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The Z Transform: Definition

Module by: Benjamin Fite

Summary: A brief definition of the z-transform, explaining its relationship with the Fourier transform and its region of convergence, ROC.

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Basic Definition of the Z-Transform

The z-transform of a sequence is defined as

Xz=∑n=-∞∞xnz-n Xz n x n z n (1)
Sometimes this equation is referred to as the bilateral z-transform. At times the z-transform is defined as
Xz=∑n=0∞xnz-n 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 defined as

Xⅇⅈω=∑n=-∞∞xnⅇ-ⅈωn X ω n x n ω n (3)
Notice that that when the z-n z n is replaced with ⅇ-ⅈωn ω n the z-transform reduces to the Fourier Transform. When the Fourier Transform exists, z=ⅇⅈω z ω , which is to have the magnitude of zz equal to unity.

The Complex Plane

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

Figure 1
Z-Plane
Z-Plane (zplane1.jpg)

The Z-plane is a complex plane with an imaginary and real axis referring to the complex-valued variable zz. The position on the complex plane is given by rⅇⅈω r ω , and the angle from the positive, real axis around the plane is denoted by ωω. XzXz is defined everywhere on this plane. Xⅇⅈω Xω on the other hand is defined only where |z|=1 z1 , which is referred to as the unit circle. So for example, ω=1ω1 at z=1z1 and ω=πω at z=-1z-1. This is useful because, by representing the Fourier transform as the z-transform on the unit circle, the periodicity of Fourier transform is easily seen.

Region of Convergence

The region of convergence, known as the ROC, is important to understand because it defines the region where the z-transform exists. The ROC for a given xn x n , is defined as the range of z z for which the z-transform converges. Since the z-transform is a power series, it converges when xnz-n x n z n is absolutely summable. Stated differently,

∑n=-∞∞|xnz-n|<∞ n x n z n (4)
must be satisfied for convergence. This is best illustrated by looking at the different ROC's of the z-transforms of αnun α n u n and αnun−1 α n u n 1 .

Example 1

For

xn=αnun x n α n u n (5)

Figure 2: xn=αnun x n α n u n where α=0.5α0.5.
Figure 2 (sig1.png)

Xz=∑n=-∞∞xnz-n=∑n=-∞∞αnunz-n=∑n=0∞αnz-n=∑n=0∞αz-1n Xz n x n z n n α n u n z n n 0 α n z n n 0 α z 1 n (6)
This sequence is an example of a right-sided exponential sequence because it is nonzero for n≥0 n 0 . It only converges when |αz-1|<1 α z 1 . When it converges,
Xz=11−αz-1=zz−α Xz 1 1 α z z z α (7)
If |αz-1|≥1 α z 1 , then the series, ∑n=0∞αz-1n n 0 α z n does not converge. Thus the ROC is the range of values where
|αz-1|<1 α z 1 (8)
or, equivalently,
|z|>|α| z α (9)

Figure 3: ROC for xn=αnun x n α n u n where α=0.5 α 0.5
Figure 3 (ROC1.jpg)

Example 2

For

xn=-αnu-n−1 x n α n u n 1 (10)

Figure 4: xn=-αnu-n−1 x n α n u n 1 where α=0.5α0.5.
Figure 4 (sig2_2.png)

Xz=∑n=-∞∞xnz-n=∑n=-∞∞-αnu-n−1z-n=-∑n=-∞-1αnz-n=-∑n=-∞-1α-1z-n=-∑n=1∞α-1zn=1−∑n=0∞α-1zn Xz n x n z n n α n u -n 1 z n n -1 α n z n n -1 α -1 z n n 1 α -1 z n 1 n 0 α -1 z n (11)
The ROC in this case is the range of values where
|α-1z|<1 α -1 z 1 (12)
or, equivalently,
|z|<|α| z α (13)
If the ROC is satisfied, then
Xz=1−11−α-1z=zz−α Xz 1 1 1 α -1 z z z α (14)

Figure 5: ROC for xn=-αnu-n−1 x n α n u n 1
Figure 5 (ROC2.jpg)

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