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 […]

# The Z-Transform

## 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 […]

## 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 […]

## Introduction to the -Transform

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.

## 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 […]

## 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 […]