Unformatted text preview: Digital Modulation 1 - Problem Set 6
Problem 1 (MPSK)
1Â· Derive an upper bound for the symbol error probability of 8PSK by using the union bound and assuming that X = s1 was transmitted. a) Determine the squared Euclidean distance between the transmitted symbol s1 and the other symbols sm , m = 1. b) For each sm , m = 1, write the symbol error probability Ë† P (X = sm |X = s1 ) = P (Y âˆˆ Dm |X = s1 ) using the Q-function (similar to BPSK). c) Use the result from b) to express the union bound for the symbol error probability. Hint: The union bound for P (Y âˆˆ D1 |X = s1 ) is P (Y âˆˆ D1 |X = s1 ) â‰¤
i=1 P (Y âˆˆ Di |X = s1 ). d) Why is the union bound an upper bound? 2Â· Derive an approximation of the bit error probability of MPSK for high values of Eb when Gray codes are used. N0 a) Determine a simple union bound for the symbol error probability of MPSK. b) Take into account the Gray encoding and relate Pb and Ps . 3Â· Simulate the BER of 8PSK versus obtained in 2.
Eb N0 and compare it with the approximation Problem 2 (QPSK)
The symbol error probability of QPSK reads Ps = 2Q Es N0 âˆ’ Q2 Es . N0 Relate the two summands in the right-hand expression to the conditional probabilities that Y lies in some sub-regions outside the decision region of the transmitted symbol. BFl, TPe. Digital Modulation 1 NavCom ...
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