
2/80

Channel Coding

Channel coding refers to the class of signal

transformations designed to improvecommunications performance by enabling the

transformed signals to better withstand the effects of

various channel impairments, such as

Noise Interference

Fading

These signal processing techniques can be thought

of as vehicles for accomplishing desirable systemtrade-offs such as

Error performance versus bandwidth

Power versus bandwidth


3/80

Waveform Coding

Channel coding can be partitioned into two study

areas:

Waveform coding or signal design coding

Waveform coding deals with transforming waveform

into better waveforms to make the detectionprocess less subject to errors.

Structure sequence or structure redundancy

Structured sequences deals with transforming data

sequences into better sequences , havingstructured redundancy (redundant bits). The

redundant bits can then be used for the detectionand correction of errors.

The encoding procedure provides the coded signal

(whether waveforms or structured sequences) with

better distance properties than those of their uncoded

counterparts.


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Generalized One Dimensional Signals

One Dimensional Signal Constellation


6/80

Binary Baseband Orthogonal Signals

Binary Antipodal Signals

Binary orthogonal Signals


7/80

Constellation Diagram

Is a method of representing the symbol states of modulatedbandpass signals in terms of their amplitude and phase

In other words, it is a geometric representation of signals There are three types of binary signals:

Antipodal

Two signals are said to be antipodal if one signal is thenegative of the other

The signal have equal energy with signal point on thereal line

ON-OFF Are one dimensional signals either ON or OFF with

signaling points falling on the real line

)()( 01 tsts


8/80

8

With OOK, there are just 2 symbol states to map onto the

constellation space

a(t) = 0 (no carrier amplitude, giving a point at the origin)

a(t) = A cos wct (giving a point on the positive horizontal

axis at a distance A from the origin)

Orthogonal

Requires a two dimensional geometric representation since

there are two linearly independent functions s1(t) and s0(t )


9/80

6.1.1 Antipodal and Orthogonal Signals

Antipodal: s1(t) = - s2(t)

Orthogonal pulse waveforms:

s1(t) = p(t) 0 < t < T

s2(t) = p(t - T/2) 0 < t < T

where p(t) is a pulse with duration is the symbol duration.

Another orthogonal waveform set frequently used in

communication systems is sin x and cos x.

In general, a set of equal-energy signals si(t) , i = 1, 2, M,constitutes an orthonormal (orthogonal, normalized to unity)

set if and only if:

The signal energy is

T

jiijotherwise

jif ordttsts

Ez

0 0

1)()(

1(6.1)

T

i dttsE0

2 )( (6.2)


10/80

Figure 6.2: Antipodal signal set

Figure 6.3: Binary Orthogonal signal set


11/80

6.1.3 Waveform coding

Waveform coding procedures transform a waveform set

(representing a message) into an improved waveform set.

The most popular of such waveform codes are referred to asorthogonal and bi-orthogonal codes

The goal is to render the cross-correlation coefficient among all

pairs of signals as small as possible

The smallest possible value of the cross-correlation coefficient

occurs when the signals are anticorrelated ( zij= -1); this can beachieved only when the number of symbols in the set is two

(M=2) and the symbols are antipodal.

In general, it is possible to make all the cross-correlation

coefficients equal to zero

The set is then said to be orthogonal

Antipodal signal sets are optimum in the sense that each signal

is most distant from the other signal in the set (as shown in the

figure); where the distance d between signal vectors is seen to

be d = 2 E, where E represents the signal energy during asymbol duration T.


12/80

6.1.3 Waveform coding

Compared with antipodal signals, the distance properties of

orthogonal signal sets can be thought as prettygood(for a

given level of waveform energy). As shown in figure, thedistance between the orthogonal signal vectors is seen to be

d = 2E The cross-correlation between two signals is a measure of the

distance between the signal vectors.

The smaller the cross-correlation, the more distant are the

vectors from each other.


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Another form of orthogonality condition for sequence of pulses:

Orthogonal Codes

A one-bit data set can be transformed, using orthogonal codewordsof two digits each, described by the rows of matrix H1as follows

Data set Orthogonal codeword set

otherwise

jifor

sequencetheindigitsofnumbertotal

ntsdisagreemedigitofnumberagreementdigitofnumerzij

0

1 (6.3)

1

0

10

001H


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11

01

10

00

1 1

21 1

0 0 | 0 0

0 1 | 0 1

0 0 | 1 1

0 1 | 1 0

H H

H H H

111

011

101001

110

010

100

000

2 2

3

2 2

0 0 0 0 | 0 0 0 0

0 1 0 1 | 0 1 0 1

0 0 1 1 | 0 0 1 1

0 1 1 0 | 0 1 1 0

0 0 0 0 | 1 1 1 1

0 1 0 1 | 1 0 1 0

0 0 1 1 | 1 1 0 0

0 1 1 0 | 1 0 0 1

H HH

H H


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In general, we can construct a codeword set Hk, of dimension 2kx 2k,

called a Hadamard matrix, for a k-bit data set from the Hk-1matrix,

as follows:

Each pair of words in each codeword set H1, H2, . Hk, has as

many digit agreements as disagreements . Hence, in accordance

with equation 6.3, zij =0 (for I j), and each of the set is orthogonal.

Just as in M-ary signaling with an orthogonal format (such as MFSK),

the error performance improves. The probability of codeword error,

PE, can be upper bounded as

where Es=kEb

11

11

kk

kk

kHH

HHH (6.4)

0

)1()(N

EQMMP sE (6.5)


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Biorthogonal Codes

A biothogonal signal set of M total signals or codewords can be

obtained from an orthogonal set of M/2 signals by augmenting it with

the negative of each signals as

Data set Biorthogonal codeword set

111

011

101

001

110

010

100

000

1001

0011

0101

1111

0110

1100

10100000

3B

1

1

k

k

k

H

HB


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The biorthogonal set is really two sets of orthogonal codes such that

each codeword in one set has its antipodal codeword in the other

set

Advantages

The biorthogonal code requires one-half as many code bits per codeword as required by orthogonal codes

Since the antipodal signal vectors have better distance properties

than orthogonal ones, biothogonal codes perform slightly better than

orthogonal ones

Probability of codeword error can be upper bounded as:

2/||,0

2/||,11

Mjijifor

Mjijiforjifor

zij (6.8)

00

2)2()(

N

EQ

N

EQMMP ssE (6.9)


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6.1.4 Waveform Coding System Example

Figure 6.4: Waveform-encoded system (transmitter)

Tcis code bit duration


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Figure 6.4 illustrates the example of assigning a k-bit

message from a message set of M=2k

, with a coded-pulse sequence from a code set of the same size.

Each k-bit message chooses one of the generatorsyielding a coded-pulse sequence or codeword.

The sequences in the coded set that replace the

messages from a wave-form set with good distanceproperties (e.g., orthogonal, biorthogonal).

For the orthogonal code, each codeword consists ofM=2kpulses (representing code bits).

Hence, 2k

code bits replace k message bits.The chosen sequence then modulates a carrier waveusing binary PSK, such that the phase (= 0, ) ofthe carrier during each code-bit duration, 0 t Tc,corresponds to the amplitude ( j = -1 or 1) of the jth

bipolar pulse in the codeword.



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Figure 6.5: Waveform-encoded system with coherent detection (receiver)


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At the receiver in Figure 6.5, the signal isdemodulated to baseband and fed to M correlators(or matched filters).

For orthogonal codes, such as those characterizedby Hadamard matrix, correlation is performed over a

codeword duration that can be expressed as T = 2k

Tc.For a real-time communication system, messagesmay not be delayed; hence, the codeword durationmust be the same as the message duration, and thus,T can also be expressed as T (log2M)Tb= kTb, where

Tbis the message-bit duration.Time duration of a message bit is M/k times longerthan that of a code bit.



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In other words, the code bits or coded pulses (whichare PSK modulated) must move at a rate M/k fasterthan the message bits.

For such orthogonally coded waveforms and anAWGN channel, the expected value at the output of

each correlator, at time T, is zero, except for thecorrelator corresponding to the transmittedcodeword.



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Error control Coding

Errors are introduced in the data when it passes through the

channel. The channel noise interfere the signal. The signal

power is reduced. Hence errors are introduced.

Rationale for Coding and Types of codes

Transmission of data over the channel depends upon following

two parameters:

transmitter power channel bandwidth

The power spectral densityof channel noise andthese two

parameters determine signal to noise power ratio.

The signal to noise power ratio determine the probability of error

of the modulation scheme. For the given signal to noise ratio, the error probability can be

reduced further by using coding techniques.

The coding techniques also reduce signal to noise power ratio

for fixed probability of error.


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Error control Coding

Figure: Digital communication system with channel encoding

Channel encoder

The channel encoder adds extra bits (redundancy) to the message bits.The encoded signal is then transmitted over the noisy channel.

Channel decoder

The channel decoder identifies the redundant bits and uses them to detectand correct the errors in the message bits if any.

Thus the number of errors introduced due to channel noise are minimizedby encoder and decoder.

Due to the redundant bits, the overall data rate increases. Hence channelhas to accommodate this increased data rate.

The systems become slightly complex because of coding techniques.


25/80

6.2 Types of Error control

There are two basic ways for using redundancy for controlling

errors:1. Error detection and retransmission

2. Forward error correction (FEC)

Error detection and retransmission

It utilizes parity bits (redundant bits added to the data) to

detect that an error has been made. The receiving terminal does notattempt to correct the error; it simply requests that the transmitter

retransmit the data

Note that a two-way link is required for such dialogue

between the transmitter and receiver.

Forward error correction (FEC)

FEC requires a one-way link only, since in this case the

parity bits are designed for both the detection and correction of

errors.


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6.2 Types of Error control

Terminal Connectivity

Simplex

Half duplex

Full duplex

Figure 6.6: Terminal connectivity classifications (a) Simplex (b) Half-duplex(c) Full-duplex


27/80

Automatic Repeat Request

When the error control consists of error detection

only, the communication system generally needs to

provide a means of alerting the transmitter that anerror has been detected and that a retransmission is

necessary.

ARQ vs. FEC

ARQ is much simpler than FEC and need noredundancy.

ARQ is sometimes not possible if

A reverse channel is not available or the delay with

ARQ would be excessive The retransmission strategy is not conveniently

implemented

The expected number of errors, without corrections,

would require excessive retransmissions


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Figure 6.7: Automatic Repeat Request (ARQ) (a) Stop and wait ARQ (b)Continuous ARQ with pullback (c) Continuous ARQ with selective repeat

(a) Stop and wait ARQ

(b) Continuous ARQ with pullback

(c) Continuous ARQ with selective repeat (full duplex)


29/80

6.3 Structured Sequences

Parity-check coding procedures are classified as

structured sequences because they represent

methods of inserting structured redundancy into the

source data so that the presence of errors can be

detected or the errors corrected.

Structured sequences are partitioned into three sub-

categorries:

Block codes

Convolutional codes

Turbo codes


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Block codes In case of block codes, encoder transforms each k-bit data

block into a larger block or codeword of n-bits called code bits

or or channel symbol

The block or codeword consists of k message bits and (n-k)

redundant bits.

The (n-k)-bits added to each data block are called redundant

bits, parity bitsor check bits. Such block codes are called (n, k)

block codes.

They carry no new information.

Ratio of redundant bits to data bits: (n-k)/kis called redundancy

of code

Ratio of data bits to total bits, k /nis called code rate

6.3.2 Code Rate and Redundancy


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Block codes



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Block codes


Illustration:

The generating set for the (7,4)

code:

1000 ===> 1101000

0100 ===> 0110100

0010 ===> 1110010

0001 ===> 1010001


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Convolutional codes The coding operation is discrete time convolution of input

sequence with the impulse response of the encoder.

The convolutional encoder accepts the message bits

continuously and generates the encoded sequence

continuously.

The codes can be classified as linear or nonlinear codes.

Linear code:

If the two codewords of the linear code are added by modulo-2

arithmetic, then it produces third codeword in the code.

This is very important property of the codes, since other codewords

can be obtained by addition of existing codewords.

Nonlinear code:

Addition of the nonlinear codewords does not necessarily produce

third codeword.



34/80

6.3.3 Parity-Check Codes

Single-parity-Check Code

Parity check codes use linear sums of the information bits, called

parity symbols or parity bits, for error detection or correction.

Even Parity Example:

Two-dimensional

Parity-check Code


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Rectangular Code

Also called a product code, can be thought of as a parallel code

structure. The rate of the code k/n is

( 1)( 1)

MNkn M N

Figure 6.8: Parity checks for parallel structure


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6.3.4 Why Use Error Correction Coding

Ideal PB(Probability ofbit error) versus Eb/N0

Bit error probability for coherently

detected M-ary orthogonal signalingBit error probability of coherently

detected multiple phase signaling


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38/80


Trade-Off 1: Error Performance vs. Bandwidth (A to C rather than B)

Error performance improvement can be achieved at the expense of

bandwidth.

Aside from the new components (encoder and decoder) needed, the

price is more transmission bandwidth.

Error-correction coding needs redundancy.

If we assume that the system is a real-time communication system (such

that the message may not be delayed), the addition of redundant bits

dictates a faster rate of transmission, which of course means morebandwidth.


39/80


Trade-Off 2: Power versus Bandwidth (D to E)

If error correction coding is introduced, a reduction in the required Eb/N0

can be achieved.

Thus, the trade-off is one in which the same quality of data is achieved,

but the coding allows for a reduction in power or Eb/N0.

What is the cost? The same as before- more bandwidth.


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42/80



43/80



44/80

6.4 Linear block Codes

Linear block codes are a class of parity-check codes thatcan be characterized by the (n, k) notation.

The encoder transforms a block of kmessage digits (a

message vector) into a longer block of ncodeword digit

(a code vector) constructed from a given alphabet ofelements.

When the alphabet consists of two elements (0 and 1),

the code is a binary code comprising binary digits (bits).

The k-bit messages form 2kdistinct messagesequences, referred to as k-tuples(sequences of k

digits).

The n-bit blocks can form as many as 2ndistinct

sequences, referred to as n-tuples.


45/80


The encoding procedure assigns to each of the 2kmessage k-tuples one of the 2nn-tuples.

A block code represents a one-to-one assignment,

whereby the 2kmessage k-tuples are uniquely mapped

into a new set of 2kcodeword n-tuples. The mapping can be accomplished via a look-up table.


46/80


6.4.1 Vector Spaces

011101

110

000

111001

010

000


47/80



48/80


6.4.2 Vector Subspaces For example, the vector space V4is totally populated by the

following 24 = sixteen 4-tuples:

0000 0001 0010 0011 0100 0101 0110 0111

1000 1001 1010 1011 1100 1101 1110 1111

An example of a subset of V4that forms a subspace is

0000 0101 1010 1111


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The subset chosen for the code should include as many as

elements to reduce the redundancy but they should be as apart as

possible to maintain good error performances

Figure 6.10: Linear block-code structure


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6.4.3 A (6.3) Linear Block Code Examples

Message Vector Codeword

000 000000

100 110100

010 011010

110 101110

001 101001

101 011101

011 110011

111 000111

Table 6.1: Assignment of Codewords to Messages

Examine the following coding assignment that describes a (6, 3)code. There are 2k = 23 = 8 message vectors, and therefore eight

codewords. There are 2k= 26= sixty-four 6-tuples in the V6vector space.

6 4 4 G M i


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6.4.4 Generator Matrix

If k is large, a lookup table implementation of the encoder becomes

prohibitive.

For a (127, 92) code there are 292or approximately 5 x 1027code

vectors.

If the encoding procedure consists of a simple look-up table, imagine

the size of the memory necessary to contain such a large number of

codewords. Fortunately, it is possible to reduce complexity by

generating the required codeword as needed, instead of storing them.

Since a set of codewords that forms a linear block code is a k-

dimensional subspace of the n-dimensional binary vector space (k < n),

it is always possible to find a set of n-tuples, fewer than 2k, that can

generate all the 2kcodewords of the subspace.

The generating set of vectors is said to span the subspace.

The smallest linearly independent set that spans the subspace is called

a basis of the subspace, and the number of vectors in this basis set is

the dimension of the subspace.

6 4 4 G M i


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Any basis set of k linearly independent n-tuples V1, V2, ..,

Vkcan be used to generate the required linear block code

vectors, since each code vector is a linear combination of

V1, V2, , Vk.

Let the set of 2kcodewords {U} be described as:

U = m1V1+m2V2+..+mkVk

where mi= (0 or 1) are the message digits and i = 1,, k.

In general, generator matrix can be defined by the following

k x n array:

knkk

n

n

k vvv

vvvvvv

V

VV

G

21

22221

11211

2

1

(6.24)

6 4 4 G M i


53/80


Code vectors are usually designated as row vectors.

Thus, the message m, a sequence of k message bits, is

shown below as a row vector (1xk matrix having one row and

k columns):

m= m1, m2, .., mk

Generation of codeword Uis written in matrix notation as theproduct of m and G:

U = m G (6.25)


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100101

010110

001011

3

2

1

V

V

V

G (6.26)

6.4.5 Systematic Linear Block codes


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A systematic (n,k) linear block code is a mapping from a k-

dimensional message vector to an n-dimensional codeword in such

a way that part of the sequence generated coincides with the kmessage digits. Thus remaining (n-k) digits are parity digits.

A systematic linear code will have a generator matrix of the form

Where P is the parity array portion of the generator matrix

pij= (0 or 1), andIkis the kx kidentity matrix (ones on themain diagonal and zeros elsewhere).

100

010

001

)(,21

)(,22221

)(,11211

knkkk

kn

kn

k

ppp

ppp

ppp

IPG (6.27)

G P | Ik

1 1 0

0 1 1

1 1 1

1 0 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1k n k k k ( )

Storage requirement reduced from2k(n+k) to k(n-k).

6 4 5 S t ti Li Bl k d


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Combining (6.26) and (6.27):

where

ui = m1p1i+ m2p2i+ .mkpki for i=1,(n-k)

= mi-n+k for i=(n-k+1).n

Given the message k-tuple

m= m1, m2, .., mk

and the general code vector n-tuple

U= u1, u2, .., un

100

010

001

].m,m,[mu,....u,u

)(,21

)(,22221

)(,11211

k2121

knkkk

kn

kn

n

ppp

ppp

ppp


57/80

Example


58/80

Example

For (6,3) code example in sec.6.4.3, the codewords can be described

as:

Equation (6.31) depicts that the first parity bit is the sum of the first andthird message bits; the second parity bit is the sum of the first and

second message bits; and the third parity bit is the sum of the second

and third message bits.

Such structure, compared with single-parity checks or simple digit-

repeat procedures, may provide greater ability to detect and correcterrors.

3

100

010

001

101

110

011

,, 321

IP

mmmU

654321

321322131 ,,,,,

uuuuuu

mmmmmmmmm

(6.30)

(6.31)


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Example

We see structure of linear block code (Equation (6.31)) that the redundant

digits are produced in a variety of ways.

The first parity bit is the sum of the first and third message bits; the second

parity bit is the sum of the first and second bits; and the third parity bit is

the sum of the second and third message bits.

Intuition tells us that such structure, compared with single-parity checks or

simple digit-repeat procedures, may provide greater ability to detect and

correct errors.

6 4 6 Parity Check Matrix


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6.4.6 Parity-Check Matrix

Let H denote the parity check matrix, that will enable us to decode

the received vectors

For each (k x n) generator matrix G, there exists an (n-k) x n matrix

H, such that rows of Gare orthogonal to the rows of Hi.e., GHT=0

Fulfilling the orthogonality requirements:

and

T

kn PIH | (6.32)

(6.33)

)(,21

)(,22221

)(,11211

100

010

001

knkkk

kn

kn

kn

T

ppp

ppp

pppP

I

H


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It is easy to verify that the product UHTof each codeword U,generated by G and the HTmatrix, yields

UHT

=p1+p1, p2+p2, . Pn-k+ Pn-k = 0where the parity bits p1, p2, .pn-kare defined earlier in Eq.6.29. Uis a code word generated by matrix Giff UHT= 0


62/80

6.4.7 Syndrome Testing

Let r = r1, r2, .., rn be received vector (one of 2n

n-tuples)resulting from the transmission of vector U = u1, u2, .., un(one of 2kn-tuples). We can therefore describer as

r = U + e

where e =e1, e2, .., en is an error vector or error pattern

introduced by the channel. There are a total of 2n-1 potential nonzero error patterns in in the

space of 2nn-tuples.

The syndrome of r is defined as:

S = r HT

The syndrome is the result of a parity check performed on rtodetermine whether ris a valid member of the codeword set.

If, in fact,r is a member, the syndrome Shas a value 0. If r containsdetectable errors, the syndrome has some nonzero value.

(6.34)

(6.35)


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Requirements of the parity-check matrix


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Requirements of the parity-check matrix

No column of H can be all zeros, or else an error in the

corresponding codeword position would not affect the syndrome

and would be undetectable

All columns of H must be unique. If two columns of H were

identical, errors in these two corresponding codeword positions

would be indistinguishable

Example

Codeword U = 1 0 1 1 1 0 , and r = 0 0 1 1 1 0 Find S=rHT

00111,11,1101

110011

100

010

001

011100

TrHS

001

000001

T

T

H

eHS

6 4 8 Error Correction


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6.4.8 Error Correction

Arranging 2n n-tuples; representing possible received vectors, in an

array is called standard array. Standard array for (n,k) code is:

Each row, called a cosetconsists of an error pattern in the first

column called coset leader

If error pattern is not a coset leader, erroneous decoding will result

knkkni

knkn

j

k

jijj

k

i

k

i

k

i

eUeUeUe

eUeUeUe

eUeUeUe

eUeUeUe

UUUU

222222

22

323323

222222

221

(6.38)


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The syndrome of a Coset

Coset is a short for a set of numbers having a common feature

If ej is the coset leader then Ui+ejis an n-tuple in this coset.

Syndrome of this n-tuple is:S = (Ui+ ej) H

T= ejHT

The syndrome must be unique to estimate the error pattern

Error Correction Decoding Calculate the syndrome of R using S=rHT

Locate the coset leader ( error pattern) ej, whose syndrome

equals rHT

This error pattern is assumed to be the corruption caused by the

channel The corrected received vector, or code word, is identified as

U=r+ej. We retrieve the valid codeword by subtracting(adding) the

identified error

Locating the Error Pattern


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Locating the Error Pattern

Example of a standard array for a (6,3) code is shown:

Computing ejHT for each coset leader

101

110

011100

010

001

jeS


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Error Correction Example

Error pattern is an estimate of error, the decoder addes the

estimated error to received signal to obtain an estimate of

transmitted code word as:

Example: let U=101110, and r=001110, then show how the

decoder can correct the error using look-up table 6.2

eeUeeUerU (6.40)

011101

000001011100

:byestimatedisvectorcorrectedThe

000001

:errorestimated

001011100

erU

e

HS T

Implementation of decoder


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Implementation of decoder

Figure 6.12: Implementation of the (6,3) decoder


71/80

6.5 Error Detection and Correcting Capability

6.5.1 Weigh and Distance of Binary Vector

Hamming distance between two codewords is the number of

elements in which they differ

Hamming weight is the number of nonzero elements

Example:

U = 1 0 0 1 0 1 1 0 1V = 0 1 1 1 1 0 1 0 0

w(U) = 5

d(U,V) = w(U+V) = 6

6.5.2 Minimum Distance of a Linear Code


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6.5.2 Minimum Distance of a Linear Code

The minimum distance (dmin)among all the distances between each

pair of codes in the code set

6.5.3 Error Detection and Correction

The error-correcting capability t of a code: the maximum number of

guaranteed correctable errors per codeword

Error detecting capability defined in terms of dmin

2

1mindt

1minde

6.7 Cyclic Codes


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6.7 Cyclic Codes

An (n,k) linear code is called a cyclic code if it can be described by

the following property: the code with shifted version being in the

code set, so if we have a code word:

Then its shifted version:

is also a code word.

1

1

2

210)(

n

n XuXuXuuXU

n

nn XuXuXuXuuXU 13

2

2

101)(

6.7.2 Binary Cyclic Codes Properties


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6.7. a y Cyc c Codes ope t es

Cyclic code can be generated using a generator polynomial:

where g0and gpmust be equal to 1. (n-p=k)

If the message polynomial is given by:

The codeword polynomial in (n,k) cyclic code can be expressed as:

U is said to be a valid codeword of the subspace S iff g(X) dividesinto U(X) without a remainder. A generator polynomial g(X) of an

(n,k) cyclic code is a factor of Xn+1, that is:

Xn+1=g(X)h(X)

p

pXgXgXggXg 2

210)( (6.57)

1

1

2

210)(

pn

pn XmXmXmmXm (6.58)

XgXmXmXmmXU kk 112210 (6.59)

6.7.3 Encoding in Systematic Form


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g y

Message vector:

Multiply m(k) with Xn-k

Divide by g(X)

Adding p(X) to both sides, we have:

Where:

112

210

k

k XmXmXmmXm

11

1

10

nk

knknknXmXmXmXmX

XpXgXqXmX kn

XUXgXqXmXXp kn

),,,,,,,( 110)(

110

bitsmessagek

k

bitsparitykn

kn mmmpppU

Example (on the board)


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p ( )

6.8 Well-Known Block Codes


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6.8.1 Hamming Codes

Simple class of block codes characterized by the structure:

These codes have a minimum distance of 3 and are capable of

correcting single errors

For hard-decision decoding, bit error probability is:

Where p is channel symbol error probability

Equivalent equation:

For performance over Gaussian channel using coherently

demodulated BPSK, channel symbol error probability is:

mkn mm 12,12,

n

j

jnj

B ppjnj

nP

2

)1(1

1)1( nB pppP

0

2

N

EQp c

6.8.2 BCH Codes


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Bose-Chadhuri-Hocquenghem (BCH) codes are generalization of

Hamming Codes

Powerful class of cyclic codes that provide large selection of block

lengths, code rates, alphabet sizes, and error correcting capability

Table 6.4 lists some code generators g(X) for various values of n,k

and t upto block length of 255


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Table 6.4: Generators of BCH Codes


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DC Channel Coding

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