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 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

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