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!


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:


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:


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!


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

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io


Previous articles of the series

Basics

LTI Systems and Convolution

Fourier Series and Transform

Filtering, Sampling, Modulation, Interpolation

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

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!

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