**Update June 2020**

I’ll be adding new papers to this post as I find them. At the end of the original post there is a sequence of date-labeled updates that briefly describe the relevant aspects of the newly found papers.

Skip to content
# Tag: parameter estimation

## All BPSK Signals

## SPTK: The Fourier Series

## CSP Resources: The Ultimate Guides to Cyclostationary Random Processes by Professor Napolitano

## A Gallery of Cyclic Correlations

## Simple Synchronization Using CSP

## Can a Machine Learn a Power Spectrum Estimator?

## Data Set for the Machine-Learning Challenge

### Overview of Data Set

### The Label and Parameter File

### signal_1.tim

### signal_4000.tim

### Final Thoughts

### Additional Batches of Signals:

## MATLAB’s SSCA: commP25ssca.m

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

## A Challenge for the Machine Learners

## UPDATE

## CSP Patent: Tunneling

## CSP Estimators: Cyclic Temporal Moments and Cumulants

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

## Radio-Frequency Scene Analysis

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

Cyclostationary Signal Processing

Understanding and Using the Statistics of Communication Signals

**Update June 2020**

I’ll be adding new papers to this post as I find them. At the end of the original post there is a sequence of date-labeled updates that briefly describe the relevant aspects of the newly found papers.

Previous SPTK Post: Signal Representations Next SPTK Post: The Fourier Transform

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.

My friend and colleague Antonio Napolitano has just published a new book on cyclostationary signals and cyclostationary signal processing:

*Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations*, Academic Press/Elsevier, 2020, ISBN: 978-0-08-102708-0. The book is a comprehensive guide to the structure of cyclostationary random processes and signals, and it also provides pointers to the literature on many different applications. The book is mathematical in nature; use it to deepen your understanding of the underlying mathematics that make CSP possible.

You can check out the book on amazon.com using the following link:

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.

In this post I discuss the use of cyclostationary signal processing applied to communication-signal synchronization problems. First, just what are *synchronization problems*? Synchronize and synchronization have multiple meanings, but the meaning of synchronize that is relevant here is something like:

**syn·chro·nize**: *To cause to occur or operate with exact coincidence in time or rate*

If we have an analog amplitude-modulated (AM) signal (such as voice AM used in the AM broadcast bands) at a receiver we want to remove the effects of the carrier sine wave, resulting in an output that is only the original voice or music message. If we have a digital signal such as binary phase-shift keying (BPSK), we want to remove the effects of the carrier but also sample the message signal at the correct instants to optimally recover the transmitted bit sequence.

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!

Continue reading “Can a Machine Learn a Power Spectrum Estimator?”

Update July 2020. I originally posted signals in the posted data set. I’ve now added another for a total of 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 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.

The signals are stored in five zip files, each containing individual signal files:

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 zipped text file:

signal_record_first_20000.txt.zip

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.

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 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:

**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 th signal is contained in the file signal_n.tim. The Batch 1 zip file contains signal_1.tim through signal_4000.tim.**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, -DQPSK, 16QAM, 64QAM, 256QAM, and MSK. These are denoted by the strings bpsk, qpsk, 8psk, dqpsk, 16qam, 64qam, 256qam, and msk, respectively.**Base symbol period**. In the example above (line one of the truth file), the base symbol period is .**Carrier offset**. In this case, it is .**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 . It can be any real number between and .**Upsample factor**. The sixth field is an upsampling parameter U.**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: . So the BPSK signal in signal_1.tim has rate .**If the downsample factor is zero in the truth-parameters file, no resampling was done to the signal.****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.**Noise spectral density (dB)**. It is always 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.

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 :

8.181762695e-02 7.480e-01 5.406e+00

Conjugate :

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

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 . The carrier offset is and the excess bandwidth is . Because the signal type is 256QAM, it has a single (non-zero) non-conjugate cycle frequency of and no conjugate cycle frequencies. But the square of the signal has cycle frequencies related to the quadrupled carrier:

Is 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.

In this short post, I describe some errors that are produced by MATLAB’s strip spectral correlation analyzer function commP25ssca.m. I don’t recommend that you use it; far better to create your own function.

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.

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!

My colleague Dr. Apurva Mody (of BAE Systems, IEEE 802.22, and the WhiteSpace Alliance) and I have received a patent on a CSP-related invention we call tunneling. The US Patent is 9,755,869 and you can read it here or download it here. We’ve got a journal paper in review and a 2013 MILCOM conference paper (My Papers [38]) that discuss and illustrate the involved ideas. I’m also working on a CSP Blog post on the topic.

* Update December 28, 2017*: Our Tunneling journal paper has been accepted for publication in the journal IEEE Transactions on Cognitive Communications and Networking. You can download the pre-publication version here.

In this post we discuss ways of estimating -th order cyclic temporal moment and cumulant functions. Recall that for , 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 is greater than . Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order greater than is . Our estimation task is to estimate -th order temporal moment and cumulant functions for using a sampled-data record of length .

Continue reading “CSP Estimators: Cyclic Temporal Moments and Cumulants”

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].

So why do I obsess over cyclostationary signals and cyclostationary signal processing? What’s the big deal, in the end? In this post I discuss my view of the ultimate use of cyclostationary signal processing (CSP): Radio-Frequency Scene Analysis (RFSA). Eventually, I hope to create a kind of Star Trek Tricorder for RFSA.

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 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

When , the wavefronts first strike receiver 2, then must propagate over meters before striking receiver 1. On the other hand, when , 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 azimuthal degrees can be determined by estimating the TDOA.

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

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

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.

Continue reading “CSP-Based Time-Difference-of-Arrival Estimation”