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
xs) dx
f(x) = F(s) exp(i 2
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 and the Fourier transform is often represented
by the operator
(Bracewell, 6-8).
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
2E(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).
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) = 2f(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) = 2Fc(s)
cos 2
sx ds.
Likewise, the sine transform of f(x) is defined by
FS(s) = 2f(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).
{f(ax)} =
f(ax) exp(i 2
sx) dx
= |a|-1 f(
) exp(i 2
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).
{f(x - x0)} =
f(x - x0) exp(i 2
sx) dx
= f(
) exp(i 2
s (
+ x0)) d
= exp(i 2x0s)
f(
) exp(i 2
s
) d
= F(s) exp(i 2x0s).
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 2xs0)
{f(x) exp(i 2
xs0)} =
F(s-s0).
G(s) = {f(x)
h(x)}
= {
f(
) h(x -
) d
}
= [
f(
) h(x -
)
d
] exp(-i 2
sx) dx
= f(
) [
h(x -
) exp(-i 2
sx)
dx ] d
= H(s) f(
) exp(-i 2
s
) 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).
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.
h2(t) dt =
|H(f) |2 df
(Brigham, 23). The power spectrum, P(f), is given by
P(f) = |H(f) |2,
f
.
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
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 = fk exp(-i r
0k)
where 0 = 2
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).
Lathi, B. P., 1992, Linear Systems and Signals, Carmichael, Calif: Berkeley-Cambridge Press, 656 pp.