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 instrument recovers the spectrum by inverse Fourier transformation of the measured (interferogram or free induction decay) 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. The power spectrum retains the frequency information
but discards the phase information, so that the power
spectrum of a sine wave would be the same as that of a cosine wave
of the same frequency, even though the complete Fourier transforms
of sine and cosine waves are different in phase. In situations
where the phase components of a signal are the major
source of noise (e.g. random shifts in the horizontal xaxis
position of the signal), it can be advantageous to base
measurement on the power spectrum, which discards the phase
information, by ensemble
averaging the power spectra of repeated signals: this is
demonstrated by the Matlab/Octave scripts EnsembleAverageFFT.m and EnsembleAverageFFTGaussian.m.
A
timeseries signal with n points gives a power spectrum
with only (n/2)+1 points. The first point is the zerofrequency
(constant) component, corresponding to the DC (direct current)
component of the signal. The second point corresponds to a
frequency of 1/nΔx (whose period is exactly equal to the time
duration of the data), the next point to 2/nΔx, the next point to
3/nΔx, etc., where Δx is the interval between adjacent xaxis
values and n is the total number of points. The last (highest
frequency) point in the power spectrum (n/2)/nΔx=1/2Δx, which is
onehalf the sampling rate. The highest frequency that can be
represented in a discretelysampled waveform is onehalf the
sampling frequency, which is called the Nyquist
frequency; frequencies above the Nyquist
frequency are "folded back" to lower frequencies, severely
distorting the signal.
A pure sine or cosine wave that has an exactly integral number of cycles within the recorded signal will have a single nonzero Fourier component corresponding to its frequency. Conversely, a signal consisting of zeros everywhere except at a single point, called a delta function, has equal Fourier components at all frequencies. Random noise also has a power spectrum that is spread out over a wide frequency range, but shaped according to its noise color, with pink noise having more power at low frequencies, blue noise having more power at high frequencies, and white noise having roughly the same power at all frequencies.
For periodic waveforms that repeat over time,
a single period is the smallest repeating unit of the signal, and
the reciprocal of that period is called the fundamental
frequency. Nonsinusoidal periodic waveforms exhibit a
series of frequency components that are multiples of the
fundamental frequency; there are called "harmonics". A familiar example is the electrical recording of
a heartbeat, call an electrocardiograph
(ECG), which consists of a highly repeatable series of
waveforms, as in the real data example on the left, which shows a
fundamental frequency of 0.6685 Hz with multiple harmonics
at frequencies that are ×2, ×3, ×4...,
etc, times the fundamental frequency. The waveform is shown in
blue in the top panel and its frequency spectrum is shown in red
in the bottom panel. The fundamental and the harmonics are sharp
peaks, labeled with their frequencies. The spectrum is
qualitatively similar to what is obtained for perfectly regular identical peaks.
Recorded vocal sounds, especially vowels, also have a periodic waveform with harmonics.
(The
sharpness of the peaks in these spectra shows that the
amplitude and the frequency are very constant over the
recording interval in this example. Changes
in amplitude or frequency over the recording interval will
produce clusters or bands of
Fourier
components rather than sharp peaks,
as in this example).
Another familiar example of periodic change is the seasonal
variation in temperature, for example the average daily
temperature measured in New York City between 1995 and 2015,
shown in the figure on the right. (The negative spikes are missing
data points  power outages?) In this example the spectrum in the
lower panel is plotted with time (the reciprocal of
frequency) on the xaxis (called a periodogram)
which, despite the considerable random noise due to local
weather variations and missing data, shows the expected peak at
exactly 1 year; that peak is sharp because the
periodicity is extremely (in fact, astronomically) precise. In
contrast, the random noise is not periodic and is spread
out roughly equally over the entire periodogram.
The figure on the right is a
simulation that shows how hard it is to see a periodic component
in the presence of random noise, and yet how easy it is to pick it
out in the frequency spectrum. In this example, the signal (top
panel) contains an equal mixture of random white noise and
a single sine wave; the sine wave is almost completely obscured by
the random noise. The frequency spectrum (created
using the downloadable Matlab/Octave function "PlotFrequencySpectrum") is
shown in the bottom panel. The frequency spectrum of the white
noise is spread out evenly over the entire spectrum, whereas the
sine wave is concentrated into a single spectral element,
where it stands out clearly. Here is the Matlab/Octave code that
generated that figure; you can Copy and Paste it into
Matlab/Octave:
x=[0:.01:2*pi]';
y=sin(200*x)+randn(size(x));
subplot(2,1,1);
plot(x,y);
subplot(2,1,2);
PowerSpectrum=PlotFrequencySpectrum(x,y,1,0,1);
Data from an
audio recording, zoomed in to the period immediately before
(left) and after (right) the actual sound, shows a regular
sinusoidal oscillation
(x = time in seconds). In the lower panel,
the power spectrum of each signal (x =
frequency in Hz) shows a strong sharp peak very near 60 Hz,
suggesting that the oscillation is caused by stray pickup
from the 60
Hz power line in the USA (it would be 50 Hz in
Europe). Improved shielding and grounding of the equipment
might reduce this interference.
There are several
Web sites that can compute Fourier transforms interactively
(e.g. WolframAlpha).

iPower: Keyboardcontrolled
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 Xaxis scale of power spectrum........H Print keyboard commands......................K Play signal as sound................Enter or P PREPROGRAMMED 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 nonlinear 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 slideroperated version (see left) for Matlab version 6.5. Click here to view or download. 