Rectification of two signals in random noise

The spectrum of the output of a half-wave rectifier is derived for an input which is the sum of random noise and two sinusoidal signals of different frequencies. The method used is the characteristic function method described by Rice. The components of the output spectrum are given as infinite series of hypergeometric functions. If both the input signals .are small compared with the noise, it is shown that the ratio of the output signal power at the difference frequency to the output noise power is proportional to the product of the input signal-to-noise power ratios at the two frequencies. If one of the input signals is very large compared with the noise, it is shown that the other signal and the noise are translated in frequency without alteration of the signal-to-noise ratio. A correction factor is obtained for the case where the large signal is not quite large enough. Finally, the output signal-to-noise ratio of a single-sideband detector is calculated as a function of the input signal-to-noise ratio, when the sideband amplitude is one-half the carrier amplitude.