Turbo 338

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ECED 4504Digital Transmission Theory

Jacek Ilow

(based on the slides by Matthew Valenti)

Soft Decision Decoding of Convolutional Codes

Intro to Turbo Codes

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Performance of

Hard Decision Decoding

Consider the K=7 Odenwalder codes

r=, generator (133,171), dfree = 10

r=, generator (133,145,171), dfree = 14

Distance spectra (first six non-zero terms)

Assume BPSK modulation.

Find the performance of hard decision decoding

d ad d

10 11 36

12 38 211

14 193 1,404

16 1,331 11,633

18 7,275 77,433

20 40,406 502,690

d ad d

14 1 1

16 7 20

18 11 53

20 35 184

22 90 555

24 279 1,961

r=1/2 r=1/3

Number of paths with

output weight d

Sum of the input

weights for all

paths with output

weight d

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Bit Error Rate of

Hard Decision Decoding

Bit error rate:

Pairwise error probability:

Since BPSK modulation:

p Qr

N

b

o

HG

2

P d

d

jp p

d

jp p

d

dp p

j d j

jd

d

j d j

jd

dd d

2

1

2

21

2 2

1

11

2 21

( )

/

/ /

HG

KJ

F

HGI

KJ F

HG I

KJ

S||

T

|

|

a f

a f a f

odd d

even d

P P d P d b dd d

d free

free

free

2 2( ) ( )Free distance asymptote

only valid for small pUnion bound

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System Model:

Soft Decision Decoding

estimates of

data bits

L data bits (L+m)/r

code bits

rate r = k/n

convolutional

encoder

BPSK

modulator

+AWGN

n(t)

s(t)

decisionstatistic

Viterbi

decoder

r(t)

x c

x r

x

f(t)

zdtTb

0

correlator

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Using the Viterbi Algorithm

The Viterbi algorithm can be used to performsoft decision decoding.

The only difference is with the branch metric. With hard decision decoding, we used Hamming

distance.

With soft-decision decoding, we use the squaredEuclidian distance.

Soft decision decoding outperforms harddecision decoding by about 2 dB. One of the biggest benefits of convolutional codes.

d wH c r c r,

d r ce i i2 22

c r r c, g

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Performance of


For soft decision decoding, a different pairwise error

probability is used.

Depends on modulation and type of channel.

Based on Euclidian distance

For BPSK over an AWGN channel:

Where de is the Euclidian distance between the all-zeros

code word and the code word of Hamming weight d.

So the analysis is the same as for hard-decision, only

now we use the above expression for the pairwise

error probabilty

P d Qd

NQ

rd

N

e b

o

2

02

2( ) HG KJ HG

E

P P d QE rd

Nb d

d d

db

od dfree free

H

G

22

( )

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Comparison:

Soft-Decision vs. Hard-Decision

0 1 2 3 4 5 6 7 8 9 10

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No in dB

BER

Pb of uncodedBPSK

r=1/2r=1/3

Soft-decision

Coding gain @ BER = 10-5

5.4 dB for r=1/2

5.7 dB for r=1/3

Hard-decisiondecoding

Soft-decision

decoding

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Principles of Good Code Design

Consider the BER of a block code:

Where is the average number of information

bits associated with weight d code words. How can we minimize Pb?

For k = 10,000 which distance spectrum is better?

We want the probabilityof a low weight code word tobe small!

Pa f

kQ

rd

N kQ

rd

Nb

d d b

od d

nd b

od d

n

HG KJ HG 2 2

min min

f ad d d /

d ad d

3 2 6

6 4 12

9 10 40

12 100 450

15 1,000 5,000

18 5,000 40,000

d ad d

5 10 100

6 50 500

7 100 1000

8 350 3500

9 1000 10,000

10 10000 100,000

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Review: Convolutional Codes

A convolutional encoderencodes a streamof data.

The size of the code word is

unbounded.

The encoder is a Finite ImpulseResponse(FIR) filter.

k binary inputs

n binary outputs

K -1 delay elements Coefficients are either 1 or 0

All operations over GF(2)

Addition: XOR

Constraint Length K = 3

D Dim

)0(

ix

)1(

ix

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Recursive SystematicConvolutional (RSC) Encoding

An RSCencoder is constructedfrom a standard convolutionalencoder by feeding back one of

the outputs. An RSC code is systematic.

The input bits appear directly inthe output.

An RSC encoder is an Infinite

Impulse Response(IIR) Filter. An arbitrary input will cause a

good (high weight) output withhigh probability.

Some inputs will cause bad (low

weight) outputs.

D D

im

)0(

ix

)1(ix

ix

ir

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Parallel Concatenated Codeswith Nonuniform Interleaving

A stronger code can be created by encoding inparallel.

A nonuniform interleaverscrambles the ordering ofbits at the input of the second encoder.

Uses a pseudo-random interleaving pattern. It is very unlikely that both encoders produce low

weight code words.

MUX increases code rate from 1/3 to 1/2.

RSC#1

RSC#2

NonuniformInterleaver

MUX

Input

ParityOutput

Systematic Output

ix

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Random Coding Interpretation

of Turbo Codes

Random codesachieve the best performance.

Shannon showed that as n, random codes achieve

channel capacity.

However, random codes are not feasible.

The code must contain enough structure so that decodingcan be realized with actual hardware.

Coding dilemma:

All codes are good, except those that we can think of.

With turbo codes: The nonuniform interleaver adds apparentrandomness to

the code.

Yet, they contain enough structure so that decoding is

feasible.

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Comparison of a Turbo Code

and a Convolutional Code First consider a K=12 convolutional code.

dmin = 18

d = 187

Now consider the original turbo code.

C. Berrou, A. Glavieux, and P. Thitimasjshima, NearShannon limit error-correcting coding and decoding: Turbo-codes, in Proc. IEEE Int. Conf. on Commun., Geneva,Switzerland, May 1993, pp. 1064-1070.

Same complexity as the K=12 convolutional code

Constraint length 5 RSC encoders

k = 65,536 bit interleaver

Minimum distance dmin = 6

ad = 3 minimum distance code words

Minimum distance code words have average informationweight of only f

d

2

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0.5 1 1.5 2 2.5 3 3.5 4

10-8

10-6

10-4

10-2

100

Eb/N

oin dB

BER

Convolutional CodeCC free distance asymptoteTurbo CodeTC free distance asymptote

Comparison of

Minimum-distance Asymptotes

Convolutional code:

Turbo code:

o

bb

N

EQP 18187

o

b

o

b

o

bdd

b

N

EQ

N

EQ

N

rdEQ

k

faP

6102.9

)6)(5.0(2

536,65

)2)(3(

2

5

minmi nmi n

187mindc

18min d

6min d

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The Turbo-Principle

Turbo codes get their name because the

decoder uses feedback, like a turbo engine.

We will go over the decoding algorithm next time.

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0.5 1 1.5 210

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No in dB

BER

1 iteration

2 iterations

3 iterations6 iterations

10 iterations

18 iterations

Performance as a Function of

Number of Iterations K = 5

r = 1/2

k = 65,536

Log-MAP algorithm

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Summary of Performance Factors

and Tradeoffs

Latency vs. performance Frame (interleaver) size k

Complexity vs. performance Decoding algorithm

Number of iterations

Encoder constraint length K

Spectral efficiency vs. performance

Overall code rate r Other factors

Interleaver design

Puncture pattern

Trellis termination

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0.5 1 1.5 2 2.510

-7

10-6

10-5

10-4

10-3

10-2

10-1

Tradeoff: BER Performance versusFrame Size (Latency)

K = 5

Rate r = 1/2

18 decoder iterations

AWGN Channel

0.5 1 1.5 2 2.5 310

-8

10-6

10-4

10-2

100

Eb/N

oin dB

BER

K=1024

K=4096

K=16384

K=65536

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Turbo 338