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 lower-case Z) is used as the transformation variable.

z Transform DefinitionEdit

For a given sequence x[n], we can define the z-transform X(z) as such:


[Z Transform]

 it is important to note that z is a continuous complex variable defined as such:
  where   is the imaginary unit.

There can be several sequences   which will generate the same z-transform   with the different functions being differentiated by the convergence region of   for which the summation in the z-transform will converge. These convergence regions are annuli centered at the orgin. In a given convergence region, only one   will converge to a given  .

Example 1Edit

 
  for  

Example 2Edit

 
  for  

Note that both examples have the same function   as their z-transforms but with different convergence regions needed for the respective infinite summations in their z-transforms to converge. Many textbooks on z-transforms are only concerned with so-called right-handed functions which is to say functions   for which   for all   less than some initial start point   ; that is,   for all  . So long as the function grows at most exponentially after the start point, the z-transform of these so-called right-handed functions will converge in an open annulus going to infinity,   for some positive real  .

It is important to note that the z-transform 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 z-transform is actually a special case of the so-called Laurent series, which is a special case of the commonly used Taylor series.

The Inverse Z-TransformEdit

The inverse z-transform can be defined as such:


[Inverse Z Transform]

 

Where C is a closed-contour that lies inside the unit circle on the z-plane, and encircles the point z = {0, 0}.

The inverse z-transform is mathematically very complicated, but luckily—like the z-transform itself—the results are extensively tabulated in tables.

Equivalence to DTFTEdit

If we substitute  , where   is the frequency in radians per second, into the Z-transform, we get

 

which is equivalent to the definition of the Discrete-Time Fourier Transform. In other words, to convert from the Z-transform to the DTFT, we need to evaluate the Z-transform around the unit circle.

PropertiesEdit

Since the z-transform is equivalent to the DTFT, the z-transform has many of the same properties. Specifically, the z-transform has the property of duality, and it also has a version of the convolution theorem (discussed later).

The z-transform is a linear operator.

Convolution TheoremEdit

Since the Z-transform is equivalent to the DTFT, it also has a convolution theorem that is worth stating explicitly:

Convolution Theorem
Multiplication in the discrete-time domain becomes convolution in the z-domain. Multiplication in the z-domain becomes convolution in the discrete-time domain.

Y(s)=X(s).H(s)

Z-PlaneEdit

Since the variable z is a continuous, complex variable, we can map the z variable to a complex plane as such:

Transfer FunctionEdit

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.

ExampleEdit

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

StabilityEdit

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 unit-circle on the z-plane for the system to be stable. Zeros of the transfer function may lie inside or outside the circle.

GainEdit

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".

PropertiesEdit

Here is a listing of the most common properties of the Z transform.


Time domain Z-domain ROC
Notation     ROC:  
Linearity     At least the intersection of ROC1 and ROC2
Time shifting     ROC, except   if   and   if  
Scaling in the z-domain      
Time reversal      
Conjugation     ROC
Real part     ROC
Imaginary part     ROC
Differentiation     ROC
Convolution     At least the intersection of ROC1 and ROC2
Correlation     At least the intersection of ROC of X1(z) and X2( )
Multiplication     At least  
Parseval's relation    
  • Initial value theorem
 , If   causal
  • Final value theorem
 , Only if poles of   are inside unit circle

Phase ShiftEdit

Further readingEdit