6. (15 points) Suppose a transmitter sends S be a random variable defined as S =...

  

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6. (15 points) Suppose a transmitter sends S be a random variable defined as S = -1, w.p. 0, w.p. (+1, w.p. The receiver has a noise observation Y = S+Z where Z is a Laplacian noise with pdf of fz(z) = e-^|| for assume that S and Z are statistically independent. 18 < < . We (a) (5 points) Find fy|s(y|s) for s = -1,0,1. Sketch the conditional pdfs on the same graph. (b) (5 points) Find the optimal decoding rule for classify the signal whether S is -1, 0 or +1. Give your answer in terms of ranges of values of Y. (c) (5 points) Find the probability of decoding error in terms of A. 0.5 0.4 0.3 0.2 0.1 Figure 2: fys(y, s) for X = 1 sylo) sly) s-1) 6. (15 points) Suppose a transmitter sends S be a random variable defined as S = -1, w.p. 0, w.p. (+1, w.p. The receiver has a noise observation Y = S+Z where Z is a Laplacian noise with pdf of fz(z) = e-^|| for assume that S and Z are statistically independent. 18 < < . We (a) (5 points) Find fy|s(y|s) for s = -1,0,1. Sketch the conditional pdfs on the same graph. (b) (5 points) Find the optimal decoding rule for classify the signal whether S is -1, 0 or +1. Give your answer in terms of ranges of values of Y. (c) (5 points) Find the probability of decoding error in terms of A. 0.5 0.4 0.3 0.2 0.1 Figure 2: fys(y, s) for X = 1 sylo) sly) s-1)