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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
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
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 ω n x n ω n
Notice that that when the z n is replaced with ω n the z-transform reduces to the Fourier Transform. When the Fourier Transform exists, z ω , which is to have the magnitude of z 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 lookat the complex plane or z-plane . Take a look at the complex plane:

Z-plane

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 r ω , and the angle from the positive, real axis around the plane is denoted by ω . X z is defined everywhere on this plane. X ω on the other hand is defined only where z 1 , which is referred to as the unit circle. So for example, ω 1 at z 1 and ω at z -1 . This is useful because, by representing the Fourier transformas 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 wherethe z-transform exists. The ROC for a given x n , is defined as the range of z for which the z-transform converges. Since the z-transform is a power series , it converges when x n z n is absolutely summable. Stated differently,

n x n z n
must be satisfied for convergence. This is best illustratedby looking at the different ROC's of the z-transforms of α n u n and α n u n 1 .

For

x n α n u n

x n α n u n where α 0.5 .

X z n x n z n n α n u n z n n 0 α n z n n 0 α z 1 n
This sequence is an example of a right-sided exponential sequence because it is nonzero for n 0 . It only converges when α z 1 . When it converges,
X z 1 1 α z z z α
If α z 1 , then the series, n 0 α z n does not converge. Thus the ROC is the range of values where
α z 1
or, equivalently,
z α

ROC for x n α n u n where α 0.5
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For

x n α n u n 1

x n α n u n 1 where α 0.5 .

X z 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
The ROC in this case is the range of values where
α -1 z 1
or, equivalently,
z α
If the ROC is satisfied, then
X z 1 1 1 α -1 z z z α

ROC for x n α n u n 1
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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