# CSP Estimators: Cyclic Temporal Moments and Cumulants

In this post we discuss ways of estimating $n$-th order cyclic temporal moment and cumulant functions. Recall that for $n=2$, cyclic moments and cyclic cumulants are usually identical. They differ when the signal contains one or more finite-strength additive sine-wave components. In the common case when such components are absent (as in our recurring numerical example involving rectangular-pulse BPSK), they are equal and they are also equal to the conventional cyclic autocorrelation function provided the delay vector is chosen appropriately.

The more interesting case is when the order $n$ is greater than $2$. Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order $n$ greater than $2$ is $4$. So our estimation task is to estimate $n$-th order temporal moment and cumulant functions for $n \ge 4$ using a sampled-data record of length $T$.

# Cyclostationarity of Direct-Sequence Spread-Spectrum Signals

In this post we look at direct-sequence spread-spectrum (DSSS) signals, which can be usefully modeled as a kind of PSK signal. DSSS signals are used in a variety of real-world situations, including the familiar CDMA and WCDMA signals, covert signaling, and GPS. My colleague Antonio Napolitano has done some work on a large class of DSSS signals (The Literature [R11, R17, R95]), resulting in formulas for their spectral correlation functions, and I’ve made some remarks about their cyclostationary properties myself here and there (My Papers [16]).

A good thing, from the point of view of modulation recognition, about DSSS signals is that they are easily distinguished from other PSK and QAM signals by their spectral correlation functions. Whereas most PSK/QAM signals have only a single non-conjugate cycle frequency, and no conjugate cycle frequencies, DSSS signals have many non-conjugate cycle frequencies and in some cases also have many conjugate cycle frequencies.

# Cyclostationarity of Digital QAM and PSK

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 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 textbook.

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.

# Square-Root Raised-Cosine PSK/QAM

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.