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The Z Transform: DefinitionVersion 2.9: Jun 1, 2005 9:10 pm GMT-5

Benjamin Fite

This work is produced by The Connexions Project and licensed under theCreative Commons Attribution License ∗

AbstractA brief de�nition of the z-transform, explaining its relationship with the Fourier

transform and its region of convergence, ROC.

The Z Transform: De�nition

1 Basic De�nition of the Z-Transform

The z-transform of a sequence is de�ned as

X (z) =∞∑

n=−∞

(x [n] z−n

)(1)

Sometimes this equation is referred to as the bilateral z-transform. At times the z-transform is de�ned as

X (z) =∞∑

n=0

(x [n] z−n

)(2)

which is known as the unilateral z-transform.There is a close relationship between the z-transform and the Fourier transform of a

discrete time signal, which is de�ned as

X(eiω

)=

∞∑n=−∞

(x [n] e−(iωn)

)(3)

Notice that that when the z−n is replaced with e−(iωn) the z-transform reduces to the FourierTransform. When the Fourier Transform exists, z = eiω , which is to have the magnitudeof z equal to unity.

2 The Complex Plane

In order to get further insight into the relationship between the Fourier Transform and theZ-Transform it is useful to look at the complex plane or z-plane. Take a look at the complexplane:

The Z-plane is a complex plane with an imaginary and real axis referring to the complex-valued variable z. The position on the complex plane is given by reiω , and the angle from

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Z-Plane

Figure 1

the positive, real axis around the plane is denoted by ω. X (z) is de�ned everywhere on thisplane. X

(eiω

)on the other hand is de�ned only where |z| = 1, which is referred to as the

unit circle. So for example, ω = 1 at z = 1 and ω = π at z = −1. This is useful because, byrepresenting the Fourier transform as the z-transform on the unit circle, the periodicity ofFourier transform is easily seen.

3 Region of Convergence

The region of convergence, known as the ROC, is important to understand because itde�nes the region where the z-transform exists. The ROC for a given x [n] , is de�ned as therange of z for which the z-transform converges. Since the z-transform is a power series, itconverges when x [n] z−n is absolutely summable. Stated di�erently,

∞∑n=−∞

(|x [n] z−n|

)< ∞ (4)

must be satis�ed for convergence. This is best illustrated by looking at the di�erent ROC'sof the z-transforms of αnu [n] and αnu [n− 1].

Example 1:

Forx [n] = αnu [n] (5)

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Figure 2: x [n] = αnu [n] where α = 0.5.

X (z) =∑∞

n=−∞ (x [n] z−n)=

∑∞n=−∞ (αnu [n] z−n)

=∑∞

n=0 (αnz−n)=

∑∞n=0

((αz−1

)n) (6)

This sequence is an example of a right-sided exponential sequence because it isnonzero for n ≥ 0. It only converges when |αz−1| < 1. When it converges,

X (z) = 11−αz−1

= zz−α

(7)

If |αz−1| ≥ 1, then the series,∑∞

n=0

((αz−1

)n)does not converge. Thus the ROC

is the range of values where|αz−1| < 1 (8)

or, equivalently,|z| > |α| (9)

Example 2:

Forx [n] = (− (αn))u [−n− 1] (10)

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Figure 3: ROC for x [n] = αnu [n] where α = 0.5

Figure 4: x [n] = (− (αn)) u [−n− 1] where α = 0.5.

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Figure 5: ROC for x [n] = (− (αn)) u [−n− 1]

X (z) =∑∞

n=−∞ (x [n] z−n)=

∑∞n=−∞ ((− (αn))u [−n− 1] z−n)

= −(∑−1

n=−∞ (αnz−n))

= −(∑−1

n=−∞

((α−1z

)−n))

= −(∑∞

n=1

((α−1z

)n))= 1−

∑∞n=0

((α−1z

)n)(11)

The ROC in this case is the range of values where

|α−1z| < 1 (12)

or, equivalently,|z| < |α| (13)

If the ROC is satis�ed, then

X (z) = 1− 11−α−1z

= zz−α

(14)

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