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 ,