FM reception and the zeros of narrow-band Gaussian noise
Nelson M. Blachman · IEEE Transactions on Information Theory · 1964
Ifx(t)andy(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), thenu(t) = x(t) \cos 2\pi Ft - y(t) \sin 2 \pi Ftis a stationary Gaussian random process with correlation function\psi(\tau) \cos 2 \pi F \tauand with spectral density\frac{1}{2} \Psi(f-F) + \frac{1}{2} \Psi(f + F), symmetric aboutFfor largeF. From this representation ofu(t)it is shown that the variance of the number of zeros ofu(t)in the interval(0, T)is, for integral2FT,\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 varZfor wide-band Gaussian noise. The limit of(var Z)/TasT \rightarrow \inftyis 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 frequencyF. 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}, whereA^{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, andWis 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.