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The Autocorrelation Function of the Output of an N-Level Quantizer with an Angle-Modulated Input Signal

A readily computable formula is given for the autocorrelation function of the output of an arbitrary time-invariant nonlinearity with an arbitrary angle-modulated input. The general result is specialized to the case of an odd-symmetric, but otherwise arbitrary, quantizer. It is shown that to evaluate the output autocorrelation the second-order characteristic function (SOCF) of the modulation is required. Using previously derived SOCFs for a constant, a Gaussian random process, and a generalized random telegraph wave (RTW), the output autocorrelation is given explicitly. Using the SOCF for a binary phase-shift keyed (BPSK) signal, derived here for the first time, the output autocorrelation with BPSK modulation is also presented. Finally, the autocorrelation functions of the modulated input processes are given by reducing the quantizer to a linearity, with the results for the generalized RTW and BPSK modulations being new.