In this chapter we consider a communication system that transmits a single bit of information using one of two signals. The receiver filters the received signal, samples the filter output, and then makes a decision about which of the two signals was transmitted. We first consider an example in which the two signals are just rectangular pulses with opposite sign. For those signals in additive white Gaussian noise (AWGN) we analyze the probability of error for a receiver that uses a filter matched to the transmitted signal. Second, we consider optimizing the system over all possible filters, signals, and decision rules. The optimal filter and signals are derived for binary modulation in which one of two signals is transmitted. Finally, the effect of imperfect receivers is considered. Approaches to analyzing a system with intersymbol interference (ISI) are discussed.
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