This video presents an example of plotting the amplitude response of a discrete-time linear system using the “graphical approach” as derived in the previous video. By walking around the unit circle and keeping track of the distance between poles and zeros of the system, an approximate sketch can be made of the systems amplitude response. While the exact amplitude response could obviously be computed by a calculator or Matlab, using this technique is often helpful when trying to determine the rough characteristics of a system (e.g. is it low-pass?, is it high-pass?, etc.)

We introduce the concept of the Unilateral Z-Transform (UZT). The UZT assumes the signals under consideration are always right-sided signals, thus, keeping track of the ROC is much simpler since we will always have an ROC that is a set of points outside of a circle in the complex plane. The UZT definition looks very similar to the Bilateral Z-Transform definition we've been working with except the lower limit of the summation starts at zero instead of minus infinity.

Not surprisingly, the Unilateral Z-Transform (UZT) has properties very similar to the Bilateral Z-Transform (BZT). The only significant different between properties of these two transforms is the time-shift properties. Since the UZT assumes signals that start at time zero and go to the right, and time-shifting that occurs must account for signal samples that shift into positive time (i.e. time delays) or shift out of positive time (i.e. time advances). We present these general time shift properties and also explicitly write the properties for time-shift value of N = 1, 2, and 3.

In this example we solve a discrete-time difference equation using the unilateral Z-transform. The difference equation solved is an example of a first-order recursive system.

This video provides another example of using the Unilateral Z-Transform to solve a difference equation with initial conditions and a non-zero input. The difference equation is first transformed into the z-domain, then partial fraction expansion is used to solve for Y(z), and then finally the difference equation solution y[k] is found by using the inverse Z-transform to go from the Z-domain back into the time domain.

We summarize the different concepts related to the Z-transform that we have learned during this portion of the course.