To successfully apply the various signals and systems concepts as part of practical engineering scenarios, you need to know what analysis tools are available. The figure shows the mathematical relationships between time, frequency, and the s- and z-domains. This portrayal offers perspective on a well-rounded study of signals and systems and reveals that you can establish relationships between domains in more than one way.
When working through engineering problems, stay on a single branch when possible — but sometimes you just can’t. Fortunately, if you need to go from the time domain t to the discrete-time frequency domain
you can hop from t to n and then from
Some of the interconnects are simply conceptual; they don’t hold mathematically under all conditions. For example, Fourier transforms in the limit make it invalid to sample the s- or z-domains by letting
The region of convergence (ROCs) must include the frequency axis for this connection to be valid. Sampling theory offers the links between t and n, but a causal interpolation filter leads to less than perfect reconstruction, revealing that the idea of exact is conceptual only. An approximation is used in real-world systems.