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. But 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
over 500
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 https://www.mathworks.com/pricing-licensing.html
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
http://www.dspguru.com/dsp/links/matlab-clones.
This work is dedicated to the Joy of Uncompetitive
Purposefulness.
"...in 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
"...in 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
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Updated
October, 2021 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 toh@umd.edu.
