Z Transforms

• The bilateral $z$ transform of the discrete-time signal $x(n)$ is given by
  $$X(z) \triangleq \sum_{n=-\infty}^\infty x(n)z^{-n}.$$

• Since most signals we'll be using are causal, we will typically see the unilateral $z$ transform given by
  $$X(z) \triangleq \sum_{n=0}^\infty x(n)z^{-n}.$$

• The $z$ transform is a generalization of the DTFT (the DFT in the limit as the number of its samples approaches infinity).
  • The carrier term is a generalized sampled complex sinusoid $z^{n} = e^{(\sigma + j\omega)t}$ with an exponential envelope rather than a constant modulus.
  • The DTFT equals the $z$ transform evaluated on the unit circle in the $z$ plane.

• The $z$ transform of a signal $x$ is given by
  $$\mathcal{Z}\{x(\cdot)\} = X(z)$$   also notated$$\quad\mathcal{Z}\{x(n)\} = X(z).$$