SPTK: Linear Time-Invariant Systems

Previous SPTK Post: The Fourier Transform         Next SPTK Post: Frequency Response

In this Signal Processing Toolkit post, we’ll take a first look at arguably the most important class of system models: linear time-invariant (LTI) systems.

What do signal processors and engineers mean by system? Most generally, a system is a rule or mapping that associates one or more input signals to one or more output signals. As we did with signals, we discuss here various useful dichotomies that break up the set of all systems into different subsets with important properties–important to mathematical analysis as well as to design and implementation. Then we’ll look at time-domain input/output relationships for linear systems. In a future post we’ll look at the properties of linear systems in the frequency domain.

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SPTK: The Fourier Transform

Previous SPTK Post: The Fourier Series      Next SPTK Post: Linear Systems

This post in the Signal Processing Toolkit series deals with a key mathematical tool in CSP: The Fourier transform. Let’s try to see how the Fourier transform arises from a limiting version of the Fourier series.

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SPTK: The Fourier Series

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.

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Signal Processing Toolkit: Signals

This is the inaugural post of a new series of posts I’m calling the Signal Processing Toolkit (SPTK).  The SPTK posts will cover relatively simple topics in signal processing that are useful in the practice of cyclostationary signal processing. So, they are not CSP posts, but CSP practitioners need to know this material to be successful in CSP. The CSP Blog is branching out! (But don’t worry, there are more CSP posts coming too.)

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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!

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Data Set for the Machine-Learning Challenge

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.

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MATLAB’s SSCA: commP25ssca.m

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.

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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.

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‘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:

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Can a Machine Learn the Fourier Transform?

Update: See Part 2 of this post at this link. If you want to leave on comment, leave it on Part 2. Comments closed on this Part 1 post.

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Creating a Simple CS Signal: Rectangular-Pulse BPSK

To test the correctness of various CSP estimators, we need a sampled signal with known cyclostationary parameters. Additionally, the signal should be easy to create and understand. A good candidate for this kind of signal is the binary phase-shift keyed (BPSK) signal with rectangular pulse function.

PSK signals with rectangular pulse functions have infinite bandwidth because the signal bandwidth is determined by the Fourier transform of the pulse, which is a sinc() function for the rectangular pulse. So the rectangular pulse is not terribly practical–infinite bandwidth is bad for other users of the spectrum. However, it is easy to generate, and its statistical properties are known.

So let’s jump in. The baseband BPSK signal is simply a sequence of binary (\pm 1) symbols convolved with the rectangular pulse. The MATLAB script make_rect_bpsk.m does this and produces the following plot:

rect_bpsk_time_domain

The signal alternates between amplitudes of +1 and -1 randomly. After frequency shifting and adding white Gaussian noise, we obtain the power spectrum estimate:

rect_bpsk_psd

The power spectrum plot shows why the rectangular-pulse BPSK signal is not popular in practice. The range of frequencies for which the signal possesses non-zero average power is infinite, so it will interfere with signals “nearby” in frequency. However, it is a good signal for us to use as a test input in all of our CSP algorithms and estimators.

The MATLAB script that creates the BPSK signal and the plots above is here. It is an m-file but I’ve stored it in a .doc file due to WordPress limitations I can’t yet get around.