The amplitude distribution and false-alarm rate of filtered noise

A digital computer has been used to simulate the passage of white Gaussian noise through a narrow-band filter followed by a square-law envelope detector and low-pass postdetection filter (PDF). The amplitude distribution and false-alarm rate at the PDF output were computed at many different voltage levels for a wide range of values of the product of predetection bandwidth times postdetection time constants (Bτ
0
). Data were also obtained for the output of a linear envelope detector without postdetection filtering. The gamma probability distribution was found to be in excellent agreement with the PDF-output amplitude probability statistics for square-law detection. The relationship between the gamma distribution parameter and Bτ
0
was determined. The ratio of false-alarm rate to amplitude probability density was found to be closely approximated by the theoretical mathematical relationship for a Gaussian process for all except low values of Bτ
0
. Curves of amplitude probability density (Bτ
0
= 0 to ∞) and normalized false-alarm rate (Bτ
0
= 0 to 1000) vs threshold voltage level are provided for both single and double-section postdetection filtering. This paper thus makes available for engineering design and analysis numerical data which describe extensively the characteristics of detector and PDF noise outputs. The relationship between normalized false-alarm rate, threshold voltage, and Bτ
0
has, in particular, wide application in signal-detection problems. These data have neither been previously published nor are they easily derived from previous theoretical studies.