Definition of the z-Transform

4.1 Definition of the z-Transform

For a discrete-time sequence x[n], the bilateral (two-sided) and unilateral (one- sided) z-transforms are defined as

X [z] = Z{x[n]} = ∞ x [n]z −n X

[z] = Z{x[n]} = x [n]z −n

n=−∞

n=0

(4.1.1b) where z is a complex variable. For convenience, the z-transform relationship will

(4.1.1a)

sometimes be denoted as

x [n] ↔ X [z] x [n] ↔ X[z] Note that the unilateral z-transform of x[n] can be thought of as the bilateral

ZZ

z -transform of x[n]u s [n] (u s [n]: the unit step sequence) and therefore the two definitions will be identical for causal sequences. The unilateral z-transform is par- ticularly useful in analyzing causal systems described by linear constant-coefficient difference equations with initial conditions.

The z-transform does not necessarily exist for all values of z in the complex plane. For the z-transform to exist, the series in Eq. (4.1.1) must converge. For a given sequence x[n], the domain of the z-plane within which the series converges is called the region of convergence (ROC) of the z-transform X[z] or X [z]. Thus, the specification of the z-transform requires both the algebraic expression in z and its region of convergence.

Note that a sequence x[n] is called – a right-sided sequence if x[n] = 0 ∀ n < n 0 for some finite integer n 0 ,

– a causal sequence if x[n] = 0 ∀ n < n 0 for some nonnegative integer n 0 ≥ 0, – a left-sided sequence if x[n] = 0 ∀ n ≥ n 0 for some finite integer n 0 , and – a anti-causal sequence if x[n] = 0 ∀ n ≥ n 0 for some non-positive integer n 0 ≤ 0,

respectively. Example 4.1 The z-Transform of Exponential Sequences

(a) For a right-sided and causal sequence x 1 [n] = a n u s [n], we can use Eq. (4.1.1a) to get its bilateral z-transform as

1 [z] = Z{x 1 [n]} =

a n u s [n]z −n

a n z −n

n=0 (D.23)

This geometric sequence converges for |az −1 | < 1, which implies that the ROC of X 1 [z] is R 1 = {z : |z| > |a|} (see Fig. 4.1(a)).

4.1 Definition of the z-Transform 209 Im {z } Im {z } Im {z }

0 Re {z } |a|

0 b Re {z } 0 | | | a || b Re | {z }

R 1 = {z: |z| > |a|} R 2 = {z: |z| < |b|} R = R 1 ∩ R 2 = {z: |a| < |z| < |b|}

(a) ROC for a right-sided

(c) ROC for a both-sided sequence

(b) ROC for a left-sided

sequence Fig. 4.1 Three forms of ROC (region of convergence)

sequence

(b) For a left-sided and anti-causal sequence x 2 [n] = −b n u s [−n − 1], we can use Eq. (4.1.1a) to get its bilateral z-transform as

X −1

[z] = Z{x n 2 [n]} = −b u s [−n − 1]z −n =− b z −n

n=−∞ ∞

n=−∞

n (D.23) −b −1 −1 z

This geometric sequence converges for |b −1 z| < 1, which implies that the ROC of X 2 [z] is R 2 = {z : |z| < |b|} (see Fig. 4.1(b)).

(c) For the both-sided sequence y

1 [n] + x 2 [n] = a u s [n] − b u s [−n − 1] (E4.1.3) we can combine the above results of (a) and (b) to obtain its bilateral z-

[n] = x n

transform as

Y[z] = X 1 [z] + X 2 [z] =

(E4.1.4)

z−b For this series to converge, both of the two series must converge and therefore,

z−a

its ROC is the intersection of the two ROCs R 1 and R 2 : R=R 1 ∩R 2 = {z : |a| < |z| < |b|}. This is an annular region |a| < |z| < |b| (Fig. 4.1(c)) if

|b| > |a| and it will be an empty set if |b| ≤ |a|. (cf) This example illustrates that different sequences may have the same z-transform,

but with different ROCs. This implies that a z-transform expression may correspond to different sequences depending on its ROC.

Remark 4.1 Region of Convergence (ROC) (1) The ROC for X [z] is an annular ring centered at the origin in the z-plane of

the form

210 4 The z-Transform

(4.1.2) where r + can be as large as ∞ (Fig. 4.1(a)) for causal sequences and r − can be

r − < |z| < r +

as small as zero (Fig. 4.1(b)) for anti-causal sequences. (2) As illustrated in Example 4.1, the three different forms of ROC shown in Fig. 4.1 can be associated with the corresponding three different classes of discrete-time sequences; 0 < r − < |z| for right-sided sequences, |z| < r + < ∞ for left-sided sequences, and r − < |z| < r + (annular ring) for two-sided sequences.

Example 4.2

A Causal Sequence Having Multiple-Pole z-transform

For a sum of two causal sequences x n [n] = x

1 [n] + x 2 [n] = 2u s [n] − 2(1/2) u s [n] (E4.2.1) we can use Eq. (E4.1.1) to get the z-transform for each sequence and combine the

z -transforms as

X[z] = Z{x[n]} = X 1 [z] + X 2 [z]

(E4.1.1)

−2 z−a a=1

z−a a=1/2 2z

with a=1 and a=1/2

2z

(E4.2.2) z−1

z − 1/2

(z − 1)(z − 1/2) Since both X 1 [z] = Z{x 1 [n]} and X 2 [z] = Z{x 2 [n]} must exist for this z-transform

to exist, the ROC is the intersection of the ROCs for the two z-transforms: R=R 1 ∩R 2 = {z : |z| > 1} ∩ {z : |z| > 1/2} = {z : |z| > 1}

(E4.2.3) (cf) Figure 4.2 shows the pole-zero pattern of the z-transform expression (E4.2.2)

where a pole and a zero are marked with x and o, respectively. Note that the ROC of the z-transform X[z] of a right-sided sequence is the exterior of the circle which is centered at the origin and passes through the pole farthest from the origin in the z-plane.

Im{z} z-plane

× × 1 1/2 Re{z}

Fig. 4.2 Pole-zero pattern of the z-transform (E4.2.2)

4.1 Definition of the z-Transform 211

Before working further with more examples, let us see the relationship between the z-transform and DTFT (discrete-time Fourier transform), which is summarized in the following remark:

Remark 4.2 The z-Transform and the DTFT (Discrete-Time Fourier Transform)

Comparing the bilateral z-transform (4.1.1a) with the DTFT (3.1.1), we can see their relationship that the DTFT can be obtained by substituting z = e jΩ into the

(bilateral) z-transform:

F{x[n]} = X( jΩ) ∞ =

= X[e ] = X[z]| z=e jΩ = Z{x[n]}| z=e jΩ

x [n]e − jΩn

This implies that the evaluation of the z-transform along the unit circle in the z- plane yields the DTFT. Therefore, in order for the DTFT to exist, the ROC of the z -transform must include the unit circle (see Fig. 4.3).

Example 4.3 The z-transform of a Complex Exponential Sequence Let us consider a causal sequence

x jΩ 1 [n] = e n u s [n] = cos(Ω 1 n )u s [n] + j sin(Ω 1 n )u s [n] (E4.3.1) jΩ We can find the z-transform of this exponential sequence by substituting a = e 1

into Eq. (E4.1.1) as

jΩ n (E4.1.1)

z (z − e − jΩ 1 1 )

u s [n]} = ± jΩ X[z] = Z{e 1

e (D.20) u

s [n] = cos(Ω 1 n )u s [n] + j sin(Ω 1 n )u s [n]

Im{z}

z-plane

z=e jΩ

0 1 Re{z} Fig. 4.3 The relationship

between the DTFT and z

F {x[n]} = X( j Ω) = (X [z] = Z{x [n]}) with z = e -transform jΩ

212 4 The z-Transform This implies the following two z-transform pairs:

z (z − cos Ω 1 ) cos(Ω 1 n )u s [n] ↔

with R = {z : |z| > 1}

z sin Ω 1 z sin Ω 1

sin(Ω 1 n )u s [n] ↔ (E4.3.3)

− 2z cos Ω 2 +1 (z − cos Ω

1 ) 2 + sin Ω 1

with R = {z : |z| > 1}

Example 4.4 The z-transform of an Exponentially Decreasing Sinusoidal Sequence For a causal sequence

[n] = r n cos(Ω 1 n )u s [n] =

(D.21) 1

r n (e jΩ 1 n +e − jΩ 1 n )u s [n] (E4.4.1)

we can use Eq. (E4.1.1) with a = e ± jΩ 1 to find the z-transform of this sequence as

1 ) X[z]

a=re j Ω1 and a=re − jΩ1 2 z − re jΩ 1

z − re − jΩ 1

(E4.4.2)

where the ROC is R = {z : |z| > |r e ± jΩ 1 | = |r|}.

(cf) Note that if |r| ≥ 1, the ROC does not contain the unit circle and consequently, the DTFT does not exist in the strict sense.

We could elaborate on more examples of the z-transform computation. However, we instead list a number of useful z-transform pairs in Table B.9 in Appendix B so that the readers can use them for finding the z-transform or inverse z-transform.

In many cases, we deal with the z-transform that is expressed in the form of a rational function, i.e., a ratio of polynomials in z. For such a rational z-transform X[z] = Q[z]/P[z], we define its singularities, i.e., poles and zeros as follows:

Remark 4.3 Poles and Zeros of a Rational z-Transform Expression X[z] = Q[z]/ P[z]

The roots of Q[z] = 0, or the values of z at which X[z] = 0 are called the zero s of X[z]. The poles are the roots of P[z] = 0, or the values of z for which

X[z] = ∞. If we count the singularities at z = ∞, the number of poles will be equal to that of zeros. Notice that, for polynomials with real-valued coefficients, the

complex-valued singularities must occur in complex-conjugate pairs. The ROC of a z-transform X[z] can be described in terms of its poles as follows:

4.2 Properties of the z-Transform 213 Remark 4.4 Pole Locations and Region of Convergence (ROC)

(1) It is obvious that there can be no poles of a z-transform X[z] inside the ROC since the z-transform does not converge at its poles. In fact, the ROC is bounded by poles or infinity as illustrated by Examples 4.1 and 4.2.

(2) For a right-sided sequence x a [n] and left-sided sequence x b [n], the ROCs for their z-transforms X a [z] and X b [z] are outside/inside the outermost/innermost pole, respectively:

R 1 = {z : |z| > the maximum of |a i | ′ s}(z = a i : the poles of X a [z]) (4.1.4a) (as illustrated by Example 4.2) or R 2 = {z : |z| < the minimum of |b i | ′ s}(z = b i : the poles of X b [z]) (4.1.4b) For a two-sided sequence x[n] = x[n] + x b [n], the ROC is an annular region

given by R=R 1 ∩R 2 = {z : max i |a i | < |z| < min j |b j |}

(4.1.4c)