NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
<< Previous
page TOC INDEX Next
page >>
As a preview of things to come, note that one signal
5.10 is projected onto another
signal 
using an inner
product. The inner product 
computes the coefficient of projection5.11 of 
onto
. If
(a sampled, unit-amplitude,
zero-phase, complex sinusoid),
then the inner product computes the Discrete Fourier
Transform (DFT), provided the
frequencies are chosen to be 
. For the DFT, the inner product is
specifically
Another commonly used case is the Discrete Time Fourier
Transform(DTFT) which is like the DFT, except that the
transform accepts an infinite number of samples instead of only
. In this
case, frequency
is continuous, and
The DTFT is what you get in the limit as the number of samples in
the DFT approaches infinity. The lower limit of summation remains
zero because we are assuming all signals are zero for negative
time. This means we are working with unilateral Fourier
transforms. There are also corresponding bilateral
transforms for which the lower summation limit is 
.
If, more generally, 
(a sampled complex sinusoid withexponential
growth or decay), then the inner product becomes
and this is the definition of the 
transform. It is a generalization of the DTFT: The DTFT equals
the transform
evaluated on the unit circle in the plane. In
principle, the
transform can also be recovered from the DTFT by means of
''analytic continuation'' from the unit circle to the entire
plane
(subject to mathematical disclaimers which are unnecessary in
practical applications since they are always finite).
Why have a
tranform when it seems to contain no more information than the
DTFT? It is useful to generalize from the unit circle (where the
DFT and DTFT live) to the entire complex
plane (the
transform's domain) for a number of reasons. First, it allows
transformation of growingfunctions of time such as unstable
exponentials; the only limitation on growth is that it cannot be
faster than exponential. Secondly, the transform
has a deeper algebraic structure over the complex plane as a whole
than it does only over the unit circle. For example, the
transform
of any finite signal is simply a polynomial in . As such,
it can be fully characterized (up to a constant scale factor) by
itszeros in the plane. Similarly, the transform
of an exponential can be characterized by a single point of
the transform (the point which generates the exponential);
since the
transform goes to infinity at that point, it is called a
pole of the transform. More generally, the transform
of any generalized complex sinusoidis simply a pole
located at the point which generates the sinusoid. Poles and zeros
are used extensively in the analysis of recursivedigital
filters. On the most general level, every finite-order,
linear, time-invariant, discrete-time system is fully specified (up
to a scale factor) by its poles and zeros in the
plane.
In the continuous-time case, we have the Fourier
transformwhich projects onto the
continuous-time sinusoids defined by 
, and the appropriate inner product is
Finally, the Laplace transform is the continuous-time
counterpart of the transform, and it projects signals onto exponentially
growing or decaying complex sinusoids:
The Fourier transform equals the Laplace transform evaluated along
the ''
axis'' in the 
plane, i.e., along the line 
, for which 
. Also, the Laplace transform is obtainable from the
Fourier transform via analytic continuation. The usefulness of the
Laplace transform relative to the Fourier transform is exactly
analogous to that of the transform outlined above.
<< Previous
page TOC INDEX Next
page >>
