In this post, we look at the ability of various CSP estimators to distinguish cycle frequencies, temporal changes in cyclostationarity, and spectral features. These abilities are quantified by the resolution properties of CSP estimators.
Resolution Parameters in CSP: Preview
Consider performing some CSP estimation task, such as using the frequency-smoothing method, time-smoothing method, or strip spectral correlation analyzer method of estimating the spectral correlation function. The estimate employs seconds of data.
Then the temporal resolution of the estimate is approximately , the cycle-frequency resolution is about , and the spectral resolution depends strongly on the particular estimator and its parameters. The resolution product was discussed in this post. The fundamental result for the resolution product is that it must be very much larger than unity in order to obtain an SCF estimate with low variance.
What do we usually mean by spectral resolution as the term is used in conventional spectrum analysis? It sums up the ability of the estimator to display distinct responses to data components that have distinct spectra. That is somewhat abstract, so spectral resolution is often described more concretely as the ability of the estimator to reveal the presence of two closely spaced spectral components from, say, two equal-strength sine waves at the estimator input.
Spectral Resolution in the Periodogram
The estimators I’m interested in are variants on the two basic non-parametric spectrum estimators that apply either time smoothing (temporal averaging) or frequency smoothing (spectral averaging) to the periodogram. The former is called the time-smoothing method and the latter the frequency-smoothing method of spectrum estimation, although there are the alternate names involving Proper Nouns that some people prefer: the Bartlett and Danielle methods.
Both methods take as the starting point a periodogram. In the case of the time-smoothing method, multiple periodograms are computed and averaged over time, so that the spectral resolution of the result must be the spectral resolution of the involved periodograms. In the case of the frequency-smoothing method, a single periodogram is convolved with a pulse-like smoothing kernel, so that the spectral resolution cannot be smaller than the native spectral resolution of the involved periodogram, and is equal to the width of the pulse-like kernel. So we must start with understanding the spectral resolution properties of the periodogram.
Let’s review the periodogram. First, we define the sliding (over time) finite-time Fourier transform as
The periodogram is the scaled magnitude-square of this quantity:
What is the frequency resolution of this estimator? Since it integrates over a total of seconds, you might guess it is approximately . Let’s find out.
Consider a noiseless input that consists of a single complex-valued sine wave with frequency , amplitude , and phase :
What is the periodogram for this input? Let’s first calculate the sliding Fourier transform ,
If then Otherwise, we evaluate the integral
Now, we can use Euler’s formula to simplify the term in the square brackets,
Application of (9) in (8) leads to
We recognize the function in the square brackets as the sinc function, which is defined by
so that our time-varying Fourier transform is given by
For , , so the result just above (6) is consistent with (13), meaning (13) is the correct result for all , including . The magnitude of this Fourier transform is plotted in the following figure:
The figure indicates that the transform peaks at the correct frequency, but that the transform is also non-zero for a great many other frequencies–frequencies that are not present in the tone itself. This frequency response arises due to the finite-time window of the transform, because we know that the Fourier transform (which is taken over all time) for the noiseless tone is an impulse function centered at .
Now, the periodogram is given by
so that the periodogram for our noiseless tone is simply
This periodogram is plotted in the following figure:
We see that the width of the main lobe of this function is .
Now, let’s get back to talking about spectral resolution. What is the periodogram for the case of a signal consisting of two tones? The model is
The Fourier integral is linear, so we can immediately generalize our previous result for to
and we can use this expression for in the definition of the periodogram to obtain
We see there is a sinc function centered at and another one centered at . But there is also a term representing a mixture of the two sinc functions. The question becomes: Under what conditions are the sinc-peaks for the two frequencies and separable (distinguishable)?
In the case of the centers of the two squared sinc functions are separated by more than the width of the main lobe of each one, and so should be distinguishable. On the other hand, when the centers of the two squared sincs are close relative to their widths, and there is no hope of distinguishing their individual contributions.
What about inbetween?
The squared-sinc terms represent the responses to each tone as if it were the only tone present, and do not depend on time . The third term is a mixture of responses to the two tones, and depends on The worst case, in the context of spectral resolution, is when and , so that the interference term becomes
The best case is when and are purely imaginary, in which case the term is zero and we’re left with the sum of the two individual responses.
We can numerically evaluate these two extremes and plot the results to determine, approximately, the resolvability of the two tones as a function of their difference and the observation interval length . In the following, and , so that the difference between the two frequencies is , which is important because we consider in our periodogram computations.
First, the best-case version of the periodogram formula (19):
Here we see that two distinct peaks in the periodogram are visible for , which is about the difference between the two frequencies, indicating that the best-case periodogram for seconds can resolve tones with frequency separation of about . For smaller than , we see only a single bump in the graph–no way to tell there are two tones present so they are unresolved.
The worst-case version of formula (19) is computed and graphed in the following figure:
Here the two tones show two distinct peaks for rather than , indicating that the resolution of the worst-case periodogram is a little worse than that for the best case. Quelle surprise.
Summing up, the resolution of the periodogram is about , but might be as bad as about .
Spectral Resolution in Non-Parametric Spectrum Estimators
Now that we know something about the spectral resolution of the periodogram, and since we know how the periodogram is related to important spectrum estimators, we can discuss the spectral resolution of these estimators.
For the frequency-smoothing method of PSD estimation, the periodogram is convolved with a smoothing function , which has unit area and width . If is greater than the native periodogram resolution , then the resolution of the PSD estimator will be approximately .
For the time-smoothing method of PSD estimation, multiple short-time periodograms are averaged over time. If blocks of data, each having duration are averaged, then the spectral resolution is the spectral resolution of the periodograms, or about . We can force the frequency-smoothing and time-smoothing methods to produce similar estimates by using in the TSM and in the FSM.
Spectral Resolution in the Cyclic Periodogram
It is probably intuitive for my readers to see that the spectral resolution of the cyclic periodogram is about equal to the spectral resolution of the periodogram. In fact, the (non-conjugate) cyclic periodogram is identically equal to the periodogram for , in which case the resolution properties match exactly.
Spectral Resolution in Non-Parametric Spectral Correlation Estimators
Like the spectral estimators, the spectral correlation estimators will have spectral resolution that is determined by the averaging that is applied to the cyclic periodogram. For the frequency-smoothing method, it will be approximately the width of the smoothing kernel function, and for the time-smoothing method, it will be the reciprocal of the periodogram length .
But let’s verify and illustrate these claims before leaving the topic of spectral resolution and moving to temporal and cycle-frequency resolution. Here it is harder to illustrate the situation using tone inputs (why?), so we look at a signal that has significant variation over frequency for both its spectrum and its spectral correlation function: the direct-sequence spread-spectrum BPSK signal. Let’s consider a DSSS BPSK signal with chip rate (normalized frequencies are used, which is typical on the CSP Blog), processing gain (code length) of , and a square-root raised-cosine pulse function. The data rate is .
We start with estimates of the PSD, the chip-rate non-conjugate SCF, and the data-rate non-conjugate SCF using a very long version of the signal and a very small smoothing-kernel width in the FSM. This produces a set of reference (high-fidelity) estimates useful for later comparison with other estimates. Here are the reference estimates:
Note here that the three SCF estimates have several very narrow spectral features (nulls). Unlike the case of the periodogram, where we looked at the ability to distinguish two peaks in the spectrum, here we look at the ability to accurately estimate a narrow valley. Either way, we are trying to resolve something.
First, let’s look at a sequence of FSM estimates, where we vary the width of the smoothing kernel :
By studying these plots, you can see that the basic claim is verified: The spectral resolution is approximately equal to the width of the smoothing kernel. That is probably unsurprising and somewhat obvious to most readers, who are likely quite familiar with convolution and its smearing/smoothing effects. Let’s repeat this experiment, then, with the TSM, which employs no convolution at all:
Temporal Resolution in Non-Parametric CSP Estimators
The temporal resolution quantifies the ability of the estimate to distinguish statistically distinct probabilistic parameters over time. In other words, if the signal changes its character over time, can the estimator reflect that?
Temporal resolution is a little hard to talk about in the context of our usual definitions of parameters such as the cyclic autocorrelation or spectral correlation function, because those definitions involve averaging over an infinite amount of time. If we consider finite-time estimates, and also consider a sequence of such estimates arising from processing a sequence of non-overlapping adjacent contiguous data blocks, then we can consider the ability of the sequence of estimates to resolve any changes in the parameter over time.
From this point of view, it seems clear that the temporal resolution $\Delta t$ is simply the block length used in creating each estimate in the sequence of estimates. Temporal variations happening faster than a block length will tend to be obscured by the averaging over the block’s samples, whereas temporal variations happening slower than a block length have the chance to be resolved by comparing the algorithm outputs for the different blocks.
Cycle-Frequency Resolution in the Context of the Cyclic Autocorrelation Function
Now let’s turn to resolution in cycle frequency. In particular, let’s first look at the cyclic autocorrelation function. I’ll be focusing on the non-conjugate CAF, but all of what follows is applicable to the conjugate CAF too.
Recall that the non-conjugate CAF is defined as a Fourier coefficient in the Fourier series expansion of the time-variant autocorrelation function. It is also equal to the following limit
A natural and effective estimator is simply the finite-time version of this idealized average.
The lag product must have a representation of the form
But this kind of representation must also be true for each cycle frequency corresponding to a non-zero cyclic autocorrelation function (21). Therefore the lag product must have a representation of the form
So an estimate of the cyclic autocorrelation function for cycle frequency using seconds of data is
Neglecting (because we’re interested in resolution not reliability here) by, say, assuming is large, we have
If , the integral is equal to , which is satisfying. Otherwise, let’s evaluate the integral. We have
Turning back to the CAF estimate, we now have
When , we have . This means the contributions from the other cycle frequencies are small when
So our first cycle-frequency resolution result is that individual components of the autocorrelation function can be distinguished from each other—resolved—provided that the minimum separation between cycle frequencies is greater than the reciprocal of the data-record length used to make the CAF estimates.
In particular, the cycle frequency of zero is always a non-conjugate cycle frequency (because we are always considering the class of signals called power signals). So to detect the presence of a small cycle frequency , we must process sufficient data to force the contribution from the nearby term to be small. The approximate amount of sufficient data in this case is .
Let’s illustrate the CAF cycle-frequency resolution with an example involving two cochannel BPSK signals. Their bit rates are quite close: and . Their power levels are equal and their carrier frequencies are and . Here is a PSD estimate for the observed sum of these two signals:
First, let’s look at the CAF maxima for a fixed block length of samples. Here we estimate the CAF for a set of uniformly spaced cycle frequencies with separation . The fixed value of is sufficient to resolve the two bit rate cycle frequencies, since their separation is larger than . However, this cycle-frequency resolution must also be accompanied by a correspondingly fine grid of cycle frequencies (unless the bit rates are known in advance, but let’s pretend they aren’t here). Once the grid spacing is approximately equal to the cycle-frequency resolution , both cycle frequencies are seen.
Next, consider variable , but fix the grid spacing at the cycle-frequency resolution, :
In each plot, we see at least one prominent feature. For greater than the separation between the two cycle frequencies, the two features get lumped together. Once is sufficiently small, the two cycle frequencies are resolved.
Cycle-Frequency Resolution in the Context of the Spectral Correlation Function
Let’s consider the time-smoothing method of spectral correlation estimation in discrete time and discrete frequency. The theoretical SCF can be obtained by first averaging over all time, and then increasing the periodogram block length without bound:
where is the closest number of FFT bins to the desired cycle frequency and is the cyclic periodogram.
(assume is even for simplicity in this exposition).
Now, for the double limit in (33) to hold true, the cyclic periodogram must have a representation of the form
But consider two distinct cycle frequencies and that map to the same sequence of cyclic periodograms This only requires that the difference between the two cycle frequencies is significantly smaller than the FFT bin width implied by .
The TSM spectral correlation function estimates for and converge to the proper SCFs, even though they use exactly the same sequence of cyclic periodograms because of the effective filtering performed by the multiplication by the complex sine wave for . So we have the representation
As a check, let’s verify that the double limit holds for and .
So, keeping in mind that , what is the function
equal to? We recognize this as a geometric series of the form
with . Such geometric series are easy to evaluate for finite ,
and by a symmetry argument the same is true for .
Now consider the general case where distinct cycle frequencies all map to the same sequence of cyclic periodograms in the TSM spectral correlation estimator. Our representation of the cyclic periodogram is then
The “noise” term does not relate to resolution, but to the variability (errors) in the estimate, so we will neglect it from now on. We have the cyclic periodogram representation
or, in terms of the desired spectral correlation estimate
If the combined contributions to from , , are small compared to , then we can say we’ve resolved the spectral correlation function for cycle frequency in the cycle-frequency dimension. So, when are those contributions small?
The contribution to from is dependent on the following geometric series
What is the width of this function? It is approximately , which is the total amount of processed data in the time-smoothing method under consideration, which we call :
So if this simplified analysis is correct, we once again see that the approximate cycle-frequency resolution for a second-order CSP parameter is the reciprocal of the data-block length used to form the estimate.
Let’s illustrate with some measurements.
First, we repeat the cyclic autocorrelation measurements above with the frequency-smoothing method:
Finally, we apply the strip spectral correlation analyzer to the single-BPSK and two-BPSK cases for various data-block lengths (recall that the two bit rates are and
There are multiple dots at the SSCA output locations because the spectral coherence can exceed threshold for multiple values of spectral frequency for each cycle frequency .
We see that once the block length is equal to samples, the two distinct cycle frequencies are detected, which is consistent with our approximate analysis since .
Resolution Properties for Second-Order Cyclostationarity: The Take-Away
If you use seconds of data to estimate the spectral correlation function, and your processing is a blind search for cycle frequencies, you must use a cycle-frequency search grid with grid fineness of about Hz. Otherwise, you run the risk of missing significant parts of the spectral correlation function.
Same for the cyclic autocorrelation function.
If you know a cycle frequency in advance, and want to detect its presence, then if the potential error in that knowledge exceeds , you may very well declare the absence of the feature when it is actually present. This becomes increasingly important as you increase to combat inband noise and interference.
Resolution Parameters in the Context of Higher-Order Cyclic Cumulants and Cyclic Polyspectra
This topic is advanced–it is much harder to analyze, understand, and illustrate the resolution properties of higher-order cyclic moments, cumulants, and polyspectra. So, we’ll leave it for a future post.
As always, I invite you to leave a comment if you have a question, correction, or relevant experience.