## Discrete-Time Signals and Systems

###### We now examine how to compute the inverse Z-Transform.  Evaluating the Z-Transform explicitly requires complex/contour integration.  Instead of evaluating this integral, we usually use Partial Fraction Expansion (PFE) to decompose a Z-domain quantity into a summation of simple terms that can be easily inverted.  Long division can also be used to compute individual samples of the time-domain signal.  The videos below examine these different approaches.

Inverse Z-Transform Introduction

7/13/15

Running Time: 8:15

This video introduces the general strategy that we'll use to invert Z-Domain signals back into the discrete-time domain.In this course we avoid directly using the inverse Z-Transform equation as it requires performing contour integration in the complex plane.Instead, our strategy is to write the Z-Transform we're working with (e.g. X(z)) as a linear combination of first-order terms that we can lookup in a table.  This process of writing X(z), which in general is a ratio of polynomials in z, as a linear combination of first-order terms is called Partial Fraction Expansion.Once expanded into first-order terms, we go back into the time-domain by picking either a right-sided or left-sided signal as appropriate based on the overall region of convergence (ROC) of the X(z).A specific example of this process is worked in the subsequent video.

Z-Transform Inversion by Partial Fraction Expansion (PFE)

7/13/15

Running Time: 11:17

The previous video outlined the general strategy we use in this course to invert a z-domain quantity back into the discrete-time domain.This video works a specific example of this process.  Starting with X(z), we decompose X(z) into a linear combination of first-order terms using partial fraction expansion (PFE).For each term we then analyze the pole location and a decide on a right-sided or left-sided time-domain signal based on the pole location relative to the region of convergence of X(z).

The Inverse Z-Transform by Long Division Example #1

4/12/19

Running Time: 7:13

The partial fraction expansion (PFE) approach to finding the inverse Z-transform is great for finding a time-domain equation that is valid for all time.  Sometimes, finding just a few values of the time-domain signal is needed.  In this case, finding the inverse Z-transform of a signal via long division can be used to compute time-domain samples of the signal one value at a time.

In this video, we show how to use the long division approach to compute several values of a right-sided time-domain signal.  For right-sided signals, this division operation should result in “z” raised to negative powers (e.g. z^-1, z^-2, etc.) since negative powers of z correspond to time-domain samples at positive values of time.

The Inverse Z-Transform by Long Division Example #2

4/12/19

Running Time: 5:46

This video also provides an example of how to use the long division approach to compute the inverse Z-transform of a discrete-time signal.  In the previous video, the inverse Z-transform of a right-sided signal was computed.  In this example, the inverse Z-transform of a left-sided signal is considered.  For left-sided signals, this division operation should result in “z” raised to positive powers (e.g. z^1, z^2, etc.) since positive powers of z correspond to time-domain samples at negative values of time.

The Transfer Function of a Discrete-Time Linear System

4/15/19

Running Time: 8:15

This video is the first of several that involve working with the Transfer Function of a discrete-time LTI system. The transfer function H(z) is Z-domain quantity equal to the ratio of Y(z)/X(z) (e.g. the ratio of the system output and system input in the Z-domain).

In this video, we start with a time-domain difference equation for a DT LTI system.  We then transform this difference equation into the Z-domain and then solve for the transfer function H(z).

Finding the Difference Equation from the System Transfer Function

4/15/19

Running Time: 3:42

In the previous video we started with a system difference equation, and then solved for the system transfer function.  The example presented here goes the other direction.  We start with the transfer function H(z) of a discrete-time LTI system, and then we find the corresponding difference equation of the system.

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