Pictures are worth N words, and M equations, where N and M are large integers.
In this post I provide plots of the spectral correlation for a variety of simulated textbook signals and several collected communication signals. The plots show the variety of cycle-frequency patterns that arise from the disparate approaches to digital communication signaling. The distinguishability of these patterns, combined with the inability to distinguish based on the power spectrum, leads to a powerful set of classification (modulation recognition) features (My Papers [16, 25, 26, 28]).
There are three categories of signal types in this gallery: textbook signals, collected signals, and feature-rich signals. The latter comprises some collected signals (e.g., LTE) and some simulated radar signals. For the first two signal categories, the three-dimensional surface plots I’ve been using will suffice for illustrating the cycle-frequency patterns and the behavior of the spectral correlation function over frequency. But for the last category, the number of cycle frequencies is so large that the three-dimensional surface is difficult to interpret–it is a visual mess. For these signals, I’ll plot the maximum spectral correlation magnitude over spectral frequency versus the detected cycle frequency (as in this post).
Yes, the CSP Blog uses the simplest idealized cyclostationary digital signal–rectangular-pulse BPSK–to connect all the different aspects of CSP. But don’t mistake these ‘textbook’ signals for the real world.
What good is having a blog if you can’t offer a rant every once in a while? In this post I talk about what I call textbook signals, which are mathematical models of communication signals that are used by many researchers in statistical signal processing for communications.
We’ve already encountered, and used frequently, the most common textbook signal of all: rectangular-pulse BPSK with independent and identically distributed (IID) bits. We’ve been using this signal to illustrate the cyclostationary signal processing concepts and estimators as they have been introduced. It’s a good choice from the point of view of consistency of all the posts and it is easy to generate and to understand. However, it is not a good choice from the perspective of realism. It is rare to encounter a textbook BPSK signal in the practice of signal processing for communications.
I use the term textbook because the textbook signals can be found in standard textbooks, such as Proakis (The Literature [R44]). Textbook signals stand in opposition to signals used in the world, such as OFDM in LTE, slotted GMSK in GSM, 8PAM VSB with synchronization bits in ATSC-DTV, etc.
Typical communication signals combine a textbook signal with an access mechanism to yield the final physical-layer signal–the signal that is actually transmitted (My Papers , ). What is important for us, here on the CSP blog, is that this combination usually results in a signal with radically different cyclostationarity than the textbook component. So it is not enough to understand textbook signals’ cyclostationarity. We must also understand the cyclostationarity of the real-world signal, which may be sufficiently complex to render mathematical modeling and analysis impossible (at least for me).
What factors influence the quality of a spectral correlation function estimate?
The two non-parametric spectral-correlation estimators we’ve looked at so far–the frequency-smoothing and time-smoothing methods–require the choice of key estimator parameters. These are the total duration of the processed data block, , and the spectral resolution .
For the frequency-smoothing method (FSM), an FFT with length equal to the data-block length is required, and the spectral resolution is equal to the width of the smoothing function . For the time-smoothing method (TSM), multiple FFTs with lengths are required, and the frequency resolution is (in normalized frequency units).
The choice for the block length is partially guided by practical concerns, such as computational cost and whether the signal is persistent or transient in nature, and partially by the desire to obtain a reliable (low-variance) spectral correlation estimate. The choice for the frequency (spectral) resolution is typically guided by the desire for a reliable estimate.
Cross correlation functions can be normalized to create correlation coefficients. The spectral correlation function is a cross correlation and its correlation coefficient is called the coherence.
In this post I introduce the spectral coherence function, or just coherence. It deserves its own post because the coherence is a useful detection statistic for blindly determining significant cycle frequencies of arbitrary data records. See the posts on the strip spectral correlation analyzer and the FFT accumulation method for examples.
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.
Spectral correlation in CSP means that distinct narrowband spectral components of a signal are correlated-they contain either identical information or some degree of redundant information.
Spectral correlation is perhaps the most widely used characterization of the cyclostationarity property. The main reason is that the computational efficiency of the FFT can be harnessed to characterize the cyclostationarity of a given signal or data set in an efficient manner. And not just efficient, but with a reasonable total computational cost, so that one doesn’t have to wait too long for the result.
Just as the normal power spectrum is actually the power spectral density, or more accurately, the spectral density of time-averaged power (variance), the spectral correlation function is the spectral density of time-averaged correlation (covariance). What does this mean? Consider the following schematic showing two narrowband spectral components of an arbitrary signal:
The sequence of shaded rectangles on the left are meant to imply a time-series corresponding to the output of a bandpass filter centered at with bandwidth Similarly, the sequence of shaded rectangles on the right imply a time-series corresponding to the output of a bandpass filter centered at with bandwidth
We’ll use this simple textbook signal throughout the CSP Blog to illustrate and tie together all the different aspects of CSP.
To test the correctness of various CSP estimators, we need a sampled signal with known cyclostationary parameters. Additionally, the signal should be easy to create and understand. A good candidate for this kind of signal is the binary phase-shift keyed (BPSK) signal with rectangular pulse function.
PSK signals with rectangular pulse functions have infinite bandwidth because the signal bandwidth is determined by the Fourier transform of the pulse, which is a sinc() function for the rectangular pulse. So the rectangular pulse is not terribly practical–infinite bandwidth is bad for other users of the spectrum. However, it is easy to generate, and its statistical properties are known.
So let’s jump in. The baseband BPSK signal is simply a sequence of binary ( 1) symbols convolved with the rectangular pulse. The MATLAB script make_rect_bpsk.m does this and produces the following plot:
The signal alternates between amplitudes of +1 and -1 randomly. After frequency shifting and adding white Gaussian noise, we obtain the power spectrum estimate:
The power spectrum plot shows why the rectangular-pulse BPSK signal is not popular in practice. The range of frequencies for which the signal possesses non-zero average power is infinite, so it will interfere with signals “nearby” in frequency. However, it is a good signal for us to use as a test input in all of our CSP algorithms and estimators.
The MATLAB script that creates the BPSK signal and the plots above is here. It is an m-file but I’ve stored it in a .doc file due to WordPress limitations I can’t yet get around.