Question

1) Handout problem 7.7. (Hint: Write down the expression for the average probability of error P,, which will involve Po,P1,fo 1),。。), and the threshold λ, and then minimize P, by setting dP/άλ-O. Use the Fundamental Theorem of Calculus to compute dP/d.) Problem 7.7 In a binary PCM system, symbols 0 and 1 have a priori probabilities Po and p1, respectively. The conditional probability density function of the ran- dom variable Y (with sample value y) obtained by sampling the matched filter BASEBAND PULSE TRANSMISSION 468 output in the receiver of Fig. 7.4 at the end of a signaling interval, given that symbol 0 was transmitted, is denoted by fy (yl0). Similarly, fy()11) denotes the conditional probability density function of Y given that symbol 1 was transmitted. Let λ denote the threshold used in the receiver, so that if the sample value y exceeds λ, the receiver decides in favor of symbol 1; otherwise, it decides in favor of symbol 0. Show that the optimum threshold Nopt for which the average prob- ability of error is a minimum, is given by the solution of 0

Answer 1

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