There are some situations in which the spectral correlation function is not the preferred measure of (second-order) cyclostationarity. In these situations, the cyclic autocorrelation (non-conjugate and conjugate versions) may be much simpler to estimate and work with in terms of detector, classifier, and estimator structures. So in this post, I’m going to provide plots of the cyclic autocorrelation for each of the signals in the spectral correlation gallery post. The exceptions are those signals I called feature-rich in the spectral correlation gallery post, such as LTE and radar. Recall that such signals possess a large number of cycle frequencies, and plotting their three-dimensional spectral correlation surface is not helpful as it is difficult to interpret with the human eye. So for the cycle-frequency patterns of feature-rich signals, we’ll rely on the stem-style (cyclic-domain profile) plots in the gallery post.

In this post I discuss the use of cyclostationary signal processing applied to communication-signal synchronization problems. First, just what are synchronization problems? Synchronize and synchronization have multiple meanings, but the meaning of synchronize that is relevant here is something like:

syn·chro·nize: To cause to occur or operate with exact coincidence in time or rate

If we have an analog amplitude-modulated (AM) signal (such as voice AM used in the AM broadcast bands) at a receiver we want to remove the effects of the carrier sine wave, resulting in an output that is only the original voice or music message. If we have a digital signal such as binary phase-shift keying (BPSK), we want to remove the effects of the carrier but also sample the message signal at the correct instants to optimally recover the transmitted bit sequence.

In this post, we look at the ability of various CSP estimators to distinguish cycle frequencies, temporal changes in cyclostationarity, and spectral features. These abilities are quantified by the resolution properties of CSP estimators.

Then the temporal resolution of the estimate is approximately , the cycle-frequency resolution is about , and the spectral resolution depends strongly on the particular estimator and its parameters. The resolution product was discussed in this post. The fundamental result for the resolution product is that it must be very much larger than unity in order to obtain an SCF estimate with low variance.

The cyclic autocorrelation for rectangular-pulse BPSK can be derived as a relatively simple closed-form expression (see My Papers [6] for example or The Literature [R1]). It can be estimated in a variety of ways, which we will discuss in future posts. The non-conjugate cycle frequencies for the signal are harmonics of the bit rate, , and the conjugate cycle frequencies are the non-conjugate cycle frequencies offset by the doubled carrier, or .

Recall that the simulated rectangular-pulse BPSK signal has samples per bit, or a bit rate of , and a carrier offset of , all in normalized units (meaning the sampling rate is unity). We’ve previously selected a sampling rate of MHz to provide a little physical realism. This means the bit rate is kHz and the carrier offset frequency is kHz. From these numbers, we see that the non-conjugate cycle frequencies are kHz, and that the conjugate cycle frequencies are kHz, or kHz.

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