x, 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.
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.
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.
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.
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iPower: Keyboard-controlled
interactive power spectrum demonstrator, useful for teaching and
learning about the power spectra of different types of signals and the
effect of signal duration and sampling rate. Single keystrokes allow
you to select the type of signal (12 different signals included), the
total duration of the signal, the sampling rate, and the global
variables f1 and f2 which are used in different ways in the different
signals. When the Enter key is pressed, the signal (y) is sent to the
Windows WAVE audio device. Press K to see a list of all the keyboard
commands. Tested in Matlab version 7.8 (R2009a). Click here to view or download. You can also download it from the Matlab File Exchange. T. C. O'Haver (toh@umd.edu), version 2, October 2011 KEYBOARD CONTROLS: Adjust signal duration 10% up/down.........A,Z
Adjust sampling rate 10% up/down...........S,X Adjust first variable 10% up/down......... D,C Adjust second variable 10% up/down........ F,V Cycle through Linear/Log plot modes..........L Switch X-axis scale of power spectrum........H Print keyboard commands......................K Play signal as sound................Enter or P PRE-PROGRAMMED SIGNAL TYPES *Sine wave, frequency f1 (Hz), phase f2
*Square wave, frequency f1 (Hz), phase f2 *Sawtooth wave, frequency Ff1(Hz) *Triangle wave, frequency f1 (Hz), phase f2 *Sine wave burst of frequency f1 (Hz) and length f2 sec *440 Hz carrier amplitude modulated by sine wave, frequency f1 (Hz) and amplitude f2 *440 Hz carrier frequency modulated by sine wave of frequency f1 (Hz) and amplitude f2 *Sine wave, frequency f1 (Hz), modulated with Gaussian of width f2 sec *Sine wave, frequency f1 (Hz) with non-linear transfer function f2 *Sine wave sweep from 0 to f1 (Hz) *Sine wave of frequency f1 (Hz) and amplitude f2 plus random white noise *Pink (1/f) noise *Sine wave, frequency f1 (Hz), amplitude f2 plus pink noise There is also an older slider-operated version (see left) for Matlab version 6.5. Click here to view or download. |
