HKN ECE 310 Exam Review Session - PowerPoint Presentation

Slide1

HKN ECE 310Exam Review Session

Kanad

Sarkar

Corey Snyder

Topics

LSIC Systems

BIBO Stability

Impulse Response and Convolution

Z-Transform

LSIC Systems

Linearity

Satisfy Homogeneity and Additivity: Let

be our system

Homogeneity:

Additivity: Can be summarized by SuperpositionIf and then Shift InvarianceAny arbitrary shift in the input simply leads to the same shift in the outputIfthen andCausalityOutput cannot depend on future input values

BIBO Stability

Three ways to check for BIBO Stability:

Pole-Zero Plot (more on this later)

Absolute summability

of the impulse response:

System Definition:Given then the system is BIBO stableA bounded input yields a bounded output Ex: vs.  

Impulse Response

Let

be the input to an LSI system identified by its impulse response

. Then, the output

is given by

].System output to an inputConvolution in the time/sample domain is multiplication in the transformed domain, both the z-domain and frequency domain. 

Convolution

System must be:

Linear

Shift Invariant

If

is of length and is of length , must be of length .Can be done graphically or algebraically. 

Z-Transform

Typically perform inverse z-transform by inspection or by Partial Fraction Decomposition

Important properties:

Multiplication by n:

Delay Property #1:

Make sure to note the Region of Convergence (ROC) for your transforms!More in the next slide! 

BIBO Stability Revisited: Pole-Zero Plots

For an LSI system: if the ROC contains the unit circle, this system is BIBO stable

The ROC is anything greater than the outermost pole if the system/signal is causal or “right-handed”

The ROC is anything less than the innermost pole if the system/signal is anti-causal or “left-handed”

If we sum multiple signals, the ROC is the

intersection of each signal’s ROC

BIBO Stability Revisited: Pole-Zero Plots

What if the ROC is

or

?

This is

marginally stable, but unstable for ECE 310 purposesFor unstable systems, you are commonly asked to find a bounded input that yields an unbounded output. Few ways to do this:Pick an input that excites the poles of the system.If the system’s impulse response is not absolutely summable, will work frequently works too, like when is unbounded, e.g. . 

LSIC Examples

For the following systems, determine whether it is linear, shift-invariant, and causal:

BIBO Stability Example

For each of the following systems defined either by an input-output relationship or impulse response, determine whether the system is BIBO stable or not.

Impulse Response and Convolution Examples

Given

and

, compute the system output.

What does this filter do?

Suppose we have a digital filter with an unknown impulse response. We do know the system output to the follow two input signals. Determine the impulse response in terms of the two system outputs. 

Combinations of Systems

Given the following two LSI systems

Suppose we pass input

to a system

where

is the connection of and in series. Write the output in terms of and .Suppose the two systems are still connected together in series. What is the resulting transfer function and impulse response of [n]?Suppose the two systems are now connected in parallel to form system . What is the resulting transfer function and impulse response of ? 

Marginal Stability Example

Suppose we have a system response given by

. Which of the following bounded inputs would cause this system to have an unbounded output? There may be more than one!