By Mischa Schwartz
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Application to Binary Transmission The probability distributions of the envelope of narrowband noise with or without a sinusoidal carrier again can be used to calculate the probability of error due to noise when transmitting binary information. Signal phase coherence is not necessary in this case, and the receiver requires only a simple envelope detector following the i-f strip. This contrasts with in-phase local carrier injection required to implement the binary system of Sec. 1-3. Here we assume the 0 and 1 symbols to be carried by on-off bursts of r-f power.
O. Rice and appears as Fig. 7, Sec. 3-10 in Rice's classic paper, previously cited. The solution to Eq. (1-5-1) provides the error curve of Fig. 1-5-2. Superimposed is the plot from Fig. 1-3-3 for the case of coherent binary 1 J. I. Marcum, A Statistical Theory of Target Detection by Pulsed Radar, IRE Tram. Inform. Theory, vol. IT-6, pp. 59-267, April, 1960. This is a reprint of an earlier Rand Corp. Memo RM-753, July, 1948, and RM-754, December, 1947. 2 J. I. Marcum, Tables of the Q-function, Rand Corp.
G(t - r ) Re [z(r)] dr [J-"'", g(t - r)z(r) dr ] = Re [J-"'", e;wIG(w)Z(w) df ] = Re = Re [fo'" e;wIG(w)Z(w) df] (1-8-9a) since Z(w) = 0, f < O. 211" = Re [e iwot co «-u (x) U(x) dX] co eiztH(x)U(x) -wo -00 dX] (1-8-9b) since Hand U = 0, f < - to, by hypothesis. But note that the integral in Eq. (1-8-9b) is just the Fourier-transform formulation of Eq. (1-8-4). We thus get, finally, (1-8-3) q(t) = Re [v(t)eiwot] with 1 vet) = 211" = foo_ co 1-'' ", h(t eiwtH(w) U(w) dw - r)u(r) dr (1-8-4) as was to be proven.