# Z Transform and LTI System Properties Study Guide

## Introduction

This page will go over some of my conclusions about properties of the z-transform and discuss some examples of how they may be used to draw conclusions about LTI systems and digital filters. This topic assumes some basic knowledge of the z-transform and signal processing, however some definitions are provided below as a refresher or for reference. The information provided is intended to clarify or augment some of the z-transform properties presented in class. Some of the information includes statements or arguments to validate or clarify claims; these are intended to aid understanding but not substitute for a proof.

## definitions

z-transform: $X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$ z ∈ ℂ

DTFT: $X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n}$

right sided: $x[n]$ such that $x[n] = 0$ ∀ n < n0

left sided: $x[n]$ such that $x[n] = 0$ ∀ n > n0

both sided: $x[n]$ such that $x[n] = 0$ ∀ n > n1 and n < n2 ,n1 < n2

neither sided: $x[n]$ has finite duration, meaning $x[n] = 0$ ∀ n < n1 and n > n2, n1 < n2

Absolutely Summable: $x[n]$ is absolutely summable if: $\sum_{n=-\infty}^{\infty} |x[n]| = c$, where $c$ is a constant.

z-plane: the complex plane.

## Relationship Between Fourier Transform and Z Transform

In the z-transform definition above, z is any complex number, which can be represented in polar coordinates by $z = re^{j\omega}$ Then the z-transform can be written as: $X(z) = X(re^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n](re)^{-j \omega n}$. Thus if r = 1, meaning that z is constrained to z = $e^{j\omega}$, then the z-transform is equivalent to the DTFT, and the DTFT is the z-transform constrained with z constrained to the unit circle in the z-plane.

Since the z-transform exists outside the unit circle, it is useful for analyzing signal that don't have a DTFT, analyzing LTI system stability, and looking at the transfer function characteristics among other uses.

## Summary of Z Transform Properties

(a) The DTFT(x[n]) converges if the ROC of X(z) contains the unit circle on the z-plane.
(b) No poles of X(z) are included in the ROC of X(z), poles of X(z) shape the ROC. E.g. if X(z) has a pole at 0, then z = 0 is not included in the ROC.
(c) For Right Sided x[n], the ROC of X(z):
(i) is generally defined by |z| > z0 (outside a circle in the z-plane)
(ii) includes $\infty$ if n0 ≥ 0
(iii) z = 0 could possibly be included in the ROC
(iv) the farthest pole from the origin lies just inside |z0|
(d) For Left Sided x[n], the ROC of X(z):
(i) is generally defined by |z| < z0 (inside a circle in the z-plane)
(ii) includes z = 0 if n0 ≤ 0
(iii) x = $\infty$ could possibly be included
(iv) the nearest pole to the origin lies just outside of |z0|
(e) For Both Sided x[n], the ROC of X(z):
(i) is generally a ring in the z-plane such as z1 < |z| < z2
(ii) does not include z = 0 or z = $\infty$
(f) For finite duration x[n] (neither sided), the ROC of X(z):
(i) includes the entire complex plane
(ii) x[n] has no poles except maybe z = 0 or z = $\infty$
(iii) if for x[n], n1 ≥ 0, z = $\infty$ is included
(iv) if for x[n], n2 ≤ 0, z = 0 is included
(v) if n1 < 0 and n2 > 0, then neither z = 0 or z = $\infty$ is included

## LTI Systems and Z Transform Properties

(1) System Stability An LTI System with impulse response h[n] is BIBO stable if h[n] is absolutely summable.

(a) This is a condition for the DTFT to exist, so it must also be true that the ROC of H(z) contains the unit circle in the z-plane.
(b) This implies that the poles of a right sided h[n] must be inside the unit circle and poles of a left sided h[n] must be outside of the unit circle in the z-plane for h[n] to be BIBO stable.

(2) Causal LTI Systems From statements in (1) and (c) above, a few conclusions about causal system can be made:

(a) The ROC of H(z) includes z = $\infty$ and |z| > z0
(b) For H(z) to be BIBO stable, it must have poles inside the unit circle in the z-plane.
(c) H(z) must have more poles than zeros.
One possible reasoning: Assuming that all causal systems have the ROC defined in (a), then assuming H(Z) has more zeros than poles would be a contradiction, since H(z) can could then be decomposed by long division or partial fractions into a function of z that diverges in the region |z| > z0 and $\infty$. Simply taking the limit of H(z) as z approaches $\infty$ would show a contradiction since the ROC would not include z = $\infty$.

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