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 projection4.16 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
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 [
68].
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.
Next Section: Comparing Analog and Digital Complex PlanesPrevious Section: Phasor and Carrier Components of Sinusoids