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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!