## Discrete-Time Signals and Systems

###### We introduce the Z-Transform, a general transform that takes discrete-time signals and transforms them into function of the complex variable z.  The continuous-time counterpart of the Z-Transform is the Laplace Transform.  The videos below introduce the Z-Transform, discuss its convergence properties, and then work a variety of examples of how to compute the Z-Transform for different signals.

Introduction to the Z-Transform

6/30/14

Running Time: 4:18

We introduce the concept of the Z-Transform.  The Z-Transform is a general transform used for discrete-time signals.  It's continuous-time counterpart is the Laplace Transform.

Eigenfunctions of Discrete-Time LTI Systems

6/30/14

Running Time: 7:56

A key component of the Z-Transform is the complex exponential signal z^k.  We analyze this class of signal and show that it is an eigenfunction of discrete-time linear time-invariant (DT LTI) systems.  The output of a DT LTI system due to input z^k is just H(z)z^k, i.e. the output is the same as the input except it has been scaled by the complex number H(z).

Definition of the Bilateral Z-Transform

6/30/14

Running Time: 5:31

The previous two videos introduced and provided motivation for use of the Z-Transform.  We now formally define both the Bilateral Z-Transform and Bilateral Inverse Z-Transform.

Convergence of the Z-Transform

7/2/14

Running Time: 6:48

The Z-transform definition, which includes an infinite summation over time, doesn't necessarily converge.  This video discusses conditions on x[k]z^-k for which it does converge.  The values of z for which the Z-transform converges is called the Region of Convergence (ROC).conditions

The Z-Transform in the Complex Plane

7/2/14

Running Time: 4:00

The Z-Transform is a function of the complex number z.  This video sketches/represents the complex quantity z in the complex plane in polar format, i.e. z = re^(j\Omega).  We also show that for r = 1, the set of points z = e^(j\Omega) is the unit circle, and the Z-Transform evaluated on the unit circle is equivalent to the Discrete-Time Fourier Transform (DTFT).

Z-Transform Example #1

7/2/14

Running Time: 5:33

Given the discrete-time signal x[k], we use the definition of the Z-Transform to compute its Z-Transform X(z) and region of convergence (ROC).  The discrete-time signal x[k] in this problem is a finite length signal.

Z-Transform Example #2

7/3/14

Running Time: 5:35

Given the discrete-time signal x[k], we use the definition of the Z-Transform to compute its Z-Transform X(z) and region of convergence (ROC).  The discrete-time signal x[k] in this problem is a right-sided signal, i.e. it has an infinite number of non-zero samples for positive time.

Z-Transform Example #3

7/3/14

Running Time: 7:07

Given the discrete-time signal x[k], we use the definition of the Z-Transform to compute its Z-Transform X(z) and region of convergence (ROC).  The discrete-time signal x[k] in this problem is a left-sided signal, i.e. it has an infinite number of non-zero samples for negative time.

Z-Transform Example Summary

7/3/14

Running Time: 5:30

This video summarizes the Z-Transform Example #1, #2, and #3 worked in the previous videos.  We see that one must specify both X(z) and the region of convergence to uniquely determine the discrete-time signal x[k].  Also, finite-length signals, right-sided signals, and left-sided signals will always have a ROC that consists of the entire complex plane (except possibly z = 0 or infinity), points outside of a circle, or points inside of a circle, respectively.