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SAMPLING
Ha(jR)= 1Ql = T Y
otherwise
3.41 3.42 3.43 H (el") = jw/T, for l < n. o 1
Both are linear and shiftvarying. Upsample by L = 6, filter with a lowpass filter that has a cutoff frequency of o,. ~ / and a gain of 6, and = 6 downsample by M = 5. 4
The *Transform
4.1 INTRODUCTION
The ztransform is a useful tool in the analysis of discretetime signals and systems and is the discretetime counterpart of the Laplace transform for continuoustime signals and systems. The ztransform may be used to solve constant coefficient difference equations, evaluate the response of a linear timeinvariant system to a given input, and design linear filters. In this chapter, we will look at the ztransform and examine how it may be used to solve a variety of different problems. 4.2 DEFINITION OF THE ZTRANSFORM
In Chap. 2, we saw that the discretetime 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 ztransform is a generalization of the DTFT that allows one to deal with such sequences and is defined as follows: Definition: The ztransform of a discretetime 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 zplane referred to as the region of convergence (ROC). Notationally, if x(n) has a ztransform X(z), we write The ztransform 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 discretetime 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 2TRANSFORM
Because the ztransform is a function of a complex variable, it is convenient to describe it using the complex zplane. With z = Re(z) jIm(z) = rejU the axes of the zplane 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 ztransform 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. 41. The unit circle in the complex zplane.
Many of the signals of interest in digital signal processing have ztransforms that are rational functions of z : Factoring the numerator and denominator polynomials, a rational ztransform 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 ztransform to within a constant. Therefore, they provide a concise representation for X(z) that is often represented pictorially in terms of apolezero plot in the zplane. With a polezero 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 zplane. 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:

