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CHAP. 31
SAMPLING
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otherwise
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4
The *Transform
4.1 INTRODUCTION
The z-transform is a useful tool in the analysis of discrete-time signals and systems and is the discrete-time counterpart of the Laplace transform for continuous-time signals and systems. The z-transform may be used to solve constant coefficient difference equations, evaluate the response of a linear time-invariant system to a given input, and design linear filters. In this chapter, we will look at the z-transform and examine how it may be used to solve a variety of different problems.
4.2 DEFINITION OF THE Z-TRANSFORM
In Chap. 2, we saw that the discrete-time Fourier transform (DTFT) of a sequence .c(n) is equal to the sum
However. in order for this series to converge, it is necessary that the signal be absolutely summable. Unfortunately, many of the signals that we would like to consider are not absolutely summable and, therefore, d o not have a DTFT. Some examples include x(n) = u(n) x(n) = (OS)"u(-n) x(n) = sin n q
The z-transform is a generalization of the DTFT that allows one to deal with such sequences and is defined as follows:
Definition: The z-transform of a discrete-time signal x(n) is defined by'
where z = reJ" is a complex variable. The values of z for which the sum converges define a region in the z-plane referred to as the region of convergence (ROC). Notationally, if x(n) has a z-transform X(z), we write
The z-transform may be viewed as the DTFT of an exponentially weighted sequence. Specifically, note that with z = rejo,
and we see that X(z) is the discrete-time Fourier transform of the sequence r-"x(n). Furthermore, the ROC is determined by the range of values of r for which
h he reader should note that in many mathematics books, and in some engineering books, X ( z ) is defined in terms of a sum using positive powers of z.
CHAP. 41
THE 2-TRANSFORM
Because the z-transform is a function of a complex variable, it is convenient to describe it using the complex z-plane. With z = Re(z) jIm(z) = rejU
the axes of the z-plane are the real and imaginary parts of z as illustrated in Fig. 4- 1, and the contour corresponding to Izl = 1 is a circle of unit radius referred to as the unit circle. The z-transform evaluated on the unit circle corresponds to the DTFT, = (4.2) ~(ej") X(Z)I~=~,~ More specifically, evaluating X(z) at points around the unit circle, beginning at z = l(w = 0), through z = j (W = n/2), to z = - 1 ( = n ) , we obtain the values of X(el") for 0 5 w 5 n . Note that in order for the DTFT ~ of a signal to exist, the unit circle must be within the region of convergence of X(z).
Im(z>
Unit circle
Fig. 4-1. The unit circle in the complex z-plane.
Many of the signals of interest in digital signal processing have z-transforms that are rational functions of z :
Factoring the numerator and denominator polynomials, a rational z-transform may be expressed as follows:
The roots of the numerator polynomial, Bk, are referred to as the zeros of X(z), and the roots of the denominator polynomial, ak, are referred to as the poles. The poles and zeros uniquely define the functional form of a rational z-transform to within a constant. Therefore, they provide a concise representation for X(z) that is often represented pictorially in terms of apole-zero plot in the z-plane. With a pole-zero plot, the location of each pole is indicated by an " x " and the location of each zero is indicated by an " o w ,with the region of convergence indicated by shading the appropriate region of the z-plane. The region of convergence is, in general, an annulus of the form If a = 0, the ROC may also include the point z = 0, and if B = oo,the ROC may also include infinity. For a rational X(z), the region of convergence will contain no poles. Listed below are three properties of the region of convergence:
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