The Spectral Correlation Function

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:

scf_schematic

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 f-A/2 with bandwidth B. Similarly, the sequence of shaded rectangles on the right imply a time-series corresponding to the output of a bandpass filter centered at f+A/2 with bandwidth B.

Let’s call the first time-series Y_1(t, f-A/2) and the second one Y_2(t, f+A/2). Since these time-series, or signals, are bandpass in general, if we attempt to measure their correlation we will get a small value if A \neq 0. However, if we downconvert each of them to baseband (zero frequency), we will obtain lowpass signals, and there is the possibility that these new signals are correlated to some degree.

As a limiting case, suppose each of the Y_j(t, \cdot) signals were noiseless sine waves. By the construction of the figure, their frequencies must be different if A \neq 0, and so their correlation will be zero. However, if each sine wave is perfectly downconverted to baseband, the resulting signals are simply complex-valued constants, and the correlation between the two constant time-series is high.

In general, the narrowband time-series Y_1(t, \cdot) and Y_2(t, \cdot) are not simple sine waves, but complicated random processes. But the correlation between separated spectral components–spectral correlation–is still a highly useful characterization of the signal for a large class of interesting signals.

The spectral components (the individual downconverted narrowband spectral components of the signal) are most often obtained through the use of the Fourier transform. As the transform of length T slides along the signal, if produces a number of downconverted spectral components with approximate bandwidth 1/T.  The two involved time-series of interest are then renamed as X_T(t, f-A/2) and X_T(t, f+A/2). A measure of the spectral correlation is given by the limiting average of the cyclic periodogram, which is defined by

\displaystyle I_T^A (t, f) = \frac{1}{T} X_T(t, f-A/2) X_T^*(t, f+A/2), \hfill (1)

as the amount of processed data increases without bound, and then the spectral resolution (B = 1/T) is allowed to decrease to zero,

\displaystyle S_x^A (f) = \lim_{T\rightarrow\infty} \lim_{U\rightarrow\infty} \displaystyle\frac{1}{U} \int_{-U/2}^{U/2} I_T^A(t, f) \, dt. \hfill (2)

The limit spectral correlation function we just wrote down is a time-smoothed (time-averaged) cyclic periodogram. But the limit function can also be obtained by frequency smoothing the cyclic periodogram

\displaystyle S_x^A(f) = \lim_{\Delta\rightarrow 0} \lim_{T\rightarrow\infty} g_\Delta(f) \otimes I_T^A(t, f), \hfill (3)

where g_\Delta(f) is a unit-area pulse-like smoothing kernel (such as a rectangle). In (3), the symbol \otimes denotes convolution.

The Significance of the Frequency A

The spectral correlation function (SCF) S_x^A(f) is typically zero for almost all real numbers A. Those A for which the SCF is not identically zero are called cycle frequencies (CFs). The set of SCF CFs is exactly the same as the set of cycle frequencies for the cyclic autocorrelation function (CAF)! That is, the separation between correlated narrowband signal components of x(t) is the same as a frequency of a sine wave that can be generated by a quadratic nonlinearity (for example, a squarer or a delay-and-multiply device) operating on the original time-series data x(t).

The Cyclic Wiener Relationship

It can be shown that the Fourier transform of the CAF is equal to the SCF (The Literature [R1], My Papers [5,6]):

\displaystyle S_x^\alpha(f) = \int_{-\infty}^\infty R_x^\alpha(\tau) e^{-i 2 \pi f \tau}\, d\tau, \hfill (4)

which is called the cyclic Wiener relationship. The Wiener relationship (sometimes called the Wiener-Khintchine theorem)  is a name given to the familiar Fourier transform relation between the conventional power spectral density and the autocorrelation

\displaystyle S_x^0 (f) = \int_{-\infty}^\infty R_x^0(\tau) e^{-i 2 \pi f \tau} \, d\tau, \hfill (5)

where S_x^0(f) is the conventional power spectrum and R_x^0(\tau) is the conventional autocorrelation function.

Conjugate Spectral Correlation

This post has defined the non-conjugate spectral correlation function, which is the correlation between X_T(t, f-\alpha/2) and X_T(t, f+\alpha/2). (The correlation between random variables X and Y is defined as E[XY^*]; that is, the standard correlation includes a conjugation.)

The conjugate SCF is defined as the Fourier transform of the conjugate cyclic autocorrelation function,

\displaystyle S_{x^*}^\alpha (f) = \int_{-\infty}^\infty R_{x^*}^\alpha (\tau) e^{-i 2 \pi f \tau}\, d\tau . \hfill (6)

From this definition, it can be shown that the conjugate SCF is the density of time-averaged correlation between X_T(t, f+\alpha/2) and X_T^*(t, \alpha/2-f),

\displaystyle S_{x^*}^\alpha (f) = \lim_{T\rightarrow\infty} \lim_{U\rightarrow\infty} \displaystyle\frac{1}{U} \int_{-U/2}^{U/2} J_T^\alpha(t, f) \, dt, \hfill (7)

where J_T^\alpha(t, f) is the conjugate cyclic periodogram

J_T^\alpha(t,f) = \displaystyle \frac{1}{T} X_T(t, f+\alpha/2) X_T(t, \alpha/2-f). \hfill (8)

The detailed explanation for why we need two kinds of spectral correlation functions (and, correspondingly, two kinds of cyclic autocorrelation functions) can be found in this post.

Illustrations

The SCF below is estimated (more on that estimation in another post) from a simulated BPSK signal having bit rate of 333.3 kHz and carrier frequency of 100 kHz. A small amount of noise is added to the signal prior to SCF estimation. The (non-conjugate) SCF shows the power spectrum for \alpha = 0 and the bit-rate SCF for \alpha = 333.3 kHz. The conjugate SCF plot shows the prominent feature for the doubled-carrier cycle frequency \alpha = 200.0 kHz, and features offset from the doubled-carrier feature by \pm 333.3 kHz. More on the spectral correlation of the BPSK signal can be found here and here. The spectral correlation surfaces for a variety of communication signals can be found in this gallery post.

A closely related function called the spectral coherence function is useful for blindly detecting cycle frequencies exhibited by arbitrary data sets.

ww_1

Now consider a similar signal: QPSK with rectangular pulses. Let’s switch to normalized frequencies here for convenience. The signal has a symbol rate of 1/10, a carrier frequency of 0.05, unit power, and a small amount of additive white Gaussian noise. A power spectrum estimate is shown in the following figure:

spec_corr_example_psds

Consider also four distinct narrowband (NB) components of this QPSK signal as shown in the figure. The center frequencies are 0.0, 0.1, 0.08, and 0.18. We know that this signal has non-conjugate cycle frequencies that are equal to harmonics of the symbol rate, or k/10 for k = 0, \pm 1, \pm 2, \ldots. This means that the NB components with separations k/10 are correlated. So if we extract such NB components “by hand” and calculate their correlation coefficients as a function of relative delay, we should see large results for the pairs (0.0, 0.1) and (0.08, 0.18), and small results for all other pairs drawn from the four frequencies.

So let’s do that. We apply a simple Fourier-based ideal filter (ideal meaning rectangular pass band) with center frequency f_0 \in \{0.0, 0.1, 0.08, 0.18\}, frequency shift to complex baseband (zero center frequency) and decimate. The results are our narrowband signal components. Are they correlated when they should be and uncorrelated when they should be?

Here are the correlation-coefficient results:

spec_corr_example_cc

Here the signals y_j(t) arise from the frequencies \{0.0, 0.1, 0.08, 0.18\}. So the spectral correlation concept is verified here. One can also simply plot the decimated shifted narrowband components and assess correlation visually:

spec_corr_example_time

Estimators

I’ve written several posts on estimators for the spectral correlation function; they are listed below. I think of them as falling into two categories: exhaustive and focused. For exhaustive spectral correlation estimators, the goal is to estimate the function over its entire (non-redundant) domain of definition as efficiently as possible. For focused estimators, the goal is to estimate the spectral correlation function for one or a small number of cycle frequencies with high accuracy and selectable frequency resolution.

Exhaustive Estimators

Strip Spectral Correlation Analyzer (SSCA)

FFT Accumulation Method (FAM)

Focused Estimators

The Frequency-Smoothing Method (FSM)

The Time-Smoothing Method (TSM)

 

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