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## Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about **Signals and Systems** in to get into the **Z Transform**.

So, without further ado, let's dive straight into it!

## Getting into the Z Transform

Similar to how the Laplace Transform has been developed as an generalization of the continuous-time Fourier Transform, the Z Transform is the corresponding equivalent for discrete-time. In discrete-time Fourier analysis, complex exponentials where used as basic building blocks for signals.

Those exponentials of the form are now replaced with the more general form , where *z* is a complex number.

As with the continuous-time Fourier Transform and Laplace Transform, a similar close relationship also exists in the case of the Z Transform and discrete-time Fourier Transform.
To get even more specific, in the case where *z = e ^{jΩ}*, which is basically a magnitude of unity (as we will see in the ROC section), the Z Transform recudes to the Fourier Transform.
So, similar to the Laplace Transform, the Z Transform can again be viewed as some kind of exponential weighting.
And because of this exponential weighting sequences/signals that don't converge using the Fourier Transform may converge with the Z Transform, thus allowing for more signals and systems to be analyzed.

## Z Transform

Mathematically, the Z Transform is defined as:

And denoting the Transform by its letter *Z*, the Z Transform is basically the following function:

Using the Z Transform, the impulse response (system response for a complex exponential) can be defiend as follows:

In this previous equation the following substitution has been made:

which can also be used to relate the Fourier and Z Transform, as follows:

Thus, its now clear as day that the Z Transform applies exponential weighting of the form *r ^{-n}*, allowing this Transform to converge in cases where the Fourier Transform might not.

### Example

Let's get into a simple example, to understand the concept better...

Consider the following commonly used signal:

Its a known case of the Fourier Transform and gives the following result:

Let's note that *X(Ω)* is the same as *X(e ^{jΩ})*, and so referring to the Fourier Transform.

Substituting the corresponding *z*, yields the following result:

In the case of the Z Transform, the conditions are far more important and play a key role in the so called Region of Convergence (ROC) of the Transform. More on that later on, but let's quickly summarize the Z Transform for the case of the exponential signal:

## Region of Convergence (ROC)

The algebraic expression of the Z Transform can be similar for different signals, and so the range of values of *z*, referred to as the region of convergence (ROC), for which the given expression is valid, is also important to specify.

Similar to the Laplace Transform, the Z Transform can again by described as a ratio of polynomials in *z*.

As such, its convenient to describe the signal by the location of poles (roots of the denominator polynomial) and zeros (roots of the numerator polynomial) in the complex plane or z-plane. This complex z-plane is reduced to a unit circle (circle of radius 1) concentric with the origin, when the Z Transform reduces to the Fourier Transform. In the case of the Laplace Transform this was a little different as the imaginary axis in the s-plane played that role. Either way, the pole-zero pattern in the z-plane can be used to specify the algebraic expression for the Z Transform, but its also important to implicitly or explicitly indicate the ROC.

The z-plane and the corresponding unit circle are visualized below.

### Example (Continued)

Well, let's get back to the previous example, to see how the ROC and the pole-zero pattern is visualized.

In the case of this exponential signal, there are no zeros (roots of the numerator), and the pole is basically defining a concentric circle of radius *a*.
Thus, the whole region around that circle is the ROC of the signal.
Below is a visualization of the ROC and pole-zero pattern in the z-plane.

Its easy to notice that the signal still converges even for values of *|a| > 1* (outside of the unit circle), which don't converge in the case of the Fourier Transform.

The properties of the Z Transform and its ROC will be covered in-depth next time!

## Inverse Z Transform

Lastly, let's quickly mention how the inverse Z Transform is calculated.

Mathematically, the inverse Z Transform equation is as follows:

Of course, Transform tables are used, and such contour integrals don't have to be calculated.

### Exercise for the viewer

Try to come up with this equation starting of with the notation:

and substituting *z = re ^{jω}*.

## RESOURCES:

### References

### Images

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io and GeoGebra

## Previous articles of the series

### Basics

- Introduction → Signals, Systems
- Signal Basics → Signal Categorization, Basic Signal Types
- Signal Operations with Examples → Amplitude and Time Operations, Examples
- System Classification with Examples → System Classifications and Properties, Examples
- Sinusoidal and Complex Exponential Signals → Sinusoidal and Exponential Signals in Continuous and Discrete Time

### LTI Systems and Convolution

- LTI System Response and Convolution → Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral
- LTI Convolution Properties → Commutative, Associative and Distributive Properties of LTI Convolution
- System Representation in Discrete-Time using Difference Equations → Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)
- System Representation in Continuous-Time using Differential Equations → Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)
- Exercises on LTI System Properties → Superposition, Impulse Response and System Classification Examples
- Exercise on Convolution → Discrete-Time Convolution Example with the help of visualizations
- Exercises on System Representation using Difference Equations → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example
- Exercises on System Representation using Differential Equations → Equation to Block Diagram Example, Direct Form I to Equation Example

### Fourier Series and Transform

- Continuous-Time Periodic Signals & Fourier Series → Input Decomposition, Fourier Series, Analysis and Synthesis
- Continuous-Time Aperiodic Signals & Fourier Transform → Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals
- Continuous-Time Fourier Transform Properties → Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties
- Discrete-Time Fourier Series & Transform → Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations
- Discrete-Time Fourier Transform Properties → Differences with Continuous-Time, Periodicity, Linearity, Time and Frequency Shifting, Conjugate Summetry, Differencing and Accumulation, Time Reversal and Expansion, Differentation in Frequency, Convolution and Multiplication, Dualities
- Exercises on Continuous-Time Fourier Series → Fourier Series Coefficients Calculation from Signal Equation, Signal Graph
- Exercises on Continuous-Time Fourier Transform → Fourier Transform from Signal Graph and Equation, Output of LTI System
- Exercises on Discrete-Time Fourier Series and Transform → Fourier Series Coefficient, Fourier Transform Calculation and LTI System Output

### Filtering, Sampling, Modulation, Interpolation

- Filtering → Convolution Property, Ideal Filters, Series R-C Circuit and Moving Average Filter Approximations
- Continuous-Time Modulation → Getting into Modulation, AM and FM, Demodulation
- Discrete-Time Modulation → Applications, Carriers, Modulation/Demodulation, Time-Division Multiplexing
- Sampling → Sampling Theorem, Sampling, Reconstruction and Aliasing
- Interpolation → Reconstruction Procedure, Interpolation (Band-limited, Zero-order hold, First-order hold)
- Processing Continuous-Time Signals as Discrete-Time Signals → C/D and D/C Conversion, Discrete-Time Processing
- Discrete-Time Sampling → Discrete-Time (or Frequency Domain) Sampling, Downsampling / Decimation, Upsampling
- Exercises on Filtering → Filter Properties, Type and Output
- Exercises on Modulation → CT and DT Modulation Examples
- Exercises on Sampling and Interpolation → Graphical/Visual Sampling and Interpolation Examples

### Laplace and Z Transforms

- Laplace Transform → Laplace Transform, Region of Convergence (ROC)
- Laplace Transform Properties → Linearity, Time- and Frequency-Shifting, Time-Scaling, Complex Conjugation, Multiplication and Convolution, Differentation in Time- and Frequency-Domain, Integration in Time-Domain, Initial and Final Value Theorems
- LTI System Analysis using Laplace Transform → System Properties (Causality, Stability) and ROC, LCCDE Representation and Laplace Transform, First-Order and Second-Order System Analysis
- Exercises on the Laplace Transform → Laplace Transform and ROC Examples, LTI System Analysis Example

## Final words | Next up

And this is actually it for today's post!

Next time we will get into the various properties of the Z Transform...

See Ya!

Keep on drifting!