Zero-crossing rate for the sum of two sinusoids or a signal plus noise (Corresp.)

For the sum of two sinusoids of incommensurable frequencies, the number of zero-crossings per second
Z
is calculated. By averaging this result over a Rayleigh distribution for the amplitude of one of the sinusoids, Rice's formula is obtained for the number
z
of zero-crossings per second of a sinusoid plus Ganssian noise of any spectrum. Simple approximations are obtained for
z
in the high-signal-to-noise-ratio and narrow-band cases. In addition, alternative derivations of
Z
and
z
are presented that further illuminate the significance of the foregoing results.