# Control Systems/Transforms Appendix

## Laplace Transform

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:

$$F(s) 
        = \mathcal{L}[f(t)] 
        = \int_{-\infty}^\infty x(t)e^{-st}dt$$

And the Inverse Laplace Transform is defined as:

$$f(t) 
        = \mathcal{L}^{-1} \left\{F(s)\right\}
        = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s)e^{st}ds$$

### Table of Laplace Transforms

This is a table of common Laplace Transforms.

:   
### Properties of the Laplace Transform

This is a table of the most important properties of the laplace
transform.

:   
### Convergence of the Laplace Integral

### Properties of the Laplace Transform

## Fourier Transform

The Fourier Transform is used to break a time-domain 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:

$$F(j\omega) 
        = \mathcal{F}[f(t)] 
        = \int_0^\infty f(t) e^{-j\omega t} dt$$

And the Inverse Fourier Transform is defined as:

$$f(t) 
        = \mathcal{F}^{-1}\left\{F(j\omega)\right\}     
        = \frac{1}{2\pi}\int_{-\infty}^\infty F(j\omega) e^{-j\omega t} d\omega$$

### Table of Fourier Transforms

This is a table of common fourier transforms.

:   
### Table of Fourier Transform Properties

This is a table of common properties of the fourier transform.

:   
### Convergence of the Fourier Integral

### Properties of the Fourier Transform

## Z-Transform

The Z-transform is used primarily to convert discrete data sets into a
continuous representation. The Z-transform 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 Z-transform results
are extensively tabulated, whereas star-transform results are not.

The Z Transform is defined as:

$$X(z) 
        = \mathcal{Z}[x[n]] 
        = \sum_{n = -\infty}^\infty x[n] z^{-n}$$

### Inverse Z Transform

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.

$$x[n] = \frac{1}{2 \pi j} \oint_C X(z) z^{n-1} dz$$

### Z-Transform Tables

## Modified Z-Transform

The Modified Z-Transform is similar to the Z-transform, except that the
modified version allows for the system to be subjected to any arbitrary
delay, by design. The Modified Z-Transform 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:

$$X(z, m) 
        = X(z, \Delta)|_{\Delta \to 1 - m} 
        = \mathcal{Z} \left\{ X(s)e^{-\Delta T s} \right\}|_{\Delta \to 1 - m}$$

## Star Transform

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:

$$F^*(s) 
        = \mathcal{L}^*[f(t)] 
        = \sum_{k = 0}^\infty f(kT)e^{-skT}$$

Star transform pairs can be obtained by plugging $z = e^{sT}$ into the
Z-transform pairs, above.

## Bilinear Transform

The bilinear transform is used to convert an equation in the Z domain
into the arbitrary W domain, with the following properties:

1.  roots inside the unit circle in the Z-domain will be mapped to roots
    on the left-half of the W plane.
2.  roots outside the unit circle in the Z-domain will be mapped to
    roots on the right-half of the W plane
3.  roots on the unit circle in the Z-domain will be mapped onto the
    vertical axis in the W domain.

The bilinear transform can therefore be used to convert a Z-domain
equation into a form that can be analyzed using the Routh-Hurwitz
criteria. However, it is important to note that the W-domain is not the
same as the complex Laplace S-domain. To make the output of the bilinear
transform equal to the S-domain, the signal must be prewarped, to
account for the non-linear nature of the bilinear transform.

The Bilinear transform can also be used to convert an S-domain 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:

$$z = \frac{(T/2) + w}{(T/2) - w},\quad w = \frac{2}{T} \frac{z - 1}{z + 1}$$

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:

$$\omega_w = \frac{2}{T} \tan \left( \frac{ \omega_s T}{2} \right)$$

This relationship is called the **frequency warping characteristic** of
the bilinear transform. To counter-act the effects of frequency warping,
we can **pre-warp** the Z-domain 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 s-domain.

$$\omega = \frac{2}{T} \arctan \left( \omega_a \frac{T}{2} \right).$$

Applying these transformations before applying the bilinear transform
actually enables direct conversions between the S-Domain and the
Z-Domain. The act of applying one of these frequency warping
characteristics to a function before transforming is called
**prewarping**.

## Wikipedia Resources

-   w:Laplace transform
-   w:Fourier transform
-   w:Z-transform
-   w:Star transform
-   w:Bilinear transform

