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As a preview of things to come, note that one signal
4.16 is
projected onto another signal
using an inner
product. The inner product
computes the coefficient
of projection4.17 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 case of importance 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 (such
signals are said to be causal). This means we are working with
unilateral Fourier transforms. There are also corresponding
bilateral transforms for which the lower summation limit is
. The DTFT is discussed further in
§B.1.
If, more generally,
(a sampled complex sinusoid with
exponential 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
transform 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 growing
functions of time such as growing 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 its
zeros in the
plane. Similarly, the
transform of an
exponential can be characterized to within a scale factor
by a single point in the
plane (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 sinusoid
is simply a pole located at the point which generates the sinusoid.
Poles and zeros are used extensively in the analysis of recursive
digital 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. This topic will be taken
up in detail in Book II [71].
In the continuous-time case, we have the Fourier transform
which 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.
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