Bandwidth of a sinusoidal carrier wave, frequency-modulated by a rectangular wave with half-sine-wave build-up

The problem of determining the frequency spectrum of a sinusoidal carrier, frequency modulated by a modified rectangular wave, is considered and a general solution obtained. The modification of the rectangular wave consists in replacing the vertical sides by a non-instantaneous transition in the form of a half-cycle sine wave. It is shown that for the determination of bandwidth it is reasonable to use, instead of the spectrum proper, a function which forms an envelope to it; this function is simpler and, in particular, it is independent of the on/off ratio of the rectangular wave. curves are given for various values of the paramenters involved.