The expected number of peaks/sec. (nσ+N), caused by noise or by a signal and noise, in excess of a given amplitude, or triggering level, is determined for broad‐band noise (a) and for the envelope of a narrow‐band disturbance (b). The signals considered here are pulses of various shapes (rectangular, triangular, trapezoidal), or a carrier modulated by these pulses. The expected number of peaks per second is directly proportional to the band width of the noise, and depends in some degree (20–30 percent) on filter shape, the gaussian response yielding a larger value of nσ+N than the rectangular, for the same total input power. Further, nσ+N is greatest when the triggering level is the most probable amplitude. Pulse shape is not significant when the pulses are on only a small fraction of the total period of observation, but becomes so if (1) the time of observation is comparable to the pulse length, and (2) the pulses have sharp leading edges. The latter gives rise to a transient effect which is not ignorable. This transient contribution does not appear in the envelope of narrow‐band waves, as long as the maximum time‐rate of change of the wave form of the envelope is much less than the central or resonant frequency of the IF, i.e., as long as the concept of ``envelope'' is meaningful. Rectification does not change nσ+N, provided that there is no appreciable distortion of the envelope by video‐ or audio‐filters and that the triggering level of the input is passed by the detector.
Improvement, defined in ways appropriate to the systems in question, of one circuit which can tolerate A1 spurious signals/sec. in excess of a given triggering level over another circuit, which can tolerate only A2 such signals (A1>A2), is calculated for broad‐band signal and noise, and for narrow‐band waves following half‐wave linear or quadratic rectification. In general, the improvement is slight (a few db) if the trigger levels of the two circuits are not close to the most probable value of the amplitude and if the ratio of the number of spurious signals allowed/sec. to the band width of the disturbance is small. Significant improvement follows only when the maximum tolerable number of triggering peaks/sec. is the same order of magnitude as the band width of the noise. Tables and figures illustrating the above are included in Part I; the mathematical derivations are contained in Part II.
