US N: M.S. RANIAIAH INSTITUTE OF TECHNOLOGY (Autonomous Institute, Affiliated to VTU) Semester End Examinations -January 2009 Course : M.Tech (Digital Electronics & Communication) Subject Code : MDLC 15 Subject : Error Control Coding Maximum !Marks : 100 Duration : 3 Hours Instructions to the Candidates :. 1) Answer any 5 full questions 2) All questions carry equal marks 3) Answer must be brief and to the point 1. a) Define with suitable examples, the following (6) i) Irreducible polynomial ii) Primitive polynomial iii) Minimal polynomial b) Let a be a primitive element in GF (24). Find the roots of the polynomial (8) F(x)=x3+a2X+Q(13 Table of GF(2) generated b y (x) = 1+ x + x -0 0 0 0 7 1 1 0 1 0 1 0 0 0 8 1 0 1 0 T0 1 0 0 0 1 0 1 19 2 0 3 0 4 1 5 0 6 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 10 11 12 13 14 1 1 1 0 0. 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1c) Let a be a primitive element in GF(24). Solve the following simultaneous (6) equations for X, Y and Z. X+a10Y+a3Z =a11 a6X+Y+a8Z =06 X+a2Y+(X6Z =a1I 2. a) The parity check bits of a (8, 4) Linear Block Code are generated by (15) V5=u1 + U2+ U4 V6=U1+U3+U4 V7 = u1 + u2 + u3 vs= U2+ U3+114 where u1, u2, u3 and u4 are message bits. i) Find the generator matrix [G] ii) Find the parity check matrix [H] iii)Find the minimum weight of this code iv)With suitable examples, show that it is a SEC and DED code. v) Draw the encoding circuit vi) Draw the syndrome circuit b) Describe "HAMMING BOUND". Under what condition is a code called (5) "PERFECT CODE"?3 a) Form the generator matrix of the first order RM code RM (1, 3). (12) Ii) What is the minimum distance of this code? ii) Determine its parity check sums and devise a majority logic decoder for the code, iii) Decode the received vector r = (01000101) b) Suppose that (24, 12) Golay code is used for error correction . Decode the (8 ) following received sequences i) r = [101101110010000011000011] 11)r=[001111110010000000000001] 4. a) Discuss the various properties of cyclic codes. (5) b) Consider a (15, 11) cyclic code generated by g(x) = 1 + x + x4. Devise a (7) feedback shift register encoder circuit. Illustrate the encoding procedure with the message vector 10100011011 by listing the states of the register. c) The generator polynomial for a (15, 7) cyclic x8 (8) i) Find the code - vector in systematic form for ii) Draw the syndrome calculator circuit for the determine the syndrome for the received vector r code is g(x) = 1+ x4 +x6 + x7+ the message u= [1001011] (15, 7) cyclic code and = [100011110011100]5. a) Let a be a primitive element of the Galois Field GF (24) such that l+a+ a4 =0. (8) Using this find the generator polynomials for i) (15, 11) single-error-correcting BCH code. ii) (15 , 7) double-error-correcting BCH code. iii) (15 , 5) triple-error-correcting BCH code. Table of Minimal Polynomials of GF (2 ) Power of a Coefficients Power of a Coefficients 1 (0, 1, 4) 5 (0, 1, 2) 3 (0,1,2,3,4) 7 (0,3, 4) b) For (15, 7) double-error-correcting BCI-I code, obtain the parity check matri x (4) H. c) Using (15, 5) BCH code, the following vector is received (8) R =[001000001000001] Using BERLEKAMPS iterative procedure, detect and correct all the errors. 6. a) Consider the (3, 1,2) convolutional code with g(l) = (1, 0, 1), g(2) = (1, 1, 0) (8) and g(3) = (1, 1, 1). i) Draw the encoder block diagram. ii) Find the generator matrix iii) Find the code-word corresponding to the information sequence (10111) using time - domain and transform - domain approach. b) For the convolutional encoder shown in figure below. (12) i) Draw state transition table ii) Draw state diagram, iii) Draw code-tree iv) Find the encoder output produced by the message sequence 11101 v) Draw trellis diagram 27. a) With a neat block diagram, explain the principle of concatenation with sin gle- (10) level concatenated codes and interleaved concatenated coding system. b) With a neat block diagram explain the basic turbo encoding structure. Also (1 0) draw a turbo encoder using two (2, 1, v) convolutional encoders and an i,nterleaver and explain its operation. 8. a) Draw the diagram of an error-trapping-decoder for an 1-burst-error-correct ing (12) cyclic code and discuss the step-by-step decoding procedure. b) What are Fire Codes? How are they generated and decoded? (8) 3