Leo Lam © 2010-2011

Signals and Systems

EE235

Leo Lam © 2010-2011

Pet

Q: Has the biomedical imaging engineer done anything useful lately?

A: No, he's mostly been working on PET projects.

Leo Lam © 2010-2011

Today’s menu

• Homework due now!• Tomorrow: The Hossein Lecture• Friday: Lecture will be online for download• System properties examples

– Stability– Time invariance– Linearity

Leo Lam © 2010-2011

System properties

• Time-invariance: A System is Time-Invariant if it meets this criterion

“System Response is the same no matter when you run the system.”

Leo Lam © 2010-2011

Time invariance

• The system behaves the same no matter when you use it

• Input is delayed by t0 seconds, output is the same but delayed t0 seconds

{ ( )} ( )T x t y t 0 0{ ( )} ( )T x t t y t t If then

SystemT

Delayt0

SystemT

Delayt0

x(t)

x(t-t0)

y(t)y(t-t0)

T[x(t-t0)]

System 1st

Delay 1st

=

Leo Lam © 2010-2011

Time invariance example

• T{x(t)}=2x(t)

x(t) y(t)= 2x(t) y(t-t0)T Delay

x(t-t0)2x(t-t0)

Delay T

Identical time invariant!

Leo Lam © 2010-2011

Time invariance test

• Test steps:1. Find y(t)2. Find y(t-t0)

3. Find T{x(t-t0)}

4. Compare!• IIf y(t-t0) = T{x(t-t0)} Time invariant!

Leo Lam © 2010-2011

Time invariance example

• T(x(t)) = x2(t)1. y(t) = x2(t)2. y(t-t0) =x2(t-t0)

3. T(x(t-t0)) = x2(t-t0)

4. y(t-t0) = T(x(t-t0))

• Time invariant!

KEY: In step 2 you replace t by t-t0.In step 3 you replace x(t) by x(t-t0).

Leo Lam © 2010-2011

Time invariance example

• Your turn!• T{(x(t)} = t x(t)

1. y(t) = t*x(t)2. y(t-t0) =(t-t0) x(t-t0)

3. T(x(t-t0)) = t x(t-t0)

4. y(t-t0)) != T(x(t-t0))

• Not time invariant!

KEY: In step 2 you replace t by t-t0.In step 3 you replace x(t) by x(t-t0).

Leo Lam © 2010-2011

Time invariance example

• Still you…• T(x(t)) = 3x(t - 5)

1. y(t) = 3x(t-5)2. y(t – t0) = 3x(t-t0-5)

3. T(x(t – t0)) = 3x(t-t0-5)

4. y(t-t0)) = T(x(t-t0))

• Time invariant!

KEY: In step 2 you replace t by t-t0.In step 3 you replace x(t) by x(t-t0).

Leo Lam © 2010-2011

Time invariance example

• Still you…• T(x(t)) = x(5t)

1. y(t) = x(5t)2. y(t – 3) = x(5(t-3)) = x(5t – 15)3. T(x(t-3)) = x(5t- 3)4. Oops…

• Not time invariant!• Does it make sense?

KEY: In step 2 you replace t by t-t0.In step 3 you replace x(t) by x(t-t0).

Shift then scale

Leo Lam © 2010-2011

Time invariance example

• Graphically: T(x(t)) = x(5t)1. y(t) = x(5t)2. y(t – 3) = x(5(t-3)) = x(5t – 15)3. T(x(t-3)) = x(5t- 3)

t0

system inputx(t)

5

t0

system outputy(t) = x(5t)

1

t0 3 4

shifted system outputy(t-3) = x(5(t-3))

t0 3 8

shifted system inputx(t-3)

0.6 1.6 t

system outputfor shifted system inputT(x(t-3)) = x(5t-3)

Leo Lam © 2010-2011

Time invariance example

• Integral

1. First:2. Second:

3. Third:

4. Lastly:

• Time invariant!

KEY: In step 2 you replace t by t-t0.In step 3 you replace x(t) by x(t-t0).[ ( )] ( )

t

T x t x d

( ) ( )

t

y t x d

0

0( ) ( )t t

y t t x d

0

0 0 0[ ( )] ( ) ( )t tt

T x t t x t d x v dv v t

0 0

( ) ( )t t t t

x v dv x d