Rice's formula^1for the "envelope" of a given signal is very cumbersome; in any case where the signal is not a single sine wave, the analytical use and explicit calculation of the envelope is practically prohibitive. A different formula for the envelope is given herein which is much simpler and easier to handle analytically. We show precisely that if\hat{u}(t)is the Hilbert transform ofu(t), then Rice's envelope ofu(t)is the absolute value of the complex-valued functionu(t) + i \hat{u}(t). The functionu + i\hat{u}is called the pre-envelope ofuand is shown to be involved implicitly in some other usual engineering practices. The Hilbert transform\hat{u}is then studied; it is shown that\hat{u}has the same power spectrum asuand is uncorrelated withuat the same time instant. Further, the autocorrelation of the pre-envelope ofuis twice the pre-envelope of the autocorrelation ofu. By using the pre-envelope, the envelope of the output of a linear filter is easily calculated, and this is used to compute the first probability density for the envelope of the output of an arbitrary linear filter when the input is an arbitrary signal plus Gaussian noise. An application of pre-envelopes to the frequency modulation of an arbitrary waveform by another arbitrary waveform is also given.

A method for automatically controlling the threshold bias in a detector is described and analyzed. In Section I, the threshold bias problem is described: Bias is set for a constant false alarm rate (or a constant false alarm time). By "standard biasing" is meant the common practice of adjusting the required bias under the assumption that the noise is Gaussian and has a flat power spectrum. In Section II, the error that is made by standard biasing, if it turns out the Gaussian noise does not have a flat power spectrum, is given. In Section III, the automatic biasing method is given in the case where a constant false alarm time is required (its operation to maintain a constant false alarm rate is analogous). The device envisioned operates as follows: One bias level\tilde{\nu}_0is used as reference; the number of crossings per false alarm timeTwith positive slope of the noise envelope through\tilde{\nu}_0is averaged over a sufficiently long time to yield a stable valueC. This countCserves to determine the threshold bias\nu. The level\nuchanges only if the "long time" average count changes; it is specifically assumed that there is no response to instantaneous changes inC. It is shown that such a biasing method automatically adjusts to give a constant false alarm time (or rate), whenever the noise is Gaussian, and so has an advantage over the standard biasing method. In Section IV, the efficiency of both methods for non-Gaussian noise, is compared. In Section V, the probability of detection of a short (relative to the averaging time) "sure" signal with Rayleigh distributed amplitude is given when automatic biasing is used; due to the complexity of the expression obtained, no direct comparison has been made with the case where standard biasing is used.

The object of this note is to show that there is no finite characteristic matrix for any one of Lewis and Langford's systems. Theorem I. There is no finite characteristic matrix for any one of the systems S1–S5. Proof. Let M be a matrix with less than n elements for which all the provable formulas in S1, or S2, or S3, or S4, or S5, are satisfied. Let Fn represent the formula where ∑ stands for a ∨-chain, and the pi, are variables in any one of the calculi. Using M, there is always at least one summand in Fn where pi and pk have the same value. Therefore, Fn can always be written in the form (a = a)∨ B, and thus will give, for any B, a “designated” value, since the formula (p = p) ∨ q is provable in any one of the systems S1–S5. Give to any one of the systems S1–S5, the following matrix, due to Henle: 1. Elements: all possible classes formed from the integers 1, 2, 3, …, n. 2. “Designated” element: the class {1, 2, 3, …,n}. 3. Boole-Schröder algebra on the elements. 4. ◊N = N (N the null class) ◊A = {1, 2, …, n} (A any non-null class).5