Second and Third Order Modulation Terms in the Distortion Produced when Modulated FM Waves are Filtered

When an angle-modulated wave (FM or PM) passes through a filter, the signal becomes distorted. For a multichannel system this distortion may produce crosstalk. In many practical cases the second and third order modulation terms give a good measure of the distortion. These terms have been studied by a number of investigators. In this paper we obtain some general expressions for them for the case in which the modulation is gaussian. Our main results includes all of the second and third order modulation products. In this respect, it is more general than some of the earlier expressions for the distortion (see Medhurst, 3 Magnusson, 4 and Liou 5 ). However, it does not give higher order modulation terms, some of which appear in earlier "first order" approximations. A first order approximation (similar to the earlier ones) occurs in our derivation of the main result. It is stated, along with the main result, in Section III. As in Ref. 1, the complex form of the filter input is s(t) = exp [iu0t + v.p(t)] (1) where the carrier frequency is w 0 = 2irf0 and the signal is carried by the real input phase angle tp(t). Let the filter have the transfer function G(j) and the impulse response g(t): (2) G(f) = Q( ) exp (-iwt) dt, l