Lesson 13

Definitions of *z*-Transforms

is the bilateral (2-sided)

which is a counterclockwise contour integral along a closed path in the -plane. We will see how to take inverse

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.

Bilateral *z*-Transform

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.

is for 2-sided, anticausal, and causal signals (i.e. all signals).

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.

Remember:

You'll use this a lot!

Find the *z* transforms of

and plot the ROCs and pole/zero diagrams.

We see that we must specify the ROC for the bilateral *z*- transform
to be unique.

Definitions and Regions of Convergence

- is right-sided if
- is left-sided if

We can write

- We've seen that
right-sided signals have an ROC of the form ,
i.e.,

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. , then

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.

**Left-sided**signals have ROC of form , i.e.,

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

does not converge at so don't include in the ROC.**2-sided**signals have ROC of the form

**Finite Duration**has ROC of entire z-plane except possibly or

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

Would this system be BIBO stable?

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):

- When the ROC includes the unit circle, then the signal is
absolutely summable. (If the signal is an impulse
response the system is stable.)

- A pole on the positive real axis corresponds to a simple decaying or
growing function (of form for a pole at ).

- Poles off the positive real axis correspond to an oscillating
signal where the frequency of oscillation is the angle from the
positive real axis. (Poles on the negative real axis have an angle
of , so the frequency of oscillation is , as in .)
When the poles are ...
- on the unit circle sinusoidal functions with constant amplitude
*not*on the unit circle sinusoidal functions with a decaying (or growing) envelope (rate of decay/growth depends on the distance from the pole to the origin).

- Poles and zeroes must come in complex conjugate pairs for the
signal to be real (consequence of the
*z*-transform property: ).