Channel Coding I Exercises â WS 2012/2013 â - UniversitÃ¤t Bremen

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4 CONVOLUTION CODES November 7, 2012 12 a) The generator polynomial g 0 (D) of the code from exercise 4.3 shall now be used for the feedback of a recursive systematic convolution code. Determine the IOWEF of the code and again draw them separately for even and odd input weights. b) Now alternatively use the polynomial g 1 (D) for feedback. Again determine the IOWEF and determine for both RSC-codes the coefficients a d and c d . Compare them with those of the nonrecursive code. c) Estimate the bit error rates with the help of the Union Bound for both RSC-codes and draw them together with the error rate of the non-recursive convolution code in a diagram. Which differences are striking? 4.4 Puncturing of convolution codes of the rate1/n Exercise 4.5 Puncturing of convolution codes* a) The non-recursive convolution code from exercise 4.3 shall now be punctured to the code rate R c = 2/3. Give several puncturing patterns and find out, if the puncturing results in a catastrophic code with the routine tkatastroph. b) Determine for the suitable puncturing schemes from item a) the coefficients a d and c d and oppose them to those of the unpunctured code. Note, that the routine iowef block conv doesn’t consider different starting instants of time and these therefore have to be entered manually (P 1a = [31] T and P 1b = [13] T yield different results). c) Calculate an estimation of the bit error rate for the different puncturing patterns from item b) with the routine pb unionbound awgn and draw it together with that of the unpunctured code in a diagram. Hand over a mean IOWEF of the under b) determined spectra to the routine. d) Now puncture the code to the rate R c = 3/4 and consider the results from item a). Check if the code is catastrophic. e) For reaching higher code rates than 2/3 with this code, the RSC-code from exercise 4.4b) shall now be punctured, where the systematic information bits are always transmitted. Determine for the code rates R c = 3/4, R c = 4/5 and R c = 5/6 the puncturing schemes and the resulting bit error rates for the AWGN-channel. Sketch them together with those of the unpunctured code and of the code that is punctured to the rate R c = 2/3.
4 CONVOLUTION CODES November 7, 2012 13 4.5 Optimal decoding with Viterbi-algorithm Exercise 4.6 Viterbi-decoding Given is a convolution code withg 0 (D) = 1+D+D 2 and g 1 (D) = 1+D 2 , where a terminated code shall be used. a) Generate the corresponding Trellis and code the information sequence u(l) = (1101). b) Conduct the Viterbi-decoding respectively for the transmitted code sequencex = (110101001011) and for the two disturbed receiving sequencesy 1 = (111101011011) andy 2 = (111110011011) and describe the differences. c) Check the results with the help of a MATLAB-program. Define the convolution code with G=[7;5], r flag=0 and term=1, generate the trellis diagram with trellis = make trellis(G,r flag) and sketch it with show trellis(trellis). Encode the information sequence u with c = conv encoder (u,G,r flag,term) and decode this sequence with viterbi omnip(c,trellis,r flag,term,length(c)/n,1). Decode now the sequences y 1 and y 2. Exercise 4.7 Viterbi-decoding with puncturing Given is a convolution code with g 0 (D) = 1+D+D 2 and g 1 (D) = 1 + D 2 , out of which shall be generated a punctured code by puncturing with the scheme ( ) 1 1 1 0 P 1 = 1 0 0 1 a) Determine the code rate of the punctured code. b) Conduct the Viterbi-decoding in the case of the undisturbed receiving sequence y = (11000101) (pay attention to the puncturing!). Exercise 4.8 Coding and decoding of a RSC code a) We now consider the recursive systematic convolution code with the generator polynomials˜g 0 (D) = 1 and ˜g 1 (D) = (1+D+D 3 )/(1+D+D 2 +D 3 ). Generate the Trellis diagram of the code with the help of the MATLAB-command make trellis([11;15],2). b) Conduct a coding for the input sequence u(l) = (11011). The coder shall be conducted to the zero state by adding tail bits (command conv encoder (u,[11;15],2,1)). What are the output- and the state sequences? c) Now the decoding of convolution codes with the help of the Viterbi-algorithm shall be considered. For the present the sequence u(l) = (1 10110011110) has to be coded (with termination), modulated with BPSK and subsequently superposed by Gaussian distributed noise. Choose a signal-to-noise ratio of E b /N 0 = 4 dB. With the help of the program viterbi the decoding now can be made. By setting the parameter demo=1 the sequence of the decoding is represented graphically by means of the Trellis diagram. What’s striking concerning the error distribution if the Trellis is considered as not terminated?
Page 1 and 2: Channel Coding I Exercises - WS 201
Page 3 and 4: November 7, 2012 II Exercise 3.16:
Page 5 and 6: 1 INTRODUCTION November 7, 2012 1 1
Page 7 and 8: 2 INFORMATION THEORY November 7, 20
Page 9 and 10: 3 LINEAR BLOCK CODES November 7, 20
Page 15: 4 CONVOLUTION CODES November 7, 201

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