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[Coverage]   [Software requirements]   [References]

The interfacing of measurement instrumentation to small computers has now become standard practice in the modern science laboratory. Computers are used for data acquisition, data, and storage, using a large number of digital computer-based numerical methods. Techniques are available that can transform signals into more useful forms, detect and measure peaks, reduce noise, improve the resolution of overlapping peaks, compensate for instrumental artifacts, test hypotheses, optimize measurement strategies, diagnose measurement difficulties, and decompose complex signals into their component parts. These techniques can often make difficult measurements easier by extracting more information from the available data. Many of these techniques are based on laborious mathematical procedures and/or analog electronics that were not really practical before the advent of computerized instrumentation.

It is important to appreciate the abilities, as well as the limitations, of these techniques. In recent decades, computers and digital storage and processing has become commonplace, much more accurate, far less costly, easier to program, and literally millions of times more capable altogether, reducing the cost of raw data and making complex computer-based signal processing techniques both more practical and necessary. Computations that were previously impractical are now common, and approximations and shortcuts that were once necessitated by mathematical convenience are no longer needed. But it's not just the growth of computer power: there are now new materials, new instruments, new fabrication techniques, new automation capabilities. We have lasers, fiber optics, superconductors, supermagnets, holograms, quantum technology, nanotechnology, and more. Sensors are now smaller and cheaper and faster than ever before; we can measure over a wider range of speeds, temperatures, pressures, and locations. There are new kinds of data that we never had before. As Erik Brynjolfsson and Andrew McAfee wrote in The Second Machine Age (W. W. Norton, 2014): "...many types of raw data are getting dramatically cheaper, and as data get cheaper, the bottleneck increasingly is the ability to interpret and use data". Kate Keahey, a Senior Scientist at Argonne National Laboratory, involved with gravitational wave research, has said that "Software is a vital part of the research landscape, and most researchers will benefit from understanding its possibilities, limitations and the requirements for building it".

This essay covers only basic topics related to one-dimensional time-series signals, not two-dimensional data such as images. It uses a pragmatic approach and is limited to mathematics only up to the most elementary aspects of calculus, statistics, and matrix math. I use logical arguments, analogies, graphics, and animation to explain ideas, rather than lots of formal mathematics. Data processing without math? Not really! Math is essential, just as it is for the technology of cell phones, GPS, digital photography, the Web, and computer games. But you can get started using these tools without understanding all the underlying math and software details. Seeing it work makes it more likely that you'll want to understand how it works. But in the long run, it's not enough just to know how to operate the software, any more than knowing how to use a word processor or a MIDI sequencer makes you a good author or musician. 

Why do I title this document "signal processing" rather than "data processing"? By "signal" I mean the continuous x,y numerical data recorded by scientific instruments as time-series, where x may be time or another quantity like energy or wavelength, as in the various forms of spectroscopy. "Data" is a more general term that includes categorical data as well. In other words, I'm oriented to data that you would plot in a spreadsheet using the scatter chart type rather than bar or pie charts. 

Some of the examples come from my own areas of research in analytical chemistry, but these techniques have been used in a wide range of application areas. My software has been cited in 750 journal papers, theses, and patents, covering fields from industrial, environmental, medical, engineering, earth science, space, military, financial, agriculture, and even music and linguistics. Suggestions and experimental data sent by hundreds of readers from their own work has helped shape my writing and software development. Much effort has gone into making this document concise and understandable; it has been highly praised by many readers.

At the present time, this work does not cover image processing, pattern recognition, or factor analysis. For more advanced topics and for a more rigorous treatment of the underlying mathematics, refer to the extensive literature on signal processing and on statistics and chemometrics.

This site had its origin in one of the experiments in a course called "Electronics and Computer Interfacing for Chemists" that I developed and taught at the University of Maryland in the 80's and 90's. The first Web-based version went up in 1995. Subsequently it has been revised and greatly expanded based on feedback from users. It is still a work in progress and, as such, benefits from continued feedback from readers and users.

This tutorial makes considerable use of Matlab, a high-performance commercial and proprietary numerical computing environment and "fourth generation" programming language that is widely used in research (14, 17, 19, 20), Octave, a free Matlab alternative that runs almost all of the programs and examples in this tutorial, and also Python, a powerful but free and open-source language. There is a good reason why Matlab is so massively popular in science and engineering; it's powerful, fast, and relatively easy to learn. A very important aspect of Matlab is the concept of functions, which are self contained modules of code that accomplish a specific task. Functions usually "take in" data, process it, and "return" a result. (A trivial example is a=sqrt(b), which takes the value of b, computes its square root, and assigns it to the variable a). Once a function is written, it can be used over and over and over again. Functions can be "called" from the inside of other functions. Matlab comes with built-in functions for doing data processing tasks like matrix math, filtering, Fourier transforms, convolution and deconvolution, multilinear regression, and optimization. You can write your own custom functions to use in your future programming projects, and you can download powerful toolboxes and free user-contributed functions. Matlab can interface to C, C++, Java, Fortran, and Python; and it's extensible to symbolic computing and model-based design for dynamic and embedded systems.

There are many code examples in this text that you can Copy and Paste (or drag and drop) into the Matlab/Octave command line to run or modify, which is especially convenient if you can split your screen between the two. If you try to run one of my scripts or functions and it gives you a "missing function" error, that means either that you have not yet downloaded that item from my web site or that you have not placed it in the "path". Look for the missing item here, download it into your path, and try again. Type "help path" at the Matlab/Octave command prompt for help and related commands.

Most of the techniques covered in this work can also be performed in spreadsheets (11, 22, 23) such as Excel or OpenOffice Calc.

Octave (currently version 6.4.0) and the OpenOffice Calc (LibreOffice Calc) spreadsheet program can be downloaded without cost from their respective web sites. Python is also a free download.

All of the Matlab/Octave scripts and functions, and all of the spreadsheets used here can all be downloaded from this site at no cost; they have received extraordinarily positive feedback from users. If you try to run one of my scripts or functions and it gives you a "missing function" error, look for the missing item on functions.html, download it into your path, and try again.

If you are unfamiliar with Matlab, read these sections about basics and functions and scripts for a quick start-up. Matlab is not really a general-purpose programming languages like C++ or Python; rather, it is specifically suited to numerical methods, matrix manipulation, plotting of functions and data, implementation of algorithms, creation of user interfaces, and deployment to portable devices such as tablets - essentially the needs of numerical computing by scientists and engineers. Matlab is more loosely typed and less well structured in a formal sense than other languages, and thus tends to be more favored by scientists and engineers and less well liked by computer scientists and professional programmers. To get a basic language like Python up to the point where Matlab starts takes a considerable effort and familiarity with computer jargon to install add-on "packages" of functions that Matlab comes with. This is not a criticism of Python, which is an extremely capable and widley-used language, just an observation of different needs for different fields.

There are several versions of Matlab, including lower-cost student and home versions. See for prices and restrictions in their use. It is possible that your workplace may have a site license for Matlab. There are also several other good free alternatives to MATLAB, in particular Octave, which is essentially a Matlab clone, but there is also Scilab, FreeMat, Julia, and Sage which are somewhat compatible with the MATLAB language and which illustrate the influence of Matlab in the scientific computing community. For a discussion of other possibilities, see

Current personal computers and laptops are now so fast at calculating and plotting that it is possible to work with data and signal processing in a new way, in real time, or interactively, pressing a key or clicking the mouse and seeing the results instantly, for example using my keystroke-driven programs, Matlab "Live scripts", Matlab "apps", or Python Jupyter Notebooks. These programming methods have made working with data a different experience.

This work is dedicated to the Joy of Uncompetitive Purposefulness.

"As we benefit from the inventions of others, we should be glad to share our own ... freely and gladly". Benjamin Franklin

" our culture of competitive self-comparison, we can choose to amplify each other's accomplishments because there is, after all, enough to go around."  Maria Popova 

"People are generally better persuaded by the reasons which they have themselves discovered than by those which have come into the mind of others."
Blaise Pascal

"...producing technologies, and then teaching them to others, ... pushes humankind ahead". David Premack

"A computer does not substitute for judgment any more than a pencil substitutes for literacy. But writing without a pencil is no particular advantage."
Robert McNamara

" the course of looking deeply within ourselves, we may challenge notions that give comfort before the terrors of the world. Supporters of superstition and pseudoscience are human beings with real feelings, who, like the skeptics, are trying to figure out how the world works and what our role in it might be. Their motives are in many cases consonant with science." Carl Sagan, in The Demon-Haunted World: Science as a Candle in the Dark.

"...[be] full of wonder, generously open to every notion, [dismiss] nothing except for good reason, but at the same time, and as second nature, [demand] stringent standards of evidence, ...[applied] with at least as much rigor to what [you] hold dear as to what [you] are tempted to reject with impunity." Carl Sagan

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57., the most popular independent internet resource for Digital Signal Processing (DSP) engineers around the world.

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UpMaydated , 2024 This page is part of "A Pragmatic Introduction to Signal Processing", created and maintained by Prof. Tom O'Haver, Department of Chemistry and Biochemistry, The University of Maryland at College Park. Comments, suggestions and questions should be directed to Prof. O'Haver at