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)