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.

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Symmetries of Second-Order Probabilistic Parameters in CSP

As you progress through the various stages of learning CSP (intimidation, frustration, elucidation, puzzlement, and finally smooth operation), the symmetries of the various functions come up over and over again. Exploiting symmetries can result in lower computational costs, quicker debugging, and easier mathematical development.

What exactly do we mean by ‘symmetries of parameters?’ I’m talking primarily about the evenness or oddness of the time-domain functions in the delay \tau and cycle frequency \alpha variables and of the frequency-domain functions in the spectral frequency f and cycle frequency \alpha variables. Or a generalized version of evenness/oddness, such as f(-x) = g(x), where f(x) and g(x) are closely related functions. We have to consider the non-conjugate and conjugate functions separately, and we’ll also consider both the auto and cross versions of the parameters. We’ll look at higher-order cyclic moments and cumulants in a future post.

You can use this post as a resource for mathematical development because I present the symmetry equations. But also each symmetry result is illustrated using estimated parameters via the frequency smoothing method (FSM) of spectral correlation function estimation. The time-domain parameters are obtained from the inverse transforms of the FSM parameters. So you can also use this post as an extension of the second-order verification guide to ensure that your estimator works for a wide variety of input parameters.

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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 nth 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:

psd_1

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

cfs_signal_1

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:

cfs_signal_4000

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.

Additional Batches of Signals:

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 27

Batch 28

Signal parameters text file

How we Learned CSP

This post is just a blog post. Just some guy on the internet thinking out loud. If you have relevant thoughts or arguments you’d like to advance, please leave them in the Comments section at the end of the post.

How did we, as people not machines, learn to do cyclostationary signal processing? We’ve successfully applied it to many real-world problems, such as weak-signal detection, interference-tolerant detection, interference-tolerant time-delay estimation, modulation recognition, joint multiple-cochannel-signal modulation recognition (My Papers [25,26,28,38,43]), synchronization (The Literature [R7]), beamforming (The Literature [R102,R103]), direction-finding (The Literature [R104-R106]), detection of imminent mechanical failures (The Literature [R017-R109]), linear time-invariant system identification (The Literature [R110-R115]), and linear periodically time-variant filtering for cochannel signal separation (FRESH filtering) (My Papers [45], The Literature [R6]).

How did this come about? Is it even interesting to ask the question? Well, it is to me. I ask it because of the current hot topic in signal processing: machine learning. And in particular, machine learning applied to modulation recognition (see here and here). The machine learners want to capitalize on the success of machine learning applied to image recognition by directly applying the same sorts of image-recognition techniques to the problem of automatic type-recognition for human-made electromagnetic waves.

Continue reading “How we Learned CSP”

A Challenge for the Machine Learners

UPDATE

I’ve decided to post the data set I discuss here to the CSP Blog for all interested parties to use. See the new post on the Data Set. If you do use it, please let me and the CSP Blog readers know how you fared with your experiments in the Comments section of either post. Thanks!

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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.

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More on Pure and Impure Sine Waves

Remember when we derived the cumulant as the solution to the pure nth-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.

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Modulation Recognition Using Cyclic Cumulants, Part I: Problem Description and Variants

In this post, we start a discussion of what I consider the ultimate application of the theory of cyclostationary signals: Automatic Modulation Recognition. My relevant papers are My Papers [16,17,25,26,28,30,32,33,38,43,44].

Continue reading “Modulation Recognition Using Cyclic Cumulants, Part I: Problem Description and Variants”

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).

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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.

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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].

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Conjugation Configurations

When we considered complex-valued signals and second-order statistics, we ended up with two kinds of parameters: non-conjugate and conjugate. So we have the non-conjugate autocorrelation, which is the expected value of the normal second-order lag product in which only one of the factors is conjugated (consistent with the normal definition of variance for complex-valued random variables),

\displaystyle R_x(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x^*(t+\tau_2) \right] \hfill (1)

and the conjugate autocorrelation, which is the expected value of the second-order lag product in which neither factor is conjugated

\displaystyle R_{x^*}(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x(t+\tau_2) \right]. \hfill (2)

The complex-valued Fourier-series amplitudes of these functions of time t are the non-conjugate and conjugate cyclic autocorrelation functions, respectively.

The Fourier transforms of the non-conjugate and conjugate cyclic autocorrelation functions are the non-conjugate and conjugate spectral correlation functions, respectively.

I never explained why both the non-conjugate and conjugate functions are needed. In this post, I rectify that omission. The reason for the many different choices of conjugated factors in higher-order cyclic moments and cumulants is also provided.

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Cyclic Temporal Cumulants

In this post I continue the development of the theory of higher-order cyclostationarity (My Papers [5,6]) that I began here. It is largely taken from my doctoral work (download my dissertation here).

This is a long post. To make it worthwhile, I’ve placed some movies of cyclic-cumulant estimates at the end. Or just skip to the end now if you’re impatient!

In my work on cyclostationary signal processing (CSP), the most useful tools are those for estimating second-order statistics, such as the cyclic autocorrelation, spectral correlation function, and spectral coherence function. However, as we discussed in the post on Textbook Signals, there are some situations (perhaps only academic; see my question in the Textbook post) for which higher-order cyclostationarity is required. In particular, a probabilistic approach to blind modulation recognition for ideal (textbook) digital QAM, PSK, and CPM requires higher-order cyclostationarity because such signals have similar or identical spectral correlation functions and PSDs. (Other high-SNR non-probabilistic approaches can still work, such as blind constellation extraction.)

Recall that in the post introducing higher-order cyclostationarity, I mentioned that one encounters a bit of a puzzle when attempting to generalize experience with second-order cyclostationarity to higher orders. This is the puzzle of pure sine waves (My Papers [5]). Let’s look at pure and impure sine waves, and see how they lead to the probabilistic parameters widely known as cyclic cumulants.

Continue reading “Cyclic Temporal Cumulants”

Introduction to Higher-Order Cyclostationarity

We’ve seen how to define second-order cyclostationarity in the time- and frequency-domains, and we’ve looked at ideal and estimated spectral correlation functions for a synthetic rectangular-pulse BPSK signal. In future posts, we’ll look at how to create simple spectral correlation estimators, but in this post I want to introduce the topic of higher-order cyclostationarity (HOCS).  This post is more conceptual in nature; for mathematical details about HOCS, see the posts on cyclic cumulants and cyclic polyspectra. Estimators of higher-order parameters, such as cyclic cumulants and cyclic moments, are discussed in this post.

To contrast with HOCS, we’ll refer to second-order parameters such as the cyclic autocorrelation and the spectral correlation function as parameters of second-order cyclostationarity (SOCS).

The first question we might ask is Why do we care about HOCS? And one answer is that SOCS does not provide all the statistical information about a signal that we might need to perform some signal-processing task. There are two main limitations of SOCS that drive us to HOCS.

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