1) Consider a (15,5) linear block code (cyclic) in systematic form. The generator polynomial is given as g(x) = 1 + X + X^2 + X^5 + X^8 + X^10. a. Design and draw the circuit of the feedback shift register encoder and decoder. (6 Marks) b. Use the encoder obtained in part a to find the code word for the message [x x x x x]. (Assume the right most bit is the earliest bit) (5 Marks) c. Repeat the steps of part b for decoding. (5 Marks) d. Verify the codeword obtained in part b polynomial division method (5 Marks) e. Consider a codeword C = [y y y y y y y y y y y y y y y]. Is this a codeword of the above system? Provide suitable justification for your answer. (4 Marks)
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Since it is a (15,5) linear block code, the message length is 5 and the codeword length is 15. The feedback shift register encoder will have 10 flip-flops (15 - 5). The feedback connections are determined by the generator polynomial. The feedback connections will Show more…
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Consider a (15,5) linear block code (cyclic) in systematic form. The generator polynomial is given as g(x) = 1 + X + X^2 + X^5 + X^8 + X^10. Design and draw the circuit of the feedback shift register encoder and decoder. (6 Marks) Use the encoder obtained in part a to find the code word for the message [x x x x x]. (Assume the right most bit is the earliest bit) (5 Marks) Repeat the steps of part b for decoding. (5 Marks) Verify the codeword obtained in part b polynomial division method (5 Marks) Consider a codeword C = [y y y y y y y y y y y y y y y]. Is this a codeword of the above system? Provide suitable justification for your answer. (4 Marks)
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Consider the systematic (7, 4) Hamming code. The parity-bit generator matrix P is shown below. i) Determine the corresponding generator matrix G and the parity check matrix H. (2 points) ii) Generate the codeword for the following message: 1001. (2 points) iii) Determine whether the received words 0111100, 1000111 are valid codewords using the syndrome decoding and correct if necessary. Decode the codewords to recover the original messages. (4 points) iv) Calculate the bit error rate (BER) for the above Hamming code in a forward error correction (FEC) system if a binary symmetric channel (BSC) channel with a crossover probability p = 0.001 is used. (6 points) v) Calculate the channel capacity and compare it with the basic rate of the above Hamming code. (4 points)
Question 1: Linear Block Code (20 marks) Consider a (6,3) linear block code C with generator matrix G = [P: I3], where I3 = [1 0 0; 0 1 0; 0 0 1] and P is a 3 x 3 matrix. Given three codewords: C1 = 110110; C2 = 011011; C3 = 000111. (a) [5 marks] Based on the above given linear block codewords, find the generator matrix G. (b) [2 marks] Determine the parity check matrix H of C. (c) [5 marks] Use pictorial representation to decode the codeword r = [1 1 1 1 1 0]. (d) [8 marks] Construct the following table: Syndrome | Error Pattern
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