Definitions of z-Transforms
We can also define a unilateral z-transform as
IMPORTANT: The textbook uses the notation and to denote the two-sided or bilateral z- Transform of a function , but since we will only look at the bilateral transform (and NOT the unilateral), we will drop the subscript here.
Since our focus is on the Bilateral z-Transform, let's jump ahead to that section, and then we'll come back to some of the earlier sections.
The bilateral (2-sided) z-transform.
Note that, whereas for Laplace Transform we considered where the integral converges, here we consider where the sum converges.
We must consider the Region of Convergence (ROC) of the z-transform for the bilateral z-transform because left-sided and right-sided time functions will have the same z-transform and only the ROC will distinguish between the two possible time functions.
You'll use this a lot!
Find the z transforms of
We see that we must specify the ROC for the bilateral z- transform to be unique.
Definitions and Regions of Convergence
We can write
Examine for right-sided
As , need for sum to converge.
This will happen for values of outside rather than inside the pole, i.e.
What about ?
If is not causal but is still right-sided, e.g.
Will not converge at , and we won't include it in the ROC.
Thus we can tell if a system is causal from the ROC of the z-transform of its impulse response.
Examine for left-sided
As , need or
This happens for values of z inside rather than outside the poles.
What about ?
If is left-sided but not strictly anticausal
( for but )
e.g. , then
FACT: An ROC must contain the unit circle for stability - this holds for causal, anticausal, and two-sided signals.
Find the z-Transform of for .
Find the z-Transform of
Find the z-transform of .
What is its ROC?
Find the z-transform of
Find the z-transform of using Euler's rule.
Note that the previous examples are for a unilateral z-transform but if you add a to all the time functions, you will get the same answer as for the bilateral transform. You can derive all these transform pairs for practice in taking z-transforms.
Insights from the Pole-Zero Plot and ROC
Things that you can tell about a signal from its pole-zero plot (and ROC):