Category: Signal Processing Toolkit

SPTK: The Matched Filter

Matchmaker, Matchmaker,
Make me a match,
Find me a find,
catch me a catch!
–“Matchmaker” from Fiddler on the Roof

Previous SPTK Post: The Characteristic Function Next SPTK Post: Wavelets

In this post, we take a look at a special linear time-invariant system called the matched filter. It is used to detect the presence of a known signal. In practice, it is often applied to the detection of a periodically repeated known portion of a communication signal, such as a channel-estimation frame or frequency-correction burst. It is also widely used in the detection of radar pulses, where the matched-filtering operation is renamed to pulse compression. Filtering, which implies a convolution operation, and correlation are nearly the same thing. Therefore applying a matched filter is sometimes referred to as applying a correlator.

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SPTK: The Characteristic Function

The collision of probability, Fourier analysis, and communication-signal models.

Previous SPTK Post: I and Q Next SPTK Post: The Matched Filter

Let’s return to the probability section of the Signal Processing ToolKit posts with a look at the characteristic function, which is the Fourier transform of the probability density function. We will see it has a deep connection to the central mathematical entities of CSP, which are moments and cumulants.

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SPTK: I and Q

Where does IQ (or I/Q) data come from?

Previous SPTK Post: Digital Filters Next SPTK Post: The Characteristic Function

Let’s really get into the mathematical details of “IQ data,” a phrase that appears in many CSP Blog posts and an awful lot of machine-learning papers on modulation recognition. Just what are “I” and “Q” anyway?

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SPTK: Digital Filters

A look at general linear time-invariant filtering in the discrete-time domain.

Previous SPTK Post: The Z Transform   Next SPTK Post: IQ Data

Linear shift-invariant systems are often called digital filters when they are designed objects as opposed to found objects, which are models, really, of systems occurring in the natural world. A basic goal of digital filtering is to perform the same kind of function as does an analog filter, but it is used after sampling rather than before. In some cases, the digitally filtered signal is then converted to an analog signal. These ideas are illustrated in Figure 1.

Figure 1. A typical role for a linear shift-invariant system, or digital filter, in signal processing.
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SPTK: The Z Transform

I think of the Z transform as the Laplace transform for discrete-time signals and systems.

Previous SPTK Post: Practical Filters Next SPTK Post: Digital Filters

In this Signal Processing ToolKit post, we look at the discrete-time version of the Laplace Transform: The Z Transform.

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SPTK: Practical Filters

We know that ideal filters are not physically possible. Here we take our first steps toward practical–buildable–linear time-invariant systems.

Previous SPTK Post: The Laplace Transform Next SPTK Post: The Z Transform

Before we translate the Laplace transform from continuous time to discrete time, deriving the Z transform, let’s take a step back and look at practical filters in continuous time. Practical here stands in opposition to ideal as in the ideal lowpass, highpass, and bandpass filters we studied earlier in the SPTK thread.

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SPTK: The Laplace Transform

The Laplace transform easily handles signals that are not Fourier transformable by introducing an exponential damping function inside the transform integral.

Previous SPTK Post: MATLAB’s resample.m Next SPTK Post: Practical Filters

In this Signal Processing ToolKit post, we look at a generalization of the Fourier transform called the Laplace Transform. This is a stepping stone on the way to the Z Transform, which is widely used in discrete-time signal processing, especially in control theory.

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SPTK Addendum: Problems with resampling using MATLAB’s resample.m

Sometimes MATLAB’s resample.m gives results that can be trouble for subsequent CSP.

Previous SPTK Post: Echo Detection Next SPTK Post: The Laplace Transform

In this brief Signal Processing Toolkit note, I warn you about relying on resample.m to increase the sampling rate of your data. It works fine a lot of the time, but when the signal has significant energy near the band edges, it does not.

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SPTK: Echo Detection and the Prisoner’s Dilemma

Let’s apply some of our Signal Processing ToolKit tools to a problem in forensic signal processing!

Previous SPTK Post: The Sampling Theorem Next SPTK Post: Resampling in MATLAB

No, not that prisoner’s dilemma. The dilemma of a prisoner that claims, steadfastly, innocence. Even in the face of strong evidence and a fair jury trial.

In this Signal Processing ToolKit cul-de-sac of a post, we’ll look into a signal-processing adventure involving a digital sting recording and a claim of evidence tampering. We’ll be able to use some of our SPTK tools to investigate a real-world data record that might, just might, have been tampered with. (But most probably not!)

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SPTK: Sampling and The Sampling Theorem

The basics of how to convert a continuous-time signal into a discrete-time signal without losing information in the process. Plus, how the choice of sampling rate influences CSP.

Previous SPTK Post: Random Processes Next SPTK Post: Echo Detection

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 Discman Itunes Ipod DVD Smartphone Cloud Neuralink 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?

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SPTK (and CSP): Random Processes

The merging of conventional probability theory with signal theory leads to random processes, also known as stochastic processes. The ideas involved with random processes are central to cyclostationary signal processing.

Previous SPTK Post: Examples of Random Variables Next SPTK Post: The Sampling Theorem

In this Signal Processing ToolKit post, I provide an introduction to the concept and use of random processes (also called stochastic processes). This is my perspective on random processes, so although I’ll introduce and use the conventional concepts of stationarity and ergodicity, I’ll end up focusing on the differences between stationary and cyclostationary random processes. The goal is to illustrate those differences with informative graphics and videos; to build intuition in the reader about how the cyclostationarity property comes about, and about how the property relates to the more abstract mathematical object of a random process on one hand and to the concrete data-centric signal on the other.

So … this is the first SPTK post that is also a CSP post.

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SPTK: Examples of Random Variables in Communication-Signal Contexts

Some examples of random variables encountered in communication systems, channels, and mathematical models.

Previous SPTK Post: Random Variables Next SPTK Post: Random Processes

In this Signal Processing ToolKit post, we continue our exploration of random variables. Here we look at specific examples of random variables, which means that we focus on concrete well-defined cumulative distribution functions (CDFs) and probability density functions (PDFs). Along the way, we show how to use some of MATLAB’s many random-number generators, which are functions that produce one or more instances of a random variable with a specified PDF.

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SPTK: Random Variables

Our toolkit expands to include basic probability theory.

Previous SPTK Post: Complex Envelopes Next SPTK Post: Examples of Random Variables

In this Signal Processing ToolKit post, we examine the concept of a random variable.

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SPTK: The Analytic Signal and Complex Envelope

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.

Previous SPTK Post: The Moving-Average Filter    Next SPTK Post: Random Variables

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.

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SPTK: The Moving-Average Filter

A simple and useful example of a linear time-invariant system. Good for smoothing and discovering trends by averaging away noise.

Previous SPTK Post: Ideal Filters             Next SPTK Post: The Complex Envelope

We continue our basic signal-processing posts with one on the moving-average, or smoothing, filter. The moving-average filter is a linear time-invariant operation that is widely used to mitigate the effects of additive noise and other random disturbances from a presumably well-behaved signal. For example, a physical phenomenon may be producing a signal that increases monotonically over time, but our measurement of that signal is corrupted by noise, interference, or flaws in the measurement process. The moving-average filter can reveal the sought-after trend by suppressing the effects of the unwanted disturbances.

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SPTK: Ideal Filters

Ideal filters have rectangular or unit-step-like transfer functions and so are not physical. But they permit much insight into the analysis and design of real-world linear systems.

Previous SPTK Post: Convolution       Next SPTK Post: The Moving-Average Filter

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.

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SPTK: Convolution and the Convolution Theorem

Convolution is an essential element in everyone’s signal-processing toolkit. We’ll look at it in detail in this post.

Previous SPTK Post: Interconnection of Linear Systems      Next SPTK Post: Ideal Filters

This installment of the Signal Processing Toolkit series of CSP Blog posts deals with the ubiquitous signal-processing operation known as convolution. We originally came across it in the context of linear time-invariant systems. In this post, we focus on the mechanics of computing convolutions and discuss their utility in signal processing and CSP.

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SPTK: Interconnection of Linear Systems

Real-world signal-processing systems often combine multiple kinds of linear time-invariant systems. We look here at the general kinds of connections.

Previous Post: Frequency Response Next Post: Convolution

It is often the case that linear time-invariant (or for discrete-time systems, linear shift-invariant) systems are connected together in various ways, so that the output of one may be the input to another, or two or more systems may share the same input. In such cases we can often find an equivalent system impulse response that takes into account all the component systems. In this post we focus on the serial and parallel connections of LTI systems in both the time and frequency domains. Much more complex interconnections can be constructed from these two basic kinds of connections.

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SPTK: Frequency Response of LTI Systems

The frequency response of a filter tells you how it scales each and every input sine-wave or spectral component.

Previous SPTK Post: LTI Systems             Next SPTK Post: Interconnection 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).

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SPTK: Linear Time-Invariant Systems

LTI systems, or filters, are everywhere in signal processing. They allow us to adjust the amplitudes and phases of spectral components of the input.

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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