12-10-2012, 04:02 PM
Z-TRANSFORM
MATHS (1).doc (Size: 145.5 KB / Downloads: 26)
INTRODUCTION
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.
It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time scale calculus.
Relationship to Laplace transform
The Bilinear transform is a useful approximation for converting continuous time filters (represented in Laplace space) into discrete time filters (represented in z space), and vice versa.
Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire axis of the s-plane onto the unit in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the axis is in the region of convergence of the Laplace transform.
MATHS (1).doc (Size: 145.5 KB / Downloads: 26)
INTRODUCTION
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.
It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time scale calculus.
Relationship to Laplace transform
The Bilinear transform is a useful approximation for converting continuous time filters (represented in Laplace space) into discrete time filters (represented in z space), and vice versa.
Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire axis of the s-plane onto the unit in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the axis is in the region of convergence of the Laplace transform.