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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:

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Definition: The z-transform of a discrete-time signal x(n) is defined by'

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Some consumers are no longer able to borrow to purchase goods and services in the current period The consumption function will shift downward from C to C Consumption is lower for each level of disposable income c Consumers reschedule future purchases to the current period because of the expected rise in prices for goods and services Consumption function C shifts upward to C Solved Problem 43 Variables other than the rate of interest affect gross investment Changes in these other variables cause investment demand to shift downward or upward What should happen to the economy s investment demand when there is a change in the following variables a There is an increase in consumer con dence b Manufacturers utilization of existing capacity declines c There is an increase in vacancy rates in commercial buildings Solution: a Investment demand should shift upward.

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.

Format: Examples:

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).

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:

Housing sales should increase as consumers become more con dent; builders would construct more new housing to meet this increased demand b Investment demand should shift downward Businesses purchases of durable equipment should fall since such purchases would expand.

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