As a preview of things to come, note that one signal
^{4.15} is
*projected* onto another signal
using an *inner
product*. The inner product
computes the *coefficient
of projection*^{4.16} 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

If, more generally, (a sampled complex sinusoid with exponential growth or decay), then the inner product becomes

and this is the definition of the

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,

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University