DocumentCode :
905361
Title :
FM reception and the zeros of narrow-band Gaussian noise
Author :
Blachman, Nelson M.
Volume :
10
Issue :
3
fYear :
1964
fDate :
7/1/1964 12:00:00 AM
Firstpage :
235
Lastpage :
241
Abstract :
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 {\\pi^{2}(1 - exp - A^{2}/2\\sigma^{2})^{2}D^{2}_{rms} \\over W \\int_{0}^{\\infty}{\\psi^{\\prime 2}(\\tau ) \\over \\sigma^{4} - \\psi^{2}(\\tau )} exp - {A^{2} \\over \\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.
Keywords :
FM modulation/demodulation; Gaussian processes; Additive noise; Bandwidth; Frequency; Gaussian noise; Gaussian processes; Low-frequency noise; Narrowband; Random processes; Reactive power; Signal to noise ratio;
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/TIT.1964.1053674
Filename :
1053674
Link To Document :
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