**Lesson 15**

LTI System Applications

Transfer Functions

The Z-transform properties are particularly useful when you have an LTI system described by an LCCDE.

We can use this to determine
outputs of LTI systems by multiplying the Z-transform with the input
with to get the Z-transform of the output. Then we can recover
the time domain output using the *Inverse Z-transform*.

Given a difference equation,

find the Z-transform of the equation and then find the response of the system to an input .

What if you wanted to find the response in the time domain?

We can use **Partial Fraction Expansion** to invert the Z-transform.

As we saw for Laplace Transforms,

where

Then use tables to invert the Z-transform, e.g.

Find Inverse Z-Transform of

Expand:

Given () and , find

What if ?

Find the output to an input and an LTI
system with impulse response

Another method to invert Z-transforms is the **Power Series Expansion**.
Using

So if you can expand like this as a series in , you can pick off as the coefficients of the series.

Find the Inverse Z-Transform of

Find the Inverse Z-Transform of

Divide into :

Find the inverse Z-transform
of

Find the inverse Z-transform of

Would a system having this Z-transform be BIBO stable?

Find the inverse Z-transform of

**Stability**

As we saw earlier, for BIBO stability of a causal LTI system, all roots of the system characteristic equation lie within the unit circle in the -plane.

This is equivalent to stating that all *poles* of the
transfer function must lie within the unit circle on the z-plane.
We point out that does not converge at its poles.

Because causal systems have Regions of Convergence that lie *outside*
the largest magnitude pole, an equivalent condition for BIBO stability
is that the ROC must contain the unit circle.

Find the Z-Transform of the unit step . Would an LTI system with as its system function be BIBO stable?

Find the Z-transform of Would an LTI system with as its system function be BIBO stable?

**Invertibility**

Ex. Find the inverse system of

Check your results by taking the convolution of with .

Find the inverse system of where

For BIBO stability of both systems (assuming they are both causal), where must all poles and zeros of lie?

**Frequency Response**

If we evaluate at , i.e., on the unit circle (as long as unit circle is in the ROC), then we get the DTFT , which we call the frequency response. is periodic with period . You will discuss the DTFT in Chapter 12.

Ex. Find the magnitude of for ,