27-10-2012, 05:13 PM
The z-Transform
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z-Transform
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
Region of Convergence
The z-transform of x(n) can be viewed as the Fourier transform of x(n) multiplied by an exponential sequence r-n, and the z-transform may converge even when the Fourier transform does not.
By redefining convergence, it is possible that the Fourier transform may converge when the z-transform does not.
For the Fourier transform to converge, the sequence must have finite energy, or:
Poles and Zeros
When X(z) is a rational function, i.e., a ration of polynomials in z, then:
The roots of the numerator polynomial are referred to as the zeros of X(z), and
The roots of the denominator polynomial are referred to as the poles of X(z).
Note that no poles of X(z) can occur within the region of convergence since the z-transform does not converge at a pole.
Furthermore, the region of convergence is bounded by poles.