## SPTK: Frequency Response of LTI Systems

We continue our progression of Signal-Processing ToolKit posts by looking at the frequency-domain behavior of linear time-invariant (LTI) systems. In the previous post, we established that the time-domain output of an LTI system is completely determined by the input and by the response of the system to an impulse input applied at time zero. This response is called the impulse response and is typically denoted by $h(t)$.

## Symmetries of Higher-Order Temporal Probabilistic Parameters in CSP

In this post, we continue our study of the symmetries of CSP parameters. The second-order parameters–spectral correlation and cyclic correlation–are covered in detail in the companion post, including the symmetries for ‘auto’ and ‘cross’ versions of those parameters.

Here we tackle the generalizations of cyclic correlation: cyclic temporal moments and cumulants. We’ll deal with the generalization of the spectral correlation function, the  cyclic polyspectra, in a subsequent post. It is reasonable to me to focus first on the higher-order temporal parameters, because I consider the temporal parameters to be much more useful in practice than the spectral parameters.

This topic is somewhat harder and more abstract than the second-order topic, but perhaps there are bigger payoffs in algorithm development for exploiting symmetries in higher-order parameters than in second-order parameters because the parameters are multidimensional. So it could be worthwhile to sally forth.

## A Gallery of Cyclic Correlations

There are some situations in which the spectral correlation function is not the preferred measure of (second-order) cyclostationarity. In these situations, the cyclic autocorrelation (non-conjugate and conjugate versions) may be much simpler to estimate and work with in terms of detector, classifier, and estimator structures. So in this post, I’m going to provide plots of the cyclic autocorrelation for each of the signals in the spectral correlation gallery post. The exceptions are those signals I called feature-rich in the spectral correlation gallery post, such as LTE and radar. Recall that such signals possess a large number of cycle frequencies, and plotting their three-dimensional spectral correlation surface is not helpful as it is difficult to interpret with the human eye. So for the cycle-frequency patterns of feature-rich signals, we’ll rely on the stem-style (cyclic-domain profile) plots in the gallery post.

## 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!

## Data Set for the Machine-Learning Challenge

Update September 2020. I made a mistake when I created the signal-parameter “truth” files signal_record.txt and signal_record_first_20000.txt. Like the DeepSig RML data sets that I analyzed on the CSP Blog here and here, the SNR parameter in the truth files did not match the actual SNR of the signals in the data files. I’ve updated the truth files and the links below. You can still use the original files for all other signal parameters, but the SNR parameter was in error.

Update July 2020. I originally posted $20,000$ signals in the posted data set. I’ve now added another $92,000$ for a total of $112,000$ signals. The original signals are contained in Batches 1-5, the additional signals in Batches 6-28. I’ve placed these additional Batches at the end of the post to preserve the original post’s content.

I’ve posted $20000$ PSK/QAM signals to the CSP Blog. These are the signals I refer to in the post I wrote challenging the machine-learners. In this brief post, I provide links to the data and describe how to interpret the text file containing the signal-type labels and signal parameters.

### Overview of Data Set

The $20,000$ signals are stored in five zip files, each containing $4000$ individual signal files:

Batch 1

Batch 2

Batch 3

Batch 4

Batch 5

The zip files are each about 1 GB in size.

The modulation-type labels for the signals, such as “BPSK” or “MSK,” are contained in the text file:

signal_record_first_20000.txt

Each signal file is stored in a binary format involving interleaved real and imaginary parts, which I call ‘.tim’ files. You can read a .tim file into MATLAB using read_binary.m. Or use the code inside read_binary.m to write your own data-reader; the format is quite simple.

### The Label and Parameter File

Let’s look at the format of the truth/label file. The first line of signal_record_first_20000.txt is

1 bpsk  11  -7.4433467080e-04  9.8977795076e-01  10  9  5.4532617590e+00  0.0

which comprises $9$ fields. All temporal and spectral parameters (times and frequencies) are normalized with respect to the sampling rate. In other words, the sampling rate can be taken to be unity in this data set. These fields are described in the following list:

1. Signal index. In the case above this is `1′ and that means the file containing the signal is called signal_1.tim. In general, the $n$th signal is contained in the file signal_n.tim. The Batch 1 zip file contains signal_1.tim through signal_4000.tim.
2. Signal type. A string indicating the modulation format of the signal in the file. For this data set, I’ve only got eight modulation types: BPSK, QPSK, 8PSK, $\pi/4$-DQPSK, 16QAM, 64QAM, 256QAM, and MSK. These are denoted by the strings bpsk, qpsk, 8psk, dqpsk, 16qam, 64qam, 256qam, and msk, respectively.
3. Base symbol period. In the example above (line one of the truth file), the base symbol period is $T_0 = 11$.
4. Carrier offset. In this case, it is $-7.4433467080\times 10^{-4}$.
5. Excess bandwidth. The excess bandwidth parameter, or square-root raised-cosine roll-off parameter, applies to all of the signal types except MSK. Here it is $9.8977795076\times 10^{-1}$. It can be any real number between $0.1$ and $1.0$.
6. Upsample factor. The sixth field is an upsampling parameter U.
7. Downsample factor. The seventh field is a downsampling parameter D. The actual symbol rate of the signal in the file is computed from the base symbol period, upsample factor, and downsample factor: $\displaystyle f_{sym} = (1/T_0)*(D/U)$. So the BPSK signal in signal_1.tim has rate $0.08181818$. If the downsample factor is zero in the truth-parameters file, no resampling was done to the signal.
8. Inband SNR (dB). The ratio of the signal power to the noise power within the signal’s bandwidth, taking into account the signal type and the excess bandwidth parameter.
9. Noise spectral density (dB). It is always $0$ dB. So the various SNRs are generated by varying the signal power.

To ensure that you have correctly downloaded and interpreted my data files, I’m going to provide some PSD plots and a couple of the actual sample values for a couple of the files.

### signal_1.tim

The line from the truth file is:

1 bpsk  11  -7.4433467080e-04  9.8977795076e-01  10  9  5.4532617590e+00  0.0

The first ten samples of the file are:

-5.703014e-02   -6.163056e-01
-1.285231e-01   -6.318392e-01
6.664069e-01    -7.007506e-02
7.731103e-01    -1.164615e+00
3.502680e-01    -1.097872e+00
7.825349e-01    -3.721564e-01
1.094809e+00    -3.123962e-01
4.146149e-01    -5.890701e-01
1.444665e+00    7.358724e-01
-2.217039e-01   -1.305001e+00

An FSM-based PSD estimate for signal_1.tim is:

And the blindly estimated cycle frequencies (using the SSCA) are:

The previous plot corresponds to the numerical values:

Non-conjugate $(\alpha, C, S)$:

8.181762695e-02  7.480e-01  5.406e+00

Conjugate $(\alpha, C, S)$:

8.032470942e-02  7.800e-01  4.978e+00
-1.493096002e-03  8.576e-01  1.098e+01
-8.331298083e-02  7.090e-01  5.039e+00

### signal_4000.tim

The line from the truth file is

4000 256qam  9  8.3914849139e-04  7.2367959637e-01  9  8  1.0566301192e+01  0.0

which means the symbol rate is given by $(1/9)*(8/9) = 0.09876543209$. The carrier offset is $0.000839$ and the excess bandwidth is $0.723$. Because the signal type is 256QAM, it has a single (non-zero) non-conjugate cycle frequency of $0.098765$ and no conjugate cycle frequencies. But the square of the signal has cycle frequencies related to the quadrupled carrier:

### Final Thoughts

Is $20000$ waveforms a large enough data set? Maybe not. I have generated tens of thousands more, but will not post until there is a good reason to do so. And that, my friends, is up to you!

That’s about it. I think that gives you enough information to ensure that you’ve interpreted the data and the labels correctly. What remains is experimentation, machine-learning or otherwise I suppose. Please get back to me and the readers of the CSP Blog with any interesting results using the Comments section of this post or the Challenge post.

For my analysis of a commonly used machine-learning modulation-recognition data set (RML), see the All BPSK Signals post.

Batch 6

Batch 7

Batch 8

Batch 9

Batch 10

Batch 11

Batch 12

Batch 13

Batch 14

Batch 15

Batch 16

Batch 17

Batch 18

Batch 19

Batch 20

Batch 21

Batch 22

Batch 23

Batch 24

Batch 25

Batch 26

Batch 28

Signal parameters text file

## Comments on “Detection of Almost-Cyclostationarity: An Approach Based on a Multiple Hypothesis Test” by S. Horstmann et al

I recently came across the conference paper in the post title (The Literature [R101]). Let’s take a look.

The paper is concerned with “detect[ing] the presence of ACS signals with unknown cycle period.” In other words, blind cyclostationary-signal detection and cycle-frequency estimation. Of particular importance to the authors is the case in which the “period of cyclostationarity” is not equal to an integer number of samples. They seem to think this is a new and difficult problem. By my lights, it isn’t. But maybe I’m missing something. Let me know in the Comments.

## 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.

## 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 $T$.

## More on Pure and Impure Sine Waves

Remember when we derived the cumulant as the solution to the pure $n$th-order sine-wave problem? It sounded good at the time, I hope. But here I describe a curious special case where the interpretation of the cumulant as the pure component of a nonlinearly generated sine wave seems to break down.

## Cyclostationarity of Direct-Sequence Spread-Spectrum Signals

In this post we look at direct-sequence spread-spectrum (DSSS) signals, which can be usefully modeled as a kind of PSK signal. DSSS signals are used in a variety of real-world situations, including the familiar CDMA and WCDMA signals, covert signaling, and GPS. My colleague Antonio Napolitano has done some work on a large class of DSSS signals (The Literature [R11, R17, R95]), resulting in formulas for their spectral correlation functions, and I’ve made some remarks about their cyclostationary properties myself here and there (My Papers [16]).

A good thing, from the point of view of modulation recognition, about DSSS signals is that they are easily distinguished from other PSK and QAM signals by their spectral correlation functions. Whereas most PSK/QAM signals have only a single non-conjugate cycle frequency, and no conjugate cycle frequencies, DSSS signals have many non-conjugate cycle frequencies and in some cases also have many conjugate cycle frequencies.

## Cumulant (4, 2) is a Good Discriminator? Comments on “Energy-Efficient Processor for Blind Signal Classification in Cognitive Radio Networks,” by E. Rebeiz et al.

Let’s talk about another published paper on signal detection involving cyclostationarity and/or cumulants. This one is called “Energy-Efficient Processor for Blind Signal Classification in Cognitive Radio Networks,” (The Literature [R69]), and is authored by UCLA researchers E. Rebeiz and four colleagues.

My focus on this paper it its idea that broad signal-type classes, such as direct-sequence spread-spectrum (DSSS), QAM, and OFDM can be reliably distinguished by the use of a single number: the fourth-order cumulant with two conjugated terms. This kind of cumulant is referred to as the $(4, 2)$ cumulant here at the CSP Blog, and in the paper, because the order is $n=4$ and the number of conjugated terms is $m=2$.

## 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).

## Comments on “Cyclostationary Correntropy: Definition and Application” by Fontes et al

I recently came across a published paper with the title Cyclostationary Correntropy: Definition and Application, by Aluisio Fontes et al. It is published in a journal called Expert Systems with Applications (Elsevier). Actually, it wasn’t the first time I’d seen this work by these authors. I had reviewed a similar paper in 2015 for a different journal.

I was surprised to see the paper published because I had a lot of criticisms of the original paper, and the other reviewers agreed since the paper was rejected. So I did my job, as did the other reviewers, and we tried to keep a flawed paper from entering the literature, where it would stay forever causing problems for readers.

The editor(s) of the journal Expert Systems with Applications did not ask me to review the paper, so I couldn’t give them the benefit of the work I already put into the manuscript, and apparently the editor(s) did not themselves see sufficient flaws in the paper to merit rejection.

It stings, of course, when you submit a paper that you think is good, and it is rejected. But it also stings when a paper you’ve carefully reviewed, and rejected, is published anyway.

Fortunately I have the CSP Blog, so I’m going on another rant. After all, I already did this the conventional rant-free way.

## 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.

## Cyclostationarity of Digital QAM and PSK

Let’s look into the statistical properties of a class of textbook signals that encompasses digital quadrature amplitude modulation (QAM), phase-shift keying (PSK), and pulse-amplitude modulation (PAM). I’ll call the class simply digital QAM (DQAM), and all of its members have an analytical-signal mathematical representation of the form

$\displaystyle s(t) = \sum_{k=-\infty}^\infty a_k p(t - kT_0 - t_0) e^{i2\pi f_0 t + i \phi_0}. \hfill (1)$

In this model, $k$ is the symbol index, $1/T_0 = f_{sym}$ is the symbol rate, $f_0$ is the carrier frequency (sometimes called the frequency offset), $t_0$ is the symbol-clock phase, and $\phi_0$ is the carrier phase. The finite-energy function $p(t)$ is the pulse function (sometimes called the pulse-shaping function). Finally, the random variable $a_k$ is called the symbol, and has a discrete distribution that is called the constellation.

Model (1) is a textbook signal when the sequence of symbols is independent and identically distributed (IID). This condition rules out real-world communication aids such as periodically transmitted bursts of known symbols, adaptive modulation (where the constellation may change in response to the vagaries of the propagation channel), some forms of coding, etc. Also, when the pulse function $p(t)$ is a rectangle (with width $T_0$), the signal is even less realistic, and therefore more textbook.

We will look at the moments and cumulants of this general model in this post. Although the model is textbook, we could use it as a building block to form more realistic, less textbooky, signal models. Then we could find the cyclostationarity of those models by applying signal-processing transformation rules that define how the cumulants of the output of a signal processor relate to those for the input.

## 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].

## CSP-Based Time-Difference-of-Arrival Estimation

Let’s discuss an application of cyclostationary signal processing (CSP): time-delay estimation. The idea is that sampled data is available from two antennas (sensors), and there is a common signal component in each data set. The signal component in one data set is the time-delayed or time-advanced version of the component in the other set. This can happen when a plane-wave radio frequency (RF) signal propagates and impinges on the two antennas. In such a case, the RF signal arrives at the sensors with a time difference proportional to the distance between the sensors along the direction of propagation, and so the time-delay estimation is also commonly referred to as time-difference-of-arrival (TDOA) estimation.

Consider the diagram shown to the right. A distant transmitter emits a signal that is well-modeled as a plane wave once it reaches our two receivers. An individual wavefront of the signal arrives at the two sensors at different times.

The line segment AB is perpendicular to the direction of propagation for the RF signal. The angle $\theta$ is called the angle of arrival (AOA). If we could estimate the AOA, we can tell the direction from which the signal arrives, which could be useful in a variety of settings. Since the triangle ABC is a right triangle, we have

$\displaystyle \cos (\theta) = \frac{x}{d}. \hfill (1)$

When $\theta = 0$, the wavefronts first strike receiver 2, then must propagate over $x$ meters before striking receiver 1. On the other hand, when $\theta = 90^\circ$, each wavefront strikes the two receivers simultaneously. In the former case, the TDOA is maximum, and in the latter it is zero. The TDOA can be negative too, so that $180^\circ$ azimuthal degrees can be determined by estimating the TDOA.

In general, the wavefront must traverse $x$ meters between striking receiver 2 and striking receiver 1,

$\displaystyle x = d \cos(\theta). \hfill (2)$

Assuming the speed of propagation is $c$ meters/sec, the TDOA is given by

$\displaystyle D = \frac{x}{c} = \frac{d\cos{\theta}}{c} \mbox{\rm \ \ seconds}. \hfill (3)$

In this post I’ll review several methods of TDOA estimation, some of which employ CSP and some of which do not. We’ll see some of the advantages and disadvantages of the various classes of methods through inspection, simulation, and application to collected data.