# CSP Estimators: The Frequency-Smoothing Method

In this post I describe a basic estimator for the spectral correlation function (SCF): the frequency-smoothing method (FSM). The FSM is a way to estimate the SCF for a single value of cycle frequency. Recall from the basic theory of the cyclic autocorrelation and SCF that the SCF is obtained by infinite-time averaging of the cyclic periodogram or by infinitesimal-resolution frequency averaging of the cyclic periodogram. The FSM is merely a finite-time/finite-resolution approximation to the SCF definition.

One place the FSM can be found is in (My Papers [6]), where I introduce time-smoothed and frequency-smoothed higher-order cyclic periodograms as estimators of the cyclic polyspectrum. When the cyclic polyspectrum order is set to $n = 2$, the cyclic polyspectrum becomes the spectral correlation function, so the FSM discussed in this post is just a special case of the more general estimator in [6, Section VI.B].

Let’s start by reviewing the FSM for power-spectrum measurement. That is, let’s begin by looking at an estimator for the power spectral density (PSD). We’ll use discrete time and discrete frequencies in this post, in contrast to My Papers [6], because almost everybody who might read this post will be interested in sampled data. Moreover, discrete frequencies lead to discrete cycle frequencies in the method, which causes a significant complication in practice. We’ll highlight that complication, and show how it can be mitigated.

### The Frequency-Smoothed Periodogram

The power spectrum estimate is obtained by smoothing the periodogram. The discrete Fourier transform (DFT) of $x(t)$ is defined by

$X(f) = \displaystyle \sum_{t=0}^{N-1} x(t) e^{-i2\pi f t}, \hfill (1)$

where $f = n/N$ for $n = 0,1, \ldots, N-1$. The periodogram is the normalized squared magnitude of the DFT,

$I(f) = \displaystyle\frac{1}{N} \left| X(f) \right|^2. \hfill (2)$

The periodogram is not a particularly good spectrum estimator when the data is random; it is quite useful when the data is nonrandom and contains periodic components. The ML estimator for the frequency of a tone in WGN is the maximum of the Fourier transform for all $f$, which is usefully approximated by the periodogram. For random data (like communication signals), the periodogram is erratic and does not converge to the true PSD even as $N\rightarrow\infty$.

The PSD can be accurately estimated by smoothing the periodogram,

$\hat{S}(f) = g(f) \otimes I(f), \hfill (3)$

where $g(f)$ is some unit-area pulse-like function, such as a rectangle. The smoothing implied by this convolution allows the estimate to converge to a biased (distorted) estimate of the true PSD. If $N$ is large and the width of $g(f)$ is small relative to the fluctuations over frequency in the true PSD, then the bias is minimized and the estimator converges to very nearly the true PSD. More explicitly, the FSM for PSD estimation is

$\hat{S}(f) = g(f) \otimes I(f) \hfill (4)$

$= \displaystyle \sum_\nu g(f-\nu) I(\nu) \hfill (5)$

$= \displaystyle\frac{1}{N} \displaystyle \sum_\nu g(f-\nu) \left| X(\nu) \right|^2 \hfill (6)$

$= \displaystyle\frac{1}{N} \displaystyle \sum_{n=0}^{N-1} g(f - n/N) \left| X(n/N) \right|^2, \hfill (7)$

for $f = m/N$. So to implement this FSM, we need only use a FFT algorithm and a general convolution algorithm capable of handling complex inputs. A FSM-based PSD estimate for our rectangular-pulse BPSK signal is shown here:

This estimate is formed from a data record with length $32,768$ samples and a rectangular smoothing window $g(f)$ with width $164$ frequency bins, which is about $0.005$ (normalized) Hz.

The frequency-smoothed periodogram method of spectrum estimation is referred to as the Daniell method after P. J. Daniell (The Literature [R42]).

### The Frequency-Smoothed Cyclic Periodogram

For the spectral correlation function, we modify the FSM for the PSD
by replacing the periodogram with the cyclic periodogram, which is defined
by

$I^\alpha(f) = \displaystyle\frac{1}{N} X(f + \alpha/2) X^*(f - \alpha/2), \hfill (8)$

where $\alpha$ is a cycle frequency of interest. Since $X(f)$ is a discrete-frequency function, the arguments $f \pm \alpha/2$ are valid only if $\pm \alpha/2 = k/N$. That is, the discrete-frequency function $X(f)$ can be shifted to the left and right only by multiples of the discrete frequency $1/N$. It may happen that the true cycle frequency does not satisfy this constraint. In practice, the closest discrete shifts are chosen for $\pm \alpha/2$. When the two frequencies do not correspond to multiples of $1/N$, then zero-padding the data prior to FSM estimation is advisable; this will allow a closer match to discrete frequencies. In general, the $+\alpha/2$ and $-\alpha/2$ shifts don’t have to be negatives of each other. The idea is to choose the shifts $a_1$ and $a_2$ such that $\left|\alpha - |a_1/N| - |a_2/N|\right|$ is minimized (My Papers [16]).

To estimate the SCF, then, we convolve the cyclic periodogram with a smoothing window as before,

$\hat{S}^\alpha (f) = g(f) \otimes I^\alpha(f) \hfill (9)$

$= \displaystyle\frac{1}{N} \displaystyle \sum_{n=0}^{N-1} g(f - n/N) X(n/N + a_1/N) X^*(n/N - a_2/N) \hfill (10)$

for $f = m/N$.  All that is required to implement this algorithm is an FFT, complex multiplication, and a general convolution algorithm (such as MATLAB’s conv.m or filter.m). The conjugate SCF is estimated by frequency smoothing the conjugate cyclic periodogram given by

$I_*^\alpha(f) = \displaystyle\frac{1}{N} X(f + \alpha/2) X(\alpha/2-f). \hfill (11)$

Applying this algorithm to our rectangular-pulse BPSK signal, we obtain the following estimates for the non-conjugate SCF:

The conjugate SCF estimates are:

To obtain these estimates, we used a data-record length of $32,768$ samples and a rectangular $g(f)$ with width $164$ frequency points, or about $0.005$ Hz. Recall the sampling frequency is set to unity here, the signal’s bit rate is $1/T_0 = 0.1$, and the carrier frequency is $f_c = 0.05$. The $32,768$-sample data record is zero-padded by a factor of two prior to forming the cyclic periodogram.

### Choice for the Frequency-Smoothing Window $g(f)$$g(f)$

We used the rectangular pulse $g(f)$ in our estimator examples for a good reason: computational cost. For arbitrary $g(f)$, a general-purpose convolution routine can be used to find the SCF estimate, but the convolutions can be expensive. Computational cost can be reduced by selecting only a subset of all possible output frequencies, and computing the convolution just for those particular spectral frequencies.

Arbitrary output frequencies can be accommodated, however, by using a rectangular smoothing window $g(f)$ because the convolution can be efficiently performed by a running sum over the cyclic periodogram. The estimates for successive spectral frequencies can be obtained by subtracting the cyclic-periodogram value that is no longer covered by the support of $g(f)$ (left edge of the window) and adding the one that is newly included in the support (right edge of the window).

In the final analysis, the FSM offers good control over spectral resolution, but requires an FFT operation on the entire data record. When the signal of interest has low signal-to-interference-and-noise ratio (SINR), the required data-record length can be very large. For non-real-time CSP applications running on modern general-purpose computers, this isn’t usually a problem. But when attempting to transition CSP to hardware, long complex-valued data vectors are problematic. For such cases, we can turn to the time-smoothing method (TSM) of SCF estimation, which we discuss in another post.

I'm a signal processing researcher specializing in cyclostationary signal processing (CSP) for communication signals. I hope to use this blog to help others with their cyclo-projects and to learn more about how CSP is being used and extended worldwide.

## 30 thoughts on “CSP Estimators: The Frequency-Smoothing Method”

1. AN says:

Hey! This is a great blog! Learning a lot here. Can you please provide MATLAB codes to generate these plots?
Thanks!

1. Thanks AN. The CSP Blog is a self-help blog. I don’t give out much code. Let me know if you get stuck, though, trying to write your own. That’s the only way you’ll really understand CSP.

2. Chen says:

I think you chose complex-valued waveform of BPSK signal to estimate the PSD and cyclic spectrum, right?
And how can I judge the simulation results of the FSM priodogram and cyclic priodogram of BPSK signal correct or not?
The value range of n in Equation (10) can not reach 0 and N. The reason is that for certain cycle frequency, “n+a1=0” , right?

1. Yes, the BPSK signal I use throughout the CSP Blog is complex-valued.

And how can I judge the simulation results of the FSM priodogram and cyclic priodogram of BPSK signal correct or not?

Do you mean how can you judge whether my plots are correct? Or your plots? My plots are consistent with theory in the cases where I know the theory, such as PSK and QAM signals. So you can use my plots in the BPSK posts (here and here) or the gallery posts (here and here) to compare with yours.

The value range of n in Equation (10) can not reach 0 and N. The reason is that for certain cycle frequency, “n+a1=0” , right?

Yes, when $\alpha \neq 0$ in the non-conjugate SCF, you cannot use all the indices in the sum over $n$ unless you are willing to view the FFT output $X$ as a periodic function (with period $N$). But generally you don’t want to do that, and the edges of the cyclic periodogram will contain zeros. The larger $\alpha$ becomes, the more zeros you’ll see at the ends (this also corresponds to the diamond shape of the support for the SCF).

1. Chen says:

But why the estimation performance of FSM method decreased as I increased the length of data record? The simulation parameters I chose were the same as yours.

1. It isn’t possible to substantively answer this question. Not enough information!

3. Chen says:

According to the equation (25) in Gardner’s paper “Measurement of Spectral Correlation”, I think the result of equation (3) should be divided by the frequency bins number of smoothing window. Do as what I advised, the FSM-based PSD matches well with the theoretical PSD for BPSK signal.

1. My (3) is a bit more general than Gardner’s (25), which smooths using a unit-area rectangle. My $g(f)$ can be any pulse-like unit-area window function, but I do use a rectangle a lot too. The factor you say is missing is actually embedded in the definition of $g(f)$: It has unit area. Therefore if it is a rectangle, it has height equal to the reciprocal of the width. I’ve added “unit area” to the post to clarify.

1. Mimi says:

The Figure 1 in this post is similar to what your code make_rect_bpsk.m. But you did not use the convolution after the FFT, but the plot looks same.

So my question is related to this: To estimate the SCF, we should take the FFT of non conjugate CAF and then we need to convolve this with a smoothing window g(f). Does it give any extra benefit of using the g(f)? What is a good size of g(f)?

Is the FFT of non conjugate CAF called the cyclic Periodogram?

1. The Figure 1 in this post is similar to what your code make_rect_bpsk.m. But you did not use the convolution after the FFT, but the plot looks same.

I don’t quite understand this. The plot in Figure 1 looks the same as what?

So my question is related to this: To estimate the SCF, we should take the FFT of non conjugate CAF and then we need to convolve this with a smoothing window g(f).

No, to estimate the SCF, take the FFT of the data, shift it up and down by $\alpha/2$, multiply, divide by the length of the data, then convolve the result with a smoothing window. See the posts on the frequency-smoothing and time-smoothing methods of spectral correlation estimation for details and examples.

Does it give any extra benefit of using the g(f)?

Yes. The main point of the Periodogram post is that if you don’t average (“smooth”) the periodogram in either time or frequency, you will have a highly erratic function of frequency. Note also that the reliability (variance) of a spectral correlation estimate is inversely proportional to the product of the data-block length and the frequency resolution. For the frequency-smoothing method, the frequency resolution is the width of $g(f)$. So, to generate a reliable (low variance) estimate, you want to use large data-block lengths and large (coarse) frequency resolution (wide $g(f)$).

What is a good size of g(f)?

A good width for $g(f)$ is typically 1-5% of the sampling rate.

Is the FFT of non conjugate CAF called the cyclic Periodogram?

No. The cyclic periodogram is defined in (3) and (5) in the Periodogram post. The Fourier transform of the non-conjugate CAF is the non-conjugate spectral correlation function.

1. Mimi says:

Thanks for the reply. I meant, the Figure 1 in this post is similar to what your code make_rect_bpsk.m. generates.

2. Mimi says:

Thanks for the reply. I need some more clarification. You said “To estimate the SCF, take the FFT of the data, shift it up and down by \alpha/2, multiply, divide by the length of the data”.

1. Shift it up and down by alpha/2: Is it not giving back the same matrix if I shift it up by alpha/2 and then down by the same amount. Also is this shifting the same what fftshift does? If not same, then what this shift up and down by alpha/2 is doing here?

2. Then you said “multiply”: what do I multiply with what? I understood the divide by the length of the data.

3. Is the cyclic periodogram calculated as: FT[x(t)*exp(j*2*pi*alpha/2*t)] * conj{FT[x(t)*exp(-j*2*pi*alpha/2*t)]}

3. 1. Shift it up and down by alpha/2: Is it not giving back the same matrix if I shift it up by alpha/2 and then down by the same amount.

Well, you shift the FFT up by $\alpha/2$ to get $X(f-\alpha/2)$ and shift it down by $\alpha/2$ to get $X(f+\alpha/2)$. Those are the two quantities you need to multiply together to get the cyclic periodogram.

Also is this shifting the same what fftshift does?

No. MATLAB’s fftshift.m swaps the second half of a vector for the first half: fftshift([1 2 3 4]) = [3 4 1 2]. This is identical to circshift([1 2 3 4], 2). The shifting we need to do in CSP depends on the value of the cycle frequency $\alpha$.

If not same, then what this shift up and down by alpha/2 is doing here?

The mathematics of CSP says that you can estimate the spectral correlation function by averaging the cyclic periodogram, which is the product of two frequency-shifted Fourier transforms. More physically, the spectral correlation function is the temporal correlation between two narrowband spectral components of a signal, after those components have been isolated and frequency-shifted to zero frequency. And those two frequency components have frequencies specified by $f_1 = f + \alpha/2$ and $f_2 = f - \alpha/2$ in the usual parameterization.

2. Then you said “multiply”: what do I multiply with what? I understood the divide by the length of the data.

See answer to Question 1 above.

3. Is the cyclic periodogram calculated as: FT[x(t)*exp(j*2*pi*alpha/2*t)] * conj{FT[x(t)*exp(-j*2*pi*alpha/2*t)]}

I’m not sure about the plus and minus signs in the exp() functions, and you’ll need to divide by length(x(t)), but, yes, this is one way to obtain the cyclic periodogram. I typically don’t use this method, because if you’re careful, you can do the frequency shifting in the frequency domain using only things like circshift.m, which is much cheaper than creating the complex sine waves and multiplying them by the signal x(t) and taking two (not one) Fourier transforms.

4. Chen says:

I find an interesting phenomenon. To meet the condition that the approximated value, obtained by integrating PSD values over frequency and multiplying that sum by the frequency increment, must be equal to the estimate of power. Therefore, different PSD estimating methods have different frequency increment and different PSD value. For example, periodogram based PSD values are much greater than that based on TSM and FSM method.

1. Chen says:

How do you explain that phenomenon?

1. Therefore, different PSD estimating methods have different frequency increment and different PSD value.

Yes, the different methods can have different frequency-bin increments as well as different spectral resolutions. This means that for any particular given frequency $f$, the PSD estimates from the various methods will likely have different values. But for properly set up measurements (the measurement spectral resolution is well-matched to the data’s spectral characteristics, the data-block length is sufficient to allow averaging away erratic components), the values will be quite similar, as you can see from my comparisons between the FSM and TSM outputs.

And the sum of the PSD estimate values, multiplied by the frequency increment, should be very close for each method.

For example, periodogram based PSD values are much greater than that based on TSM and FSM method.

Yes, the periodogram is highly erratic, and is a terrible estimator of the PSD, so there will be periodogram values that are very much greater than the true PSD value and periodogram values that are very much less than the true PSD value–that’s why we need the averaging.

5. Chen says:

Convolving the periodogram of x(t) with g(f) would obtain PSD estimate with length of N+164-1. So what is the frequency increment between the adjacent frequency bins of that PSD estimates? Is the frequency increment fs/(N+164-1)?

1. This is a convolution question, not a CSP question! The spacing between the samples does not change as a result of convolution.

6. Smoon says:

The posts are very helpful for the beginners like me. It would be better if you supply more detailed unit labels on the figures. I think the unit of the nomalized frequency is [cycle/sample], and of the PSD is [dB/(rad/sample)]. I am not sure whether the units I added here is correct? Or maybe it is a traditional label manner in the CPS society.

1. Thanks for visiting the CSP Blog and leaving a comment Smoon! I appreciate it.

I agree with you about normalized frequency–strictly speaking I should label axes with [Cycles/Sample] or the like. But when the actual physical sampling rate really is $1.0$, then the unit is also Cycles/Second, or Hz. Mostly I’m following convention, which is to use ‘Hz’ for both physical frequency and normalized frequency. To translate any frequency axis from normalized frequency to physical frequency, just multiply by the actual sampling rate.

And yes, the physical unit of a power spectrum is Watts/Hz. Normally expressed in decibels, but often in decibels relative to some basic power level, such as one Watt (dBW) or 1 milli Watt (dBm). In almost all of my posts, I’m unconcerned with the actual power level of the data–even when I capture data using my lab equipment I don’t keep track of the relationship between the integers I obtain from a sampler and the power level of the electromagnetic wave incident on the attached antenna. So I rather lazily just use decibels and the relationship to actual physical power is suppressed.

What is the ‘CPS society’?

7. Clint says:

This feels like a dumb question, but I’m completely failing to re-create your plots for the conjugate SCF. (Using what should be identical to your textbook BPSK signal example.)

I have another function that calls the function below and convolves a smoothing window with the result. The results match your FSM plots quite well for the non-conjugate SCF, but not for the conjugate version.

I’ve tried computing the SCF for all possible values of alpha for the signal under consideration, and none of the results match your plot for the conjugate SCF; in fact, they’re all pretty noisy and remain below 0dB.

I’ve also implemented the time-smoothing method, and I can replicate your plots for the non-conjugate SCF, but again not for the conjugate SCF. (I still have a bug somewhere such that my phase compensation factor only works when alpha*N is an integer, but I’m working on fixing my frequency-smoothed conjugate SCF before going back to that.)

Below is my python function computing the cyclic periodogram; my understanding is that the only difference is whether X2 should have the conjugate operation applied (yes for the non-conjugate SCF, no for the conjugate SCF).

def cyclic_periodogram(x, alpha=0, conjugate=False):
“””Compute the cyclic periodogram from the given signal x

I^A(f) = 1/N * X(f + alpha/2) * conj(X(f – alpha/2))
where X(f) is the Fourier transform of x(n) and x(n) has N total samples

Assumes that alpha is in normalized frequency; i.e. alpha = alpha_absolute/sampling_frequency

Note that the conjugate cyclic periodogram does not include the conjugation operation in its definition:
I^A_*(f) = 1/N * X(f + alpha/2) * X(f – alpha/2)
“””

X = numpy.fft.fftshift(numpy.fft.fft(x))

# shift X left by # of bins corresponding to alpha/2 and another copy right by the same amount
# eventually can slightly improve the resolution by allowing slightly different left/right shifts
offset = int(numpy.round(alpha/2 * len(X)))
X1 = numpy.roll(X, offset)
X2 = numpy.roll(X, -offset)

if not conjugate:
X2 = numpy.conj(X2)
print(f’Computing non-conjugate cyclic periodogram for alpha = {alpha}, offset = {offset}’)
else:
print(f’Computing conjugate cyclic periodogram for alpha = {alpha}, offset = {offset}’)

return 1/len(X) * X1 * X2

Can you see something I’m missing here? Except for the conjugation of X2 (and the values of alpha where the SCF is meaningful), all my code is identical for the non-conjugate and conjugate SCF, and it matches your plots perfectly for the non-conjugate case.

Thanks for any help!

1. Clint says:

Ugh, it got rid of all the indentation, making it much less readable. I’ll clarify if needed…

2. Clint says:

To clarify one point, – when I tried to estimate the conjugate SCF for “all possible values of alpha” I did reduce the total length from 32,768 to 4,096 samples. My understanding is that that should just increase the variance of the resulting SCF estimate, but shouldn’t fundamentally change the behavior.

Also, there’s raw LaTeX in this post, as well as the post for the TSM.

1. Thanks very much for pointing out the raw latex! Fixed it.

Yes, reducing the amount of processed data does increase the variance of the estimate, and also impacts the all-important cycle-frequency resolution parameter. Which means that one may also experience cycle leakage when the block length is too small–energy from a nearby slice of the spectral correlation function can leak into the slice you are focusing on. So, variance and leakage are both increased in general.

3. Hey Clint! Sorry about the loss of the indentation when you posted your code. When I comment on a comment, I have several markup tool buttons available, including “code,” which allows posting of code that should be formatted reasonably by WordPress. Do you see that when you comment?

One major problem you are having is your statement:

my understanding is that the only difference is whether X2 should have the conjugate operation applied (yes for the non-conjugate SCF, no for the conjugate SCF).

That is not the only difference; see Equation (11) in the FSM post and Equation (8) in the SCF post. Instead of the non-conjugate cyclic periodogram

$\frac{1}{T} X_T(t, f+\alpha/2) X_T^*(f-\alpha/2)$

you remove the conjugate and negate the argument of the second factor

$\frac{1}{T} X_T(t, f+\alpha/2) X_T(\alpha/2 -f )$

Try implementing that and re-running the estimates.

8. Clint says:

Thanks for clarifying, Chad, in spite of my apparent inability to read equations.

I don’t see any formatting/markup options when commenting; just raw text.

I don’t have it working yet, although I have something implemented that at least corrects for the problem you mentioned, as well as limiting the domain of the SCF by padding with zeros beyond f = +/-0.5 (my previous code treated the X(t, f) as being infinitely periodic in frequency, though I came across this comment – https://cyclostationary.blog/2018/06/01/csp-estimators-the-fft-accumulation-method/#comment-2131 – which makes me think that you had a discussion of that with plots of the SCF domain elsewhere, though I can’t remember where at this point. Is there a way to view a list of all blog posts, instead of either searching (and hoping I stumble upon the right term) or viewing the archives a month at a time? (Those are the only options I’ve found, other than just going through articles sequentially.)

1. I haven’t yet published a post on the spectral correlation “principal domain,” but I may have a plot or two showing the diamond or part of it. Perhaps in the SSCA post.

I’ve created a new page for the CSP Blog. Look at the top of the home page for “All CSP Blog Posts” and click that to get a simple list, in chronological order, of all posts. Is that what you were looking for? Once I hear back from you about whether or not that is what you desired, and perhaps suggestions for improvement, I’ll do a little post announcing it. You aren’t the first to suggest the navigation on the site is cumbersome.

Thanks Clint!

1. Clint says:

The only additional suggestion I might have would be a list of all tags, but that would be somewhat redundant with the list of categories, and thus much less useful.

2. Clint says:

Got to work on this some more today. Finally got it working!

Turns out the remaining problem was an off-by-one error in the indexing for the conjugate SCF case. Now I just need to look at it in more detail to be sure of _why_ the indexing needs to be the way it is.

Thanks again for the help Chad!

1. Clint says:

In case it helps others trying to implement this, I’ll describe what I was seeing before I found my bug.

My problematic curves were generally the same shape as Chad’s, but 10-15dB lower and they had fairly high variance/noise. After thinking about cycle leakage, I wondered if my offset was close but not quite right, which turned out to be the case.

Consistent with Chad’s comments above about cycle leakage and block length, the level of my curves went down as I increased the total length of random data that I was processing.