Question Bank

Question Bank for Optical communication I Sem M. TECH. (ECE)

Tulsiramji Gaikwad Patil College of Engineering And Technology

Wardha Road, Mohgaon, Nagpur.Department of Computer Science & Engineering

(PG Courses)I Semester M. TECH. (ECE)

Question Bank

(Error Control Coding)

UNIT I:

1) Construct the group under modulo-6 addition. 2) Construct the group under modulo-3 multiplication. 3) Let m be the positive integer. If m is not a prime, prove that the set {1,2.,.,.,,m-1} is

not a group under modulo-m multiplication. 4) Construct the prime field GF(11) with modulo -11 addition and multiplication. Find

all the primitive elements and determine the order of other elements. 5) Let m be the positive integer. If m is not a prime, prove that the set {0,1,2.,.,.,,m-1} is

not a field under modulo-m addition and multiplication. 6) Prove that every finite field has a primitive element. 7) Solve the following simultaneous equations of X,Y,Z and W with modulo-2

arithmetic : X+Y+W=1, X+W+Z=0, Y+Z+W=1,Y+Z+W=0 8) Show thatX5+X3+1 is irreducible over GF(2)

9) Find all irreducible polynomials of degree 5 over GF(2).

10) Prove that GF(2m)is a vector space over GF(2)

11) Construct the vector space V5 of all 5-tuples over GF(2).Find the three dimensional

subspace and determine its null space.

12) Let v be a vector space over a field F. for any element c in F, prove that c 0=0.

13) Let S1 S2 be the two subspaces of a vector v. show that intersection of S1 S2 is also a subspace in v.

UNIT II:

1) Consider a systematic (8,4) code whose parity check equations are vo=u1+u2+u3, v1=uo+u1+u2, V2=uo+u1+u3, v3=uo+u2+u3 where uo,u1, u2 and u3 are message digits and vo,v1,v2,and v3 are parity check digits. Find the generator and parity check matrices for this code. show analytically that minimum distance of this code is 4.

2) Construct the encoder for the code given in problem 14.. 3) Construct the syndrome circuit for the problem in 14. 4) Let c be a linear code with both even weight and odd weight code vectors. Show that

number of even weight code vectors is equal to odd weight code vectors.

Asst. Prof Prashant Yelekar, Dept. of CSE M.TECH. (ECE), TGPCET 1





5) Prove that hamming distance satisfies triangular in equality that is let x,y and z three n tuples over GF(2), and show that d(x,y)+d(y,z)>=d(x,z).

6) Prove that a linear code is capable of correcting λ or fewer errors and simultaneously detecting l (l>λ) or fewer errors if its minimum distance dmin>= λ+l+1.

7) Determine the weight distribution of the (8,4) linear code given in problem 14 .let the transition probability of a BSC be p=10-2. Compute the probability of an undetected error of this code

8) Because the (8,4) linear code given in problem 14 has minimum distance 4, it is capable of correcting all the single –error patterns and simultaneously detecting any combination of double errors. Construct a decoder for this code. The decoder must be capable of correcting any single error and detecting any double errors.

9) The (8,4) linear code given in problem 14. Is capable of correcting 16 error patterns. Suppose that this code is used for a BSC. Devise a decoder for this code based on the table-lookup decoding scheme. The decoder is designed to correct the 16 most probable error patterns.

UNIT III

1) Consider (15,11) cyclic hamming code generated by g(x) =1+X+X4 a. Determine the parity polynomial h(x) of this code. b. Determine the generator polynomial of its dual code. c. Find the generator and parity matrices in systematic form for this code.

2) Devise an encoder and a decoder for the (15,11) cyclic hamming code generated by g(X)= 1+X+X4

3) Show that g(X) =1+x2+X4+X6+X7+X10 generates a (21,11) cyclic code. Devise a syndrome computation circuit for this code. Let r(X) =1+X5+X17 be a received polynomial. Compute the syndrome of r(X).display the contents of the syndrome register after each digit of r has been shifted in to the syndrome computation circuit.

4) Shorten this (15,11) cyclic hamming by deleting the seven leading high order message digits. the resultant code is an (8,4) shortened cyclic code. design a decoder for this code that eliminates the extra shifts of the syndrome register.

5) Shorten the (31,26) cyclic hamming by deleting the 11 leading high order message digits. The resultant code is a (20,15) shortened cyclic code. devise a decoding circuit for this code that requires no extra shifts of the syndrome register.

6) Suppose that the (15,10) cyclic hamming code of minimum distance 4 is used for error detection over a BSC with transition probability p=10-2.compute the probability of an undetected error ,pu(E), for this code.

7) Devise a decoding circuit for the (7,3) hamming code generated by g(X)=(X+1)(X3+X+1).The decoding circuits corrects all the single error patterns and all the double adjacent error patterns.






8) For a cyclic code, if an error pattern e(X) is detectable ,show that its ith cyclic shift e(i)(X) is also detectable.

9) Let g(X) be the generator polynomial of an (n,k)cyclic code c. suppose c is interleaved to a depth of λ.prove that the interleaved code cλ is also cyclic and its generator polynomial is g(Xλ).

10) Show that the probability of an undetected error for the distance -4 cyclic hamming codes is upper bounded by 2-(m+1).

UNIT IV:

1) Determine the generator polynomials of all the primitive BCH codes of length 31.use the Galois field GF(25) generated by p(X) =1+X2+X5.

2) Suppose that the double error correcting BCH code of length 31 constructed in problem 33 is used for error correction on a BSC. Decode the received polynomials r1(X)=X7+X30 and r2(X)=1+X17+X28.

3) Devise a chien’s searching circuit for the binary double error correcting (31, 21)BCH code.

4) Devise a syndrome computation circuit for the binary double error correcting (31,21) BCH code.

5) Is there a binary t-error correcting BCH code of length 2m+1 for m>=3 and t<2m-1? if there is such a code, determine its generator polynomial.

6) Devise a circuit that is capable of multiplying any two elements in GF(25).Use p(X)=1+X2+X5 to generate GF(25).

UNIT V:

1) Consider the (31,5) maximum –length code whose parity check polynomial is p(X)=1+X+X2+X5.Find all the polynomials orthogonal on the digit position X30.Devise both type-I and type –II majority –logic decoders for this code.

2) Example 39 shows that the (15,7)BCH code is one step majority logic decodable and is capable of correcting any combination of two or fewer errors.show that the code is also capable of correcting some error patterns of three errors and some error patterns of four errors. List some of these error patterns.

3) Show that the extended cyclic hamming code is invariant under the affine permutations.

4) Show that the extended primitive BCH code is invariant under the affine permutations.

5) Prove that if J parity check sums orthogonal on any digit position can be formed for a linear code(cyclic or noncyclic), the minimum distance of the code is at least J+1.

6) Find the generator polynomial of the first order cyclic RM code of length






25-1.describe how to decode this code. 7) Find the generator polynomial of the third order cyclic RM code of length

26-1.describe how to decode this code.

UNIT VI:

1) Explain the difference between convolution codes and block codes.2) A rate 1/3 convolution code is described by; g1=[1 0 1], g2=[1 0 0],g3=[1 1 1]

a. Draw the encoder corresponding to this codeb. Draw code trellis diagram.c. Draw state diagram for this code.d. O/P of demodulator detector is (101001011110111),using viterbi algorithm,

find surviving path.3) Explain FANO metric 4) Explain FANO algorithm with the help of flowchart.5) Explain the distance properties of convolution encoder.

6) Draw and explain code tree and trellis structure of 2/3 convolution encoder.

7) Write short note on Asymptotic coding gain, bit error rate.

8) Explain the stack algorithm in detail with example. Compare stack and Fano algorithm

9) A rate 1/3 convolution code is described by; g1=[1 1 1], g2=[1 0 0],g3=[1 0 1] 10) Draw the encoder & code tree corresponding to this code.11) What is constraint length of convolution code.12) A rate 2/3 convolution code is described by; g1=[1 0 1 1, g2=[1 1 0 1], g3=[1 0 1 0]

Draw the encoder ,code tree ,code trellis & state diagram corresponding to this code13) A convolutional code is described by g1=[1 0 0], g2=[1 0 1],g3=[1 1 1] Find the

advance transfer function and the free distance of this code 14) Write short note on Sequential decoding15) Write short note on viterbi decoding16) A rate 2/3 convolution code is described by; g1=[1 0 1 1, g2=[1 1 0 1], g3=[1 0 1 0]

. O/P of demodulator detector is (01011111110111),using viterbi algorithm, find surviving path.

17) A convolutional code is described by g1=[1 0 1], g2=[1 0 0],g3=[1 1 1] Find the advance transfer function and the free distance of this code

18) Explain free distance & coding gain19) Give advantages of convolution code20) Explain trellis coded modulation21) Explain set partitioning of 8PSK signal constellation.22) Explain set partitioning of 16 QAM signal constellations.23) Draw and explain undgerboeck modulation technique.

a. Draw trellis diagram for ungerboeck modulation






UNIT VI:

1. write shotes notes on a) concatenation b) introduction to turbo codes and design of turbo codes

2. explain the series and parallel concatenated block codes in detail3. explain turbo codes and its properties in detail4. write a short notes on concatenated codes. Also explain distance properties of turbo

codes.5. Write shorts note on

a) need of interleavingb) fire codes, role of fire code in BEC(burst error correcting codes)c) convolution code for BEC and application

6. The output codeword of a block code Cb(6, 3) generated by the generator matrix Gis then input to a convolutional encoder like that seen in Figure P.7.1, operating inpseudo-block form. This means that after inputting the 6 bits of the codeword of theblock code, additional bits are also input to clear the registers of the convolutionalencoder.

(a) Determine the transpose of the parity check matrix of the block code, the syndrome-error pattern table, the trellis diagram of the convolutional code and its minimum free distance.

Fig: convolutional encoder in serial concatenatenation with a block code(b) Determine the minimum distance of the concatenated code.(c) Decode the received sequence r = (01 10 10 00 11 11 00 00) to find Possible errors, and the transmitted sequence.

7. The cyclic code Ccyc(3, 1) generated by the polynomial g(X) = 1 + X + X2 is appliedon a given bit ‘horizontally’ and then a second cyclic code Ccyc(7, 3) generatedby the polynomial g(X) = 1 + X + X2 + X4 is applied ‘vertically’ over thecodeword in an array code format, as seen in Figure

(a) Determine the rate and error-correction capability of this array code.(b) What is the relationship between the error-correction capability of this Array code and the individual error-correction capabilities of each cyclic code?






(c) Construct the array code by applying first the cyclic code Ccyc(7, 3) and Then the cyclic code Ccyc(3, 1) in order to compare with the result of item

Fig: an array code

Asst. Prof. Prashant Yelekar

(Dept.Of ECE M.TECH.ECE)


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