The Spectrum of Frequency-modulated Waves after Reception in Random Noise—ii* By

In our earlier paper [l]f expressions for the spectrum of an FM wave received in the presence of random (fluctuation) noise were obtained in the two extreme cases of no limiting and "super"-limiting.** It is the purpose here to develop the completelygeneral theory for the demodulated wave, taking into account the effects of arbitrary amounts of limiting on the noise and signal spectra and power of the low-frequency output of the receiver. We are interested in the spectrum because the signal-to-noise ratio determined at the output depends noticeably on the spectral distribution after demodulation when broad-band FM (in which the IF filter width is large compared with the audio response) is used. In the case of narrow-band FM (where the IF width is comparable with the audio) we are concerned mainly with the integrated spectrum or power output, so that spectral shape is not so significant (cf. [2]). The general theory is outlined in Part II, including an examination of important special cases, and a detailed study of noise alone is given in Part III; the Gaussian filter amplitude response is assumed. Remarks on signal and noise are contained in Part IV; applications to evaluation of the signal-to-noise ratio are discussed more fully in [2]. A satisfactory analytical model of the actual nonlinear elements in the receiver— the limiter followed by the discriminator—can be constructed if we assume: (1) that the physical discriminator is replaced by an "ideal" one which responds everywhere linearly with frequency; accordingly, the output current (or voltage) is directly proportional to the instantaneous difference frequency between the wave and the central or resonant frequency of the (symmetrical) IF, limiter, and discriminator bands; (2) that the filter response of the limiter is taken to be wide enough to pass the IF portion of the limited signal and noise without distortion due to frequency selection. In practise this means that a limiter band width several times the IF spread is needed; it can easily be obtained, since the limiter circuit is of necessity a low Q device. (If it were not, filtering would restore randomness to the noise, and the limiting would be nullified.) The case of the discriminator's response is more critical, but if the linear portion of the actual characteristic is at least twice the r-m-s frequency deviation, distortion will not be serious, and our idealized model is then a satisfactory substitute (cf. [3, Chs. 4 and 5] for a treatment of an actual discriminator when there is no limiting).