ON THE FREQUENCY FILTERING OF TRANSIENT NOISE SIGNALS
NAVAL ORDNANCE LAB WHITE OAK MD
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A class of transient random signals is modeled as the product of a deterministic, square integrable envelope function and a Gaussian random process having a well-defined power spectrum. The passage of an ensemble of such random signals through a linear filter is studied with particular emphasis on the mean and variance of the total output energy. It is found that an important role is played in these considerations by the covariance function between values of the energy density spectrum of a sample function evaluated at different frequency arguments. Accordingly, the form of this function is derived and portrayed as a surface lying above a two-dimensional frequency plane. Examples of these spectral covariance surfaces are presented and discussed for both rectangular and decaying exponential pulses of both broad and narrow band Gaussian noise, and their general characteristics are identified. Finally, the problem of idealized narrow band filtering is specifically approached and approximate expressions derived for the mean, variance, and normalized standard deviation of the output energy of a narrow band filter excited by rectangular pulses of narrow band Gaussian noise. The implications of these findings for spectral analysis and monopulse signal processing are discussed in the light of this uncertainty principle and the limitations it imposes on the simultaneous precision of frequency resolution and spectral amplitude.
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