The distribution of intervals between zero crossings of sine wave plus random noise and allied topics

Using the first term of a series given by Longuet-Higgins [5] for the general case an approximate solution is found in the form of a double integral for the distribution of intervals between zero crossings of a waveform consisting of sine wave plus random noise. For the important case of fairly large signal to noise ratio (SNR) this integral is evaluated approximately in closed form. In the limit as the SNR becomes very large indeed, this tends to a Gaussian distribution. Comparison is made between the approximate forms and numerical evaluation of the double integral. Good agreement results. An example of the use of this distribution is the transmission of binary data over telephone lines by frequency modulation. With a choice of two out of three possible frequencies to use, it is shown that a satisfactory error rate can only be obtained with the use of the two extremes. This is in accordance with the practical experiments of Jenks and Hannon [4]. Also given is an asymptotic form for the average rate of crossing of any level. This is somewhat different from that of Rice [9] and has been found easier to use in suitable conditions. For the zero level another simple form is given valid when the first is not.