In this Signal Processing ToolKit post we take a close look at the basic sampling theorem used daily by signal-processing engineers. Application of the sampling theorem is a way to choose a sampling rate for converting an analog continuous-time signal to a digital discrete-time signal. The former is ubiquitous in the physical world–for example all the radio-frequency signals whizzing around in the air and through your body right now. The latter is ubiquitous in the computing-device world–for example all those digital-audio files on your DiscmanItunesIpodDVDSmartphoneCloudNeuralink Singularity.
So how are those physical real-world analog signals converted to convenient lists of finite-precision numbers that we can apply arithmetic to? For that’s all [digital or cyclostationary] signal processing is at bottom: arithmetic. You might know the basic rule-of-thumb for choosing a sampling rate: Make sure it is at least twice as big as the largest frequency component in the analog signal undergoing the sampling. But why, exactly, and what does ‘largest frequency component’ mean?
Does the use of ‘total SNR’ mislead when the fractional bandwidth is very small? What constitutes ‘weak-signal processing?’
Or maybe “Comments on” here should be “Questions on.”
In a recent paper in EURASIP Journal on Advances in Signal Processing (The Literature [R165]), the authors tackle the problem of machine-learning-based modulation recognition for highly oversampled rectangular-pulse digital signals. They don’t use the DeepSig data sets, but their data-set description and use of ‘signal-to-noise ratio’ leaves a lot to be desired. Let’s take a brief look. See if you agree with me that the touting of their results as evidence that they can reliably classify signals with ‘SNRs of dB’ is unwarranted and misleading.
In signal processing, and in CSP, we often have to convert real-valued data into complex-valued data and vice versa. Real-valued data is in the real world, but complex-valued data is easier to process due to the use of a substantially lower sampling rate.
In this Signal-Processing Toolkit post, we review the signal-processing steps needed to convert a real-valued sampled-data bandpass signal to a complex-valued sampled-data lowpass signal. The former can arise from sampling a signal that has been downconverted from its radio-frequency spectral band to a much lower intermediate-frequency spectral band. So we want to convert such data to complex samples at zero frequency (‘complex baseband’) so we can decimate them and thereby match the sample rate to the signal’s baseband bandwidth. Subsequent signal-processing algorithms (including CSP of course) can then operate on the relatively low-rate complex-envelope data, which is beneficial because the same number of seconds of data can be processed using fewer samples, and computational cost is determined by the number of samples, not the number of seconds.
Let’s take a brief look at the cyclostationarity of a captured DMR signal. It’s more complicated than one might think.
In this post I look at the cyclostationarity of a digital mobile radio (DMR) signal empirically. That is, I have a captured DMR signal from sigidwiki.com, and I apply blind CSP to it to determine its cycle frequencies and spectral correlation function. The signal is arranged in frames or slots, with gaps between successive slots, so there is the chance that we’ll see cyclostationarity due to the on-burst (or on-frame) signaling and cyclostationarity due to the framing itself.
We continue with our non-CSP signal-processing tool-kit series with this post on ideal filtering. Ideal filters are those filters with transfer functions that are rectangular, step-function-like, or combinations of rectangles and step functions.
Another post-publication review of a paper that is weak on the ‘RF’ in RF machine learning.
Let’s take a look at a recently published paper (The Literature [R148]) on machine-learning-based modulation-recognition to get a data point on how some electrical engineers–these are more on the side of computer science I believe–use mathematics when they turn to radio-frequency problems. You can guess it isn’t pretty, and that I’m not here to exalt their acumen.
What happens when a cyclostationary time-series is treated as if it were stationary?
In this post let’s consider the difference between modeling a communication signal as stationary or as cyclostationary.
There are two contexts for this kind of issue. The first is when someone recognizes that a particular signal model is cyclostationary, and then takes some action to render it stationary (sometimes called ‘stationarizing the signal’). They then proceed with their analysis or algorithm development using the stationary signal model. The second context is when someone applies stationary-signal processing to a cyclostationary signal model, either without knowing that the signal is cyclostationary, or perhaps knowing but not caring.
At the center of this topic is the difference between the mathematical object known as a random process (or stochastic process) and the mathematical object that is a single infinite-time function (or signal or time-series).
A related paper is The Literature [R68], which discusses the pitfalls of applying tools meant for stationary signals to the samples of cyclostationary signals.
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.
Let’s talk about ambiguity and correlation. The ambiguity function is a core component of radar signal processing practice and theory. The autocorrelation function and the cyclic autocorrelation function, are key elements of generic signal processing and cyclostationary signal processing, respectively. Ambiguity and correlation both apply a quadratic functional to the data or signal of interest, and they both weight that quadratic functional by a complex exponential (sine wave) prior to integration or summation.
Are they the same thing? Well, my answer is both yes and no.
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 surface 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 DSSS, 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 that I used in the spectral correlation gallery post.
An alternative to the strip spectral correlation analyzer.
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 values.
Reconsidering my first attempt at teaching a machine the Fourier transform with the help of a CSP Blog reader. Also, the Fourier transform is viewed by Machine Learners as an input data representation, and that representation matters.
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:
Tunneling == Purposeful severe undersampling of wideband communication signals. If some of the cyclostationarity property remains, we can exploit it at a lower cost.
My colleague Dr. Apurva Mody (of BAE Systems, AiRANACULUS, 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 ) 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.
Radio-frequency scene analysis is much more complex than modulation recognition. A good first step is to blindly identify the frequency intervals for which significant non-noise energy exists.
In this post, I discuss a signal-processing algorithm that has almost nothing to do with cyclostationary signal processing (CSP). Almost. The topic is automatic 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 number of distinct frequency subbands. Moreover, these signals may be turning on and 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 using CSP.
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 than one cochannel signal.
The basics of my BOI-detection approach are published in a 2007 conference paper (My Papers ). 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.
Spread-spectrum signals are used to enable shared-bandwidth communication systems (CDMA), precision position estimation (GPS), and secure wireless data transmission.
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 ).
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 is 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 cumulant here at the CSP Blog, and in the paper, because the order is and the number of conjugated terms is .
Modulation recognition is the process of assigning one or more modulation-class labels to a provided time-series data sequence.
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]. See also my machine-learning modulation-recognition critiques by clicking on Machine Learning in the CSP Blog Categories on the right side of any post or page.