20.Channel Coding

8/3/2019 20.Channel Coding

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Satellite Communications

Home work: Error Correcting Control

Submitted by: Farqad Yasin Farhan


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OutlineOutline

Channel coding protects digital data from errors by selectivelyintroducing redundancies in the transmitted data.

block codes: Hamming codes, Cyclic Codes, BCH codes,Reed-Solomon Codes

convolutional codes

Trellis Coded ModulationCombine both coding and modulation to achieve significant coding gains

without compromising bandwidth efficiency


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Types of Error Correcting Codes

Repetition Code Linear Block Code, e.g. Hamming

Cyclic Code, e.g. CRC

BCH and RS Code

Convolutional Code

Viterbi Decoding

Turbo Code

Coded Modulation TCM


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ParityCheck

Add one bit so that xor of all bit is zero Send, correction, miss

Applications: ASCII, Serial port transmission


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Hammings Solution

A type of Linear Block Code

Encoding: H(7,4)

Multiple Checksums

Message=[a b c d]

r= (a+b+d) mod 2

s= (a+b+c) mod 2

t= (b+c+d) mod 2

Code=[r s a tb c d]

Coding rate: 4/7 Smaller, more redundancy, the better protection.

Difference between detection and correction

Message=[1 0 1 0]

r=(1+0+0) mod 2 =1

s=(1+0+1) mod 2 =0

t=(0+1+0) mod 2 =1

Code=[ 1 0 1 1 0 1 0 ]


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Coding Gain

Coding Rate R=k/n, k, no. of message symbol, n overall symbol

For a coding scheme, the coding gain at a given bit error

probability is defined as the difference between the energy per

information bit required by the coding scheme to achieve the

given bit error probability and that by un-coded transmission.


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Coding Gain Example


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Cyclic code

Cyclic codes are of interest and importance because They posses rich algebraic structure that can be utilized in a

variety of ways.

They have extremely concise specifications.

They can be efficiently implemented using simpleshift register Many practically important codes are cyclic

In practice, cyclic codes are often used for error detection

(Cyclic redundancy check, CRC)

Used for packet networks When an error is detected by the receiver, it requests

retransmission

ARQ


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BCHPerformance


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Reed-Solomon Codes

An important subclass of non-binary BCH Wide range of applications

Storage devices (tape, CD, DVD)

Wireless or mobile communication

Satellite communication Digital television/Digital Video Broadcast(DVB)

High-speed modems (ADSL, xDSL)


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Convolutional Code Introduction

Convolutional codes map information to code bits sequentiallyby convolving a sequence of information bits with generatorsequences

A convolutional encoder encodesKinformation bits toN>Kcode bits at one time step

Convolutional codes can be regarded as block codes for which

the encoder has a certain structure such that we can express the

encoding operation as convolution


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Encoder Convolutional codes are applied in applications that require good

performance with low implementation cost. They operate on code streams(not in blocks)

Convolution codes have memory that utilizes previous bits to encode ordecode following bits (block codes are memoryless)

Convolutional codes achieve good performance by expanding their memory

depth

Convolutional codes are denoted by (n,k,L), where L is code (or encoder)Memory depth (number of register stages)

Constraint length C=n(L+1) is defined as the number of encoded bits amessage bit can influence to


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Example Convolutional encoder, k= 1, n= 2, L=2

Convolutional encoder is a finite state machine (FSM) processing

information bits in a serial manner

Thus the generated code is a function of input and the state of the FSM

In this (n,k,L) = (2,1,2) encoder each message bit influences a span ofC=

n(L+1)=6successive output bits = constraint length C

Thus, for generation ofn-bit output, we require n shift registers in k= 1

convolutional encoders


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Example: Using generatormatrix

(1)

( 2)

[1 0 11]

[11 1 1]

! !

g

g


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Example of exhaustive maximal likelihood detection

Assume a three bit message is transmitted [and encoded by (2,1,2)

convolutional encoder]. To clear the decoder, two zero-bits are appendedafter message. Thus 5 bits are encoded resulting 10 bits of code. Assumechannel error probability is p= 0.1. After the channel 10,01,10,11,00 is

produced (including some errors). What comes after the decoder, e.g.what was most likely the transmitted code and what were the respective

message bits?

a

b

c

d

states

decoder outputs

if this path is selected

DAYANANDA SAGAR COLLEGE OF ENGINEERING, BANGALORE


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Note also the Hamming distances!

correct:1+1+2+2+2=8;8 ( 0.11) 0.88

false:1+1+0+0+0=2;2 ( 2.30) 4.6

total path metric: 5.48

!

!

The largest metric, verify

that you get the same result!



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The Viterbialgorithm

Problem of optimum decoding is to find the minimum distance

path from the initial state back to initial state (below from S0 toS0). The minimum distance is the sum of all path metrics

that is maximized by the correct path

Exhaustive maximum likelihood

method must search all the paths

in phase trellis (2kpaths emerging/

entering from 2 L+1 states foran (n,k,L) code)

The Viterbi algorithm gets its

efficiency via concentrating intosurvivor paths of the trellis

0ln ( , ) ln ( | )jm j mj p p y xg

!!y x

Received code

sequence

Decoders output sequence

for the m:th path



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The survivorpath Assume for simplicity a convolutional code with k=1, and up to 2k= 2

branches can enter each state in trellis diagram Assume optimal path passes S. Metric comparison is done by adding the

metric ofS into S1 and S2. At the survivor path the accumulated metricis naturally smaller (otherwise it could not be the optimum path)

For this reason the non-survived path can

be discarded -> all path alternatives need notto be considered

Note that in principle whole transmittedsequence must be received before decision.However, in practice storing of states forinput length of 5L is quite adequate 2 branches enter each nodek

2 nodes, determinedL

by memory depth


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Example of using the Viterbialgorithm

Assume the received sequence is

and the (n,k,L)=(2,1,2) encoder shown below. Determine the

Viterbi decoded output sequence!

01101111010001y !

(Note that for this encoder code rate is 1/2 and memory depth L= 2)

states


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The maximum likelihoodpath

The decoded ML code sequence is 11 10 10 11 00 00 00 whose Hamming

distance to the received sequence is 4 and the respective decoded

sequence is 1 1 0 0 0 0 0 (why?). Note that this is the minimum distance path.

(Black circles denote the deleted branches, dashed lines: '1' was applied)

(1)

(1)

(0)

(2)

(1)

(1)

1

1

Smaller accumulated

metric selected

First depth with two entries to the node

After register length L+1=3

branch pattern begins to repeat

(Branch Hamming distancesin parenthesis)


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Turbo Codes

Backgound Turbo codes were proposed by Berrou and Glavieux in the 1993

International Conference in Communications.

Performance within 0.5 dB of the channel capacity limit for BPSK

was demonstrated.

Features of turbo codes

Parallel concatenated coding

Recursive convolutional encoders

Pseudo-random interleaving

Iterative decoding



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otivation: Performance ofTurbo Codes

Comparison: Rate 1/2 Codes.

K=5 turbo code.

K=14 convolutional

code.

Plot is from: L. Perez, Turbo Codes,

chapter 8 of Trellis Coding

by C. Schlegel. IEEE Press,

1997

Gain of almost 2 dB!

Theoretical Limit!


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The Turbo-Principle

Turbo codes get their name because the decoder uses feedback,like a turbo engine.



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Performance as a Function of Number of Iterations

K=5, r=1/2, L=65,536

0 .5 1 1 . 5 21 0

-7

1 0-6

1 0-5

1 0-4

1 0-3

1 0

-2

1 0-1

1 00

Eb

/No

in dB

B

ER

1 i teration

2 iterations

3 i terations6 i terations

1 0 i terations

1 8 i terations


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Turbo Code Summary

Turbo code advantages: Remarkable power efficiency in AWGN and flat-fading channelsfor moderately low BER.

Deign tradeoffs suitable for delivery of multimedia services.

Turbo code disadvantages:

Long latency.

Poor performance at very low BER.

Because turbo codes operate at very low SNR, channel estimationand tracking is a critical issue.

The principle of iterative or turbo processing can be applied toother problems.

Turbo-multiuser detection can improve performance of codedmultiple-access systems.


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Trellis Coded Modulation

1. Combine both encoding and modulation. (using Euclideandistance only)

2. Allow parallel transition in the trellis.

3. Has significant coding gain (3~4dB) without bandwidth

compromise.4. Has the same complexity (same amount of computation, same

decoding time and same amount of memory needed).

5. Has great potential for fading channel.

6. Widely used in Modem


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SetPartitioning

1. Branches diverging from the same state must have the largest distance.2. Branches merging into the same state must have the largest distance.

3. Codes should be designed to maximize the length of the shortest error eventpath for fading channel (equivalent to maximizing diversity).

4. By satisfying the above two criterion, coding gain can be increased.


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Coding Gain

About 3dB

20.Channel Coding

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