been shown (Betts, 1970; Brown and Glazier, 1974; to retransmit pulses of either polarity. Schwartz, 1970; Taub and Schilling, 1986) that the impulse re- sponse of the required filter is given by Equation 19.34, where T is the duration of the signal. 19.2.3 Optimal detection h(1)=s,(T-1) (19.34) In a digital transmission system, the receiver attempts, at the re- quired instants of time, to determine whether a pulse is present that represents valid information or just noise. Optimum strategies for The impulse response of the matched filter is thus the signal doing this have been developed. waveform with time reversed. It must be zero for T, at which time If a digital signal uses symbols S (t), 2(t) (t) S (t), each of a decision is made as to whether S (t) was present. duration T, these should ideally be orthogonal; i.e. there should be The output voltage, V (t), due to the signal S (t) is given by its no correlation between them. This can be expressed mathematically convolution with h(t) as in Equation 19.35. as in Equation 19.33, where Q is a cross correlation function and (19.35) Qrr is an autocorrelation function. Digital modulation of a carrier 19/7 Since this integral is zero outside the interval for which S (t) and transmitted power. Therefore, S = S./N; and the error prob- S exist, Equation 19.35 corresponds to the autocorrelation ability is given by Equation 19.27, where S/N is the input sig- function Q of S (t). Consequently, the matched filter is equivalent nal/noise ratio of the coherent demodulator. to a correlation detector. The output voltage from a coherent demodulator is proportional Since the impulse response of the matched filter is the time inverse to cos 0, where 0 is the phase angle between the incoming carrier of the input signal, if the latter is symmetrical in time, both are of and the local carrier. Thus, ideal demodulation depends on keeping the same shape. For example, if the signal is a pulse of (sin x)/x the local carrier in phase with the incoming carrier at all times. This is not obt