題組內容

2. (Total=18%) A pulse communication system is being designed that will transmit one of two messages, mi and m₂, which occur with equal a priori probability. These messages will be signaled using the following waveforms, and we would like to select sets of transmitted signals that will either minimize the average transmitted energy or will minimize the probability of error. Assume that the channel is corrupted by an additive white Gaussian random process. Assume here that the inner product of two real-valued function of duration T = 2 is defined as

Four possible signals, s_A(t), sB(t), sc(t), and s_D(t) are being considered, and two of them will be selected as the signal set that will indicate which message had been sent. The signals are sketched in Fig. 2.

(a) (4%) Compute a set of orthonormal basis functionsusing the Gram-Schmidt procedure, whereis proportional to sA(r). The signals s_A(X), s_B(t), s_c(t), and s_D(t) can be expressed as a linear combination of these basis functions as

where the basis functions are

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題組內容

(a) (4%) Compute a set of orthonormal basis functionsusing the Gram-Schmidt procedure, whereis proportional to sA(r). The signals s_A(X), s_B(t), s_c(t), and s_D(t) can be expressed as a linear combination of these basis functions as

where the basis functions are

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