THE BILATERAL z-TRANSFORM

4.1 THE BILATERAL z-TRANSFORM

The z-transform of a sequence x(n) is given by

X(z) −n = Z[x(n)] = x(n)z

n =−∞

Chapter 4

THE z -TRANSFORM

where z is a complex variable. The set of z values for which X(z) exists is called the region of convergence (ROC) and is given by

R x − < |z| < R x +

for some non-negative numbers R x − and R x + .

The inverse z-transform of a complex function X(z) is given by

x(n) n =Z 1 1 −

[X(z)] =

X(z)z 1 dz

2πj C

where C is a counterclockwise contour encircling the origin and lying in the ROC.

Comments:

1. The complex variable z is called the complex frequency given by z = |z|e jω , where |z| is the magnitude and ω is the real frequency.

2. Since the ROC (4.2) is defined in terms of the magnitude |z|, the shape of the ROC is an open ring, as shown in Figure 4.1. Note that R x − may be equal to zero and/or R x + could possibly be ∞.

3. If R x + <R x − , then the ROC is a null space and the z-transform does not exist .

4. The function |z| = 1 (or z = e jω ) is a circle of unit radius in the z-plane and is called the unit circle. If the ROC contains the unit circle, then we can evaluate X(z) on the unit circle.

X(z)| z =e jω = X(e jω )=

x(n)e −jω = F[x(n)]

n =−∞

Therefore the discrete-time Fourier transform X(e jω ) may be viewed as a special case of the z-transform X(z).

Im{z}

Rx+ Re {z} Rx –

FIGURE 4.1

A general region of convergence

The Bilateral z -Transform 105

Im{z}

a Re{z}

FIGURE 4.2 The ROC in Example 4.1

EXAMPLE 4.1

Let x 1 (n) = a n u(n),

0 < |a| < ∞. (This sequence is called a positive-time

sequence). Then

Note: X 1 (z) in this example is a rational function; that is,

△ B(z)

where B(z) = z is the numerator polynomial and A(z) = z−a is the denominator polynomial . The roots of B(z) are called the zeros of X(z), whereas the roots

of A(z) are called the poles of X(z). In this example X 1 (z) has a zero at the origin z = 0 and a pole at z = a. Hence x 1 (n) can also be represented by a pole-zero diagram in the z-plane in which zeros are denoted by ◦ and poles by × as shown in Figure 4.2.

EXAMPLE 4.2 Let x 2 (n) = −b n u(−n−1), 0 < |b| < ∞. (This sequence is called a negative-time

sequence .) Then

&'() < |z| < |b|

The ROC 2 and the pole-zero plot for this x 2 (n) are shown in Figure 4.3.

Im{z}

Re{z}

b FIGURE 4.3 The ROC in Example 4.2

Chapter 4

THE z -TRANSFORM

Note:

If b = a in this example, then X 2 (z) = X 1 (z) except for their respective ROCs; that is, ROC 1 ̸= ROC 2 . This implies that the ROC is a distinguishing feature that guarantees the uniqueness of the z-transform. Hence it plays a very important role in system analysis.

EXAMPLE 4.3

Let x 3 (n) = x 1 (n) + x 2 (n) = a n

u(n) − b n u(−n − 1) (This sequence is called a

two-sided sequence .) Then using the preceding two examples,

If |b| < |a|, than ROC 3 is a null space, and X 3 (z) does not exist. If |a| < |b|, then the ROC 3 is |a| < |z| < |b|, and X 3 (z) exists in this region as shown in

Figure 4.4.

4.1.1 PROPERTIES OF THE ROC From the observation of the ROCs in the preceding three examples, we state the following properties.

1. The ROC is always bounded by a circle since the convergence condition is on the magnitude |z|.

2. The sequence x 1 (n) = a n u(n) in Example 4.1 is a special case of a right- sided sequence , defined as a sequence x(n) that is zero for some n < n 0 . From Example 4.1, the ROC for right-sided sequences is always

outside of a circle of radius R x− . If n 0 ≥

0, then the right-sided

sequence is also called a causal sequence.

3. The sequence x 2 (n) = −b n u(−n−1) in Example 4.2 is a special case of a left-sided sequence, defined as a sequence x(n) that is zero for some n >

0, the resulting sequence is called an anticausal sequence. From Example 4.2, the ROC for left-sided sequences is always inside of a circle of radius R x + .

n 0 . If n 0 ≤

Im{z}

Im{z}

aa

0 Re{z}

0 Re{z}

FIGURE 4.4 The ROC in Example 4.3

Important Properties of the z -Transform 107

4. The sequence x 3 (n) in Example 4.3 is a two-sided sequence. The ROC for two-sided sequences is always an open ring R x− < |z| < R x + , if it exists.

5. The sequences that are zero for n < n 1 and n > n 2 are called finite-duration sequences . The ROC for such sequences is the entire z -plane . If n 1 < 0, then z = ∞ is not in the ROC. If n 2 > 0, then

z = 0 is not in the ROC.

6. The ROC cannot include a pole since X(z) converges uniformly in there.

7. There is at least one pole on the boundary of a ROC of a rational X(z).

8. The ROC is one contiguous region; that is, the ROC does not come in pieces.

In digital signal processing, signals are assumed to be causal since almost every digital data is acquired in real time. Therefore the only ROC of interest to us is the one given in statement 2.