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.

This paper presents a simple formula for the capacity of a coherent digital communication channel operating near threshold [i.e., with very low signal-to-noise ratio (SNR)] with additive noise from a certain class of arbitrary but known probability distributions. For the noncoherent threshold channel, the optimun receiver is derived for arbitrary modulations and fading. This receiver is shown to specialize to previously derived canonical froms for unmodulated signals.

Newton's method for root finding is shown to be an effective algorithm for computing maximum likelihood estimates of the bias parameter in Hall's optimum receiver for digital singals in impulse noise. Use of a bias estimator allows the receiver to be adaptively instrumented. A simulation indicates that the number of independent samples of the impulse noise, as modeled by Hall, should be around 20 000 for satisfactory parameter estimates.