Recall: RC circuit example

x y

x(t)R

C y(t)

+

_

+

_

,dy

y xdt

1

RC

( )

0

( ) ( ) ,t

ty t e x d assuming (0) 0.y

( ) ( )y t T x t

Properties of Input-Output Systems

x y ( ) ( )y t T x t

Linearity. The system is linear if

( )

0

( ) ( ) ,t

ty t e x d

Example: RC circuit is linear. Follows from equation

1 2 1 2[ ( ) ( )] [ ( )] [ ( )]

[ ( )] [ ( )]

T x t x t T x t T x t

T k x t k T x t

Time Invariance Property:

( ) ( )y t T x t ( ) ( )y t T x t If , then

In words, a system is T.I. when: given an input-output pair, if we apply a delayed version of the input, the new output is the delayed version ofthe original output.

( )x t

t

t

t

t

x

y

( )y t

Time invariance of RC circuit

Assume all signals are zero for t < 0.

0

( ) ( ) ( .)t

y t h t x d

Introduce the notation ( ) th t e

x(t)R

C y(t)

+

_

+

_

Seems intuitive basedon physical grounds

( )

0

( ) ( )t

ty t e x d Let’s prove it using the formula

0

( ) ( ) ( ) ( ) .t

y t T x t h t x d

Now, apply the delayed input ( )x t x t

0 0

( ) ( ) ( ) ( ) ( )t t

T x t h t x d h t x d

( ) ( )t

h t u x u du

u

du d

0

( ) ( )t

h t u x u du

x(u) = 0 for u < 0

y t Remark: proof works for any h(t)!

Another example: y(t) = t x(t)

[ ( )] ( )T x t t x t

( ) ( ) ( )y t t x t

Linear? Yes, easy to show.

Time invariant? No:

t

x(t)

y(t)

They are different! Compare for a particular x(t)

1 2

x(t-1)

y(t-1)

T[x(t-1)]

1

1

1

1

1

2

Time-varying system

Example: amplitude modulation

x y 0( ) cosy t x t t

+x(t) y(t)

0cos t

Used in AM radio!

0( ) cosy t x t t

Again, this is a linear system.

Modulator

Is it time invariant?

0( ) ( )cos ( )y t x t t

0( ) ( )cosT x t x t t

Only equal if 0 2k

Therefore, it is a time varying system

Notation: LTI = linear, time invariant LTV= linear, time varying

Causality and memory

• A system is causal if y(t_0) depends only on x(t), t <=t_0 (present output only depends on past and present inputs)• A system is memoryless if y(t_0) depends only on x(t_0) (present output only depends on present input).• Causal, not memoryless: we say it “has memory”.Examples: Delay system

is causal, and has memory.( ) ( ), 0y t x t

( ) ( ), 0y t x t Backward shift systemis non-causal: output anticipates the input.

Non-causal systems are not physically realizable

Recap: properties of main examples

EXAMPLE RC Circuit Modulator

Linear? Y Y

Time

Invariant?

Y N

Causal? Y Y

Memoryless? N Y

0( ) cosy t x t t y T x ( )

0

( ) ( )t

ty t e x d