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4. Let X1, X2, ,XM represent a random sample from an exponential distribution with parameter 01 and let Y1, Y2, YN represent a random sample from an exponential distribution with parameter Q2. Assume that Xi are independent of Yj. (a) Find the generalized likelihood ratio test (GLRT) for testing H0: 01 = Q2 versus H1: 01 # Q2. (b) Show that the GLRT can be expressed as in terms of the statistic T = 5. A five-signal configuration in a two-dimensional signal space uses the following waveforms: S1(t) = 0 S2(t) = 241(t) + Z42(t) S3(t) = 1241 (t) S4(t) = S5(t) = - where P1 (t) and 42(t) are orthonormal basis signals. Assume that all the signals are equally probable. (a) Represent these signals in the signal space and determine the optimum decision regions. (b) Sketch the optimum receiver in additive white Gaussian noise (AWGN). (c) Derive an expression for the probability of error for the optimum receiver.

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