The most basic signal processing functions are those that involve simple signal arithmetic: point-by-point addition, subtraction, multiplication, or division of two signals or of one signal and a constant. Despite their mathematical simplicity, these functions can be very useful. For example, in the left part of Figure 1 (Window 1) the top curve is the absorption spectrum of an extract of a sample of oil shale, a kind of rock that is is a source of petroleum.

*Figure 1. A simple point-by--point subtraction of two
signals allows the background (bottom curve on the left) to be
subtracted from a complex sample (top curve on the left),
resulting in a clearer picture of what is really in the sample
(right). (X-axis = wavelength in nm; Y-axis = absorbance).*

This spectrum exhibits two absorption
bands, at about 515 nm and 550 nm, that are due to a class of
molecular fossils of chlorophyll called *porphyrins.*
(Porphyrins are used as geomarkers in oil exploration). These
bands are superimposed on a background absorption caused by the
extracting solvents and by non-porphyrin compounds extracted from
the shale. The bottom curve is the spectrum of an extract of a
non-porphyrin-bearing shale, showing only the background
absorption. To obtain the spectrum of the shale extract without
the background, the background (bottom curve) is simply subtracted
from the sample spectrum (top curve). The difference is shown in
the right in Window 2 (note the change in Y-axis scale). In this
case the removal of the background is not perfect, because the
background spectrum is measured on a separate shale sample.
However, it works well enough that the two bands are now seen more
clearly and it is easier to measure precisely their absorbances
and wavelengths. (Thanks to Prof. David Freeman for the spectra of
oil shale extracts).

In this example and the one below, the assumption is being made that the two signals in Window 1 have the same x-axis values, that is, that both spectra are digitized at the same set of wavelengths. Strictly speaking this operation would not be valid if two spectra were digitized over different wavelength ranges or with different intervals between adjacent points. The x-axis values much match up point for point. In practice, this is very often the case with data sets acquired within one experiment on one instrument, but the experimenter must take care if the instruments settings are changed or if data from two experiments or two instrument are combined. (Note: It is possible to use the mathematical technique of interpolation to change the number of points or the x-axis intervals of signals; the results are only approximate but often close enough in practice).

Sometimes one needs to know whether two signals have the same shape, for example in comparing the spectrum of an unknown to a stored reference spectrum. Most likely the concentrations of the unknown and reference, and therefore the amplitudes of the spectra, will be different. Therefore a direct overlay or subtraction of the two spectra will not be useful. One possibility is to compute the point-by-point ratio of the two signals; if they have the same shape, the ratio will be a constant. For example, examine Figure 2.

*Figure 2. Do the two spectra on the left have the same
shape? They certainly do not look the same, but that may simply
be due to that fact that one is much weaker that the other. The
ratio of the two spectra, shown in the right part (Window 2), is
relatively constant from 300 to 440 nm, with a value of 10 +/-
0.2. This means that the shape of these two signals is very
nearly identical over this wavelength range.*

The left part (Window 1) shows two superimposed spectra, one of
which is much weaker than the other. But do they have the same
shape? The ratio of the two spectra, shown in the right part
(Window 2), is relatively constant from 300 to 440 nm, with a
value of 10 +/- 0.2. This means that the shape of these two
signals is the same, within about +/-2 %, over this wavelength
range, and that top curve is about 10 times more intense than the
bottom one. Above 440 nm the ratio is not even approximately
constant; this is caused by *noise*, which is the subject of
the next section.

SPECTRUM, the freeware signal-processing application for Macintosh OS8, includes the following signal arithmetic functions: addition and multiplication with constant; addition, subtraction, multiplication, and division of two signals, normalization, and a large number of other basic functions (common and natural log and antilog, reciprocal, square root, absolute value, standard deviation, etc.) in the

Popular spreadsheets, such as Excel or Open Office Calc, have built-in functions for all common math operations, named variables, x,y plotting, text formatting, matrix math, etc. Cells can contain numerical values, text, mathematical expression, or references to other cells. A vector of values such as a spectrum can be represented as a row or column of cells; a rectangular array of values such as a set of spectra can be represented as a rectangular block of cells. User-created names can be assigned to individual cells or to ranges of cells, then referred to in mathematical expression by name. Mathematical expressions can be easily copied across a range of cells, with the cell references changing or not as desired. Plots of various types (including the all-important

Matlab is a commercial numerical computing environment and programming language in which a single variable can represent either a single "scalar" value, a vector of values (such as a spectrum or a chromatogram), or a matrix (a rectangular array of values, such as a set of spectra). This greatly facilitates mathematical operations on signals. For example, if you have a spectrum in the variable

You can easily import your own data into Matlab by using the load command or by using the convenient Matlab Import Wizard by clicking File > Import Data. Data can be imported from text files and from spreadsheets.

GNU Octave is a free alternative to Matlab that is "mostly compatible". In fact, almost all of the Matlab functions, scripts, demos, and examples in this document will work in the latest version of Octave without change. (The exceptions are the keystroke-operated interactive iPeak, iSignal, and ipf.m). There is a FAQ that may help in porting Matlab programs to Octave. See Key Differences Between Octave & Matlab. There are Windows, Mac, and Unix versions of Octave; the Windows version can be downloaded from Octave Forge; be sure to install all the "packages". There is lots of help online: Google "GNU Octave" or see the YouTube videos for help. The current version of Octave as of December 2013 (Octave 3.8.0) does not use a graphical user interface by default as Matlab does, which makes operation less convenient. GNU has promised that the upcoming version 4 will have a GUI. Installation of Octave is somewhat more laborious than installing a commercial package like Matlab. More seriously, Octave is also slower that Matlab computationally - about 2 to 20 times slower, depending on the task (for some specific example, see TimeTrial.txt). I am working to make all my Matlab scripts and functions compatible with Octave. Bottom line: Matlab is better, but if you can't afford Matlab, Octave provides perhaps 95% of the functionality for 0% of the cost.

Both spreadsheets and Matlab/Octave programs have a huge advantage over commercial end-user programs and compiled freeware programs such as SPECTRUM; they can be inspected and modified by the user to customize the routines for specific needs. Simple changes are easy to make with only modest knowledge of programming. For example, you could easily change the titles or colors or line style of graphs (in Matlab or Octave programs, search for "

This page is 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.

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