# Z-transform

The Z-transform converts a discrete time signal in the time domain, that is, a time sequence of complex numbers, generally, in a complex discrete signal in the frequency domain. The discrete-time z-transform is the analogue of the Laplace transform, continuous-time signals. The connection of the two transformations can be produced by the bilinear transformation, thus continuous-time system can be converted to discrete-time systems, and vice versa.

The Z-transform in a similar relationship to the discrete-time Fourier transform is ( not to be confused with the similar discrete Fourier transform ) as the Laplace transform to the Fourier transform.

- 3.1 Additional features of the unilateral Z-transform

- 4.1 Inverse unilateral Z-transform 4.1.1 Residual
- 4.1.2 With Laurent series 4.1.2.1 Example 1
- 4.1.2.2 Example 2

## Historical Development

The basic ideas for the Z-transform back to Pierre- Simon Laplace in 1947 and used by Witold Hurewicz for solving linear differential equations with constant coefficients. It was originally introduced as the " Laplace transform of sampling functions ", took place in 1952, the now common term defining Z-transform by John R. Lotfi A. Zadeh Ragazzini and when working with discrete-time data in the context of control engineering at Columbia University. The Modified Z-transform goes back to work by Eliahu Ibrahim Jury of 1958.

## Definition

### Bilateral Z transform

The bilateral Z transform of a signal x [n] is the formal Laurent series X (z):

Where n runs through all integers, and z, in general, a complex number of the form:

Is. A is the absolute value of z and the angle φ of the complex number in polar coordinates. Alternatively, σ z is the real part and the imaginary part in Cartesian form are described ω.

Under certain convergence conditions, the Z-transform is a holomorphic function on an annulus in the complex plane, under weaker conditions nevertheless still a square integrable function on the unit circle.

### Unilateral Z transform

Where x [n] only non-negative values of n, has the unilateral Z-transform can be defined:

In the signal processing, the Z-transform for unilateral causal signals is used.

## Properties

- Linearity. The Z- transform of two linearly related signals, the linear combination of the z- transformed signals.

- Shift. The signal is shifted in the time domain k to the right, then the Z-transform to be multiplied by z -k. With the shift to the left, further terms are added.

- Folding. The convolution of two signals in the time domain corresponds to the product in the frequency domain.

- Differentiation.

### Additional features of the unilateral Z-transform

It should be and whose Z-transform. Furthermore, the following notation for the transformation of discrete- time function is defined in the image plane.

Then, the following rules apply:

## Inverse Z-transform

The inverse Z-transform can be represented by the formula

Is calculated, wherein C is any closed curve to the origin of which lies in the region of convergence of the X (z).

The (unilateral ) Z-transform is discrete in time and corresponds to the Laplace transformation to continuous-time signals.

### Inverse unilateral Z-transform

Requirements: F (z) is holomorphic in an area and.

#### With residual

#### With Laurent series

The integrand is expanded in a Laurent series. The time function is then the coefficient -1 of the Laurent series, ie.

When developing in a number of the binomial theorem and basic properties of the binomial coefficients are useful.

##### Example 1

##### Example 2

#### With essential singularity

## Calculation method

Z- transformation with a limited range of n, and a limited number of z- values can be computed efficiently using the Bluestein - FFT algorithm. The Discrete Fourier transform (abbreviated DFT) is a special case of the Z-transform in the z is on the unit circle.

## Application

In the digital control technology, the Z-transform on the exact design of controllers used. The sampling time and the computing dead are taken into account in the discrete time domain, which can not be accurately modeled in the continuous domain. The usual P, I and D controllers are carrying their digital equivalent in the form of a difference equation. In addition, the digital controller can also be any, the controlled system behavior have adapted, without being limited to the continuous controller.

## Correspondences

Hereinafter own correspondence of the z transform are shown. is the delta function, stands for the step function.