Digital Signal Processing/Z Transform
The Z Transform has a strong relationship to the DTFT, and is incredibly useful in transforming, analyzing, and manipulating discrete calculus equations. The Z transform is named such because the letter 'z' (a lowercase Z) is used as the transformation variable.
Contents
z Transform Definition[edit]
For a given sequence x[n], we can define the ztransform X(z) as such:
[Z Transform]
 it is important to note that z is a continuous complex variable defined as such:
It is important to note that the ztransform rarely needs to be computed manually, because many common results have already been tabulated extensively in tables, and control system software includes it (MatLab,Octave,SciLab).
The Inverse ZTransform[edit]
The inverse ztransform can be defined as such:
[Inverse Z Transform]
Where C is a closedcontour that lies inside the unit circle on the zplane, and encircles the point z = {0, 0}.
The inverse ztransform is mathematically very complicated, but luckily—like the ztransform itself—the results are extensively tabulated in tables.
Equivalence to DTFT[edit]
If we substitute , where is the frequency in radians per second, into the Ztransform, we get
which is equivalent to the definition of the DiscreteTime Fourier Transform. In other words, to convert from the Ztransform to the DTFT, we need to evaluate the Ztransform around the unit circle.
Properties[edit]
Since the ztransform is equivalent to the DTFT, the ztransform has many of the same properties. Specifically, the ztransform has the property of duality, and it also has a version of the convolution theorem (discussed later).
The ztransform is a linear operator.
Convolution Theorem[edit]
Since the Ztransform is equivalent to the DTFT, it also has a convolution theorem that is worth stating explicitly:
 Convolution Theorem
 Multiplication in the discretetime domain becomes convolution in the zdomain. Multiplication in the zdomain becomes convolution in the discretetime domain.
Y(s)=X(s).H(s)
ZPlane[edit]
Since the variable z is a continuous, complex variable, we can map the z variable to a complex plane as such:
Transfer Function[edit]
Let's say we have a system with an input/output relationship defined as such:
Y(z) = H(z)X(z)
We can define the transfer function of the system as being the term H(z). If we have a basic transfer function, we can break it down into parts:
Where H(z) is the transfer function, N(z) is the numerator of H(z) and D(z) is the denominator of H(z). If we set N(z)=0, the solutions to that equation are called the zeros of the transfer function. If we set D(z)=0, the solutions to that equation are called the poles of the transfer function.
The poles of the transfer function amplify the frequency response while the zero's attenuate it. This is important because when you design a filter you can place poles and zero's on the unit circle and quickly evaluate your filters frequency response.
Example[edit]
Here is an example:
So by dividing through by X(z), we can show that the transfer function is defined as such:
We can also find the D(z) and N(z) equations as such:
And from those equations, we can find the poles and zeros:
 Zeros
 z → 0
 Poles
 z → 1/2
Stability[edit]
It can be shown that for any causal system with a transfer function H(z), all the poles of H(z) must lie within the unitcircle on the zplane for the system to be stable. Zeros of the transfer function may lie inside or outside the circle.
Gain[edit]
Gain is the factor by which the output magnitude is different from the input magnitude. If the input magnitude is the same as the output magnitude at a given frequency, the filter is said to have "unity gain".
Properties[edit]
Here is a listing of the most common properties of the Z transform.
Time domain  Zdomain  ROC  

Notation  ROC:  
Linearity  At least the intersection of ROC_{1} and ROC_{2}  
Time shifting  ROC, except if and if  
Scaling in the zdomain  
Time reversal  
Conjugation  ROC  
Real part  ROC  
Imaginary part  ROC  
Differentiation  ROC  
Convolution  At least the intersection of ROC_{1} and ROC_{2}  
Correlation  At least the intersection of ROC of X_{1}(z) and X_{2}()  
Multiplication  At least  
Parseval's relation 
 Initial value theorem

 , If causal
 Final value theorem

 , Only if poles of are inside unit circle