In this post I present a very useful blind cycle-frequency estimator known in the literature as the strip spectral correlation analyzer (SSCA) (The Literature [R3-R5]). We’ve covered the basics of the frequency-smoothing method (FSM) and the time-smoothing method (TSM) of estimating the spectral correlation function (SCF) in previous posts. The TSM and FSM are efficient estimators of the SCF when it is desired to estimate it for one or a few cycle frequencies (CFs). The SSCA, on the other hand, is efficient when we want to estimate the SCF for all CFs.
See also an alternate method of exhaustive SCF estimation: The FFT Accumulation Method.
The SSCA is a kind of time-smoothing method and a detailed examination of the method and its properties can be found in The Literature [R3-R5]. Here I’ll briefly describe the estimator mathematically, then try to explain why the rather odd-looking estimator form produces estimates of the spectral correlation function. Finally, I’ll provide a few processing examples. In future posts I’ll contrast the SSCA performance and computational cost with those for the FSM and TSM.
The block diagram for the SSCA looks like this:
Essentially, the input data is first channelized, and then each of the narrowband outputs of the channelizer is multiplied by the original input signal and Fourier transformed. The channelizer is usually implemented by a sliding FFT. So the method clearly harnesses the computational power and efficiency of the FFT, leading to its overall low cost compared to other exhaustive-CF spectral correlation estimators (such as the FSM and TSM when they are placed in a loop over cycle frequency).
Mathematically, a particular cross (between arbitrary signals and ) spectral correlation function point estimate is given by
where the demodulates are defined as
Here we use the subscript to denote the total averaging time in seconds, so that , where is the sampling rate. And this notational choice connects well to other CSP Blog posts’ notations. Similarly, the subscript denotes the length of time used in each of the channelizer’s FFTs, so that . But we don’t really care about the absolute value of , so the time-referencing subscripts aren’t so important going forward.
Let’s try to show that the SSCA point-estimate expression (1) can be manipulated into a form such that we can more easily see that the spectral correlation function is estimated. The first step is to substitute the expression for the demodulate into the SSCA point estimate and simplify,
Define . Then
Now, substitute , rename to , substitute , and finally rename to :
Assume that . The inner sum is just an estimate of the cyclic autocorrelation, with data tapering (windowing) and with limits on the sum that depend on the lag . This is the cyclic correlogram (The Literature [R5]). Thus, we have the following relation,
Here we can identify the pair from our usual parameterization of the spectral correlation function. First, we have . The spectral frequency is the argument of , and is given by .
Therefore, the point estimate is confirmed. We see that the time-smoothed SSCA estimate is approximately equal to a Fourier-transformed cyclic autocorrelation estimate. The requirement for reliability on this estimate is that the length of time used to estimate the cyclic correlogram, , is sufficiently large and much larger than the approximate span of the function, which is contained in the window , which itself has width . That is, or once again.
The demodulates are formed by short-duration () FFTs, which are often called the channelizer FFTs, and the final point-estimate outputs are formed by long () FFTs, which are often called the strip FFTs, because each one of those FFTs produces a set of point estimates that lie on strip parallel to the line in the plane.
Numerical Example: Rectangular-Pulse BPSK
Let’s turn to our usual example: the textbook rectangular-pulse BPSK signal. This signal has normalized bit rate of and normalized carrier frequency of . The non-conjugate cycle frequencies are harmonics of the bit rate, and so are equal to , and the conjugate cycle frequencies are the non-conjugate cycle frequencies plus the doubled carrier .
We process samples of the rectangular-pulse signal using a straightforward C-language implementation of the SSCA. The number of strips, , is set to . This SSCA processing results in SCF point estimates for the non-conjugate SCF and another for the conjugate SCF. However, symmetries allow us to look at the non-conjugate function for only the non-negative cycle frequencies. Nevertheless, there are a lot of point estimates in the example. Plotting all of them is a difficult exercise in visualization. Here I plot only the pairs that correspond to the largest values of spectral correlation magnitude:
And the 500 largest values of conjugate spectral correlation magnitude:
These two plots show that the algorithm is producing relatively large SCF values for the signal’s true cycle frequencies. There are also some relatively strong SCF magnitudes for cycle frequencies near zero (non-conjugate) and the doubled carrier (conjugate) that are not actual cycle frequencies of the textbook rectangular-pulse BPSK signal. These high values are due to the general and important phenomenon of cycle leakage, which we’ll tackle in a future post. For now, think of cycle leakage as akin to spectral leakage in a normal spectrum analysis setting: the PSD estimate for frequencies near to a powerful sine wave signal will be inflated due to proximity to the sine wave and the imperfect frequency response of the estimator.
So, if the SSCA is applied to a new signal, whose cycle frequencies are not yet known, it is clear from the two graphs above that it could be used to blindly find the significant cycle frequencies, simply by thresholding the spectral correlation magnitude. However, there is a better way to sort through the large number of SSCA-produced spectral correlation point estimates, and that is by using the spectral coherence.
Computing and Using the Coherence in the SSCA
Recall from the post on the coherence that it is simply a normalized version of the spectral correlation, in exactly the same way that the correlation coefficient from statistics is a normalized version of the correlation. The non-conjugate coherence is given by
and the conjugate coherence is
The SSCA produces the values needed for the numerators of the coherences; all we need is the power spectrum values in the denominators. In principle, these can be obtained from the values of the power spectrum automatically produced during SSCA computation. However, it is often better to use a more finely sampled PSD estimate, such as can be obtained through use of the frequency-smoothing method or time-smoothing method of spectral correlation estimation, with the cycle frequency set to zero.
Once the coherence is computed in the SSCA implementation, it can be thresholded to yield the significant cycle frequencies. Moreover, a cycle-frequency binning operation can be applied to those cycle frequencies that pass threshold, limiting the output of the algorithm to only unique values of cycle frequency. For our rectangular-pulse BPSK signal, the top non-conjugate coherence values correspond to the following pairs:
and for the conjugate coherence, we obtain
The false cycle frequencies do not pass through the thresholding process when the coherence is used!
After coherence-based thresholding and cycle-frequency binning, we obtain the following output for the non-conjugate cycle frequencies of our textbook rectangular-pulse BPSK signal:
Frequency Cycle Frequency Coherence SCF
and for the conjugate cycle frequencies:
Frequency Cycle Frequency Coherence SCF
Final Example: Captured CDMA-EVDO
There is an EVDO downlink signal at MHz where I live, and I have recorded several seconds of it for analysis purposes. A typical PSD for the captured signal is shown here:
After applying the SSCA to this EVDO data, we obtain the non-conjugate and conjugate cycle frequency sets shown in the following graphs:
We conclude that the signal has five strong non-conjugate cyclic features for cycle frequencies that are harmonics of MHz (not MHz), and no significant conjugate cyclic features. The CDMA chip rate of MHz is also present, but has smaller spectral correlation and spectral coherence than the feature at MHz. This kind of analysis illustrates the power of the SSCA: Efficient (fast) blind estimation of all second-order cycle frequencies.