A Derivation of the Spectra of N-Ary Orthogonal Comtinuous-Phase FSK Waveforms for ELF/VLF Communications

The power spectral density of continuous-phase minimum-bandwidth waveforms composed from a set of
N
mutuallyorthogonal chips (pulsed sinusoids of equal amplitude and duration and constant frequency) is derived. The spectrum and its derivation for this class of signals are of particular interest in highpower extremely low frequency/very low frequency (ELF/VLF) communications where noise immunity is a prime consideration, but the signal bandwidth and transients at the transmitting antenna must be minimized. Exact formulas are also given without derivation for wider bandwidth signals in the same class but of less practical importance. Although the power spectra of more general binary signals are already known, the special case presented here permits a much simpler derivation than that given in an earlier paper [1]. In the
N
-ary case a previously derived formula for the spectra of continuous-phase multilevel FM signals [2] is augmented and evaluated to explicitly yield new spectral formulas useful in describing the frequency-hopping mode.