FM reception and the zeros of narrow-band Gaussian noise

If
x(t)
and
y(t)
are statistically independent stationary Gaussian random processes, each having correlation function
\psi(\tau)
, mean squared value
\delta^{2} = \psi(0)
, and spectral density
\Psi(f)
, then
u(t) = x(t) \cos 2\pi Ft - y(t) \sin 2 \pi Ft
is a stationary Gaussian random process with correlation function
\psi(\tau) \cos 2 \pi F \tau
and with spectral density
\frac{1}{2} \Psi(f-F) + \frac{1}{2} \Psi(f + F)
, symmetric about
F
for large
F
. From this representation of
u(t)
it is shown that the variance of the number of zeros of
u(t)
in the interval
(0, T)
is, for integral
2FT
,
\mbox{var} Z=\frac{1}{4}-\frac{1}{\pi^{2}}\arcsin^{2}\frac{\psi(T)}{\sigma^{2}}+\frac{2}{\pi^{2}}\int_{0}^{T} \frac{(T- \tau)\psi'^{2}(\tau)}{\sigma^{4}-\psi^{2}(\tau)}d \tau + O(1/F)
. This result complements that of Steinberg, {\em et al.}, giving var
Z
for wide-band Gaussian noise. The limit of
(var Z)/T
as
T \rightarrow \infty
is evaluated for several spectra, and expressions are found for the variance of the number of zeros of the sum of the foregoing narrowband noise plus a sinusoid of frequency
F
. From these results the low-frequency output spectral density of an FM receiver is obtained. Below the threshold the output signal-to-noise ratio is found to be
\frac{\pi^{2}(1-\exp-A^{2}/2\sigma^{2})^{2}D^{2}_{rms}}{W \int_{0}^{\infty} \frac{\psi"^{2}(\tau)}{\sigma^{4} - \psi^{2}(\tau)} \exp - \frac{A^{2}}{\sigma^{2}- \psi(\tau)}d \tau}
, where
A^{2}/2\sigma^{2}
is the input signal-to-noise ratio,
D_{rms}
is the rms frequency deviation, assumed small enough not to affect the output noise, and
W
is the output bandwidth, assumed small compared to the input bandwidth. By the addition of the well known "triangular" noise, this expression is made valid through and above the threshold, thus unifying various results of Rice. The quieting of a wide-band FM receiver by a signal is also considered.