## Cyclostationarity of Digital QAM and PSK

PSK and QAM signals form the building blocks for a large number of practical real-world signals. Understanding their probability structure is crucial to understanding those more complicated signals.

Let’s look into the statistical properties of a class of textbook signals that encompasses digital quadrature amplitude modulation (QAM), phase-shift keying (PSK), and pulse-amplitude modulation (PAM). I’ll call the class simply digital QAM (DQAM), and all of its members have an analytical-signal mathematical representation of the form

$\displaystyle s(t) = \sum_{k=-\infty}^\infty a_k p(t - kT_0 - t_0) e^{i2\pi f_0 t + i \phi_0}. \hfill (1)$

In this model, $k$ is the symbol index, $1/T_0 = f_{sym}$ is the symbol rate, $f_0$ is the carrier frequency (sometimes called the carrier frequency offset), $t_0$ is the symbol-clock phase, and $\phi_0$ is the carrier phase. The finite-energy function $p(t)$ is the pulse function (sometimes called the pulse-shaping function). Finally, the random variable $a_k$ is called the symbol, and has a discrete distribution that is called the constellation.

Model (1) is a textbook signal when the sequence of symbols is independent and identically distributed (IID). This condition rules out real-world communication aids such as periodically transmitted bursts of known symbols, adaptive modulation (where the constellation may change in response to the vagaries of the propagation channel), some forms of coding, etc. Also, when the pulse function $p(t)$ is a rectangle (with width $T_0$), the signal is even less realistic, and therefore more textbooky.

We will look at the moments and cumulants of this general model in this post. Although the model is textbook, we could use it as a building block to form more realistic, less textbooky, signal models. Then we could find the cyclostationarity of those models by applying signal-processing transformation rules that define how the cumulants of the output of a signal processor relate to those for the input.

## Signal Processing Operations and CSP

How does the cyclostationarity of a signal change when it is subjected to common signal-processing operations like addition, multiplication, and convolution?

It is often useful to know how a signal processing operation affects the probabilistic parameters of a random signal. For example, if I know the power spectral density (PSD) of some signal $x(t)$, and I filter it using a linear time-invariant transformation with impulse response function $h(t)$, producing the output $y(t)$, then what is the PSD of $y(t)$? This input-output relationship is well known and quite useful. The relationship is

$\displaystyle S_y^0(f) = \left| H(f) \right|^2 S_x^0(f). \hfill (1)$

In (1), the function $H(f)$ is the transfer function of the filter, which is the Fourier transform of the impulse-response function $h(t)$.

Because the mathematical models of real-world communication signals can be constructed by subjecting idealized textbook signals to various signal-processing operations, such as filtering, it is of interest to us here at the CSP Blog to know how the spectral correlation function of the output of a signal processor is related to the spectral correlation function for the input. Similarly, we’d like to know such input-output relationships for the cyclic cumulants and the cyclic polyspectra.

Another benefit of knowing these CSP input-output relationships is that they tend to build insight into the meaning of the probabilistic parameters. For example, in the PSD input-output relationship (1), we already know that the transfer function at $f = f_0$ scales the input frequency component at $f_0$ by the complex number $H(f_0)$. So it makes sense that the PSD at $f_0$ is scaled by the squared magnitude of $H(f_0)$. If the filter transfer function is zero at $f_0$, then the density of averaged power at $f_0$ should vanish too.

So, let’s look at this kind of relationship for CSP parameters. All of these results can be found, usually with more mathematical detail, in My Papers [6, 13].

## The Cycle Detectors

CSP shines when the problem involves strong noise or cochannel interference. Here we look at CSP-based signal-presence detection as a function of SNR and SIR.

Let’s take a look at a class of signal-presence detectors that exploit cyclostationarity and in doing so illustrate the good things that can happen with CSP whenever cochannel interference is present, or noise models deviate from simple additive white Gaussian noise (AWGN). I’m referring to the cycle detectors, the first CSP algorithms I ever studied (My Papers [1,4]).

## CSP-Based Time-Difference-of-Arrival Estimation

Time-delay estimation can be used to determine the angle-of-arrival of a signal impinging on two spatially separated signals. This estimation problem gets hard when there is cochannel interference present.

Let’s discuss an application of cyclostationary signal processing (CSP): time-delay estimation. The idea is that sampled data is available from two antennas (sensors), and there is a common signal component in each data set. The signal component in one data set is the time-delayed or time-advanced version of the component in the other set. This can happen when a plane-wave radio frequency (RF) signal propagates and impinges on the two antennas. In such a case, the RF signal arrives at the sensors with a time difference proportional to the distance between the sensors along the direction of propagation, and so the time-delay estimation is also commonly referred to as time-difference-of-arrival (TDOA) estimation.

Consider the diagram shown in Figure 1. A distant transmitter emits a signal that is well-modeled as a plane wave once it reaches our two receivers. An individual wavefront of the signal arrives at the two sensors at different times.

The line segment AB is perpendicular to the direction of propagation for the RF signal. The angle $\theta$ is called the angle of arrival (AOA). If we could estimate the AOA, we can tell the direction from which the signal arrives, which could be useful in a variety of settings. Since the triangle ABC is a right triangle, we have

$\displaystyle \cos (\theta) = \frac{x}{d}. \hfill (1)$

When $\theta = 0$, the wavefronts first strike receiver 2, then must propagate over $x=d$ meters before striking receiver 1. On the other hand, when $\theta = 90^\circ$, each wavefront strikes the two receivers simultaneously. In the former case, the TDOA is maximum, and in the latter it is zero. The TDOA can be negative too, so that $180^\circ$ azimuthal degrees can be determined by estimating the TDOA.

In general, the wavefront must traverse $x$ meters between striking receiver 2 and striking receiver 1,

$\displaystyle x = d \cos(\theta). \hfill (2)$

Assuming the speed of propagation is $c$ meters/sec, the TDOA is given by

$\displaystyle D = \frac{x}{c} = \frac{d\cos{\theta}}{c} \mbox{\rm \ \ seconds}. \hfill (3)$

In this post I’ll review several methods of TDOA estimation, some of which employ CSP and some of which do not. We’ll see some of the advantages and disadvantages of the various classes of methods through inspection, simulation, and application to captured data. Consider this post as a starting point to a study or development effort rather than as a definitive performance characterization.

## Square-Root Raised-Cosine PSK/QAM

SRRC PSK and QAM signals form important components of more complicated real-world communication signals. Let’s look at their second-order cyclostationarity here.

Let’s look at a somewhat more realistic textbook signal: The PSK/QAM signal with independent and identically distributed symbols (IID) and a square-root raised-cosine (SRRC) pulse function. The SRRC pulse is used in many practical systems and in many theoretical and simulation studies. In this post, we’ll look at how the free parameter of the pulse function, called the roll-off parameter or excess bandwidth parameter, affects the power spectrum and the spectral correlation function.

## Second-Order Estimator Verification Guide

Use this post to help check the accuracy of your second-order CSP estimators.

Update September 2022: New section on the non-conjugate and conjugate coherence function.

***

In this post I provide some tools for the do-it-yourself CSP practitioner. One of the goals of this blog is to help new CSP researchers and students to write their own estimators and algorithms. This post contains some spectral correlation function and cyclic autocorrelation function estimates and numerically evaluated formulas that can be compared to those produced by anybody’s code.

The signal of interest is, of course, our rectangular-pulse BPSK signal with symbol rate $0.1$ (normalized frequency units) and carrier offset $0.05$. You can download a MATLAB script for creating such a signal here.

The formula for the SCF for a textbook BPSK signal is published in several places (The Literature [R47], My Papers [6]) and depends mainly on the Fourier transform of the pulse function used by the textbook signal.

We’ll compare the numerically evaluated spectral correlation formula with estimates produced by my version of the frequency-smoothing method (FSM). The FSM estimates and the theoretical functions are contained in a MATLAB mat file here. (I had to change the extension of the mat file from .mat to .doc to allow posting it to WordPress–change it back after downloading. It is a zipped .mat file as of 12/2/22.) In all the results shown here and that you can download, the processed data-block length is $65536$ samples and the FSM smoothing width is $0.02$ Hz. A rectangular smoothing window is used. For all cycle frequencies except zero (non-conjugate), a zero-padding factor of two is used in the FSM.

For the cyclic autocorrelation, we provide estimates using two methods: inverse Fourier transformation of the spectral correlation estimate and direct averaging of the second-order lag product in the time domain.

## A Gallery of Spectral Correlation

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 captured 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]).

In all cases, the cycle frequencies are blindly estimated by the strip spectral correlation analyzer (The Literature [R3, R4]) and the estimates used by the FSM to compute the spectral correlation function. MATLAB is then used to plot the magnitude of the spectral correlation and conjugate spectral correlation, as specified by the determined non-conjugate and conjugate cycle frequencies.

There are three categories of signal types in this gallery: textbook signals, captured signals, and feature-rich signals. The latter comprises some captured 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 $f$ versus the detected cycle frequency $\alpha$ (as in this post).

A complementary gallery of cyclic autocorrelation functions can be found here.

Continue reading “A Gallery of Spectral Correlation”

## Textbook Signals

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 [11], [16]). What is important for us, here at 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). (See also some relevant examples of real-world signals here and here.)