2Lecture Outline

Classification of Continuous Time Systems Continuous-time vs. Discrete-time systems Linear and nonlinear systems Time-varying and time-invariant systems System with and without memory Causal systems Invertibility and inverse systems Stable Systems

3Continuous-time vs. Discrete-time systems

CTx(t) y(t)

Both input and output of the system are continuous-time

DTX[n] Y[n]

Both input and output of the system are discrete-time

4Linear Systems

Let y1(t) be the response of a system to an input x1(t) andy2(t) be the response to the input x2(t). Then the system is

linear if the response to x1(t)+x2(t) is y1(t)+y2(t)

property of superposition the response to x1(t) is y1(t), where is any arbitrary

constant property of scaling or homogeneity

Property of superposition implies that the response resulting from several input signals can be computed as the sum ofthe responses resulting from each input signal alone

Property of scaling implies that linear systems scale the output by the same amount as the input gets scaled.

The two properties can be combined into a single statement1 2 1 2( ) ( ) ( ) ( )x t x t y t y t + +

6Time-Varying and Time-Invariant Systems

A system is time invariant if a time shift in the input signalcauses an identical time shift in the output signal.

If y(t) is the output corresponding to input x(t), a timeinvariantsystem will have y(t-t0) as the output whenx(t-t0) is the input

The procedure for testing a system is time invariant is asfollows:

Let y1(t) be the output corresponding to x1(t) Consider a second input, x2(t), obtained by shifting x1(t)

is x2(t) = x1(t t0) and find the output y2(t)corresponding to the input x2(t)

From step 1, find y1(t t0) and compare with y2(t) If y2(t) = y1(t t0), then the system is time invariant;

otherwise it is a time-varying system See Example 2.2.4

7Systems with and without Memory

A system is memoryless, or instantaneous, if the presentvalue of the output depends only on the present value ofthe input.

A resistor is a memoryless system, since with input x(t)taken as the current and output y(t) taken as thevoltage, the input/output relationship is y(t) = Rx(t)where R is the resistance.

A capacitor is a system with memory. With input takenas the current x(t) and output as the voltage y(t), theinput/output relationship is given by

The output at any time t depends on the entire past history ofthe input

See Example 2.2.5

1( ) ( )t

y t x dC

=

8Causal Systems

A causal system is one that is nonanticipative; that is, the output may depend on current and past inputs, but not future inputs.

All "realtime" systems must be causal, since they can not have future inputs available to them

if x1(t)=x2(t); t

9Inverse Systems

( )x t ( ) ( )z t x t=( )y tInverse systemsystem

The inverse system undoes what the givensystem does to input x(t) when the inversesystem cascaded to the given system.

If two different inputs result in the same output,then the system in not invertible.

Example 2.2.8

10

Stable System: BIBO Stability

A system in bounded-input bounded-output(BIBO) stable if, for any bounded input x(t) with|x(t)| < B1 < , the response y(t) is alsobounded, i.e., |y(t)| < B2 < . All bounded inputs must give rise to bounded

outputs. If we can find even one bounded input forwhich the output is not bounded, the system isunstable.

Example 2.2.9