GENERALIZATION OF THE DTFT TO THE z-TRANSFORM

9.6 GENERALIZATION OF THE DTFT TO THE z-TRANSFORM

LTID systems can be analyzed by using the DTFT. This method, however, has the following limitations:

1. Existence of the DTFT is guaranteed only for absolutely summable signals. The DTFT does not exist for exponentially or even linearly growing signals. This means that the DTFT method is applicable only for a limited class of inputs.

2. Moreover, this method can be applied only to asymptotically or BIBO stable systems; it cannot be used for unstable or even marginally stable systems.

These are serious limitations in the study of LTID system analysis. Actually it is the first limitation that is also the cause of the second limitation. Because the DTFT is incapable of handling growing signals, it is incapable of handling unstable or marginally stable These are serious limitations in the study of LTID system analysis. Actually it is the first limitation that is also the cause of the second limitation. Because the DTFT is incapable of handling growing signals, it is incapable of handling unstable or marginally stable

DTFT, we are using sinusoids or exponentials of the form e j Ωn to synthesize an arbitrary signal x[n]. These signals are sinusoids with constant amplitudes. They are incapable of synthesizing exponentially growing signals no matter how many such components we add. Our hope, therefore, lies in trying to synthesize x[n] by using exponentially growing sinusoids or exponentials. This goal can be accomplished by generalizing the frequency variable j Ω to σ + jΩ, that is, by using exponentials of the form e (σ + j Ω)n instead of

exponentials e j Ωn . The procedure is almost identical to that used in extending the Fourier transform to the Laplace transform. Let us define a new variable

. Hence

and

Consider now the DTFT of x[n]e −σn ( σ real)

It follows from Eq. (9.79) that the sum in Eq. (9.82) is

.Thus

Hence, the inverse DTFT of

is x[n]e −σn . Therefore

Multiplying both sides of Eq. (9.84) by e σn yields

Let us define a new variable as

Because = e σ+j Ω is complex, we can express it as = re j Ω , where r = e σ . Thus, lies on a circle of radius r, and as Ω varies from −σ to σ, circumambulates along this circle, completing exactly one counterclockwise rotation, as illustrated in Fig. 9.16 . Changing to variable in Eq. (9.85) yields

Figure 9.16: Contour of integration for the -transform.

and form Eq. (9.83) we obtain

where the integral ∡ indicates a contour integral around a circle of radius r in the counterclockwise direction. Equations (9.87a) and (9.87b) are the desired extensions. They are, however, in a clumsy form. For the sake of convenience, we make

another notational change by observing that (In ) is a function of .Let us denote it by a simpler notation X[ ]. Thus, Eqs. (9.87) become

and

This is the (bilateral) transform pair. Equation (9.88) expresses x[n] as a continuous sum of exponentials of the form n =e (σ+j Ω)n =r n

e j Ω)n . Thus, by selecting a proper value for r (or σ), we can make the exponential grow (or decay) at any exponential rate we desire. If we let σ = 0, we have = e j Ω and

Thus, the familiar DTFT is just a special case of the -transform X[ ] obtained by letting = e j Ω and assuming that the sum on the right- hand side of Eq. (9.89) converges when = e j Ω . This also implies that the ROC for X[ ] includes the unit circle.

[ † ] Recall that the output of an unstable system grows exponentially. Also, the output of a marginally stable system to characteristic mode input grows with time.