In this post I describe and illustrate the most important property of cyclostationary statistics: signal selectivity. The idea is that the cyclostationary parameters for a single signal can be estimated for that signal even when it is corrupted by strong noise and cochannel interferers. Cochannel means that the interferer occupies a frequency band that partially or completely overlaps the frequency band for the signal of interest.
A mixture of signals, whether cochannel or not, is modeled by the simple sum of the signals, as in
where is additive noise. We can write this more compactly as
A noisy cochannel mixture of four signals is shown here:
If the involved signals are mutually independent (or at least uncorrelated), and are also independent from the noise , then it is easy to show that the spectral correlation function for the sum is equal to the sum of the spectral correlation functions for each of the involved signals,
When the non-conjugate spectral correlation function is considered, and the cycle frequency , then the spectral correlation function is identical to the power spectrum, and we have
or in conventional notation,
Now each signal has some finite power, and so has a non-zero power spectrum, including the noise signal. So the power spectrum for the sum of the signals is equal to the sum of the power spectra. When the signals are cochannel, the individual power spectra add together across a common band of frequencies, and we end up getting a mixture that cannot be unmixed. That is, the individual power spectra cannot be obtained from the sum-signal power spectrum. On the other hand, if the signals were not cochannel, but instead occupied disjoint frequency bands, the power spectrum for the sum signal would still be additive, but the individual power spectra could be isolated by linear time-invariant filtering and each associated with one or another of the component signals.
Now suppose the noise is stationary, and choose a cycle frequency . Then the spectral correlation for the sum signal is given by
since the noise is not cyclostationary. This is the well-known noise tolerance of cyclostationary-signal parameters. In practice, we use estimates of the spectral correlation function, which are in fact influenced by the noise, but as the length of the processed data-block increases, the effect drops in size.
On to signal selectivity. Suppose that one of the signals, say $s_j(t)$, possesses a cycle frequency that is possessed by no other signal in the sum. Then,
Because we can operate on the sum data, yet obtain the spectral correlation for a selected signal, we say that the spectral correlation function is signal selective.
Consider the example shown above, where four signals are spectrally (and temporally) overlapped and added to noise. Blindly estimated spectral correlation functions for each of the individual signals are shown below, arranged around a plot of the blindly estimated spectral correlation function for the sum signal:
The individual slices of the spectral correlation function are labeled so that the contribution to the sum-signal spectral correlation can be identified for each signal in the sum.
The example above used only synthetic (simulated) signals and noise. To convince the reader that CSP and signal-selectivity work in the real world on real signals collected by real receivers, consider the following example, which is the result of processing a scene containing multiple collected signals added together:
Why do we Care About Signal Selectivity?
One reason is that the separation of the features for the individual signals in the spectral correlation plane allows the possibility of jointly detecting and classifying all cochannel signals (My Papers [25,26]). A related reason is that signal selectivity enables detecting a particular signal of interest in spite of strong cochannel interference, which is not of interest.
Another reason is that signal selectivity allows the creation of various interference-tolerant parameter estimators, such as time-difference-of-arrival (TDOA) estimators, which use the data from two widely separated sensors to estimate the relative delay of a signal impinging upon both. Typical cross-correlation-based TDOA estimators are highly vulnerable to the presence of cochannel interference, but cyclic (cyclostationarity-exploiting) estimators are not (My Papers [2,4,21,23]).
In later posts, we’ll see that the signal-selectivity property of the spectral correlation function (and cyclic autocorrelation function) extends to higher-order statistics only if we use higher-order cyclic cumulants, but not if we use cyclic moments.