## SPTK: The Fourier Series

This installment of the Signal Processing Toolkit shows how the Fourier series arises from a consideration of representing arbitrary signals as vectors in a signal space. We also provide several examples of Fourier series calculations, interpret the Fourier series, and discuss its relevance to cyclostationary signal processing.

## On Impulsive Noise, CSP, and Correntropy

I’ve seen several published and pre-published (arXiv.org) technical papers over the past couple of years on the topic of cyclic correntropy (The Literature [R123-R127]). I first criticized such a paper ([R123]) here, but the substance of that review was about my problems with the presented mathematics, not impulsive noise and its effects on CSP. Since the papers keep coming, apparently, I’m going to put down some thoughts on impulsive noise and some evidence regarding simple means of mitigation in the context of CSP. Preview: I don’t think we need to go to the trouble of investigating cyclic correntropy as a means of salvaging CSP from the clutches of impulsive noise.

## On The Shoulders

What modest academic success I’ve had in the area of cyclostationary signal theory and cyclostationary signal processing is largely due to the patient mentorship of my doctoral adviser, William (Bill) Gardner, and the fact that I was able to build on an excellent foundation put in place by Gardner, his advisor Lewis Franks, and key Gardner students such as William (Bill) Brown.

## Can a Machine Learn a Power Spectrum Estimator?

I continue with my foray into machine learning (ML) by considering whether we can use widely available ML tools to create a machine that can output accurate power spectrum estimates. Previously we considered the perhaps simpler problem of learning the Fourier transform. See here and here.

Along the way I’ll expose my ignorance of the intricacies of machine learning and my apparent inability to find the correct hyperparameter settings for any problem I look at. But, that’s where you come in, dear reader. Let me know what to do!

## CSP Estimators: The FFT Accumulation Method

Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper $\displaystyle (f, \alpha)$ values.

## ‘Can a Machine Learn the Fourier Transform?’ Redux, Plus Relevant Comments on a Machine-Learning Paper by M. Kulin et al.

I first considered whether a machine (neural network) could learn the (64-point, complex-valued)  Fourier transform in this post. I used MATLAB’s Neural Network Toolbox and I failed to get good learning results because I did not properly set the machine’s hyperparameters. A kind reader named Vito Dantona provided a comment to that original post that contained good hyperparameter selections, and I’m going to report the new results here in this post.

Since the Fourier transform is linear, the machine should be set up to do linear processing. It can’t just figure that out for itself. Once I used Vito’s suggested hyperparameters to force the machine to be linear, the results became much better:

## Computational Costs for Spectral Correlation Estimators

Let’s look at the computational costs for spectral-correlation analysis using the three main estimators I’ve previously described on the CSP Blog: the frequency-smoothing method (FSM), the time-smoothing method (TSM), and the strip spectral correlation analyzer (SSCA).

We’ll see that the FSM and TSM are the low-cost options when estimating the spectral correlation function for a few cycle frequencies and that the SSCA is the low-cost option when estimating the spectral correlation function for many cycle frequencies. That is, the TSM and FSM are good options for directed analysis using prior information (values of cycle frequencies) and the SSCA is a good option for exhaustive blind analysis, for which there is no prior information available.

## Resolution in Time, Frequency, and Cycle Frequency for CSP Estimators

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 $T$ seconds of data.

Then the temporal resolution $\Delta t$ of the estimate is approximately $T$, the cycle-frequency resolution $\Delta \alpha$ is about $1/T$, and the spectral resolution $\Delta f$ depends strongly on the particular estimator and its parameters. The resolution product $\Delta f \Delta t$ 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.

## CSP Estimators: Cyclic Temporal Moments and Cumulants

In this post we discuss ways of estimating $n$-th order cyclic temporal moment and cumulant functions. Recall that for $n=2$, cyclic moments and cyclic cumulants are usually identical. They differ when the signal contains one or more finite-strength additive sine-wave components. In the common case when such components are absent (as in our recurring numerical example involving rectangular-pulse BPSK), they are equal and they are also equal to the conventional cyclic autocorrelation function provided the delay vector is chosen appropriately.

The more interesting case is when the order $n$ is greater than $2$. Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order $n$ greater than $2$ is $4$. Our estimation task is to estimate $n$-th order temporal moment and cumulant functions for $n \ge 4$ using a sampled-data record of length $T$.

## Automatic Spectral Segmentation

In this post, I discuss a signal-processing algorithm that has almost nothing to do with cyclostationary signal processing. Almost. The topic is automated spectral segmentation, which I also call band-of-interest (BOI) detection. When attempting to perform automatic radio-frequency scene analysis (RFSA), we may be confronted with a data block that contains multiple signals in a large number of distinct frequency subbands. Moreover, these signals may be turning on an off within the data block. To apply our cyclostationary signal processing tools effectively, we would like to isolate these signals in time and frequency to the greatest extent possible using linear time-invariant filtering (for separating in the frequency dimension) and time-gating (for separating in the time dimension). Then the isolated signal components can be processed serially.

It is very important to remember that even perfect spectral and temporal segmentation will not solve the cochannel-signal problem. It is perfectly possible that an isolated subband will contain more that one cochannel signal.

The basics of my BOI-detection approach are published in a 2007 conference paper (My Papers [32]). I’ll describe this basic approach, illustrate it with examples relevant to RFSA, and also provide a few extensions of interest, including one that relates to cyclostationary signal processing.

## Comments on “Blind Cyclostationary Spectrum Sensing in Cognitive Radios” by W. M. Jang

I recently came across the 2014 paper in the title of this post. I mentioned it briefly in the post on the periodogram. But I’m going to talk about it a bit more here because this is the kind of thing that makes things a bit harder for people trying to learn about cyclostationarity, which eventually leads to the need for something like the CSP Blog.

The idea behind the paper is that it would be nice to avoid the need for prior knowledge of cycle frequencies when using cycle detectors or the like. If you could just compute the entire spectral correlation function, then collapse it by integrating (summing) over frequency $f$, then you’d have a one-dimensional function of cycle frequency $\alpha$ and you could then process that function inexpensively to perform detection and classification tasks.

## Cyclic Polyspectra

In this post we take a first look at the spectral parameters of higher-order cyclostationarity (HOCS). In previous posts, I have introduced the topic of HOCS and have looked at the temporal parameters, such as cyclic cumulants and cyclic moments. Those temporal parameters have proven useful in modulation classification and parameter estimation settings, and will likely be an important part of my ultimate radio-frequency scene analyzer.

The spectral parameters of HOCS have not proven to be as useful as the temporal parameters, unless you include the trivial case where the moment/cumulant order is equal to two. In that case, the spectral parameters reduce to the spectral correlation function, which is extremely useful in CSP (see the TDOA and signal-detection posts for examples).

## 100-MHz Amplitude Modulation? Comments on “Sub-Nyquist Cyclostationary Detection for Cognitive Radio” by Cohen and Eldar

I came across a paper by Cohen and Eldar, researchers at the Technion in Israel. You can get the paper on the Arxiv site here. The title is “Sub-Nyquist Cyclostationary Detection for Cognitive Radio,” and the setting is spectrum sensing for cognitive radio. I have a question about the paper that I’ll ask below.

## Signal Processing Operations and CSP

It is often useful to know how a signal processing operation affects the probabilistic parameters of a random signal. For example, if I know the power spectral density (PSD) of some signal $x(t)$, and I filter it using a linear time-invariant transformation with impulse response function $h(t)$, producing the output $y(t)$, then what is the PSD of $y(t)$? This input-output relationship is well known and quite useful. The relationship is

$\displaystyle S_y^0(f) = \left| H(f) \right|^2 S_x^0(f). \hfill (1)$

In (1), the function $H(f)$ is the transfer function of the filter, which is the Fourier transform of the impulse-response function $h(t)$.

Because the mathematical models of real-world communication signals can be constructed by subjecting idealized textbook signals to various signal-processing operations, such as filtering, it is of interest to us here at the CSP Blog to know how the spectral correlation function of the output of a signal processor is related to the spectral correlation function for the input. Similarly, we’d like to know such input-output relationships for the cyclic cumulants and the cyclic polyspectra.

Another benefit of knowing these CSP input-output relationships is that they tend to build insight into the meaning of the probabilistic parameters. For example, in the PSD input-output relationship (1), we already know that the transfer function at $f = f_0$ scales the input frequency component at $f_0$ by the complex number $H(f_0)$. So it makes sense that the PSD at $f_0$ is scaled by the squared magnitude of $H(f_0)$. If the filter transfer function is zero at $f_0$, then the density of averaged power at $f_0$ should vanish too.

So, let’s look at this kind of relationship for CSP parameters. All of these results can be found, usually with more mathematical detail, in My Papers [6, 13].

## Square-Root Raised-Cosine PSK/QAM

Let’s look at a somewhat more realistic textbook signal: The PSK/QAM signal with independent and identically distributed symbols (IID) and a square-root raised-cosine (SRRC) pulse function. The SRRC pulse is used in many practical systems and in many theoretical and simulation studies. In this post, we’ll look at how the free parameter of the pulse function, called the roll-off parameter or excess bandwidth parameter, affects the power spectrum and the spectral correlation function.

## CSP Estimators: The Strip Spectral Correlation Analyzer

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.