Spectral density and autocorrelation functions associated with binary frequency-shift keying

General equations are derived for the spectral density and autocorrelation functions of a wave train consisting of sine-wave segments with constant amplitude. The frequency of a segment may be either f 1 or f 2 . At regularly spaced intervals the frequency is switched or not switched according to a random choice. This type of wave occurs when a random series of marks and spaces is sent by frequency-shift keying. The results fall into two main classes — namely, that of discontinuous phase at the transitions, which is the typical situation in switching between two independent oscillators; and that of continuous phase at the transitions, which is more usually applicable when the frequency of a single oscillator is changed. Individual treatment is given of the various special cases which arise when integral relationships between the marking, spacing, signaling, and shift frequencies exist. No restriction is made on the relative magnitudes of the different frequencies involved.