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]).