Control Systems/Transforms Appendix
Contents
Laplace Transform[edit]
When we talk about the Laplace transform, we are actually talking about the version of the Laplace transform known as the unilinear Laplace Transform. The other version, the Bilinear Laplace Transform (not related to the Bilinear Transform, below) is not used in this book.
The Laplace Transform is defined as:
[Laplace Transform]
And the Inverse Laplace Transform is defined as:
[Inverse Laplace Transform]
Table of Laplace Transforms[edit]
This is a table of common laplace transforms.
Time Domain  Laplace Domain  

1  
2  
3  
4  
5  
6  
7  
8  
9  
10  
11  
12  
13  
14  
15  
16  
17  
18  
19 
Properties of the Laplace Transform[edit]
This is a table of the most important properties of the laplace transform.
Property  Definition 

Linearity  
Differentiation 

Frequency Division  
Frequency Integration  
Time Integration  
Scaling  
Initial value theorem  
Final value theorem  
Frequency Shifts  
Time Shifts  
Convolution Theorem 
Where:
Convergence of the Laplace Integral[edit]
Properties of the Laplace Transform[edit]
Fourier Transform[edit]
The Fourier Transform is used to break a timedomain signal into its frequency domain components. The Fourier Transform is very closely related to the Laplace Transform, and is only used in place of the Laplace transform when the system is being analyzed in a frequency context.
The Fourier Transform is defined as:
[Fourier Transform]
And the Inverse Fourier Transform is defined as:
[Inverse Fourier Transform]
Table of Fourier Transforms[edit]
This is a table of common fourier transforms.
Time Domain  Frequency Domain  

1  
2  
3  
4  
5  
6  
7  
8  
9  
10  
11  
12  
13  
14  
15  
16  
Notes: 


Table of Fourier Transform Properties[edit]
This is a table of common properties of the fourier transform.
Signal  Fourier transform unitary, angular frequency 
Fourier transform unitary, ordinary frequency 
Remarks  

1  Linearity  
2  Shift in time domain  
3  Shift in frequency domain, dual of 2  
4  If is large, then is concentrated around 0 and spreads out and flattens  
5  Duality property of the Fourier transform. Results from swapping "dummy" variables of and .  
6  Generalized derivative property of the Fourier transform  
7  This is the dual to 6  
8  denotes the convolution of and — this rule is the convolution theorem  
9  This is the dual of 8  
10  For a purely real even function  is a purely real even function  is a purely real even function  
11  For a purely real odd function  is a purely imaginary odd function  is a purely imaginary odd function 
Convergence of the Fourier Integral[edit]
Properties of the Fourier Transform[edit]
ZTransform[edit]
The Ztransform is used primarily to convert discrete data sets into a continuous representation. The Ztransform is notationally very similar to the star transform, except that the Z transform does not take explicit account for the sampling period. The Z transform has a number of uses in the field of digital signal processing, and the study of discrete signals in general, and is useful because Ztransform results are extensively tabulated, whereas startransform results are not.
The Z Transform is defined as:
[Z Transform]
Inverse Z Transform[edit]
The inverse Z Transform is a highly complex transformation, and might be inaccessible to students without enough background in calculus. However, students who are familiar with such integrals are encouraged to perform some inverse Z transform calculations, to verify that the formula produces the tabulated results.
[Inverse Z Transform]
ZTransform Tables[edit]
Here:
 for , for
 for , otherwise
Signal,  Ztransform,  ROC  

1  
2  
3  
4  
5  
6  
7  
8  
9  
10  
11  
12  
13  
14  
15  
16  
17  
18  
19  
20 
Modified ZTransform[edit]
The Modified ZTransform is similar to the Ztransform, except that the modified version allows for the system to be subjected to any arbitrary delay, by design. The Modified ZTransform is very useful when talking about digital systems for which the processing time of the system is not negligible. For instance, a slow computer system can be modeled as being an instantaneous system with an output delay.
The modified Z transform is based off the delayed Z transform:
[Modified Z Transform]
Star Transform[edit]
The Star Transform is a discrete transform that has similarities between the Z transform and the Laplace Transform. In fact, the Star Transform can be said to be nearly analogous to the Z transform, except that the Star transform explicitly accounts for the sampling time of the sampler.
The Star Transform is defined as:
[Star Transform]
Star transform pairs can be obtained by plugging into the Ztransform pairs, above.
Bilinear Transform[edit]
The bilinear transform is used to convert an equation in the Z domain into the arbitrary W domain, with the following properties:
 roots inside the unit circle in the Zdomain will be mapped to roots on the lefthalf of the W plane.
 roots outside the unit circle in the Zdomain will be mapped to roots on the righthalf of the W plane
 roots on the unit circle in the Zdomain will be mapped onto the vertical axis in the W domain.
The bilinear transform can therefore be used to convert a Zdomain equation into a form that can be analyzed using the RouthHurwitz criteria. However, it is important to note that the Wdomain is not the same as the complex Laplace Sdomain. To make the output of the bilinear transform equal to the Sdomain, the signal must be prewarped, to account for the nonlinear nature of the bilinear transform.
The Bilinear transform can also be used to convert an Sdomain system into the Z domain. Again, the input system must be prewarped prior to applying the bilinear transform, or else the results will not be correct.
The Bilinear transform is governed by the following variable transformations:
[Bilinear Transform]
Where T is the sampling time of the discrete signal.
Frequencies in the w domain are related to frequencies in the s domain through the following relationship:
This relationship is called the frequency warping characteristic of the bilinear transform. To counteract the effects of frequency warping, we can prewarp the Zdomain equation using the inverse warping characteristic. If the equation is prewarped before it is transformed, the resulting poles of the system will line up more faithfully with those in the sdomain.
[Bilinear Frequency Prewarping]
Applying these transformations before applying the bilinear transform actually enables direct conversions between the SDomain and the ZDomain. The act of applying one of these frequency warping characteristics to a function before transforming is called prewarping.