1. Introduction

The interfacing of analytical instrumentation to small computers for the purpose of on-line data acquisition has now become almost standard practice in the modern chemistry laboratory. Using widely-available, low-cost microcomputers and off-the-shelf add-in components, it is now easier than ever to acquire large amounts of data quickly in digital form.

In what ways is on-line digital data acquisition superior to the old methods such as the chart recorder? Some of the advantages are obvious, such as archival storage and retrieval of data and post-run replotting with adjustable scale expansion. Even more important, however, there is the possibility of performing post-run data analysis and signal processing. There are a large number of computer-based numerical methods that can be used to 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 that were not practical before the advent of computerized instrumentation. It is important for chemistry students to appreciate the capabilities and the limitations of these modern signal processing techniques.

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).

3. SIGNAL AND NOISE

Experimental measurements are never perfect, even with sophisticated modern instruments. Two main types or measurement errors are recognized: systematic error, in which every measurement is either less than or greater than the "correct" value by a fixed percentage or amount, and random error, in which there are unpredictable variations in the measured signal from moment to moment or from measurement to measurement. This latter type of error is often called noise, by analogy to acoustic noise. There are many sources of noise in physical measurements, such as building vibrations, air currents, electric power fluctuations, stray radiation from nearby electrical apparatus, interference from radio and TV transmissions, random thermal motion of molecules, and even the basic quantum nature of matter and energy itself.

4. Smoothing

In many experiments in physical science, the true signal amplitudes (y-axis values) change rather smoothly as a function of the x-axis values, whereas many kinds of noise are seen as rapid, random changes in amplitude from point to point within the signal. In the latter situation it may be useful in some cases to attempt to reduce the noise by a process called smoothing. In smoothing, the data points of a signal are modified so that individual points that are higher than the immediately adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased. This naturally leads to a smoother signal. As long as the true underlying signal is actually smooth, then the true signal will not be much distorted by smoothing, but the noise will be reduced.

Ø Smoothing algorithms.

The simplest smoothing algorithm is the rectangular or unweighted sliding-average smooth; it simply replaces each point in the signal with the average of m adjacent points, where m is a positive integer called the smooth width. For example, for a 3-point smooth (m = 3):

5. Differentiation

The symbolic Differentiationof functions is a topic that is introduced in all elementary Calculus courses. The numerical differentiation of digitized signals is an application of this concept that has many uses in analytical signal processing. The first derivative of a signal is the rate of change of y with x, that is, dy/dx, which is interpreted as the slope of the tangent to the signal at each point. Assuming that the x-interval between adjacent points is constant, the simplest algorithm for computing a first derivative is:

(for 1< j <n-1).

where X'_j and Y'_j are the X and Y values of the j^th point of the derivative, n = number of points in the signal, and X is the difference between the X values of adjacent data points. A commonly used variation of this algorithm computes the average slope between three adjacent points:

6. Resolution enhancement

Figure 9 shows a spectrum on the left that consists of several poorly-resolved (that is, partly overlapping) bands. The extensive overlap of the bands makes the accurate measurement of their intensities and positions impossible, even though the signal-to-noise ratio is very good. Things would be easier if the bands were more completely resolved, that is, if the bands were narrower. 

Figure 9. A resolution enhancement algorithm has been applied to the signal on the left to artificially improve the apparent resolution of the peaks. In the resulting signal, right, the component bands are narrowed so that the intensities and positions can be measured.

7. Harmonic analysis and the Fourier Transform

Some signals exhibit periodic components that repeat at fixed intervals throughout the signal, like a sine wave. It is often useful to describe the amplitude and frequency of such periodic components exactly. Actually, it is possible to analyze any arbitrary set of data into periodic components, whether or not the data appear periodic. Harmonic analysis is conventionally based on the Fourier transform, which is a way of expressing a signal as a sum of sine and cosine waves. It can be shown that any arbitrary discretely sampled signal can be described completely by the sum of a finite number of sine and cosine components whose frequencies are 0,1,2,3 ... n/2 times the fundamental frequency f=1/nx, where x is the interval between adjacent x-axis values and n is the total number of points. The Fourier transform is simply the coefficients of these sine and cosine components.

The concept of the Fourier transform is involved in two very important instrumental methods in chemistry. In Fourier transform infrared spectroscopy (FTIR), the Fourier transform of the spectrum is measured directly by the instrument, as the interferogram formed by plotting the detector signal vs mirror displacement in a scanning Michaelson interferometer. In Fourier Transform Nuclear Magnetic Resonance spectroscopy (FTNMR), excitation of the sample by an intense, short pulse of radio frequency energy produces a free induction decay signal that is the Fourier transform of the resonance spectrum. In both cases the spectrum is recovered by inverse Fourier transformation of the measured signal.

8. CONVOLATION

Convolation is an operation performed on two signals which involves multiplying one signal by a delayed or shifted version of another signal, integrating or averaging the product, and repeating the process for different delays. Convolution is a useful process because it accurately describes some effects that occur widely in scientific measurements, such as the influence of a low-pass filter on an electrical signal or of the spectral bandpass of a spectrometer on the shape of a spectrum. 

Figure 11. Convolution is used here to determine how the atomic line spectrum in Window 1 (top left) will appear when scanned with a spectrometer whose slit function (spectral resolution) is described by the Gaussian function in Window 2 (top right). The Gaussian function has already been rotated so that its maximum falls at x=0. The resulting convoluted spectrum (bottom center) shows that the two lines near x=110 and 120 will not be resolved but the line at x=40 will be partly resolved.

9. DECONVULATION

Deconfulation is the converse of confulation in the sense that division is the converse of multiplication^*. In fact, the deconvolution of one signal from another is usually performed by dividing the two signals in the Fourier domain^**. The practical significance of deconvolution is that it can be used as an artificial (i.e. computational) way to reverse the result of a convolution occurring in the physical domain, for example, to reverse the signal distortion effect of an electrical filter or of the finite resolution of a spectrometer. Two examples of the application of deconvolution are shown in Figures 12 and 13. 

10. FOURIER FILTER

The Fourier filter is a type of filtering function that is based on manipulation of specific frequency components of a signal. It works by taking the Fourier transform of the signal, then attenuating or amplifying specific frequencies, and finally inverse transforming the result. The example shown here is a simple low-pass, sharp cut-off filter, which simply cuts off all frequencies above a user-specified limit. The assumption is made here that the frequency components of the signal fall predominantly at low frequencies and those of the noise fall predominantly at high frequencies. The user tries to find a cut-off frequency that will allow most of the noise to be eliminated while not distorting the signal significantly. An example of the application of the Fourier filter is given in Figure 14.

REFERENCES

1.     Douglas A. Skoog, Principles of Instrumental Analysis, Third Edition, Saunders, Philadelphia, 1984. Pages 73-76.

2.     Gary D. Christian and James E. O'Reilly, Instrumental Analysis, Second Edition, Allyn and Bacon, Boston, 1986. Pages 846-851.

3.     Howard V. Malmstadt, Christie G. Enke, and Gary Horlick, Electronic Measurements for Scientists, W. A. Benjamin, Menlo Park, 1974. Pages 816-870.

4.     Stephen C. Gates and Jordan Becker, Laboratory Automation using the IBM PC, Prentice Hall, Englewood Cliffs, NJ, 1989.

5.     Muhammad A. Sharaf, Deborah L Illman, and Bruce R. Kowalski, Chemometrics, John Wiley and Sons, New York, 1986.

6.     Peter D. Wentzell and Christopher D. Brown, Signal Processing in Analytical Chemistry, in Encyclopedia of Analytical Chemistry, R.A. Meyers (Ed.), p. 9764–9800, John Wiley & Sons Ltd, Chichester, 2000 (http://myweb.dal.ca/pdwentze /paper/c2..pdf).

7.     Constantinos E. Efstathiou, Educational Applets in Analytical Chemistry, Signal Processing, and Chemometrics. (http://www.chem.uoa.gr/Applets /Applet Index2. htm).

8.      A. Felinger, Data Analysis and Signal Processing in Chromatography, Elsevier Scice (19 May 1998).