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11.5. Using the generator matrix given in Eq. (11.3) find the codewords for the datawords (a) ; (b) ; and (c) . 11.6. Using the parity check matrix of Eq. (11.5) find the syndrome for the codeword (1110100). Comment on this. 11.7. Using the parity check matrix of Eq. (11.5) find the syndrome for the codeword (1000110). Comment on this. 11.8. A received codeword is 1011000. Determine, using the parity check matrix of Eq. (11.5) if this is a valid codeword, and if not, write out the error vector on the assumption that only one error is present. 11.9. Calculate the code rate for a Hamming (31, 26) code.
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11.10. Calculate the code rate and the number of errors that can be corrected with a BCH (63, 36) code. 11.11. An R-S code is byte oriented with k 8. Given that there are eight redundant symbols, calculate the number of symbol errors that can be corrected. 11.12. Determine the values for N and K for the shortened R-S codes used in (a) DirecTV and (b) DVB. 11.13. Describe how convolution coding is achieved. State some of the main advantages and disadvantages of this type of code compared with block codes. 11.14. Explain what is meant by interleaving when applied to error control coding and why this might be used. 11.15. Explain what is meant by concatenated codes and why these might be used. 11.16. Explain what is meant by a FEC code. FEC coding at a code rate of 3/4 is used in a digital system. Given that the message bit rate is 1.544 Mb/s, calculate the transmission rate. 11.17. The bit rate for a baseband signal is 1.544 Mb/s, and FEC at a code rate of 7/8 is applied before the signal is used to modulate the carrier. Given that the system uses raised-cosine filtering with a rolloff factor of 0.2, determine the bandwidth required for (a) BPSK, and (b) QPSK. 11.18. A BPSK signal provides an [Eb/N0] of 9 dB at the receiver. Calculate the probability of bit error.
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11.19. For the signal in Prob. 11.18, calculate the new value of bit error probability if FEC is applied at a code rate of 3/4, given that the carrier power remains unchanged. 11.20. Derive Eq. (11.11). 11.21. Explain what is meant by coding gain as applied to error correcting coding. When FEC coding is used on a digital link, a coding gain of 3 dB is achieved for the same BER as the uncoded case. What decibel reduction in transmitted carrier power does this imply 11.22. A certain (15, 11) block code is capable of correcting one error at most. Given that this is a perfect code (see Sec. 11.8), plot, on the same set of axes, the BER for the coded and uncoded cases for an [Eb/N0] range of 2 to 12 dB. Calculate the coding gain at a BER of 10 6. 11.23. State briefly the difference between hard and soft decision decoding. Following the description given in Sec. 11.9, determine the output produced by (a) hard decision and (b) soft decision decoding when the sampled signal from the demodulator is 0.4 V, 0.85 V, and 0.4 V for triple redundancy coding. 11.24. From Fig. 11.10 find the minimum [Eb/N0] as determined by the Shannon limit curve. Explain the significance of this. 11.25. For equiprobable bit transmission a received bit level is 0.55V. Assuming this is normalized where 1V represents a certainty of the bit being a 1, calculate the LLR. 11.26. Referring to Eq. (11.26), identify the parity equations which contain code bit c7. 11.27. Complete the Tanner graph of Fig. 11.13 for all the parity equations given in Eq. 11.26.
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1 11.28. For the parity equation for row 5, (Eq. 11.26) the probabilities are: p5 1 1 0.7, p8 0.6, p14 0.3. Calculate the estimated probability p2.
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11.29. Research the literature and write brief comparative notes on the use of turbo codes and LDPC codes in satellite communications. 11.30. A (31, 6) block code is used in an ARQ scheme. Determine the upper bound on the probability of bit error.
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