2. Signal arithmetic

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 porsibility 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.

SPECTRUM, the freeware signal-processing application that accompanies this tutorial, 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 Math menu.

Matlab is a numerical computing environment and programming language in which a single variable can represent either a single value, a vector of values (such as a spectrum), 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 a, you can plot it just by typing plot(a). And if you also had a vector w of x-axis values (such as wavelengths), you can plot a vs w by typing plot(w,a). The subtraction of two spectra a and b, as in Figure 1, can be performed simply by writing a-b. To plot the difference, you would write plot(a-b). Likewise, to plot the ratio of two spectra, as in Figure 2, you would write plot(a./b).