The -transform is widely used in signal processing to analyze the interaction between signals and systems. This lesson gives you an introduction to this important tool. You will learn the definition of the -transform, the complex plane, and the relationship between the -transform and the DTFT. Prerequisites Introduction to the System Function and System Poles Read More

## The Region of Convergence for the -Transform

Much of the power of the -transform is due to the fact that it exists for signals that have no DTFT. In this lesson you will learn how the region of convergence defines the range of values for which the -transform converges. The region of convergence differentiates causal from non causal signals and indicates whether a Read More

## Poles and Zeros of the -Transform

In this lesson you will learn how to find the poles and zeros of a rational -transform. Poles and zeros are important because they provide a very insightful characterization of systems described by linear constant coefficient difference equations. Such systems are widely used to implement filters and as mathematical models for signals. The poles and Read More

## Properties of the Region of Convergence

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. Prerequisites Poles and Zeros

## 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. Prerequisites Properties of the Region of Convergence

## 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. Prerequisites Properties of the Region Read More

## 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 Read More

## -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. Prerequisites Properties of Read More

## 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 Read More

## Inverse Systems for LTI Systems Described by Difference Equations

A key signal-processing question is: when can distortion introduced by a system be reversed? In this lesson you will learn the conditions under which a stable and causal inverse system exists for correcting the distortion introduced by an LTI system described by a difference equation. The conditions for existence of a stable and causal inverse Read More

## Minimum-Phase and All-Pass Systems

In this lesson you will learn the characteristics of minimum-phase and all-pass systems. A minimum-phase system has a stable and causal inverse. An all-pass system is does not have a stable and causal inverse, but has unit gain at all frequencies. You will see that any system can be factored into the product of a Read More

## Frequency Response Magnitude and Poles and Zeros

The magnitude of a system's frequency response is tightly coupled to the locations of the poles and zeros. In this lesson you will learn how to infer the nature of the frequency response magnitude from the poles and zeros. This skill is very useful when designing and evaluating filters. The insight linking frequency response magnitude Read More

## Impulse Response and Poles and Zeros

The locations of a system's poles and zeros also provide insight into the characteristics of the impulse response. In this lesson you will learn how to infer the nature of the impulse response from the pole and zero locations. You will be able to use the pole and zero locations to predict which of two Read More