Statistical Properties of a Sine Wave Plus Random Noise

TN SOME technical problems we are concerned with a current which consists of a sinusoidal component plus a random noise component. A number of statistical properties of such a current are given here. The present paper may be regarded as an extension of Section 3.10 of an earlier paper, 1 "Mathematical Analysis of Random Noise", where some of the simpler properties of a sine wave plus random noise are discussed. The current in which we are interested may be written as , I = Ocos at + IN , = Rcos {qt + 8) (3-4) where Q and q are constants, / is time, and IN is a random (in the sense of Section 2.8 of Reference A) noise current. When the second expression involving the envelope R and the phase angle 8 is used, the power spectrum of IN is assumed to be confined to a relatively narrow band in the neighborhood of the sine wave frequency fQ = q/(2ir). This makes R and 8 relatively slowly (usually) varying functions of time. In Section 1, the probability density and cumulative distribution of I are discussed. In Section 2, the upward "crossings" of / (i.e., the expected number of times, per second, I increases through a given value /i), are examined. The probability density and the cumulative distribution of R are given in Section 3.10 of Reference A. The crossings of R are examined in Section 4 of the present paper. The statistical properties of 8', the time derivative of the phase angle 8, are of interest because the instantaneous frequency of I may be defined to be f q + 0'/(27r).