The region of convergence (ROC) plays an important role in the use of -transforms for analysis of signals and systems. This lesson will teach you several very useful properties of the ROC. You will use these properties in finding inverse -transforms and in understanding causality and stability properties of systems.

# The Z-Transform

## Inversion of the -Transform via Power Series Expansion

There are several methods for inverting -transforms. In this lesson you will learn how to use the power series expansion technique to find the time-domain signal corresponding to a -transform. This method is particularly useful for finding inverse -transforms of transcendental functions.

## Inversion of the -Transform: Partial Fraction Expansion

In this lesson you will learn how to find inverse -transforms using partial fraction expansions. This method is normally used for finding inverse -transforms of rational functions in powers of . The partial fraction expansion method offers important insight as to how the pole locations influence the time-domain signal characteristics.

## Properties of the -Transform

The -transform has several important properties. In this lesson you will learn the most commonly used properties of the -transform in the analysis of signals and systems. In particular, the convolution-multiplication property introduces the characterization of linear time-invariant systems using the -transform of the impulse response, which is called the system function. The time-shift property […]

## -Transform Analysis of LTI Systems

The -transform provides important insight into the characteristics of linear, time-invariant (LTI) systems. This lesson introduces you to this important topic by establishing the relationship between the difference equation and system function, including the pole-zero form of the system function. The lesson concludes with questions that will be answered in subsequent lessons.

## Stability and Causality of LTI Systems Described by Difference Equations

In this lesson you will learn the requirements for a system to be stable and causal. These critical properties are dependent on the locations of the poles in the -plane. You will gain key insights about the most widely used class of systems in signal processing. Systems that are not stable are of no practical […]