SECOND AND THIRD ORDER MODULATION TERMS IN DISTORTION PRODUCED WHEN NOISE MODULATED FM WAVES ARE FILTERED

This paper is concerned with the distortion produced in a frequency modulation wave when it passes through a filter. The phase or frequency modulation representing the signal is assumed to be a band of gaussian noise. The main result is an expression for the power spectrum Wθ of the output phase angle θ(t). This expression holds for any filter, contains all of the distortion terms due to second and third order modulation, and is suited to computer evaluation. It is useful in many cases, but it has the shortcoming of not containing any modulation terms higher than the third order. A second result is an approximation to Wθ(f), based on log (1 + x) ∼ x {that is, a “first‐order” approximation), which is encountered in the derivation of the main result. Although it does not contain all of the second and third order modulation terms, it does contain higher order modulation terms which may give most of the distortion in some cases. The results given here are compared with those obtained earlier. © 1969 The Bell System Technical Journal