Spectrum Occupancy of Pulsed FSK

For signals used as Time Division Multiple Access and Slow Frequency Hopping A-J waveforms, it is desirable to be able to predict the spectrum occupancy of pulses containing a number of angle modulated symbols. It is shown that if the angle modulated signal is random, but stationary in the wide sense, the power or energy spectrum is the convolution of the corresponding spectra of the angle modulated wave and of the amplitude pulse. Spectrum occupancy can be obtained by integrating the resulting spectrum. Using a 9845S desk computer, spectrum occupancy has been estimated for FSK waves having peak to peak deviation ratios of 0.5 (MSK) and 0.7 (approximately optimum for limiter-discriminator). The modulating pulse may be a square wave of duration equal to from one to sixteen FSK symbols, or such a pulse passed through a Butterworth filter (with 2, 4 or 6 poles) or an ideal Gaussian filter. The filtered pulses have a one symbol rise time. It is concluded that square wave modulation should be avoided unless the duration is substantially longer than 16 bits. The simple 2-pole Butterworth filter confines the spectrum about as well as the ideal Gaussian, and much better than the square pulse. Pulsed FSK signals should be provided with a rise (and fall) time of the order of one modulation period. The amplitude variation should be continuous from zero to maximum, and probably the first derivative of the variation should also be continuous.