Chapter 3 CT LTI Systems Updated: 9/16/13. A Continuous-Time System How do we know the output? System X(t)y(t)

Chapter 3

CT LTI SystemsUpdated: 9/16/13

A Continuous-Time System

• How do we know the output?

System X(t) y(t)

LTI Systems

• Time Invariant – X(t) y(t) & x(t-to) y(t-to)

• Linearity– a1x1(t)+ a2x2(t) a1y1(t)+ a2y2(t)– a1y1(t)+ a2y2(t)= T[a1x1(t)+a2x2(t)]

• Meet the description of many physical systems• They can be modeled systematically

– Non-LTI systems typically have no general mathematical procedure to obtain solution

What is the input-output relationship for LTI-CT Systems?

• An approach (available tool or operation) to describe the input-output relationship for LTI Systems

• In a LTI system t) h(t) – Remember h(t) is T[t)]– Unit impulse function the impulse response

• It is possible to use h(t) to solve for any input-output relationship

• One way to do it is by using the Convolution Integral

Convolution Integral

LTI SystemX(t)=(t) y(t)=h(t)

LTI System: h(t)X(t) y(t)

Convolution Integral

• Remember

• So what is the general solution for

LTI SystemX(t)=A(t-kto) y(t)=Ah(t-kto)

LTI SystemX(t) y(t)

?

Convolution Integral

• Any input can be expressed using the unit impulse function

)()1)(()()(

)()()()(

)()()()(

)()()(

)()(

txtxdttx

dtxdttx

tttxtttx

tdtttt

tt

ooo

oo

dtxtx )()()(

LTI SystemX(t) y(t)

Sifting Property

Proof:

toand integrate by d

Convolution Integral

• Given

• We obtain Convolution Integral

• That is: A system can be characterized using its impulse response: y(t)=x(t)*h(t)

LTI SystemX(t) y(t)

dthxty

dtTxtyLinearity

dtxTty

tTth

tTth

txTty

)()()(

)()()(:

)()()(

)}({)(

)}({)(

)}({)(

LTI System: h(t)X(t) y(t)

Do not confuse convolution with multiplication!

y(t)=x(t)*h(t)

By definition

Convolution Integral

LTI System: h(t)X(t) y(t)

Convolution Integral - Properties

• Commutative• Associative • Distributive

• Thus, using commutative property:

dtxhdthxtx )()()()()(

)](*)([)](*)([)]()([*)(

)](*)([*)()(*)](*)([

)(*)()(*)(

2121

2121

thtxthtxththtx

ththtxththtx

txththtx

Next: We draw the block diagram representation!

Convolution Integral - Properties

)](*)([)](*)([)]()([*)(

)](*)([*)()(*)](*)([

)(*)()(*)(

2121

2121

thtxthtxththtx

ththtxththtx

txththtx

• Commutative

• Associative

• Distributive

Simple Example

• What if a step unit function is the input of a LTI system?

• S(t) is called the Step Response

LTI Systemu(t) y(t)=S{u(t)}=s(t)

dttsdttythNote

dh

dtuhdthutsty

tuthtxthtsty

tuSty

t

/)(/)()(:

)(

)()()()()()(

)()()()()()(

)}({)(

Step response can be obtained by integrating the impulse response!

Impulse response can be obtained by differentiating the step response

Example 1

• Consider a CT-LTI system. Assume the impulse response of the system is h(t)=e^(-at) for all a>0 and t>0 and input

x(t)=u(t). Find the output.

h(t)=e^-at u(t) y(t)

)()1(1

)1(1

)()(

)()()(

)()()()()(

0

tuea

ea

de

dtuue

dtuhty

tuthtxthty

at

att

a

a

Draw x(), h(), h(t-),etc. next slide

Because t>0

The fact that a>0 is not an issue!

Example 1 – Cont.

y(t)

t>0t<0

Remember we are plotting it over and t is the variable

U(-(-t))U(-(-t))

*

)()1(1

)1(1

)()(

)()()(

)()()()()(

0

tuea

ea

de

dtuue

dtuhty

tuthtxthty

at

att

a

a

t

y(t); for a=3

t

Example 1 – Cont. Graphical Representation (similar)

http://www.wolframalpha.com/input/?i=convolution+of+two+functions&lk=4&num=4&lk=4&num=4http://www.wolframalpha.com/input/?i=convolution+of+two+functions&lk=4&num=4&lk=4&num=4

a=1In this case!

Example 1 – Cont. Graphical Representation (similar)

http://www.jhu.edu/signals/convolve/http://www.jhu.edu/signals/convolve/

Note in our caseWe have u(t) rather than rectangular function!

Note in our caseWe have u(t) rather than rectangular function!

Example 2

• Consider a CT-LTI system. Assume the impulse response of the system is h(t)=e^-at for all a>0 and t>0 and input x(t)= e^at u(-t). Find the output.

h(t)=e^-at x(t) y(t)

02

1)(

2

1]01[

2

10

2

1

2

10

)()(

)()()(

)()()()()(

02

22

)(

aea

ty

eaa

edeet

ea

ea

edeet

dtueue

dthxty

tuthtxthty

ta

atataat

atatatt

aat

taa

Draw x(), h(), h(t-),etc. next slide

Note that for t>0; x(t) =0; so the integration

can only be valid up to t=0

Example – Cont.

02

1)(

)()()()()(

aea

ty

tuthtxthty

ta

*h(t)=e^-at u(t)

x(t)= e^at u(-t)

dtxhty

tuthtxthty

)()()(

)()()()()(

?

Another Example notes

Properties of CT LTI Systems• When is a CT LTI system memory-less (static)

• When does a CT LTI system have an inverse system (invertible)?

• When is a CT LTI system considered to be causal? Assuming the input is causal:

• When is a CT LTI system considered to be Stable?

)()()()( tKxtytKth

)()(*)( tthth i

dtxhdthxtytt

)()()()()(00

notes

dtthty )()(

Example

• Is this an stable system?

• What about this?

)()( 3 tueth t

)()( 3 tueth t

notes

3/1)(

)()(

0

33

dtedttue

dtthty

tt

Differential-Equations Models • This is a linear first order differential equation with

constant coefficients (assuming a and b are constants)

• The general nth order linear DE with constant equations is

Is the First-Order DE Linear?

• Consider

• Does a1x1(t)+ a2x2(t) a1y1(t)+ a2y2(t)?

• Is it time-invariant? Does input delay results in an output delay by the same amount?

• Is this a linear system?

notes

notes

Sum Integrator

a

-

+

X(t)Y(t)e(t)

Example

• Is this a time invariant linear system?

R

LV(t)

Ldi(t)/dt + Ri(t)= v(t)a= -R/Lb=1/L y(t)=i(t)x(t) = V(t)

Solution of DE

• A classical model for the solution of DE is called method of undermined coefficients

– yc(t) is called the complementary or natural solution

– yp(t) is called the particular or forced solution

Solution of DE

Thus, forx(t) =constant yp(t)=P

x(t) =Ce^-7t yp(t)= Pe^-7tx(t) =2cos(3t) yp(t)=P1cos(3t)+P2sin(3t)

Example

• Solve – Assume x(t) = 2 and y(0) = 4

– What happens if

notes

yc(t) = Ce^-2t; yp(t) = P; P = 1y(t) = Ce^-2t + 1 y(0) = 4 C = 3 y(t) = 3e^-2t + 1

Schaum’s Examples

• Chapter 2: – 2, 4-6, 8, 10, 11-14, 18, 19, 48, 52, 53,