Jean Baptiste Joseph Fourier
• Born: 21 March 1768
in Auxerre,
Bourgogne, France
Died: 16 May 1830 in
Paris, France
The Fourier transform, a pervasive and
versatile tool, is used in many fields of
science as a mathematical or physical tool
to alter a problem into one that can be
more easily solved. Some scientists
understand Fourier theory as a physical
phenomenon, not simply as a mathematical
tool.
Forrest Hoffman
http://mathworld.wolfram.com/FourierSeries.html
Fourier Series
A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of
sines and cosines. Fourier series make use of the orthogonality relationships of the sine
and cosine functions. The computation and study of Fourier series is known as harmonic
analysis and is extremely useful as a way to break up an arbitrary periodic function into a
a set of simple terms than can be plugged in, solved individually, and then recombined to
obtain the solution to the original problem or an approximation to it to whatever accuracy
is desired or practical.
In particular, since the superposition principle holds for solutions of a linear homogeneous
ordinary differential equation, if such an equation can be solved in the case a single
sinusoid, the solution for an arbitrary function is immediately available by expressing the
original function as a Fourier series and then plugging in the solution for each sinusoidal
component. In some special cases where the Fourier series can be summed in closed form,
this technique can even yield analytic solutions.
Any set of functions that form a complete orthogonal system have a corresponding
generalized Fourier series analogous to the Fourier series. For example, using
orthogonality of the roots of a Bessel function of the first kind gives a so-called FourierBessel series.
The computation of the (usual) Fourier series is based on the integral identities
(1)
(2)
(3)
(4)
(5)
for
, where
is the Kronecker delta.
Using the method for a generalized Fourier series, the usual Fourier series involving sines
and cosines is obtained by taking
and
functions form a complete orthogonal system over
function f(x) is given by
. Since these
, the Fourier series of a
(6)
where
(7)
(8)
(9)
and n = 1, 2, 3, .... Note that the coefficient of the constant term
has been written in a
special form compared fo the general form for a generalized Fourier series in order to
preserve symmetry with the definitions of
and
.
A Fourier series converges to the function
(equal to the original function at points of
continuity or to the average of the two limits at points of discontinuity)
(10)
if the function satisfies so-called Dirichlet conditions.
As a result, near points of discontinuity, a "ringing" known as the Gibbs phenomenon,
illustrated above, can occur.
For a function f(x) periodic on an interval
instead of
variables can be used to transform the interval of integration from
Let
, a simple change of
to
.
(11)
(12)
Solving for
gives
, and plugging this in gives
(13)
Therefore,
(14)
(15)
(16)
Similarly, the function is instead defined on the interval
simply become
, the above equations
(17)
(18)
(19)
In fact, for f(x) periodic with period
, any interval
can be used, with the
choice being one of convenience or personal preference (Arfken 1985, p. 769).
The coefficients for Fourier series expansions of a few common functions are given in
Beyer (1987, pp. 411-412) and Byerly (1959, p. 51). One of the most common functions
usually analyzed by this technique is the square wave. The Fourier series for a number of
common functions are summarized in the table below.
function
f(x)
Fourier series
Fourier series-sawtooth wave
Fourier series-square wave
Fourier series-triangle
waveFourier
Series Triangle
Wave
T(x)
If a function is even so that
since
Therefore,
, then
is odd. (This follows
is odd and an even function times an odd function is an odd function.)
for all n. Similarly, if a function is odd so that
is odd. (This follows since
odd function is an odd function.) Therefore,
, then
is even and an even function times an
for all n.
The notion of a Fourier series can also be extended to complex coefficients. Consider a
real-valued function f(x). Write
(20)
Now examine
(21)
(22)
(23)
(24)
so
(25)
The coefficients can be expressed in terms of those in the Fourier series
(26)
(27)
(28)
For a function periodic in
, these become
(29)
(30)
These equations are the basis for the extremely important Fourier transform, which is
obtained by transforming
.
from a discrete variable to a continuous one as the length
Complete Set of Functions, Dirichlet Fourier Series Conditions, Fourier-Bessel Series,
Fourier Cosine Series, Fourier-Legendre Series, Fourier Series--Power Series, Fourier
Series--Sawtooth Wave, Fourier Series--Semicircle, Fourier Series--Square Wave, Fourier
Series--Triangle Wave, Fourier Sine Series, Fourier Transform, Generalized Fourier Series,
Gibbs Phenomenon, Harmonic Addition Theorem, Harmonic Analysis, Lacunary Fourier
Series, Lebesgue Constants, Power Spectrum, Riesz-Fischer Theorem, Simple Harmonic
Motion, Superposition Principle
References
Arfken, G. "Fourier Series." Ch. 14 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 760-793, 1985.
Askey, R. and Haimo, D. T. "Similarities between Fourier and Power Series." Amer. Math.
Monthly 103, 297-304, 1996.
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC
Press, 1987.
Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value Problems, 5th ed.
New York: McGraw-Hill, 1993.
Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and
Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York:
Dover, 1959.
Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals, 3rd ed., rev.
and enl. New York: Dover, 1950.
Davis, H. F. Fourier Series and Orthogonal Functions. New York: Dover, 1963.
Dym, H. and McKean, H. P. Fourier Series and Integrals. New York: Academic Press, 1972.
Folland, G. B. Fourier Analysis and Its Applications. Pacific Grove, CA: Brooks/Cole, 1992.
Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York:
Cambridge University Press, 1996.
Körner, T. W. Fourier Analysis. Cambridge, England: Cambridge University Press, 1988.
Körner, T. W. Exercises for Fourier Analysis. New York: Cambridge University Press, 1993.
Krantz, S. G. "Fourier Series." §15.1 in Handbook of Complex Variables. Boston, MA:
Birkhäuser, pp. 195-202, 1999.
Lighthill, M. J. Introduction to Fourier Analysis and Generalised Functions. Cambridge,
England: Cambridge University Press, 1958.
Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994.
Sansone, G. "Expansions in Fourier Series." Ch. 2 in Orthogonal Functions, rev. English ed.
New York: Dover, pp. 39-168, 1991.
Weisstein, E. W. "Books about Fourier Transforms."
http://www.ericweisstein.com/encyclopedias/books/FourierTransforms.html.
Whittaker, E. T. and Robinson, G. "Practical Fourier Analysis." Ch. 10 in The Calculus of
Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 260284, 1967.
Eric W. Weisstein. "Fourier Series." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/FourierSeries.html
Listen to Fourier Series
http://www.jhu.edu/~signals/listen/music1rm.html
Audio signals describe pressure variations on the ear that are perceived as sound. We
focus on periodic audio signals, that is, on tones.
A pure tone can be written as a cosinusoidal signal of amplitude a > 0, frequency o > 0,
and phase angle :
x(t) = a cos(ot + )
The frequency o is in units of radians/second, and o/2 is the frequency in Hertz.
The perceived loudness of a tone is proportional to a0.6.
The pitch of a pure tone is logarithmically related to the frequency. Perceptually,
tones separated by an octave (factor of 2 in frequency) are very similar. For the
purpose of Western music, the octave is subdivided into 12 notes, equally spaced on a
logarithmic scale. The ordering of notes in the octave beginning with 220 Hz is shown
in the following table. Click on the waveform to listen to the corresponding tone.
Fourier Synthesis
http://www.phy.ntnu.edu.tw/java/sound/sound.html
How to play:
1. Left click and drag the [ball, green] circles to change the magnitude of each
Fourier functions [Sin nf, Cos nf].
2. Right click the mouse button to change the magnitude between 0 and 1.0
3. Click Play to turn on the sound effect, Stop to turn it off.
4. The coefficient of sin(0f) is used as amplification factor for all modes.
(Use it to change the sound level)，the coefficient of cos(0f) is the DC
component.
5. Click the checkbox at the top(after stop) will show square the the amplitude of
the signal.
The default value for base frequency is f=100Hz, you can change it from the
TextField. (20 < f < 2000). The ear is 1000 times more sensitive at 1kHz than at
100Hz.
http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node30.html
Transition from Fourier Series to Fourier Integral
We now extend the domain of definition of a linear system from a finite interval, say
, to the infinite interval
. We shall do this by means of a line of
arguments which is heuristic (``serving to discover or stimulate investigation''). Even
though it pretty much ignores issues of convergence, it has the advantage of giving us
at a glance the general direction of the arguments. Treatments which do not ignore the
questions of convergence can be found in §61-65, ``Fourier Series and Boundary
Value Problems'' by Churchill and Brown or in §4.8, ``Methods in Theoretical
Physics'' by Morse and Feshbach.
We start with the complete set of basis functions orthonormal on the interval
The Fourier series for
If
is continuous at
,
is
, then
Second, we let
and, after rearranging some factors, obtain
(220)
Third, by introducing the points
we partition the real -axis into equally spaced subintervals of size
We introduce these points into the Fourier sum, Eq.(2.20),
.
(221)
(222)
Note that this Fourier sum is, in fact, a Riemann sum, a precursor (i.e. approximation)
to the definite integral
(223)
over the limit of the interval
as
The fourth and final step is to let
in order to obtain the result
(224)
Let us summarize the chain of reasoning that leads to the result expressed by Eq.(2.24):
(i) We are given the function
, and from it compute the integral
(225)
(ii) We claim that from
one can reconstruct
by using the formula
(226)
This claim is precisely Eq.(2.24), and it is known as the Fourier Transform Theorem.
Except for the final step, each of the four steps leading to Eq.(2.24) can be verified easily.
That final step, which takes us from Eq.(2.22) to Eq.(2.24) suggests that the infinite sum
approaches an improper integral as
. In other words,
But there are problems with this suggestion, and they are
1.
becomes a different function as
, and
2. an improper integral is defined by
and not by the limit of the discrete approximation given above.
These two objections can, however, be met. Churchill and Brown in §63 accomplish this in
one way. Morse and Feshbach in §4.8 accomplish this in another way. Using their more
careful line of reasoning one, therefore, finds that a function which
1. is piecewise continuous on every bounded interval of the
2. is absolutely integrable along the -axis,
has the property that as
improper integral indeed:
-axis,
the limit of the Fourier series, Eq.(2.20) is an
This result can be restated as a linear transformation and its inverse:
Alternatively, by interchanging the order in which the integration is done, one has
This holds for all (``well behaved'') functions. Consequently, the integration kernel
is an expression for the Dirac delta function.
Either of the last two equations is a generalized completeness relation for the set of
``wave train'' functions
1. These functions are not normalizable, i.e., they
. Instead,
they are said to be `` -function normalized'', as the second equation implies.
2. That the first equation is a completeness relation can be seen from the fact that
it implies Parseval's identity. For
we have
3.
Thus we obtain Parseval's identity (= completeness relation). The only proviso
is (a) that the function
be square-inegrable and (b) that its Fourier transform
be given by the Fourier transform integral.
Remark 1: Note that the Fourier transform is a one-to-one linear transformation from the
set of square-integrable functions onto the set of square integrable functions:
. In other words, the Fourier
transform is what in linear algebra is called an ``isomorphism''.
Remark 2: The line of reasoning leading to Parseval's identity also leads to
whenever
.
Remark 3: The above two remarks imply that the Fourier transform can be viewed as a
unitary transformation in
. Unitary transformations are ``isometries''
because they preserve lengths and inner products. One says, therefore, that the space of
functions defined on the spatial domain is ``isometric'' to the space of functions defined
on the Fourier domain. Thus the Fourier transform operator is a linear isometric mapping.
This fact is depicted by Figure 2.3
Figure 2.3: The Fourier transform is an isometry between
, the Hilbert space of
square integrable functions on the spatial domain, and
, the Hilbert space of
square integrable functions on the Fourier domain.
Note, however, that even though the Fourier transform and its inverse,
(227)
take square integrable functions into square integrable functions, the ``basis elements''
are not square integrable. Instead, they are ``Dirac delta function'' normalized,
i.e.,
Thus they do not belong to
. Nevertheless linear combinations such as
Eq.(2.27) are square integrable, and that is what counts.
Exercise 23.1 (THE FOURIER TRANSFORM: ITS EIGENVALUES)
The Fourier transform, call it
, is a linear one-to-one operator from the space of
square-integrable functions onto itself. Indeed,
Note that here
and
(a)
and
are viewed as points on the common domain
of
.
Consider the linear operator
and its eigenvalue equation
What are the eigenvalues and the eigenfunctions of
?
(b)
Identify the operator
? What are its eigenvalues?
(c)
What are the eigenvalues of
Exercise 23.2 (THE FOURIER TRANSFORM: ITS EIGENVECTORS)
Recall that the Fourier transform
is a linear one-to-one transformation from
onto itself.
Let
be an element of
Let
.
, the Fourier transform of
, be defined by
It is clear that
are square-integable functions, i.e. elements of
Consider the SUBSPACE
(a)
Show that
Show that
is finite dimensional.
is finite dimensional.
.
spanned by these vectors, namely
What is
?
(Hint: Compute
, etc. in terms of
)
(b)
Exhibit a basis for
.
(c)
It is evident that
is a (unitary) transformation on
Find the representation matrix of
part b).
.
, relative to the basis
found in
(d)
Find the secular determinant, the eigenvalues and the corresponding
eigenvectors of
.
(e)
For
, exhibit an alternative basis which consists entirely of eigenvectors of
, each one labelled by its respective eigenvalue.
(f)
What can you say about the eigenvalues of
as compared to
viewed as a transformation on
which acts on a finite-dimensional
vector space?
Exercise 23.3 (EQUIVALENT WIDTHS)
Suppose we define for a square-integrable function
and its Fourier transform
the equivalent width as
and the equivalent Fourier width as
(a)
Show that
is independent of the function
(b)
, and determine the value of this
Determine the equivalent width and the equivalent Fourier width for the
unnormalized Gaussian
and compare them with its full width as defined by its inflection points.
Exercise 23.4 (AUTO CORRELATION SPECTRUM)
Consider the auto-correlation
of the function
whose Fourier transform is
Compute the Fourier transform of the auto correlation function and thereby show that
it equals the ``spectral intensity'' (a.k.a. power spectrum) of whenever is a realvalued function. This equality is known as the Wiener-Khintchine formula.
Exercise 23.5 (MATCHED FILTER)
Consider a linear time-invariant system. Assume its response to a specific driving force,
say
, can be written as
Here
, the ``unit impulse response' (a.k.a. ``Green's function'', as developed
in CHAPTER 4 and used in Section 4.2.1), is a function which characterizes the
system completely. The system is said to be matched to the particular forcing function
if
(Here the bar means complex conjugate.) In that case the system response to a generic
forcing function
is
A system characterized by such a unit impulse response is called a matched filter
because its design is matched to the particular signal
called the cross correlation between
(a)
Compute the total energy
of the cross correlation
and
. The response
is
.
in terms of the Fourier amplitudes
and
(b)
Consider the family of forcing functions
and the corresponding family of normalized cross correlations (i.e. the
corresponding responses of the system)
Show that
(i)
is the peak intensity, i.e., that
(Nota bene: The function
correlation function of
(ii)
corresponding to
). Also show that
equality holds if the forcing function
has the form
is called the auto
http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node29.html
The Fourier Integral
Question: What aspect of nature is responsible for the pervasive importance of Fourier
analysis?
Answer: Translation invariance. Suppose a linear system is invariant under time or space
translations. Then that system's behaviour becomes particularly perspicuous, physically
and mathematically, when it is described in terms of translation eigenfunctions, i.e., in
terms of exponentials which oscillate under time or space translations. (Nota bene: real
exponentials are also translation eigenfunctions, but they won't do because they blow up
at
or
.) In other words, it is the translation invariance in nature which makes
Fourier analysis possible and profitable.
http://www.ma.iup.edu/projects/CalcDEMma/fouriertrans/fouriertrans04.html
Fourier Integral Examples
Let's look at the Fourier integral representation of the pulse from -1 to 1.
FourierTransform[UnitStep[t+1]-UnitStep[t-1],t,w]
Recall that C(w) is 1/2 F(w).
f[t_]:=1/Pi Integrate[Sin[w]/w Exp[I w t],{w,-Infinity,Infinity}]
f[t]
Do not be thrown off by the output. Look at the graph.
Plot[Evaluate[f[t]], {t,-2,2}];
Does the fourier transform in Mathematica converge to the proper value? (Ignore the
warning messages you see.)
f[-1]
f[1]
Let's look at the Fourier Integral representation of the function that is sin(t) from -3Pi to Pi
and zero elsewhere.
FourierTransform[(UnitStep[t+3 Pi]-UnitStep[t-Pi])Sin[t],t,w]
Yuck! There is a valuable lesson to be learned here. It may be better to evaluate the
Fourier transform from the definition instead of using the Mathematica command.
F[w_]:=Integrate[Sin[t] Exp[- I w t],{t,-3 Pi, Pi}]
F[w]
Simplify[ComplexExpand[F[w]]]
Recall that C(w) is 1/2 F(w).
g[t_]:=1/Pi Integrate[1/2 F[w] Exp[I w t],{w,-Infinity,Infinity}]
Plot[Evaluate[g[t]], {t,-6 Pi,3 Pi}];
Another valuable lesson. Look at the scale of your graph before you judge!
Plot[Evaluate[g[t]], {t,-6 Pi,3 Pi},PlotRange->{-2,2}];
http://aurora.phys.utk.edu/~forrest/papers/fourier/
An Introduction to Fourier Theory
by Forrest Hoffman
This paper is also available in DVI, and PostScript® formats.
Table of Contents











Introduction
The Fourier Transform
The Two Domains
Fourier Transform Properties
o Scaling Property
o Shifting Property
o Convolution Theorem
o Correlation Theorem
Parseval's Theorem
Sampling Theorem
Aliasing
Discrete Fourier Transform (DFT)
Fast Fourier Transform (FFT)
Summary
References
Introduction
Linear transforms, especially Fourier and Laplace transforms, are widely used in
solving problems in science and engineering. The Fourier transform is used in linear
systems analysis, antenna studies, optics, random process modeling, probability
theory, quantum physics, and boundary-value problems (Brigham, 2-3) and has been
very successfully applied to restoration of astronomical data (Brault and White). The
Fourier transform, a pervasive and versatile tool, is used in many fields of science as a
mathematical or physical tool to alter a problem into one that can be more easily
solved. Some scientists understand Fourier theory as a physical phenomenon, not
simply as a mathematical tool. In some branches of science, the Fourier transform of
one function may yield another physical function (Bracewell, 1-2).
The Fourier Transform
The Fourier transform, in essence, decomposes or separates a waveform or function
into sinusoids of different frequency which sum to the original waveform. It identifies
or distinguishes the different frequency sinusoids and their respective amplitudes
(Brigham, 4). The Fourier transform of f(x) is defined as
F(s) =
f(x) exp(-i 2
xs) dx.
Applying the same transform to F(s) gives
f(w) =
F(s) exp(-i 2
ws) ds.
If f(x) is an even function of x, that is f(x) = f(-x), then f(w) = f(x). If f(x) is an odd
function of x, that is f(x) = -f(-x), then f(w) = f(-x). When f(x) is neither even nor odd, it
can often be split into even or odd parts.
To avoid confusion, it is customary to write the Fourier transform and its inverse so that
they exhibit reversibility:
F(s) =
f(x) exp(-i 2
f(x) =
F(s) exp(i 2
xs) dx
xs) ds
so that
f(x) =
f(x) exp(-i 2
xs) dx
exp(i 2
xs) ds
as long as the integral exists and any discontinuities, usually represented by multiple
integrals of the form ½[f(x+) + f(x-)], are finite. The transform quantity F(s) is often
represented as
(Bracewell, 6-8).
and the Fourier transform is often represented by the operator
There are functions for which the Fourier transform does not exist; however, most physical
functions have a Fourier transform, especially if the transform represents a physical
quantity. Other functions can be treated with Fourier theory as limiting cases. Many of the
common theoretical functions are actually limiting cases in Fourier theory.
Usually functions or waveforms can be split into even and odd parts as follows
f(x) = E(x) + O(x)
where
E(x) = ½ [f(x) + f(-x)]
O(x) = ½ [f(x) - f(-x)]
and E(x), O(x) are, in general, complex. In this representation, the Fourier transform of
f(x) reduces to
2
E(x) cos(2
xs) dx - 2i
O(x) sin(2
xs) dx
It follows then that an even function has an even transform and that an odd function has
an odd transform. Additional symmetry properties are shown in Table 1 (Bracewell, 14).
Table 1: Symmetry Properties of the Fourier Transform
function
transform
----------------------------------------------------------real and even
real and even
real and odd
imaginary and odd
imaginary and even
imaginary and even
complex and even
complex and even
complex and odd
complex and odd
real and asymmetrical
complex and asymmetrical
imaginary and asymmetrical
complex and asymmetrical
real even plus imaginary odd
real
real odd plus imaginary even
imaginary
even
even
odd
odd
An important case from Table 1 is that of an Hermitian function, one in which the real
part is even and the imaginary part is odd, i.e., f(x) = f*(-x). The Fourier transform of
an Hermitian function is even. In addition, the Fourier transform of the complex
conjugate of a function f(x) is F*(-s), the reflection of the conjugate of the transform.
The cosine transform of a function f(x) is defined as
Fc(s) = 2
f(x) cos 2
sx dx.
This is equivalent to the Fourier transform if f(x) is an even function. In general, the even
part of the Fourier transform of f(x) is the cosine transform of the even part of f(x). The
cosine transform has a reverse transform given by
f(x) = 2
Fc(s) cos 2
sx ds.
Likewise, the sine transform of f(x) is defined by
FS(s) = 2
f(x) sin 2
sx dx.
As a result, i times the odd part of the Fourier transform of f(x) is the sine transform of the
odd part of f(x).
Combining the sine and cosine transforms of the even and odd parts of f(x) leads to the
Fourier transform of the whole of f(x):
f(x) =
CE(x)
-i
SO(x)
where
,
C, and
S stand for -i times the Fourier transform, the cosine
transform, and the sine transform respectively, or
F(s) = ½FC(s) - ½iFS(s)
(Bracewell, 17-18).
Since the Fourier transform F(s) is a frequency domain representation of a function f(x),
the s characterizes the frequency of the decomposed cosinusoids and sinusoids and is
equal to the number of cycles per unit of x (Bracewell, 18-21). If a function or waveform is
not periodic, then the Fourier transform of the function will be a continuous function of
frequency (Brigham, 4).
The Two Domains
It is often useful to think of functions and their transforms as occupying two domains.
These domains are referred to as the upper and the lower domains in older texts, ``as
if functions circulated at ground level and their transforms in the underworld''
(Bracewell, 135). They are also referred to as the function and transform domains, but
in most physics applications they are called the time and frequency domains
respectively. Operations performed in one domain have corresponding operations in
the other. For example, as will be shown below, the convolution operation in the time
domain becomes a multiplication operation in the frequency domain, that is, f(x)
g(x)
F(s) G(s). The reverse is also true, F(s)
G(s)
f(x) g(x). Such
theorems allow one to move between domains so that operations can be performed
where they are easiest or most advantageous.
Fourier Transform Properties
Scaling Property
If
{f(x)} = F(s) and a is a real, nonzero constant, then
{f(ax)} =
= |a|-1
f(
f(ax) exp(i 2
) exp(i 2
sx) dx
s
/a) d
= |a|-1 F(s/a).
From this, the time scaling property, it is evident that if the width of a function is
decreased while its height is kept constant, then its Fourier transform becomes wider and
shorter. If its width is increased, its transform becomes narrower and taller.
A similar frequency scaling property is given by
{|a|-1 f(x/a)} = F(as).
Shifting Property
If
{f(x)} = F(s) and x0 is a real constant, then
{f(x - x0)} =
=
f(
= exp(i 2
f(x - x0) exp(i 2
) exp(i 2
s(
x0s)
= F(s) exp(i 2
f(
sx) dx
+ x0)) d
) exp(i 2
s
)d
x0s).
This time shifting property states that the Fourier transform of a shifted function is just the
transform of the unshifted function multiplied by an exponential factor having a linear
phase.
Likewise, the frequency shifting property states that if F(s) is shifted by a constant s0, its
inverse transform is multiplied by exp(i 2
{f(x) exp(i 2
xs0)} = F(s-s0).
xs0)
Convolution Theorem
We now derive the aforementioned time convolution theorem. Suppose that g(x) =
f(x)
h(x). Then, given that
= H(s),
G(s) =
{f(x)
{g(x)} = G(s),
{
f(
) h(x -
)d
}
=
[
f(
) h(x -
)d
] exp(-i 2
=
f(
)[
f(
{h(x)}
h(x)}
=
= H(s)
{f(x)} = F(s), and
h(x -
) exp(-i 2
) exp(-i 2
s
sx) dx
sx) dx ] d
)d
= F(s) H(s).
This extremely powerful result demonstrates that the Fourier transform of a convolution is
simply given by the product of the individual transforms, that is
{f(x)
h(x)} = F(s) H(s).
Using a similar derivation, it can be shown that the Fourier transform of a product is given
by the convolution of the individual transforms, that is
{f(x) h(x)} = F(s)
H(s)
This is the statement of the frequency convolution theorem (Gaskill, 194-197; Brigham,
60-65).
Correlation Theorem
The correlation integral, like the convolution integral, is important in theoretical and
practical applications. The correction integral is defined as
h(x) =
f(u) g(x+u) du
and like the convolution integral, it forms a Fourier transform pair given by
{h(x)} = F(s) G*(s).
This is the statement of the correlation theorem. If f(x) and g(x) are the same function,
the integral above is normally called the autocorrelation function, and the crosscorrelation
if they differ (Brigham, 65-69). The Fourier transform pair for the autocorrelation is simply
f(u) f(x+u) du
= |F |2.
Parseval's Theorem
Parseval's Theorem states that the power of a signal represented by a function h(t) is
the same whether computed in signal space or frequency (transform) space; that is,
h2(t) dt =
|H(f) |2 df
(Brigham, 23). The power spectrum, P(f), is given by
P(f) = |H(f) |2,
f
.
Sampling Theorem
A bandlimited signal is a signal, f(t), which has no spectral components beyond a
frequency B Hz; that is, F(s) = 0 for |s| > 2
B. The sampling theorem states that a
real signal, f(t), which is bandlimited to B Hz can be reconstructed without error from
samples taken uniformly at a rate R > 2B samples per second. This minimum
sampling frequency,
s = 2B Hz, is called the Nyquist rate or the Nyquist
frequency. The corresponding sampling interval, T = 1/2B (where t = nT), is called
the Nyquist interval. A signal bandlimited to B Hz which is sampled at less than the
Nyquist frequency of 2B, i.e., which was sampled at an interval T > 1/2B, is said to be
undersampled.
Aliasing
A number of practical difficulties are encountered in reconstructing a signal from its
samples. The sampling theorem assumes that a signal is bandlimited. In practice,
however, signals are timelimited, not bandlimited. As a result, determining an
adequate sampling frequency which does not lose desired information can be difficult.
When a signal is undersampled, its spectrum has overlapping tails; that is F(s) no
longer has complete information about the spectrum and it is no longer possible to
recover f(t) from the sampled signal. In this case, the tailing spectrum does not go to
zero, but is folded back onto the apparent spectrum. This inversion of the tail is called
spectral folding or aliasing (Lathi, 532-535).
Figure 1: Undersampled, oversampled, and critically-sampled unit area gaussian curves.
As an example, Figure 1 shows a unit gaussian curve sampled at three different rates. The
FFT (or Fast Fourier Transform) of the undersampled gaussian appears flattened and its
tails do not reach zero. This is a result of aliasing. Additional spectral components have
been folded back onto the ends of the spectrum or added to the edges to produce this
curve.
The FFT of the oversampled gaussian reaches zero very quickly. Much of its spectrum is
zero and is not needed to reconstruct the original gaussian.
Finally, the FFT of the critically-sampled gaussian has tails which reach zero at their ends.
The data in the spectrum of the critically-sampled gaussian is just sufficient to reconstruct
the original. This gaussian was sampled at the Nyquist frequency.
Figure 1 was generated using IDL with the following code:
!P.Multi=[0,3,2]
a=gauss(256,2.0,2) ; undersampled
fa=fft(a,-1)
b=gauss(256,2.0,0.1) ; oversampled
fb=fft(b,-1)
c=gauss(256,2.0,0.8) ; critically sampled
fc=fft(c,-1)
plot,a,title='!6Undersampled Gaussian'
plot,b,title='!6Oversampled Gaussian'
plot,c,title='!6Critically-Sampled Gaussian'
plot,shift(abs(fa),128),title='!6FFT of Undersampled Gaussian'
plot,shift(abs(fb),128),title='!6FFT of Oversampled Gaussian'
plot,shift(abs(fc),128),title='!6FFT of Critically-Sampled
Gaussian'
The gauss function is as follows:
function gauss,dim,fwhm,interval
;
; gauss - produce a normalized gaussian curve centered in dim data
;
points with a full width at half maximum of fwhm sampled
;
with a periodicity of interval
;
; dim
= the number of points
; fwhm
= full width half max of gaussian
; interval = sampling interval
;
center=dim/2.0 ; automatically center gaussian in dim points
x=findgen(dim)-center
sigma=fwhm/sqrt(8.0 * alog(2.0)) ; fwhm is in data points
coeff=1.0 / ( sqrt(2.0*!Pi) * (sigma/interval) )
data=coeff * exp( -(interval * x)^2.0 / (2.0*sigma^2.0) )
return,data
end
Discrete Fourier Transform (DFT)
Because a digital computer works only with discrete data, numerical computation of
the Fourier transform of f(t) requires discrete sample values of f(t), which we will call
fk. In addition, a computer can compute the transform F(s) only at discrete values of s,
that is, it can only provide discrete samples of the transform, Fr. If f(kT) and F(rs0) are
the kth and rth samples of f(t) and F(s), respectively, and N0 is the number of samples
in the signal in one period T0, then
fk = T f(kT) = T0N0-1 f(kT)
and
Fr = F(rs0)
where
s0 = 2
0
=2
T0-1.
The discrete Fourier transform (DFT) is defined as
Fr =
where
fk exp(-i r
0
=2
0k)
N0-1. Its inverse is
fk = N0-1
Fr exp(i r
0k).
These equations can be used to compute transforms and inverse transforms of
appropriately-sampled data. Proofs of these relationships are in Lathi (546-548).
Fast Fourier Transform (FFT)
The Fast Fourier Transform (FFT) is a DFT algorithm developed by Tukey and
Cooley in 1965 which reduces the number of computations from something on the
order of N02 to N0 log N0. There are basically two types of Tukey-Cooley FFT
algorithms in use: decimation-in-time and decimation-in-frequency. The algorithm is
simplified if N0 is chosen to be a power of 2, but it is not a requirement.
Summary
The Fourier transform, an invaluable tool in science and engineering, has been
introduced and defined. Its symmetry and computational properties have been
described and the significance of the time or signal space (or domain) vs. the
frequency or spectral domain has been mentioned. In addition, important concepts in
sampling required for the understanding of the sampling theorem and the problem of
aliasing have been discussed. An example of aliasing was provided along with a short
description of the discrete Fourier transform (DFT) and its popular offspring, the Fast
Fourier Transform (FFT) algorithm.
References
Blass, William E. and Halsey, George W., 1981, Deconvolution of Absorption
Spectra, New York: Academic Press, 158 pp.
Bracewell, Ron N., 1965, The Fourier Transform and Its Applications, New York: McGrawHill Book Company, 381 pp.
Brault, J. W. and White, O. R., 1971, The analysis and restoration of astronomical data via
the fast Fourier transform, Astron. & Astrophys., 13, pp. 169-189.
Brigham, E. Oren, 1988, The Fast Fourier Transform and Its Applications, Englewood Cliffs,
NJ: Prentice-Hall, Inc., 448 pp.
Cooley, J. W. and Tukey, J. W., 1965, An algorithm for the machine calculation of complex
Fourier series, Mathematics of Computation, 19, 90, pp. 297-301.
Gabel, Robert A. and Roberts, Richard A., 1973, Signals and Linear Systems, New York:
John Wiley & Sons, 415 pp.
Gaskill, Jack D., 1978, Linear Systems, Fourier Transforms, and Optics, New York: John
Wiley & Sons, 554 pp.
Lathi, B. P., 1992, Linear Systems and Signals, Carmichael, Calif: Berkeley-Cambridge
Press, 656 pp.
Physics 641- Instrument Design and Signal Enhancement / Forrest Hoffman