Wireless Communication Network Lab. Chapter 4 Channel Coding and Error Control Chih-Cheng Tseng1EE...

Wireless Communication Network Lab.

Chapter 4Channel Coding and Error Control

Chih-Cheng Tseng 1EE of NIU

Prof. Chih-Cheng Tseng

[email protected]

http://wcnlab.niu.edu.tw

mailto:[email protected]

http://wcnlab.niu.edu.tw/




Antenna

Transmitter

Receiver

Antenna

Air

Information to be transmitted Source

codingChannel coding Modulation

Source decoding

Channel decoding Demodulation

Information received


Channel Coding in A Wireless Communication Systems


Forward Error Corrections (FEC)(正向糾錯 )

3

The key idea of FEC is to transmit enough redundant data to allow receiver to recover from errors all by itself.

No sender retransmission required. The major categories of FEC codes are

• Linear block codes

• Cyclic codes

• CRC codes

• Convolutional codes

• Turbo codes, etc.

EE of NIU Chih-Cheng Tseng


Linear Block Codes (1) (線性區塊碼 )

A linear code is an error-correcting code for which any linear combination of codewords is another codeword of the code.

(n,k) linear block codes• k information bits are encoded to an n bits codeword.

• Among 2n different combinations, only 2k combinations are valid codewords.

Let the uncoded k information bits be represented by the vector m = (m1,m2,…,mk).

Let the corresponding codeword be represented by the n-bit vector c = (c1,c2,…,ck,ck+1,…,cn-1,cn).



Linear Block Codes (2)

Each parity bit consists of a weighted modulo 2 sum of the data bits represented by symbol.⊕

Example:

where pij (i=1,…k;j=k+1,…,n) is the binary weight of the particular data bit.

1 1

1 1 1( 1) 2 2( 1) ( 1)

1 1 2 2

...

...

...

...

k k

k k k k k k

n n n k kn

c m

c m

c m p m p m p

c m p m p m p



EE of NIU Chih-Cheng Tseng 6

Generator matrix G

Code Vector c

Message vector m

Transmitter

Parity check matrix HTCode

Vector cNull

vector 0

Receiver

Air

Operations of The Generator Matrix and The Parity Check Matrix


Generator Matrix (生成矩陣 )

The generator matrix G (kn matrix) is made up by concatenating the identity matrix Ik (kk matrix) and the parity matrix P (k(n-k) matrix)

1( )11 12

2( )21 22

( )1 2

1 0 0 ... 0

0 1 0 ... 0[ | ]

...... ...... ... ... ... ...

0 0 0 ... 1

n k

n k

k k n

k n kk k

pp p

pp p

pp p

G I P



Representing The Code Vector c

Representing the code vector c as a matrix operation on the uncoded message vector m

For convenience, the code vector is expressed as

• cp=mP is an (n-k)-bit parity check vector

c mG

| p c m c



The parity matrix P (k(n-k) matrix) is given by

• pi=rem [xn-k+i-1/g(x)] is the remainder of [xn-k+i-1/g(x)] for i=1, 2, .., k

• g(x) is the generator polynomial (生成多項式 ). All arithmetic is performed using modulo 2

operation.

Parity Matrix (同位矩陣 )


111 12 1( )

221 22 2( )

1 2 ( )

n k

n k

kk k k n k

p p p

p p p

p p p

p

pP

p


Modulo 2 Arithmetic

Exclusive-OR operation• Binary addition with no carries

• Binary subtraction with no borrows


Wireless Communication Network Lab. For a (7,4) code, find the generator matrix G if code

generator polynomial g(x)=1+x+x3.• n=7, k=4, n-k=3

• The generator matrix is

Linear Block Codes: Example


1000110

0100011

0010111

0001101

G

3 51 3 2

3 3

4 62 2 4 2

3 3

rem 1 [110] rem 1 [111]1 1

rem [011] rem 1 [101]1 1

x xx x x

x x x x

x xx x x

x x x x

p p

p p


Parity Check Matrix (同位檢查矩陣 )

Define a matrix HT as

Parity check matrix H(n-k)×n

• In-k is a (n-k)×(n-k) unit matrix

• PT is the transpose matrix of the parity matrix P


T

n k

PH

I

Tn k H P I


Syndrome (症狀 ) s

Syndrome: s=xHT

• x=ce is the received vector, e is the error vector.

• dimension of s is (n-k).

• s=0: the received vector x is correct.

• s0: the received vector x is incorrect and the value of s is the location of the error bit.


( )T T T T T s xH c e H cH eH eH

Tp

n k

p p p

PcH m c

I

mP c c c 0


Example

For the (7,4) linear block code, given by G as

For m=[1011] and c=mG=[1011|001], if there is no error, the received vector x=c, and s=cHT=[000]

10001111110100

0100110 1101010

00101011011001

0001011

G H



Example

Let c suffer an error such that the received vector x=ce=[ 1 0 1 1 0 0 1 ][ 0 0 1 0 0 0 0 ] =[ 1 0 0 1 0 0 1 ] The syndrome

This indicates error position, giving the corrected vector as [1011001]


111

110

101

1001001 011 [101]

100

010

001

T T

s xH eH


Cyclic Codes (1) (循環碼 )

Cyclic codes are relatively easy to encode and decode than most other types of codes.

Most of block codes used in the FEC system are cyclic codes.

It is a block code which uses a shift register to perform encoding and decoding.



Cyclic Codes (2)

The codeword with n bits is expressed as

• each coefficient ci (i = 1, 2, …, n) is either a 1 or 0.

The codeword can be expressed by the data polynomial m(x) and the check polynomial cp(x) as


1 21 2( ) n n

nc x c x c x c

( ) ( ) ( )n kpc x m x x c x


Cyclic Codes (3)

The check polynomial cp(x) is the remainder from dividing m(x)xn-k by the generator polynomial g(x)

Denoting the error polynomial by e(x), the received signal polynomial or syndrome s(x) is

If there is no error, we have s(x) = 0。


( )( ) rem

( )

n k

p

m x xc x

g x

( ) ( )( ) rem

( )

c x e xs x

g x


Example: A (7,4) Cyclic Code


3 4 5( ) ( ) ( )n kpc x m x x c x x x x x

Find the codewords c(x) if m(x)=1+x+x2 and g(x)=1+x+x3

The check polynomial is

Then, the codewords can be found as

5 4 3

3( ) rem

1p

x x xc x x

x x



Error Detection

error is inevitable detect using error-detecting code added by transmitter recalculated and checked by receiver still chance of undetected error parity

• parity bit set so character has even (even parity) or odd (odd parity) number of ones

• even number of bit errors goes undetected


Cyclic Redundancy Check

one of the most common and powerful checks for block of k bits, transmitter generates an n

bit frame check sequence (FCS) transmits (k+n) bits which are exactly divisible

by some number receiver divides frame by that number

• if no remainder, assume no error



Commonly used CRC Polynomial


Code-Parity check bits Generator polynomial g(x)

CRC-12 x12 +x11+x3+x2+x+1

CRC-16 x16 +x15+x2+1

CRC-CCITT x16 +x12+x5+1

CRC-32 x32 +x26+x23+x22+x16 +x12+x11+x10+x8+x7+x5+x4+x2+x+1



CRC---Example

101000110100000

1101010110

110101

110101111011110101

111110110101

101100110101

110010110101

111010110101

01110

101000110101110

1101010110

110101

110101111011110101

111110110101

101111110101

110101110101

111010110101

00000

k bits message (M)

FCS (R) OK!!

T=M+R(n+1) bits

n bits


Convolutional Code (1)

Most widely used channel code in practical communication systems for real-time error correction, e.g. GSM and IS-95.

The encoded bits depend not only on the current input bits but also on past input bits.• Encoding of information stream rather than information

blocks. Decoding is based on the well-known Viterbi

Algorithm.



Convolutional Codes (2)

The constraint length relates the number of bits on which the output depends and is defined as K=M+1• M is the maximum number of stages (memory size) in

any shift register The code rate r is defined as

r = k / n• k is the number of parallel information bits

• n is the number of parallel output encoded bits at one time interval



Convolutional Codes: (n=2, k=1, M=2) Encoder

Input Output

Input x: 1 1 1 0 0 0 …

Output y1,y2: 11 01 10 01 11 00 …

Input x: 1 0 1 0 0 0 … Output y1,y2: 11 10 00 10 11 00 …

D1, D2 - Registers

x

y1

y2

c0 011

1 0

0

1 0

1 0

1

1

D1 D2

y1

y2



State Diagram


01/001/1

11/1 11/0

11

10/1

00/0

00

10 0110/0

00/1


Tree Diagram


00

11

00

10

01

11

11

00

10

01

10

01

00

11

11

10

01

00

11

0001

01

10

10

11

00

01

10

01

10

11

1

0

… 1 1 0 0 1

… 10 11 11 01 11

First inputFirst output

01


Trellis Diagram

00 0000 0000 0000 0000 0000

10 10 10 10 10 10

01 01 01 01 01 01

11 11 11 11 11 11

11

10

11

11

01

… 11001

…

10 10 10

01 01 0101 01 01

11 11 1111

11

10 10 10

00 00 00



Interleaver

a1, a2, a3, a4, a5, a6, a7, a8, a9, …Input Data

a1, a2, a3, a4 a5, a6, a7, a8 a9, a10, a11, a12 a13, a14, a15, a16

Write

Rea

dInterleaving

a1, a5, a9, a13, a2, a6, a10, a14, a3, …Transmitting Data

a1, a2, a3, a4 a5, a6, a7, a8 a9, a10, a11, a12 a13, a14, a15, a16

Read

Wri

te

De-Interleaving

a1, a2, a3, a4, a5, a6, a7, a8, a9, …Output Data

a1, a5, a9, a13, a2, a6, a10, a14, a3, …Received Data

Through Air

Interleaver tries to prevent burst errors.



Example: Occurring 4 Burst Bit Errors

Pros• Disperse the burst

errors into multiple individual errors.

Cons• Additional delay is

required as the sequence needs to be processed block by block.

Notice• Interleaving does not

have error-correcting capability.


0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0,…Transmitting Data

0, 1, 0, 0 0, 1, 0, 0 0, 1, 0,

0 1, 0, 0, 0

Read

Wri

teDe-Interleaving

Burst error

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, …Output Data

Discrete errors


Information Capacity Theorem(Shannon Limit)

The information capacity (or channel capacity) C of a continuous channel with bandwidth B Hz can be perturbed by additive white Gaussian noise (AWGN) of power spectral density N0/2, provided bandwidth B satisfies

• P is the average transmitted power P = EbRb (for an ideal system, Rb = C)• Eb is the transmitted energy per bit• Rb is transmission rate (bps)

As long as the transmission rate is less than the channel capacity, there exists a coding scheme that makes the BER approaches to zero.

20

log 1 bps (for AWGN channel)P

C BN B



Shannon Limit As the bandwidth B goes infinity, the least SNR to achieve the channel

capacity is -1.6 dB


0 10 20 30Eb/N0 dB

Region for which Rb<C

Region for which Rb>C

Rb/B

-1.6

Shannon Limit

Capacity boundary Rb=C

1

10

20

0.1

For details about the channel capacity and Shannon limit, a good reference can be downloaded at:1. http://www.cteccb.org.tw/pdf/IECQ-49-4.pdf2. http://www.cteccb.org.tw/pdf/IECQ-50-6.pdf

http://www.cteccb.org.tw/pdf/IECQ-49-4.pdf

http://www.cteccb.org.tw/pdf/IECQ-50-6.pdf


Turbo Codes

The turbo code concept was first introduced by C. Berrou in 1993.

Turbo codes are considered as the most efficient coding schemes for FEC.

Scheme with known components (simple convolutional or block codes, interleaver, soft-decision decoder, etc.)

Performance close to the Shannon Limit (Eb/N0 = -1.6 dB if Rb0 or B) than any other codes so far at modest complexity!



Turbo Code Encoder

The fundamental turbo code encoder is built using two identical recursive systematic convolutional (RSC) codes with parallel concatenation.

The interleaver makes the input of the encoders uncorrelated.


Convolutional Encoder 1 (RSC 1)

Interleaving

Convolutional Encoder 2 (RSC 2)

Data Source

xx

y

y1

y2

(y1, y2)


Turbo Code Decoder

The output of the 2nd RSC decoder is feedbacked to the 1st RSC decoder TURBO


Convolutional Decoder 1

Convolutional Decoder 2

Interleaving

Interleaver

De-interleaving

De-interleavingx

y1

y2

x’x’: Decoded information


Automatic Repeat reQuest (ARQ)

ARQ is one of the error-handling mechanisms in data communication.

Three kinds of ARQ schemes• Stop-And-Wait ARQ (SAW ARQ)

• Go-Back-N ARQ (GBN ARQ)

• Selective-Repeat ARQ (SR ARQ)



Concept of ARQ


Source Transmitter Channel Receiver

EncoderTransmit Controller Modulation

Destination

Demodulation Decoder Transmit Controller

Acknowledge


Stop-And-Wait ARQ (SAW ARQ)


Transmitting Data

1 32 3Time

Received Data

1 2 3Time

AC

K

AC

K

NA

K

Output Data1 2 3

Time

Error

Retransmission

ACK: Acknowledge NAK: Negative ACK


Average Tx Time of The SAW ARQ

The average Tx time in terms of a block duration

TSAW = (1+DRb/n)PACK+2(1+DRb/n)PACK(1-PACK)

+3(1+DRb/n)PACK(1-PACK)2+…

= (1+DRb/n)PACK i(1-PACK)i-1

= (1+DRb/n)PACK/[1-(1-PACK)]2 = (1+DRb/n)/PACK


1i

• n is the number of bits in a block

• D is the round trip delay

• Rb is the bit rate

• Pb is the BER of the channel

• PACK(1-Pb)n


Throughput of The SAW ARQ

Throughput

SSAW = (1/TSAW)(k/n)

= [(1-Pb)n/(1+DRb/n)] (k/n)


• n is the number of bits in a block

• k is the number of information bits in a block

• Rb is the bit rate

• Pb is the BER of the channel

• D is the round trip delay


Go-Back-N ARQ (GBN ARQ)

A special case of the sliding window protocol with the transmit window size of N and receive window size of 1.


Transmitting Data

1Time

Received Data

NA

K

Output Data Time

Error

Go-back 3

2 3 4 5 3 44 5 6 7 5

1Time

2 3 4 5

Error

NA

K

Go-back 5

1 2 3 4

Assume N=3

http://www.youtube.com/watch?v=9BuaeEjIeQI&feature=related


Average Tx Time of The GBN ARQ

The average Tx time in terms of a single block duration is given by

TGBN=1PACK+(N+1)PACK (1- PACK)+ (2N+1)PACK (1-PACK)2+

(3N+1)PACK (1-PACK)3+….

= PACK+PACK[(1-PACK)+(1-PACK)2 +(1-PACK)3+…]+

PACK[N(1-PACK)+2N(1-PACK)2 +3N(1-PACK)3+…]

= PACK+PACK[(1-PACK)/{1-(1-PACK )}+N(1-PACK)/{1-(1-PACK)}2]

= 1+N(1-PACK)/PACK

1+(N[1-(1-Pb)n ])/(1-Pb)n



Throughput of The GBN ARQ

Throughput

SGBN=(1/TGBN)(k/n)

= [(1-Pb)n/((1-Pb)n+N(1-(1-Pb)n))](k/n)



Selective Repeat ARQ


Transmitting Data

1Time

Received Data

NA

K

Error

Retransmission

2 3 4 5 3 6 7 8 9 7

1Time

2 4 3 6 8 7Error

NA

K

Retransmission

5 9

Buffer 1Time

2 4 3 6 8 75 9

Output Data 1Time

2 4 3 6 8 75 9

Assume window size=5

A general case of the sliding window protocol with the transmit and receive window sizes are greater than 1.


Average Tx Time of The SR ARQ

The average Tx time in terms of a single block duration is given by

TSR=1PACK+2PACK(1-PACK)+3PACK (1-PACK)2+….

= PACK i(1-PACK)i-1

= PACK/[1-(1-PACK)]2

= 1/(1-Pb)n

where PACK(1-Pb)n


1i


Throughput of The SR ARQ

Throughput

SSR = (1/TSR)(k/n)

= (1- Pb)n(k/n)



Homework

P4.1 P4.4 P4.9 P4.10 P4.13


Wireless Communication Network Lab. Chapter 4 Channel Coding and Error Control Chih-Cheng Tseng1EE...

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