UNIT - IV

PART – A

1. Generate the cyclic code for (n, k) syndrome calculator. [N/D-16]

Let c be the original codeword sent and r = c + e be the received codeword, where e is the

error vector. ei = {n1, if an error has occured in the ith position0, otherwise Decoding of r

A 1 × (n − k) matrix called error syndrome matrix is calculated

S=r HT

2. What is a linear code? [N/D -07]

3. What is a linear code? [M/J-16]

A code is linear if modulo-2 sum of any two code vectors produces another code vector.

This means any code vector can be expressed as linear combination of other code vectors.

4. What is meant by constraint length of a convolution encoder? [M/J-16]

Constraint length is the number of shift over which the single message bit can influence the

encoder output. It is expressed in terms of message bits.

5. What is meant by syndrome of linear block code?

The non zero output of the produce YHT is called syndrome & it is used to detect errors in

y. Syndrome is denoted by S & given as, S=YHT.

6. What is convolution code? How is it different from block codes? [A/M -12]

Fixed number of input bits is stored in the shift register & they are combined with the help

of mod 2 adders. This operation is equivalent to binary convolution coding.

7. Write the futures of BCH Codes?

BCH codes are most extensive and powerful error correcting cyclic codes. The decoding of

BCH codes is comparatively simpler.

8. What are the advantages of convolutional codes?

Advantages:

1 .The decoding delay is small in convolutional codes since they operate o smaller blocks

of data.

2. The storage hardware required by convolutional decoder is less since the block sizes are

smaller.

Disadvantages:

1. Convolutional codes are difficult to analyze since their analysis is complex.

2. Convolutional codes are not developed much as compared to block codes.

9. What is difference between block codes and convolutional codes?

Block codes takes‟ k‟ .number of bits simultaneously form „n‟ -bit .code vector. This code

vector is also called block. Convolutional code takes one message bits at a time and

generates two or more encoded bits. Thus convolutional codes generate a string of encoded

bits for input message string.

10. What are the error detection and correction capabilities of hamming codes?

The minimum distance (dmin) of hamming codes is „3‟ . .Hence it can be used to detect

double errors or correct single errors. Hamming codes are basically linear block codes with

dmin =3.

11. What is meant by cyclic codes?

Cyclic codes are the subclasses of linear block codes. They have the property that a cyclic

shift of one codeword produces another code word.

12. Define minimum distance. (MAY-07)

It is the smallest hamming distance between the valid code vectors. The error detecting

correcting capabilities of the codes depend upon the minimum distance.

13. Define hamming weight. (MAY-09)

The number of 1’s in the code word of the hamming code is called as hamming distance.

14. What is meant by systematic and non-systematic codes?

In a Systematic block code, message bits appear first and then check bits. In the

nonsystematic code, message and check bits cannot be identified in the code vector.

PART – B

1. Describe the cyclic codes with the linear and cyclic property. Also represent the

cyclic property of a code word in polynomial notation. (16) [N/D-16]

2. Determine the generator polynomial g(X) FOR A (7, 4) cyclic code and fine the

code vector for the following data vector 1010, 1111 and 1000 (8)

n=7 k=4

q=n-k=3

To obtain the generator polynomial

(p7+1) = (p+1)(p3+p2+1)(p3+p+1)

Let G (p)= (p3+p+1)

To obtain the generator matrix in systematic form

To determine the code vector

1. code vector for M=1010

X=MG

2. Code vector for M=1111

3. Code vector for M=1000

3. Describe how the errors are corrected using Hamming code with an Example.(12)

BLOCK CODE

A block code is linear if any linear combination of two code words is also a codeword. In

the binary case this requires that if ci and c j are code words then ci ⊕c j is also a code

word, where ⊕ denotes component-wise modulo-2 addition.)

HAMMING DISTANCE

The hamming distance between two code vectors is equal to the number of elements in

which they differ. For example, let the two code words be, X = (101) and Y= (110)

These two code words differ in second and third bits. Therefore the hamming distance

between X and Y is two

The Hamming distance between two code words ci and c j is the number of components

at which the two code words differ, and is denoted by d(ci , c j ).†

The Hamming weight, or simply the weight of a code word ci is the number of nonzero

components of the code word and is denoted by w(ci ).

The minimum distance of a code is the minimum Hamming distance between any two

different code words; i.e.,

The minimum weight of a code is the minimum of the weights of the code words except

the all zero code word.

Hamming Codes. Hamming codes are a class of linear block codes with n = 2m − 1, k =

2m − m − 1 and dmin =3, for some integer m ≥ 2.this minimum distance, these codes are

capable of providing error-correction capabilities for single errors. The parity check

matrix for these codes has a very simple structure. It consists of all binary sequences of

length m except the all zero sequence. The rate of these codes is given by

A code can detect any combination of d or fewer errors if

A code can correct any combination of c or fewer errors

A code can simultaneously detect d errors and correct c < d errors if

repetition code: correct 1 error and detect up to 2 errors.

Definition: A block code is linear if any linear combination of two code words is also a

codeword. In the binary case, it is equivalent to the fact that the sum of any two code

words is also a codeword where summation is defined by component wise modulo-2

addition linear codes.)

4. What are cyclic codes. Explain the merits and demerits Give the properties of cyclic

codes. (AU- May/June 2010)

Cyclic codes forms as subclass of linear block code.

Properties of cyclic code

1. Linearity property.

The sum of two code word in the code is also a code word.

2. Cyclic property

A cyclic shift of a code word in the code is also a code word.

GENERATOR POLYNOMIAL

C(X) = a (X) g (X)

Encoding procedure

Encoder of cyclic code

Syndrome Calculator:

5. Explain viterbi decoding algorithm for convolution code. (8)

Viterbi Decoding Algorithm Maximum Likelihood (ML) decoding rule

Viterbi Decoding Algorithm

An efficient search algorithm

Performing ML decoding rule.

Reducing the computational complexity.

Basic concept

the decoder the decoder penetrates through the code

trellis level by level in search for the transmitted code sequence

metrics of all

the partial paths entering a node

es the partial path with the larger metric and eliminates all the

other partial paths. The stored partial path is called the survivor.

Viterbi Decoding Process :

6. Assume a (2,1) convolutional coder with constraint length 6.Draw the tree diagram, state diagram and

trellis diagram for the assumed coder .Design block code for a message block of size eight that can correct

for single errors Briefly discuss on various error control codes and explain in detail with one example for

convolution code. (12) [A/M -11]

N=2, K=1 AND K=6(CONSTRAIN LENGHT)

M=K/n=6/2=3, snce constrain length k=n*M

3 storage element in shift register

N=2 two output bits

One set k=1 of shift register having 3 storage element the convolutional code structure is easy to draw from

its parameters. First draw m boxes representing the m memory register. Then draw n modulo-2 adders to

represent the n output bits. Now connect the memory registers to the adders using the generator

Polynomial

Convolutional codes k = number of bits shifted into the encoder at one time

k=1 is usually used!!

n = number of encoder output bits corresponding to the k0020information bits

r = k/n = code rate

K = constraint length, encoder memory Each encoded bit is a function of the present input bits and their past ones. Generator Sequence

Convolutional Codes An Example – (rate=1/2 with K=2)

7. Explain how encoding is done by convolution codes with a suitable example [A/M - 10] 8. Design a convolution coder of constraint length and rate efficiency ½.Draw its tree diagram and trellis diagram. [N/D - 12]

Convolutional codes are widely used as channel codes in practical communication systems for

error correction. *The encoded bits depend on the current k input bits and a few past input bits.

The main decoding strategy for convolutional codes is based on the widely used Viterbi

algorithm. Convolutional codes are commonly described using two parameters: the code rate

and the constraint length. The code rate, k/n, is expressed as a ratio of the number of bits into

the convolutional encoder (k) to the number of channel symbols output by the convolutional

encoder (n) in a given encoder cycle. The constraint length parameter, K, denotes the "length" of

the convolutional encoder, i.e. how many k-bit stages are available to feed the combinatorial

logic that produces the output symbols. Closely related to K is the parameter m, which can be

thought of as the memory length of the encoder. A simple convolutional encoder is shown

below(fig 3.1). The information bits are fed in small groups of k-bits at a time to a shift register.

The output encoded bits are obtained by modulo-2 addition (EXCLUSIVE-OR operation) of the

input information bits and the contents of the shift registers which are a few previous information

bits

The operation of a convolutional encoder can be explained in several but equivalent ways such

as, by a) state diagram representation. b) tree diagram representation. c) trellis diagram

representation.

a) State Diagram Representation: A convolutional encoder may be defined as a finite state

machine. Contents of the rightmost (K-1) shift register stages define the states of the encoder.

So, the encoder in has four states. The transition of an encoder from one state to another, as

caused by input bits, is depicted in the state diagram. A new input bit causes a transition from

one state to another

b) Tree Diagram Representation: The tree diagram representation shows all possible information and encoded sequences for the convolutional encoder. The encoded bits are labeled on the branches of the tree. Given an input sequence, the encoded sequence can be directly read from the tree.

9. Consider a single error correction (7,4) linear code and the corresponding decoding

table(10)

10. Find the (7,4) linear systematic block code word corresponding to 1101.Assume a

suitable generator matrix. [AUC APR/MAY 2011]

Let

n=7 k=4

q=n-k=3

code vector G=[Ik: P]

Check matrix C=MP

C1 = m1+m2+m3

C2= m2+m3+m4

C3= m1+m2+m4

C=[010]

Complete code word can be calculated X={M:C}={1 1 0 0 0 1 0}

The parity matrix H=[pT :I] =[I: pT] =

Minimum weight W(X)=3