We’ve seen how to define second-order cyclostationarity in the time- and frequency-domains, and we’ve looked at ideal and estimated spectral correlation functions for a synthetic rectangular-pulse BPSK signal. In future posts, we’ll look at how to create simple spectral correlation estimators, but in this post I want to introduce the topic of higher-order cyclostationarity (HOCS). This post is more conceptual in nature; for mathematical details about HOCS, see the posts on cyclic cumulants and cyclic polyspectra. Estimators of higher-order parameters, such as cyclic cumulants and cyclic moments, are discussed in this post.
To contrast with HOCS, we’ll refer to second-order parameters such as the cyclic autocorrelation and the spectral correlation function as parameters of second-order cyclostationarity (SOCS).
The first question we might ask is Why do we care about HOCS? And one answer is that SOCS does not provide all the statistical information about a signal that we might need to perform some signal-processing task. There are two main limitations of SOCS that drive us to HOCS.
SOCS Limitation One: Distinct Signal Types Possess Identical SOCS
Cyclostationarity can be used to automatically recognize the modulation type of a signal present in some given sampled-data set. Ideally, the entire probability structure of the signal would be compared to the probability structure of a catalog of signal types to determine the best match. By ‘probability structure,’ I mean the set of all possible probability density functions, or the set of all possible probability distribution functions, or the set of all possible temporal moment functions, or the set of all possible temporal cumulant functions. But since most signals possess an infinite number of non-zero moments (and cumulants), we must restrict our use of the probability structure in some manner.
Many signal types possess unique SOCS, such as MSK and DSSS BPSK. However, there are some classes of signals for which the parameters of SOCS are equal up to a single scale factor. That is, their spectral correlation functions are identical except for a scale factor that applies to the entire function. Unless the transmitted power and propagation channel are accurately known in advance, there is no way to unambiguously recognize the different signals in such a class. However, since the signal types are in fact distinct (more precisely, the different random processes that define the signal types are distinct), there must be some probabilistic parameters that differ between the different signals.
An important class of signals with identical SOCS are the digital QAM signals, such as QPSK, 8PSK, -DQPSK, 8QAM, 16QAM, 256QAM, etc. To illustrate this idea, I’ve simulated four digital QAM signals and estimated their SOCS and HOCS. First, here are the PSDs for the four (noisy) signals:
These four signals have independent and identically distributed symbols and all employ the same transmitter pulse-shaping filter (SRRC, roll-off of 1.0). So, their theoretical PSDs are identical, as is verified by the estimates in Figure 1.
The four signals each possess three non-conjugate cycle frequencies and no conjugate cycle frequencies. The non-conjugate cycle frequencies are , where is the symbol rate. A symbol rate of just means that there are samples in a symbol interval. So, the cycle-frequency patterns are identical for the four signals, and cannot be used to distinguish among them (compare this to the cycle-frequency pattern we’ve seen for BPSK). Moreover, their spectral correlation functions are identical:
But we can show mathematically that their higher-order moments and cumulants do differ significantly. The following plots show the magnitudes of the th-order cyclic cumulant functions, which we define and discuss in other CSP-Blog posts:
The columns of these matrices correspond to different higher-order cumulants (think moments for now), with higher orders proceeding to the right. The rows correspond to harmonic numbers of the symbol-rate cycle frequencies. The first column is the first-order cyclostationarity (simply additive sine waves in the signal, which do not exist here), the next three columns capture the SOCS, and the remaining columns to the right capture the fourth- and sixth-order statistics.
Although the patterns for QPSK and 16QAM are the same, the particular values of the cumulants differ. So these four signals are indistinguishable using SOCS, but are clearly distinguishable using HOCS.
(The multi-colored plots of higher-order cyclic-cumulant magnitudes above [and in the banner for the CSP Blog website] correspond to the use of a single delay vector for each considered cumulant order : the all-zero delay vector . See below for how the delay vector is used with moments and the post on cyclic temporal cumulants for the general treatment.)
SOCS Limitation Two: Some Cyclostationary Signals Have no SOCS
Some signals do not possess any second-order cycle frequencies besides the non-conjugate cycle frequency of zero (all finite-power signals possess this cycle frequency), but do possess higher-order cycle frequencies. An example is duobinary signaling, which is a PSK signal type that uses a transmit pulse-shaping filter with a roll-off of zero (the excess bandwidth is zero or, viewed a bit differently, the occupied bandwidth is equal to the symbol rate). The duobinary signal has no SOCS, but does have HOCS.
How to Generalize SOCS to HOCS?
We’ve seen some motivation for generalizing SOCS to HOCS, but how exactly do we go about doing it? For a more mathematical treatment, see My Papers [5,6] and the posts on cyclic cumulants and cyclic polyspectra. Here, we provide the flavor of the approach.
The theory of SOCS usually starts out by describing the periodic components of the autocorrelation function. The autocorrelation is a second-order moment for the signal under study, so it is natural to extend SOCS by looking at third-order, fourth-order, and higher-order moments. For example, we might compute the third-order temporal moment,
It turns out that for almost all communication signals, this moment is zero. What about the fourth-order temporal moment? For example,
which is a function of the delay vector . This moment is almost always non-zero for communication signals, for some values of near the origin. But here is the problem: If the signal does have SOCS, then there are additive sine-wave components in products like and . This just means that the lag products can be represented by expressions of the form
for all real numbers .
So if the signal exhibits SOCS, there are sine-waves present in the second-order lag products, and therefore there must be additive sine-wave components in the fourth-order product. But what is new in the fourth-order product that isn’t simply a result of a product of second-order sine wave components, if anything? This is the fundamental question that drives the mathematical development of higher-order cyclic cumulants in My Papers [5,6].