Definition
The z-Transform is the discrete time analog of the C.T. Laplace Transform.
For a D.T. signal $ x[n]\, $, the z-Transform is defined as
- $ X(z) = \sum_{n = -\infty}^{\infty} x[n]z^{-n} $
Realm of Convergence
Any z-Transform will have a realm of convergence. For example, if your signal is:
- $ x[n] = 2^{n}u[-n] $
The z-Transform summation reduces to:
- $ \sum_{n = 0}^{\infty} (\frac{z}{2})^{n} $, which will converge only if $ |\frac{z}{2}| < 1 $
