[9.6 GENERALIZATION OF THE](#page-14-0) DTFT TO THE z-TRANSFORM

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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 systems.† Our goal is, therefore, to extend the DTFT concept so that it can handle exponentially growing signals.

We may wonder what causes this limitation on DTFT so that it is incapable of handling exponentially growing signals. Recall that in the DTFT, we are using sinusoids or exponentials of the form ejⁿ 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 to σ + j, that is, by using exponentials of the form e(σ+j)ⁿ instead of exponentials ejⁿ. The procedure is almost identical to that used in extending the Fourier transform to the Laplace transform.

Let us define a new variable Xˆ(j) = X(). Hence,

$$\hat{X}(j\Omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\Omega n}$$

\n(9.45)

and

$$x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} \hat{X}(j\Omega) e^{j\Omega n} d\Omega$$

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

$$\text{DTFT}\left\{x[n]e^{-\sigma n}\right\} = \sum_{n=-\infty}^{\infty} x[n]e^{-\sigma n}e^{-j\Omega n} = \sum_{n=-\infty}^{\infty} x[n]e^{-(\sigma+j\Omega)n}$$

It follows from Eq. (9.45) that this sum is Xˆ(σ +j). Thus,

$$\text{DTFT}\left\{x[n]e^{-\sigma n}\right\} = \sum_{n=-\infty}^{\infty} x[n]e^{-(\sigma+j\Omega)n} = \hat{X}(\sigma+j\Omega) \tag{9.46}$$

Hence, the inverse DTFT of Xˆ(σ +j) is x[n] e−σ*ⁿ*. Therefore,

$$x[n]e^{-\sigma n} = \frac{1}{2\pi} \int_{-\pi}^{\pi} \hat{X}(\sigma + j\Omega) e^{j\Omega n} d\Omega$$

Multiplying both sides by eσ*ⁿ* yields

$$x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} \hat{X}(\sigma + j\Omega) e^{(\sigma + j\Omega)n} d\Omega$$

\n(9.47)

Let us define a new variable z as

$$z = e^{\sigma + j\Omega}$$

so that $\ln z = \sigma + j\Omega$ and $\frac{1}{z}dz = j d\Omega$

^†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.

Figure 9.18 Contour of integration for the z-transform.

Because z = eσ+j is complex, we can express it as z = rej, where r = e^σ . Thus, z lies on a circle of radius r, and as varies from −π to π, z circumambulates along this circle, completing exactly one counterclockwise rotation, as illustrated in Fig. 9.18. Changing to variable z in Eq. (9.47) yields

$$x[n] = \frac{1}{2\pi j} \oint \hat{X}(\ln z) z^{n-1} dz$$

(9.48)

and from Eq. (9.46) we obtain

$$\hat{X}(\ln z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$

\n(9.49)

where the integral 6 indicates a contour integral around a circle of radius r in the counterclockwise direction.

Equations (9.48) and (9.49) are the desired extensions. They are, however, in a clumsy form. For the sake of convenience, we make another notational change by observing that Xˆ(lnz) is a function of z. Let us denote it by a simpler notation X[z]. Thus, Eq. (9.48) becomes

$$x[n] = \frac{1}{2\pi j} \oint X[z] z^{n-1} dz$$

\n(9.50)

and Eq. (9.49) becomes

$$X[z] = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$$

\n(9.51)

This is the (bilateral) z-transform pair. Equation (9.50) expresses x[n] as a continuous sum of exponentials of the form zⁿ = e(σ+j)ⁿ = rⁿ ejⁿ. 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 z = ej and

$$X[z]|_{z=e^{j\Omega}} = \hat{X}(\ln z)\Big|_{z=e^{j\Omega}} = \hat{X}(j\Omega) = X(\Omega)$$

Thus, the familiar DTFT is just a special case of the z-transform X[z] obtained by letting z = ej and assuming that the sum on the right-hand side of Eq. (9.51) converges when z = ej. This also implies that the ROC for X[z] includes the unit circle.

9.7 MATLAB: WORKING WITH THE DTFS AND THE DTFT

This section investigates various methods to compute the discrete-time Fourier series (DTFS). Performance of these methods is assessed by using MATLAB’s stopwatch and profiling functions. Additionally, the discrete-time Fourier transform (DTFT) is applied to the important topic of finite impulse response (FIR) filter design.