## 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]).

Next Section:
Laplace Transform Theorems
Previous Section:
Analytic Continuation