R.F. spectra of waves frequency modulated with white noise

Although comprehensive sets of curves are available giving the radio-frequency spectra of waves phase modulated with band-limited white noise (simulating f.d.m. telephony signals), the corresponding problem for frequency modulation turns out to be much more intractable. In the present paper the frequency-modulation problem is attacked using a method employed in an earlier paper, in conjunction with numerical devices for improving the convergence rate of very slowly converging series. The results are presented as a set of curves covering the range of modulation parameters likely to be encountered in trunk radio systems. This information is necessary for the evaluation of system bandwidth and interference due to unwanted carriers. To bring the analysis within reasonable bounds, the minimum modulating frequency has been taken as zero. It is known that unless the ratio between r.m.s. frequency deviation and maximum modulating frequency is large, the shape of the central portion of the spectrum changes substantially as the minimum modulating frequency departs from zero. However, for practical ranges of parameters there is evidence indicating that the shape of the spectrum tails (which is of particular importance in connection with distortion problems) is not greatly sensitive to variations in the minimum modulating frequency.