### Comparing Analog and Digital Complex Planes

In signal processing, it is customary to use as the Laplace transform variable for continuous-time analysis, and as the -transform variable for discrete-time analysis. In other words, for continuous-time systems, the frequency domain is the `` plane'', while for discrete-time systems, the frequency domain is the `` plane.'' However, both are simply complex planes.

Figure 4.18 illustrates the various sinusoids represented by points in the plane. The frequency axis is , called the `` axis,'' and points along it correspond to complex sinusoids, with dc at ( ). The upper-half plane corresponds to positive frequencies (counterclockwise circular or corkscrew motion) while the lower-half plane corresponds to negative frequencies (clockwise motion). In the left-half plane we have decaying (stable) exponential envelopes, while in the right-half plane we have growing (unstable) exponential envelopes. Along the real axis (), we have pure exponentials. Every point in the plane corresponds to a generalized complex sinusoid, , with special cases including complex sinusoids , real exponentials , and the constant function (dc).

Figure 4.19 shows examples of various sinusoids
represented by points in the plane. The frequency axis is the ``unit
circle''
, and points along it correspond to *sampled*
complex sinusoids, with dc at (
).
While the frequency axis is unbounded in the plane, it is finite
(confined to the unit circle) in the plane, which is natural because
the sampling rate is finite in the discrete-time case.
As in the
plane, the upper-half plane corresponds to positive frequencies while
the lower-half plane corresponds to negative frequencies. Inside the unit
circle, we have decaying (stable) exponential envelopes, while outside the
unit circle, we have growing (unstable) exponential envelopes. Along the
positive real axis (
re im),
we have pure exponentials, but
along the negative real axis (
re im), we have exponentially
enveloped sampled sinusoids at frequency (exponentially enveloped
alternating sequences). The negative real axis in the plane is
normally a place where all signal transforms should be zero, and all
system responses should be highly attenuated, since there should never be
any energy at exactly half the sampling rate (where amplitude and phase are
ambiguously linked). Every point in the plane can be said to
correspond to sampled generalized complex sinusoids of the form
, with special cases being sampled complex
sinusoids
, sampled real exponentials
,
and the constant sequence
(dc).

In summary, the exponentially enveloped (``generalized'') complex sinusoid
is the fundamental signal upon which other signals are ``projected'' in
order to compute a Laplace transform in the continuous-time case, or a
transform in the discrete-time case. As a special case, if the exponential
envelope is eliminated (set to ), leaving only a complex sinusoid, then
the projection reduces to the Fourier transform in the continuous-time
case, and either the DFT (finite length) or DTFT (infinite length) in the
discrete-time case. Finally, there are still other variations, such as
short-time Fourier transforms (STFT) and wavelet transforms, which utilize
further modifications such as projecting onto *windowed* complex
sinusoids.

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Importance of Generalized Complex Sinusoids