Distortion and crosstalk of linearly filtered, angle-modulated signals

An important problem in the theory and practice of receiving angle-modulated signals is the design of the filtering elements which must be employed. It has long been known that filtering introduces distortion and crosstalk into the signal. However, the computation of these effects is difficult. The methods customarily used employ approximations of one kind or another, and the equations used do not apply to all cases of practical interest. Here formulas are presented which enlarge somewhat the domain of cases amenable to calculation. In this analysis, an angle-modulated signal having an arbitrary phase function is applied to a general linear filter, and the phase of the output is expanded in a series having the linearly filtered input as the leading term. The expansion is then specialized to the case of a narrowband signal applied to a narrow, symmetrical, bandpass filter. A spectral analysis is performed by assuming a Gaussian input phase and examining terms through fifth order in the output phase expansion. This leads to the main results of the paper, namely expressions for the leading terms in the output spectrum. It is argued that these terms represent the principal contribution in the case where the distortion is small. To demonstrate their application to a practical problem, the formulas are used to calculate the distortion and crosstalk produced when an FM signal, having a flat baseband spectrum, is passed through a single-pole filter. This example is of some current interest because such a filter has been employed in the forward path of a feedback FM receiver used for satellite communication. A number of cases are considered, and the results of the computations are plotted.