In this post, I introduce the cyclic autocorrelation function (CAF). The easiest way to do this is to first review the conventional autocorrelation function. Suppose we have a complex-valued signal defined on a suitable probability space. Then the mean value of is given by
For stationary signals, and many cyclostationary signals, this mean value is independent of the lag parameter , so that
The autocorrelation function is the correlation between the random variables corresponding to two time instants of the random signal, or
To see how the autocorrelation varies with some particular central time , we can use a more convenient parameterization of the two time instants and , such as
So time represents the center point of the two time instants and is their separation. If the autocorrelation depends only on the separation between the two time instants , and not their center point , the signal is stationary of order two, or just stationary, and we have
For nonstationary signals, on the other hand, the autocorrelation does depend on central time . For the special case of nonstationary signals called cyclostationary signals, the autocorrelation is either a periodic function or an almost periodic function. In either case, it can be represented by a Fourier series
where is a Fourier-series coefficient called the cyclic autocorrelation function. The Fourier frequencies are called cycle frequencies (CFs). The CAFs are obtained in the usual way for Fourier coefficients,
If the signal is a cycloergodic signal (or we are using fraction-of-time probability), then the CAFs can be obtained directly from a sample path (the signal itself),
For many cyclostationary signals, such as BPSK, the conjugate autocorrelation function is also non-zero (and also useful). This function is defined by
and is represented by its own Fourier series
I explain in detail why we need two autocorrelation functions in the post on conjugation configurations. The problem is worse when we look at higher-order moments and cumulants, where we need functions to properly characterize a signal at order .
And that is it! These are the basic definitions for the (second-order) probabilistic parameters of cyclostationary signals in the time-domain. In later posts, I’ll have much to say about their utility, their estimation, their connection to the frequency-domain parameters, and their generalization to higher-order parameters.