26-11-2012, 06:13 PM
z-Transform
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Introduction
The Laplace transform and its discrete-time counterpart the z-transform are essential
mathematical tools for system design and analysis, and for monitoring the stability of
a system. A working knowledge of the z-transform is essential to the study of discretetime
filters and systems. It is through the use of these transforms that we formulate a
closed-form mathematical description of a system in the frequency domain, design the
system, and then analyse the stability, the transient response and the steady state
characteristics of the system.
A mathematical description of the input-output relation of a system can be formulated
either in the time domain or in the frequency domain. Time-domain and frequency
domain representation methods offer alternative insights into a system, and depending
on the application it may be more convenient to use one method in preference to the
other. Time domain system analysis methods are based on differential equations which
describe the system output as a weighted combination of the differentials (i.e. the rates
of change) of the system input and output signals. Frequency domain methods, mainly
the Laplace transform, the Fourier transform, and the z-transform, describe a system in
terms of its response to the individual frequency constituents of the input signal. In
section 4.? we explore the close relationship between the Laplace, the Fourier and the
z-transforms, and the we observe that all these transforms employ various forms of
complex exponential as their basis functions. The description of a system in the
frequency domain can reveal valuable insight into the system behaviour and stability.
System analysis in frequency domain can also be more convenient as differentiation
and integration operations are performed through multiplication and division by the
frequency variable respectively. Furthermore the transient and the steady state
characteristics of a system can be predicted by analysing the roots of the Laplace
transform or the z-transform, the so-called poles and zeros of a system.
Derivation of the z-Transform
The z-transform is the discrete-time counterpart of the Laplace transform. In this
section we derive the z-transform from the Laplace transform a discrete-time signal.
The Laplace transform X(s), of a continuous-time signal x(t), is given by the integral.
The z-Plane and The Unit Circle
The frequency variables of the Laplace transform s=σ +jω, and the z-tranform z=rejω
are complex variables with real and imaginary parts and can be visualised in a two
dimensional plane. Figs. 4.2.a and 4.2.b shows the s-plane of the Laplace transform
and the z-plane of z-transform. In the s-plane the vertical jω−axis is the frequency
axis, and the horizontal σ-axis gives the exponential rate of decay, or the rate of
growth, of the amplitude of the complex sinusoid as also shown in Fig.
Poles and Zeros
One of the most useful aspects of the z-transform analysis is the description of a
system in terms of the so-called poles and zeros of the system. The zeros of a transfer
function H(z) are the values of the variable z for which the transfer function (or
equivalently its numerator) is zero. Therefore the zeros are the roots of numerator
polynomial in Eq. (4.49). The poles of H(z) are the values of the variable z for which
H(z) is infinite. This happens when the denominator of H(z) is zero. Therefore the
poles of H(z) are the roots of the denominator polynomial of Eq. (4.49).