MMMkhk

2

1

M

Mkkknkn xhz

nx

)()(

1Pr

1Prln

1|ˆ1|ˆ

ln)(

nnMMSEe

xL

n

n

xL

nn

nnn

MMSE

x

x

xxp

xxpxL

21

2

1

11

211

0

0

0

00

0

1

1

1

MM

M

M

MM

MMM

hh

h

h

hh

hhh

H

xn

znn

Hnn mx mzcˆ

)(2

1tanh

),cov(),cov(12

nnxn

nnzn

nnH

nnN

n

xLxEm

EE

x

xHzm

xHxxHIc

18

Iterative Linear MMSE Equalization and Decoding

sHmzc

sssHHDIc

Hs

D

m

H

xn

xnn

Hnn

xn

Hn

Nn

TMNxMNx

xNMn

xNMn

xNMnn

TxNMn

xNMn

xNMn

xn

nxn

mx

m

mmmdiag

mmm

xLm

ˆ

)(

010

)(1)(1)(1

)(2

1tanh

122

)(1)(1

221

2

1

2211

221111

221111

Steps to compute symbol estimates with the Linear MMSE equalizer:

Soft output calculation assuming Gaussian distributed estimates:

nHn

n

nnn

MMSEe

nHnn

Hnnnn

Hnnnn

xxxL

xxE

xxE

cs

sc

sc

sc

1

ˆ2ˆ2)(

1|ˆ

1|ˆ

2

)1(

2)1(2

)1(

)1(

19

Performance of SISO MMSE Linear Iterative Equalizer

• Recursive systematic convolutional encoder with R=1/2, K=5, • Time-invariant 5 tap channel with a spectral null (Proakis C [5])• Equalizer has perfect knowledge of the channel • Block length 4096

20

Comparison of System Performances

• Recursive systematic convolutional encoder with R=1/2, K=5, • Time-invariant 5 tap channel with a spectral null (Proakis C [5])• BER results after 6 iterations• Block length 4096

21

Experimental Study of Iterative Equalizers

ChannelProbe

TrainingSymbols

InformationSymbols

DeadTime

22

Joint MMSE Equalization and Turbo Decoding

FeedbackFilter

MAPDecoder 1 Demapper

&S/P

Converter

AdaptiveAlgorithm

DecisionDevice

x(n)

x(n)~

e-j(n)

e(n)

x(n)^

y(n)

xs

xp2

Le

12

Ld

2(n)

Le

12

TrainingSymbols

ForwardFilter +x

+

xp1

MAPDecoder 2

Lp2 Lp1

23

• Soft output of the DFE:

•RLS algorithm is used to track channel variation:

Decision Feedback Equalizer (DFE)

)(~)()()()(ˆ nnnnnx HH xwyw bf

)(ξ)(~)()1()(

)(ξ)()()1()(

))(~)1()()1(()()(ξ

))1()()()1((λ)(

)()1()(λ

)()1()(

1-

nnnnn

nnnnn

nnnnnxn

nnnnn

nnn

nnn

HH

H

H

xPww

yPww

xwyw

PykPP

yPy

yPk

bb

ff

bf

• Noise variance estimate: 222 ))(~)()()(()(λ)1()1(λσ)(σ nnnnnxnn HH xwyw bf

24

MAP Decoding

• Maximize a posteriori probability:

• Decision variable written in the form of log-likelihood ratio:

1,,1,0),|( kiddP in y

S nn

S nn

pssssp

pssssp

n

n

dP

dPdn

)(/),,'(

)(/),,'(

1

1log

)|1(

)|1(log)(

yy

yy

y

y

25

MAP Decoding (BCJR Algorithm)

)|()(),(n nnn xypdPss

• State transition probability:

where ),( p

nsnn yyy ),( p

nsnn xxx

22n σexp

σ2

)(exp),(

pn

pn

snn

e

n

xyydLdss

S nenn

S nenn

ne

nn ssss

ssssdLyd

)(β),(γ)(α

)(β),(γ)(αlog)(

σ

2)(Λ

1

1

2

channelvalue

a prioriinformation

extrinsicinformation

26

• depends on the channel trellis defined by hl(n) with 2(L-1) states

•If xn-l for J<l<L-1 is known

MAP Equalizer

1

02

en )(

σ2

1exp),(γ

L

lln

ln xnhyss

11

02

en ˆ)()(

σ2

1exp),(γ

L

Jlln

lJ

lln

ln xnhxnhyss

),(γen ss

Number of states is reduced to 2(J-1)

27

Per-Survivor Processing

00

01

10

11 1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

n-4 n-3 n-2 n-1 n

33 ,Y nn

22 ,Y nn

11,Y nn

00 ,Y nn

n

Snmn

Sn

Sn

mmm SP Y|αYPath metric:

Survivor path: ]0,0,1,1[Y0 n

]1,0,1,1[Y2 n

0

1

1

0

Discarded Paths Survivor Paths

28

Channel Estimator

AdaptiveAlgorithm

+hl (n)(n)x y(n)

e(n)-

• Each survivor path has a separate channel estimator• The input to the channel estimator, , is the estimates within the survivors• RLS algorithm is employed

(n)x

29

Channel Estimator

• Initial channel estimate is based on the correlation of the preamble• RLS algorithm is employed to track the channel

• Noise variance estimate:

)(ξ)(ˆ)()1()(

))(ˆ)1(()()(ξ

))1()(ˆ)()1((λ)(

)(ˆ)1()(ˆλ

)(ˆ)1()(

1-

nnnnn

nnnyn

nnnnn

nnn

nnn

H

H

H

xPww

xw

PxkPP

xPx

xPk

222 )(ξλ)1()1(λσ)(σ nnn

30

Channel impulse response estimate for transducer seven

obtained using the channel probe

DFE results for transducer 7.Eye Pattern - Filter coefficients

PLL phase estimate - Bit error distribution

Experimental Results

31

Experimental Results

Channel impulse response estimate for transducer seven

obtained using adaptive channel estimator

Comparison of received signal with the estimated received signal based on the

channel estimate

32

Experimental Results

• Sparse Channel with multipath delay in the order of 200 symbols• Length of the DFE or channel estimator filters cannot cover the channel• Sparse processing is needed

33

Results of DFE Turbo Decoder

Event Modulation DFE Iter1

Iter2

Iter3

Iter4

Iter5

0 QPSK 334 25 4 3 3 3BPSK 70 0

1 QPSK 488 80 56 49 47 49BPSK 22 0

2 QPSK 392 43 18 15 15 15BPSK 64 4 2 2

3 QPSK 374 27 17 12 11 11BPSK 51 0

4 QPSK 290 11 7 7 7 7BPSK 38 0

5 QPSK 121 2 0BPSK 20 0

6 QPSK 165 1 2 2 2 2BPSK 7 0

7 QPSK 247 6 2 2 2 2BPSK 1 0

8 QPSK 207 7 1BPSK 17 1 0

34

Results of Iterative DFE Turbo Decoder

Event Mod. DFE Iter1 Iter2 DFE Iter1 Iter2 DFE Iter1 Iter2 DFE Iter1 Iter2 DFE Iter1 Iter20 QPSK 334 25 4 140 1 2 133 1 1 138 1 1 139 1 11 QPSK 488 80 56 203 16 9 179 8 5 171 7 4 172 5 42 QPSK 392 43 18 186 7 7 177 6 6 173 6 6 173 6 63 QPSK 374 27 17 165 5 6 148 2 04 QPSK 290 11 7 139 5 5 131 3 3 129 3 3 129 3 35 QPSK 121 2 06 QPSK 165 1 2 73 1 1 75 2 07 QPSK 247 6 2 104 08 QPSK 207 7 1 89 0

35

Results of Iterative DFE MAP Decoder

Event Mod. DFE1 MAP1

DFE2 MAP2

DFE3 MAP3

DFE4 MAP4

DFE5 MAP5

DFE6 MAP6

0 QPSK 512 136 268 49 215 31 212 31 210 31 210 31/16BPSK 46 0

1 QPSK 307 0BPSK 40 0

2 QPSK 522 167 265 13 210 0BPSK 42 0

3 QPSK 288 40 155 5 144 5 146 5 146 5 146 5/2BPSK 47 0

4 QPSK 238 5 119 0BPSK 36 0

5 QPSK 153 0BPSK 10 0

6 QPSK 208 9 81 0BPSK 6 0

7 QPSK 91 0BPSK 7 0

8 QPSK 282 15 109 5 101 0BPSK 16 0

36

Results for Iterative Map Equalizer Turbo Decoder

PK#

MOD. EquMap1

TurIter1

TurIter2

EquMap2

TurIter1

TurIter2

EquMap3

TurIter1

TurIter2

EquMap4

TurIter1

TurIter2

EquMap5

TurIter1

TurIter2

0 BPSK 21 00 QPSK 447 73 50 80 3 3 8 01 BPSK 7 01 QPSK 682 241 222 444 154 115 197 38 25 33 4 1 3 3 02 BPSK 8 02 QPSK 427 64 45 76 1 3 6 03 BPSK 14 03 QPSK 392 59 35 62 2 1 2 04 BPSK 16 04 QPSK 313 14 6 10 05 BPSK 05 QPSK 34 1 06 BPSK 2 06 QPSK 81 1 1 3 07 BPSK 07 QPSK 52 08 BPSK 4 08 QPSK 130 2 2 6 1 1 4 1 1 4 1 1 4 1 1

37

Results for Iterative MAP Equalizer MAP Decoder

PK#

MOD. EquMAP 1

DecMAP 1

EquMAP 2

DecMAP 2

EquMAP 3

DecMAP 3

0 BPSK 13 00 QPSK 600 333 212 38 29 01 BPSK 10 01 QPSK 407 88 33 02 BPSK 7 02 QPSK 643 415 313 95 32 03 BPSK 20 03 QPSK 294 9 6 04 BPSK 18 04 QPSK 327 34 19 05 BPSK 1 05 QPSK 85 06 BPSK 8 06 QPSK 113 07 BPSK 2 07 QPSK 77 08 BPSK 2 08 QPSK 277 14 1 0

38

Conclusions

• Due to error propagation in the DFE, turbo decoder cannot provide performance improvement beyond the second iteration Error Floor

• Joint DFE and turbo decoding adds an additional loop to the system and lowers the error floor

• Joint channel estimator and iterative equalizer is able to decode packets with low SNR, which cannot be decoded with the DFE

• Tail cancellation is an effective way to reduce the computational complexity of the MAP equalizer

• If the channel is sparse, although the DFE filter lengths are short, the DFE is able to provide enough information to the turbo decoder

• A sparse DFE can be used to improve the performance of the DFE/MAP Decoder and the DFE/Turbo Decoder

39

References

[1] S. Benedetto, et.al., “Serial concatenation of interleaved codes: Design and performance analysis,” IEEE Trans. Info. Theory, vol. 42, pp. 409-429, April 1998[2] I. Lee, “The effect of a precoder on serially concatenated coding systems with ISI channel,” IEEE Trans. Commun., pp. 1168-1175, July 2001[3] C. Douilard, et.al., “Iterative correction of intersymbol interference: Turbo-equalization,” European Transactions on Telecommunications, vol. 6, pp. 507-511, Sep.-Oct. 1995[4] G. Bauch, H. Khorram, and J. Hagenauer, “Iterative equalization and decoding in mobile communications systems,” in Proc. European Personal Mobile Commun. Conf., pp. 307-312[5] J. Proakis, Digital Communications, McGraw-Hill Inc., 2001[6] M. Tuchler, A. Singer, and R. Koetter, “Minimum mean squared error equalization using a priori information,” IEEE Trans. Signal Proc., vol. 50, pp. 673-683, March 2002