Kamis, 02 Februari 2012

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/ndeltax, where deltax 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.


The power spectrum or frequency spectrum is a simple way of showing the total amplitude at each of these frequencies; it is calculated as the square root of the sum of the squares of the coefficients of the sine and cosine components.
A signal with n points gives a power spectrum with only (n/2)+1 points. The x-axis is the harmonic number. The first point (x=0) is the zero-frequency (constant) component. The second point (x=1) corresponds to the fundamental frequency, the next point (x=2) to twice the fundamental frequency, the next point (x=3) to three times the fundamental frequency, etc. An example of a practical application of the use of the power spectrum as a diagnostic tool is shown in Figure 10.
 Figure 10. The signal on the left (x = time; y = voltage), which was expected to contain a single peak, is clearly very noisy. The power spectrum of this signal (x-axis = frequency in Hz) shows a strong component at 60 Hz, suggesting that much of the noise is caused by stray pick-up from the 60 Hz power line. The smaller peak at 120 Hz (the second harmonic of 60 Hz) probably comes from the same source.
In the example shown here, the signal is a time-series signal with time as the independent variable. More generally, it is also possible to compute the Fourier transform and power spectrum of any signal, such as an optical spectrum, where the independent variable might be wavelength or wavenumber, or an electrochemical signal, where the independent variable might be volts. In such cases the units of the x-axis of the power spectrum are simply the reciprocal of the units of the x-axis of the original signal (e.g. nm-1 for a signal whose x-axis is in nm).

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