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 (or simply the variance when the mean is zero), 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:
Let’s define narrowband spectral component to mean the output of a bandpass filter applied to a signal, where the bandwidth of the filter is much smaller than the bandwidth of the 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
Let’s call the first time series and the second one Since these time series, or signals, are bandpass in general, if we attempt to measure their correlation we will get a small value if 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 signals were noiseless sine waves. By the construction of the figure, their frequencies must be different if 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 and are not simple sine waves, but are sample paths of complicated random processes. Nevertheless, 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 slides along the signal , it produces a number of downconverted spectral components with approximate bandwidth The two involved time series of interest are then renamed as and A measure of the spectral correlation is given by the limiting average of the cyclic periodogram, which is defined by
as the amount of processed data increases without bound, and then and only then the spectral resolution () is allowed to decrease to zero,
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,
The Significance of the Frequency A
The spectral correlation function (SCF) is typically zero for almost all real numbers That is, for almost all signal types, most pairs of narrowband frequency components are uncorrelated. Those few 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 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
The Cyclic Wiener Relationship
which is called the cyclic Wiener relationship. The Wiener relationship (sometimes called the Wiener-Khintchine theorem) is a name given to the more familiar Fourier transform relation between the conventional power spectral density and the autocorrelation
where is the conventional power spectrum and is the conventional autocorrelation function.
It follows that the cyclic autocorrelation function is the inverse Fourier transform of the spectral correlation function,
and the conventional autocorrelation is the inverse transform of the power spectral density
The mean-square (power) of the time series (or variance if the time series has a mean value of zero) is simply the autocorrelation evaluated at . This implies that the power of the time series is the integral of the power spectral density
Conjugate Spectral Correlation
This post has defined the non-conjugate spectral correlation function, which is the correlation between and . (The correlation between random variables and is defined as that is, the standard correlation includes a conjugation of one of the factors.)
The conjugate SCF is defined as the Fourier transform of the conjugate cyclic autocorrelation function,
From this definition, it can be shown that the conjugate SCF is the density of time-averaged correlation between and
where is the conjugate cyclic periodogram
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 the post on conjugation configurations.
The SCF in Figure 2 is estimated (more on that estimation in another post) from a simulated BPSK signal with a bit rate of kHz and carrier frequency of kHz. A small amount of noise is added to the signal prior to SCF estimation. The non-conjugate SCF shows the power spectrum for and the bit-rate SCF for kHz. The conjugate SCF plot shows the prominent feature for the doubled-carrier cycle frequency kHz, and features offset from the doubled-carrier feature by 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.
Now consider a similar signal: QPSK with rectangular pulses. Let’s switch to normalized frequencies (physical frequencies are divided by the sampling rate) here for convenience. The signal has a symbol rate of , a carrier frequency of unit power, and a small amount of additive white Gaussian noise. A power spectrum estimate is shown in Figure 3.
Consider also four distinct narrowband (NB) components of this QPSK signal as shown in the figure. The center frequencies are and We know that this signal has non-conjugate cycle frequencies that are equal to harmonics of the symbol rate, or for . This means that the NB components with separations 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 and and small results for all other pairs drawn from the four frequencies. The correlation coefficient for two random variables and is
where is the expectation operator, is the mean value of , is the mean value of , and and are the standard deviations of and . Typically . In the measurement below, we use temporal averaging in place of the expectation.
So let’s do that. We apply a simple Fourier-based ideal filter (ideal meaning rectangular passband) with center frequency 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:
Here the signals arise from the bandpass-filter center frequencies So the spectral correlation concept is verified using this example: the only large correlation coefficients are those corresponding to a spectral-component center-frequency difference that is equal to a known cycle frequency for the QPSK signal (). One can also simply plot the decimated shifted narrowband components and assess correlation visually:
Some readers question the need for the downconversion step before measuring correlation, so I’ve also performed our QPSK-based four-spectral-components measurement without doing the downconversion. That is, I just correlated the outputs of the four narrowband filters. The correlation coefficients and the outputs of the filters themselves (analogs of Figures 4 and 5) are shown in Figures 6 and 7. See also the discussion in the Comments section below.
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.
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