On the Distributions of Signals and Noise after Rectification and Filtering

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Abstract
The probability distributions for broad- and narrow-band signals and normal random noise, following a square-law rectifier and a video (or audio) filter of arbitrary width, are examined. The present approach is based on a method originally used by Kac and Seigert [J. Appl. Phys. 18, 383 (1949)] for noise alone (after quadratic rectification but no filter) in the case of the nth-order distribution (n≥2). The procedure requires an appropriate transformation to express the output waveform in terms of the input disturbance; the statistics of the output are now determined by a suitable transformation with respect to the original, normal statistics. Explicit solutions are obtained for an integral equation involving the autocorrelation function of the noise, the weighting function of the video (or audio) filter, and an orthonormal set of eigenfunctions. For noise alone only the eigenvalues need be determined, but for a signal and noise it is necessary to know the eigenfunctions as well. Among the new results are (1), the calculation of the general, nth-order characteristic function for the filtered output following quadratic rectification for an input noise and a signal and noise; (2), a discussion of the second-order probability density W2 in this case; (3), some examples of practical interest; and (4), the limiting cases of broad and narrow post-detection filters (vis-à-vis the predetection filter), with particular attention to approximations to W2 for signal and noise for narrow videos and an improved approximation for W1 for noise alone, as well.