Mathematics - Signals and Systems - Exercises on the Z Transform

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Introduction

Hey it's a me again @drifter1!

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

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

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Z Transform and ROC Examples [Based on 22.4 from Ref1]

Let's determine the Z Transform, pole-zero plot and associated ROC for each of the following cases:

a.

The signal x₁[n] is a common case, which is included in Z Transform tables. As such, it has a well-known Z Transform and ROC, which is given by:

The root of the denominator is a pole, and at point 1/3 in the real-axis. So, at last, the pole-zero plot and ROC can be visualized as follows:

b.

The second signal, x₂[n] has the same exact algebraic expression for the Z Transform. The only difference is the ROC, which is now the inner circle. So, the Z Transform and ROC are as follows:

And visually we have:

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ROC from Conditions [Based on 22.3 from Ref1]

Let's consider the following pole-zero plot of the Z Transform X(z) of a sequence x[n]:

Determine the associated region of convergence (ROC) for each of the following conditions:

1. x[n] is right-sided
2. x[n] is left-sided
3. The Fourier transform of x[n] converges
4. The Fourier transform of x[n] doesn't converge

1.

For a signal to be right-sided, the ROC must be of the form |z| > a. And, since the ROC can't include poles the ROC is:

2.

Now, for it to be left-sided, it's ROC must be of the form |z| < a, and not include poles, which leads to:

3.

In order to converge, the ROC must include the unit circle (|z| = 1). And, because it must be bounded by poles, the ROC can only be:

4.

For the FT of x[n] to not converge, the ROC must not contain the unit circle, which means that there are two possibilities:

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LTI System Analysis Example

Consider a discrete-time LTI system, with the following transfer function:

Let's specify and sketch the ROC (and pole-zero plot) for each of the following cases:

1. The system is causal
2. The system is stable

Poles and Zeros

First of all, in any of these cases the corresponding poles and zeros are as follows:

• The root of the numerator polynomial z = - 1/2 is a zero.
• The roots of the denominator polynomial z = 1/3 and z = 2 are the poles.

This leads to the following pole-zero diagram, which will be shared amongst the follow-up cases:

1. Causal

As we know, a system is causal when the ROC is the outside region of a circle. In the case of rational functions it must also be:

• outside of the most distant pole, and
• the degree of the numerator must not be larger then the degree of the denominator polynomial

So, in the case of this system, the ROC is:

or graphically:

2. Stable

For a system to be stable, the ROC must simply contain the unit circle.

As such, the ROC can be either of the following:

and so either the complete z-region, the region below the circle at 2 or the intermediate region of two circles (one at 1/3 and one at 2).

Graphically:

Let's note that it's impossible for this system to be both causal and stable!

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RESOURCES:

References

1. Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.

Images

1. 

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io

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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
• Z Transform → Z Transform, Region of Convergence (ROC), Inverse Z Transform
• Z Transform Properties → Linearity, Time-Shifting, Time-Scaling, Time-Reversal, z-Domain Scaling, Conjugation, Convolution, Differentation in the z-Domain, Initial and Final value Theorems
• LTI System Analysis using Z Transform → System Properties (Causality, Stability), LCCDE Representation and Z Transform

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Final words | Next up

And this is actually it for today's post!

From next time on, we will start getting into other various topics related to the Laplace and Z Transform...

See Ya!

Keep on drifting!