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Relation to the z Transform
The Laplace transform is used to analyze continuous-time
systems. Its discrete-time counterpart is the
transform:
If we define
, the
transform becomes proportional to the
Laplace transform of a sampled continuous-time signal:
As the sampling interval
goes to zero, we have
where
and
.
In summary,
Note that the
plane and
plane are generally related by
In particular, the discrete-time frequency axis
and
continuous-time frequency axis
are related
by
For the mapping
from the
plane to the
plane to be invertible, it is necessary that
be zero for all
. If this is true, we say
is
bandlimited to half the sampling rate. As is well known, this
condition is necessary to prevent aliasing when sampling the
continuous-time signal
at the rate
to produce
,
(see [84, Appendix G]).
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