2. [35pt] We have a communication system with a binary PAM scheme over a zero mean...

 

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2. [35pt] We have a communication system with a binary PAM scheme over a zero mean additive white Gaussian noise channel with the power spectral density of No/2. Consider a single pulse problem for 0t2 in which the transmitted waveform s(t) is determined by the following rule: 1, 0t <1 s(t)= Ah(t), 3Ah(t), 012 if 1 is sent 012 if 0 is sent where h(t)=-1, 112 0, otherwise The source data 1 and 0 occur with the equal probability. The detector is composed of a matched filter g(t), a sampler and a decision device. (a) Determine and draw the matched filter g(1) for 012 in the detector, where for 18 (1) | dt =1. (b) Determine the average energy per bit in terms of A. (c) Let Y denote the output of the sampler, which is a Gaussian random variable. Determine the mean and the variance of Y when the source data is 1 and 0, respectively. (d) Express the probability of bit error as a function of E/N You may use Q(x), where Q(x)= 2 dt. 2. [35pt] We have a communication system with a binary PAM scheme over a zero mean additive white Gaussian noise channel with the power spectral density of No/2. Consider a single pulse problem for 0t2 in which the transmitted waveform s(t) is determined by the following rule: 1, 0t <1 s(t)= Ah(t), 3Ah(t), 012 if 1 is sent 012 if 0 is sent where h(t)=-1, 112 0, otherwise The source data 1 and 0 occur with the equal probability. The detector is composed of a matched filter g(t), a sampler and a decision device. (a) Determine and draw the matched filter g(1) for 012 in the detector, where for 18 (1) | dt =1. (b) Determine the average energy per bit in terms of A. (c) Let Y denote the output of the sampler, which is a Gaussian random variable. Determine the mean and the variance of Y when the source data is 1 and 0, respectively. (d) Express the probability of bit error as a function of E/N You may use Q(x), where Q(x)= 2 dt.