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
weighted 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 frequency f=1/nΔx, where Δx is the
interval between adjacent x-axis values and n is the total number of
points. The Fourier transform is simply the set of amplitudes of
those sine and cosine components (or, which is mathematically
equivalent, the
frequency and phase of sine components). You could calculate
those coefficients yourself simply by multiplying the signal
point-by-point with each of those sine and cosine components and
adding up the products. The concept was originated by Carl Friedrich
Gauss and by Jean-Baptiste Joseph Fourier in the early 19th century.
The famous “Fast
Fourier Transform” (FFT) dates from 1965 and is a faster and
more efficient algorithm that makes use of the symmetry of the sine
and cosine functions and other math shortcuts to get the same result
much more quickly. The inverse Fourier
transform (IFT) is a similar algorithm that converts a Fourier
transform back into the original signal. As a mathematical
convenience, Fourier transforms are usually expressed in terms of "complex
numbers", with "real" and "imaginary" parts that combine the
sine and cosine (or amplitude and phase) information at each
frequency onto a single complex number. Many computer languages will
perform this operation automatically when the two quantities divided
are complex.
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 x-axis
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
time-series signal with n points gives a power spectrum
with only (n/2)+1 points. The first point is the zero-frequency
(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 x-axis
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
one-half the sampling rate. The highest frequency that can be
represented in a discretely-sampled waveform is one-half 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 non-zero Fourier componentcorresponding
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. Non-sinusoidal 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 ofFourier
components rather than sharp peaks,
as in this example).
Another familiar example of periodic change is the seasonal
variation in temperature, for example theaverage 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 x-axis (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:
A common practical application is
the use of the power spectrum as a diagnostic tool to distinguish
between signal and noise components. An example is the AC
power-line pickup depicted in the figure below, which has a
fundamental frequency of 60 Hz in the USA (why
that frequency?) or 50 Hz in some countries. Again,
the sharpness of the peaks in the spectrum shows that the
amplitude and the frequency are very constant; power companies
take pains to keep the frequency of the AC very
constant to avoid problems between different sections of the power
grid. Other examples of signals and their frequency spectra are shown below).
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 pick-up
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.
Peak-type signals have power spectra that are concentrated in a
range of low frequencies, whereas random noise is often spread out
over a much wider frequency range. This is why smoothing (low-pass
filtering) can make a noisy signal look nicer, but also
why smoothing does not usually help with quantitative measurement,
because most of the peak information is found at low frequencies,
where low-frequency noise remains unchanged by smoothing (See CurveFittingC.html#Smoothing).
The figures above show a classic example of harmonic analysis of
the annual variation in the number of observed sunspots, which
have been recorded since the year 1700! In this case the time axis
is in years (top window). A plot of the power spectrum
(bottom window, left) shows a strong peak at 0.09 cycles/year and
the periodogram (right) shows a peak at the well-known 11-year
cycle, plus some evidence of a weaker cycle at around a 100-year
period. (You can download this data set
or the latest yearly
sunspot data from NOAA. These frequency spectra are plotted
using the downloadable Matlab function iSignal.
In this case, the peaks
in the spectrum are not sharp single peaks, but rather
form a cluster of Fourier components, because the
amplitude and the frequency are not constant over the nearly
300 year interval of the data, as is obvious by inspecting the
data in the time domain.
An
example of a time series with complex multiple periodicity is the
world-wide daily page views (x=days,
y=page views) for this web site
over a 2070-day period (about 5.5 years). In
the periodogram plot (shown on the left) you can clearly see
at sharp peaks at 7 and 3.5 days, corresponding to the first and
second harmonics of the expected workday/weekend cycle, and
smaller peaks at 365 days (corresponding to a sharp dip each year
during the winter holidays) and at 182 days (roughly a half-year),
probably caused by increased use in the two-per-year semester
cycle at universities. (The large values at the longest times are
caused by the gradual increase in use over the entire data record,
which can be thought of as a very low-frequency component whose
period is much longer that the entire data record).
In the example shown on the right, the signal (in the top window)
contains no visually evident periodic components; it seems to
be just random noise. However, the frequency spectrum (in the
bottom window) shows that there is much more to this signal than
meets the eye. There are two major frequency components: one at
low frequencies around 0.02 and the other at high frequencies
between 0.5 and 5. (If the x-axis units of the signal plot had
been seconds, the units
of the frequency spectrum plot would be Hz; note that the x-axis is logarithmic). In
this particular case, the lower frequency component is in
fact the signal, and the higher frequency
component is residual blue
noiseleft over from previous signal processing
operations. The two components are fortunately well separated on
the frequency axis, suggesting that low-pass filtering (i.e
smoothing) will able to remove the noise without distorting the
signal.
In the examples shown above, the signals are time-series signals
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, or a spacial signal, where
the independent variable might be in length units. 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).
Analysis of
the frequency spectra of signals provides another way to
understand signal-to-noise ratio, filtering, smoothing, and differentiation. Smoothing is a
form of low-pass filtering, reducing the high-frequency
components of a signal. If a signal consists of smooth features,
such as Gaussian peaks, then its spectrum will be concentrated
mainly at low frequencies. The wider the width of the
peak, the more concentrated the frequency spectrum will be at low
frequencies (see animated picture on the right). If that signal is
contaminated with white noise (spread out evenly over all
frequencies), then smoothing will make the signal look better,
because it reduces the high-frequency components of the noise.
However, the low-frequency noise will remain in the signal after
smoothing, where it will continue to interfere with the
measurement of signal parameters such as peak heights, positions,
widths, and areas. This can be demonstrated by
least-squares measurement.
Conversely,
differentiation is a form of high-pass filtering, reducing
the low frequency components of a signal and emphasizing
any high-frequency components present in the signal. A
simple computer-generated Gaussian peak (shown by the animation on
the left) has most of its power concentrated in just a few low
frequencies, but as successive orders of differentiation are
applied, the waveform of the derivative swings from positive to
negative like a sine wave, and its frequency spectrum shifts
progressively to higher frequencies, as shown in the animation on
the left. This behavior is typical of any signal with smooth peaks.
So the optimum range for signal information of a differentiated
signal is restricted to a relatively narrow range, with
little useful information above and below that range.
Peak sharpening also
emphasizes the high frequency components by adding a portion of
the second and fourth derivatives to the original signal. You can
see this clearly in the script PeakSharpeningFrequencySpectrum.m,
which shows the frequency spectrum of the original and sharpened
version of a signal consisting of several peaks (graphic).
SineToDelta.m. A
demonstration animation (animated graphic) showing
the waveform and the power spectrum of a rectangular pulsed sine
wave of variable duration (whose power spectrum is a "sinc" function) changing
continuously from a pure sine wave at one extreme (where its
power spectrum is a delta function) to a single-point pulse at
the other extreme (where its power spectrum is a flat line). GaussianSineToDelta.m is
similar, except that it shows a Gaussian pulsed sine
wave, whose
power spectrum is a Gaussian function, but which is the same at
the two extremes of pulse duration (animated graphic).
Real experimental signals are often contaminated with drift and
baseline shift, which are essentially low-frequency
effects, and random noise, which is usually spread out over all
frequencies. For these reasons, differentiation is always
used in conjunction with smoothing. Working together, smoothing and
differentiation act as a kind of frequency-selective bandpass
filter that optimally passes the band of frequencies
containing the differentiated signal information but reduces both
the lower-frequency effects, such as slowly-changing drift
and background, as well as the high-frequency noise. An
example of this can be seen in the DerivativeDemo.m
described in a previous
section. In the set of six original signals, shown on the
right, the random noise occurs mostly in a high frequency range,
with many cycles over the x-axis range, and the baseline
shift occurs mostly in a much lower-frequency phenomenon, with
only a small fraction of one cycle occurring over that
range. In contrast, the peak of interest, in the center of the
x-range, occupies an intermediate frequency range, with a few
cycles over that range. Therefore we could predict that a
quantitative measure based on differentiation and smoothing might
work well, as was shown previously.
Smoothing and differentiation change the amplitudes of
the various frequency components of signals, but they do not
change of shift the frequencies themselves. An experiment described later
illustrates this idea using a brief recording of speech.
Interestingly, different degrees of smoothing and differentiation
will change timbre
of the voice but has little effect on the intelligibility;
the sequence of pitches is not shifted in pitch or time but
merely changed in amplitude by smoothing and differentiation.
Because of this, recorded speech can survive digitization,
transmission over long distances, and playback via tiny speakers
and headphones without significant loss of intelligibility. Music,
on the other hand, suffers greater loss under such circumstances,
as you can tell by listening to typically terrible telephone
"hold" music.
Software details
In a spreadsheet or computer language, a sine wave can be described
by the 'sin' function y=sin(2*pi*f*x+p) or
y=sin(2*pi*(1/t)*x+p), where pi is 3.14159....,
f is frequency of the waveform, t is the period
of the waveform, p is the phase, and x
is the independent variable (usually time).
Microsoft Excel has a
add-in function that makes it possible to perform Fourier
transforms relatively easily: (Click Tools > Add-Ins... > Analysis
Toolpak > Fourier Analysis).
See "Excel
and Fourier" for details. See http://www.bowdoin.edu/~rdelevie/excellaneous/
for an extensive and excellent collection of add-in functions and
macros for Excel, courtesy of Dr. Robert deLevie of Bowdoin
College.
There are a number of dedicated FFT spectral analysis programs,
including ScopeDSP (https://iowegian.com/scopedsp/)
and Audacity (http://sourceforge.net/projects/audacity/).SPECTRUM, the freeware
signal-processing application for Macintosh OS8, includes a power
spectrum function, as well as forward and reverse Fourier
transformation.
Matlab and Octave have
built-in functions for computing the Fourier transform (fft and ifft). These
function express their results as complex numbers. For example, if
we compute the Fourier transform of a simple 3-element vector, we
get 3-element result of complex numbers:
y=[0 1 0];fft(y) ans =
1.0000 -0.5000 - 0.8660i
-0.5000 + 0.8660i
where the "i" indicates the "imaginary" part. The first element of
the fft is just the sum of elements in y. The inverse fft , ifft([1.0000
-0.5000 - 0.8660i -0.5000 + 0.8660i]),
returns the original vector [0 1 0]. For another example,
the fft of [0 1 0 1] is [2
0 -2 0]. In general, the
fft of an n-element vector of real numbers returns an n-element
vector of real or complex numbers, but only the first n/2+1 elements
are unique; the remainder are a mirror image of the first.
Operations on individual elements of the fft, such as in Fourier filtering, must take this
structure into account.
The frequency spectrum of a signal vector "s" can be computed as real(sqrt(fft(s)
.* conj(fft(s)))). Here's a simple example where we
know the answer in advance, at least qualitatively: an 8-element
vector of integers that trace out a single cycle of a sine wave:
The frequency spectrum in this case is [0 39.9 0 0.201 0 0.201
0 39.9]. Again, the first element is the average (which is
zero) and elements 2 through 4 are the mirror image of the last 4.
The unique elements are the first four, which are the amplitudes of
the sine wave components whose frequencies are 0, 1, 2, 3 times the
frequency of a sine wave that would just fit a single cycle in the
period of the signal. In this case it the second element
(39.8) that is the largest by far, which is just what we would
expect for a signal that approximates a single cycle of a sine wave.
Had the signal been two cycles of a sine wave, s=[0 10 0
-10 0 10 0 -10], the third element would have been the
strongest (try it). The highest frequency that can be represented by
an 8-element vector is one that has a period equal to 2 elements.
The downloadable function FrequencySpectrum.m(syntaxfs=FrequencySpectrum(x,y)) returns real
part of the Fourier power spectrum of x,y as a matrix.PlotFrequencySpectrum.m can
plot frequency spectra and periodograms on linear or log
coordinates. Type "help PlotFrequencySpectrum" or try this example:
x=[0:.01:2*pi]';
f=25; % Frequency
y=sin(2*pi*f*x)+randn(size(x));
subplot(2,1,1);
plot(x,y);
subplot(2,1,2);
FS=PlotFrequencySpectrum(x,y,1,0,1);
The plot of the frequency spectrum FS (plotit(FS); graphic) shows a single
strong peak at 25. The frequency of the strongest peak in FS is
given by FS(val2ind(FS(:,2),max(FS(:,2))),1)
For some other examples of using FFT example, see these
examples. A
“Slow Fourier Transform” function has also
been published;
it is 8000 times slower with a 10,000-point data vector,
as can be shown by this bit of code: y=cos(.1:.01:100);tic; fft(y); ffttime=toc; tic;
sft(y); sfttime=toc; TimeRatio=sfttime/ffttime
iSignal is a
multi-purpose interactive signal processing tool that has a Frequency
Spectrum mode, toggled on and off by the Shift-S key; it computes
frequency spectrum of the segment of the signal displayed in the
upper window and displays it in the lower window (in red). You can
use the pan and zoom keys to adjust the region of the signal to be
viewed or press Ctrl-A
to select the entire signal. Press Shift-S
again to return to the normal mode.
In the frequency spectrum mode, you can press Shift-A to cycle through four
plot modes (linear, semilog X, semilog Y, or log-log). Because of
the wide range of amplitudes and frequencies exhibited by some
signals, the log plot modes often results in a clearer graph that
the linear modes. You can also press Shift-X to toggle between frequency on the x axis and time on the x-axis. The example
on the right shows the frequency mode. Press Shift-Z
to display the frequencies of the main harmonics on each peak (as in
the ECG example at the top of this page).
All signal processing functions remain active in the
frequency spectrum mode (smooth, derivative, etc), so you can
observe the effect of these functions on the frequency spectrum of
the signal. This is demonstrated by the animation on the right,
which show the effect of increasing the smooth width on the 2nd derivative of a signal
containing three weak noisy peaks. Without smoothing, the signal
seem to be all random noise; with sufficient smoothing, the three
weak peaks are clearly visible (in derivative form) and measurable.
Details and instructions are here. View
the code here or download the ZIP file with sample data for testing.
The script “iSignalDeltaTest”
demonstrates the frequency response of the smoothing and
differentiation functions of iSignal by a applying them to a delta
function. Change the smooth type, smooth width, and derivative
order and see how the power spectrum changes. Demonstration that the Fourier frequency spectrum of a
Gaussian is also a Gaussian:
Click on the figure window, press Shift-T to transfer
the frequency spectrum to the top panel, then press Shift-F,
press Enter three times, and click on the peak in the upper
window. The program computes a least-squares fit of a Gaussian model
to the frequency spectrum now in the top panel. The fit is
essentially perfect:
Least-squares fit of selected peaks to Gaussian peak model
using the peakfit function: Fitting Error =
6.4221e-007% R2 = 1
Peak# Position
Height
Width Area
1 -6.2545e-009
113.31
0.31265 17.721
If you repeat this with Gaussians of different widths (e.g.
width=1 or 4), you'll find that that the width of the frequency
spectrum peak is inversely proportional to the width of the
signal peak.
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).
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.
Last updated September, 2017. This page is part of "A Pragmatic
Introduction to Signal Processing", created and
maintained by Prof. Tom
O'Haver , Department of Chemistry and Biochemistry, The
University of Maryland at College Park. Comments, suggestions and
questions should be directed to Prof. O'Haver at toh@umd.edu.
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