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The Z Transform: Definition

Module by: Benjamin Fite. E-mail the author

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

Basic Definition of the Z-Transform

The z-transform of a sequence is defined as

X⁢z=∑ n =−∞∞x⁢n⁢z−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
X⁢z=∑ n =0∞x⁢n⁢z−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

X⁢ej⁢ω=∑ n =−∞∞x⁢n⁢e−(j⁢ω⁢n) X ω n x n ω n
(3)
Notice that that when the z−n z n is replaced with e−(j⁢ω⁢n) ω n the z-transform reduces to the Fourier Transform. When the Fourier Transform exists, z=ej⁢ω 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 r⁢ej⁢ω r ω , and the angle from the positive, real axis around the plane is denoted by ωω. X⁢zXz is defined everywhere on this plane. X⁢ej⁢ω 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 x⁢n 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 x⁢n⁢z−n x n z n is absolutely summable. Stated differently,

∑ n =−∞∞|x⁢n⁢z−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 αn⁢u⁢n α n u n and αn⁢u⁢n−1 α n u n 1 .

Example 1

For

x⁢n=αn⁢u⁢n x n α n u n
(5)

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

X⁢z=∑ n =−∞∞x⁢n⁢z−n=∑ n =−∞∞αn⁢u⁢n⁢z−n=∑ n =0∞αn⁢z−n=∑ n =0∞α⁢z−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 |α⁢z-1|<1 α z 1 . When it converges,
X⁢z=11−α⁢z-1=zz−α Xz 1 1 α z z z α
(7)
If |α⁢z-1|≥1 α z 1 , then the series, ∑ n =0∞α⁢z-1n n 0 α z n does not converge. Thus the ROC is the range of values where
|α⁢z-1|<1 α z 1
(8)
or, equivalently,
|z|>|α| z α
(9)

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

Example 2

For

x⁢n=(−αn)⁢u⁢(−n)−1 x n α n u n 1
(10)

Figure 4: x⁢n=(−αn)⁢u⁢(−n)−1 x n α n u n 1 where α=0.5α0.5.
Figure 4 (sig2_2.png)

X⁢z=∑ n =−∞∞x⁢n⁢z−n=∑ n =−∞∞(−αn)⁢u⁢-n−1⁢z−n=−∑ n =−∞-1αn⁢z−n=−∑ n =−∞-1α-1⁢z−n=−∑ n =1∞α-1⁢zn=1−∑ n =0∞α-1⁢zn 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-1⁢z|<1 α -1 z 1
(12)
or, equivalently,
|z|<|α| z α
(13)
If the ROC is satisfied, then
X⁢z=1−11−α-1⁢z=zz−α Xz 1 1 1 α -1 z z z α
(14)

Figure 5: ROC for x⁢n=(−αn)⁢u⁢(−n)−1 x n α n u n 1
Figure 5 (ROC2.jpg)

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