Leo Lam © 2010-2012 Signals and Systems EE235. Leo Lam © 2010-2012 Pet Q: Has the biomedical...

Leo Lam © 2010-2012

Signals and Systems

EE235

Leo Lam © 2010-2012

Pet

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

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

Leo Lam © 2010-2012

Today’s menu

• System properties examples– Invertibility– Stability– Time invariance– Linearity

Leo Lam © 2010-2011

Invertibility test

• Positive test: find the inverse• For some systems, you need tools that we’ll learn later in

the quarter…

• Negative test: find an output that could be generated by two different inputs(note that these two different inputs might only differ at only one time

value) • Each input signal results in a unique output

signal, and vice versa Invertible

Leo Lam © 2010-2011

Invertibility Example

1) y(t) = 4x(t)

2) y(t) = x(t –3)

3) y(t) = x2(t)

4) y(t) = x(3t)

5) y(t) = (t + 5)x(t)

6) y(t) = cos(x(t))

invertible: Ti{y(t)}=y(t)/4

invertible: Ti{y(t)}=y(t/3)

invertible: Ti{y(t)}=y(t+3)

NOT invertible: don’t know sign of x(t)

NOT invertible: can’t find x(-5)

NOT invertible: x=0,2 π,4 π,… all give cos(x)=1

Leo Lam © 2010-2012

Stability test

• For positive proof: show analytically that– a “bounded input” signal gives a “bounded output”

signal (BIBO stability)

• For negative proof: – Find one counter example, a bounded input signal

that gives an unbounded output signal– Some good things to try: 1, u(t), cos(t), 0

1 2| ( ) | | { ( )} | | ( ) |x t B T x t y t B

Leo Lam © 2010-2012

Stability test

• Is it stable?

( ) ( )v t Ri t1 1 2| ( ) | | ( ) | | ( ) | | ( ) |i t B v t Ri t R i t RB B

Bounded input results in a bounded output STABLE!

Leo Lam © 2010-2012

Stability test

• How about this?

Stable

2( ) 10 ( )y t x t

( )x t MLet ( )x t M2 2 2( ) 10 ( ) 10 ( ) 10y t x t x t M

for all t

Leo Lam © 2010-2012

Stability test

• How about this, your turn?

Not BIBO stable

( ) 5 ( )t

y t x d

Counter example:x(t)=u(t) y(t)=5tu(t)=5r(t)

Input u(t) is bounded.Output y(t) is a ramp, which is unbounded.

Leo Lam © 2010-2012

Stability test

• How about this, your turn?

2

2

( ) ( )

( ) ( )

( ) ( )

( ) ( ) cos(2 / 3)

( ) 1/ ( )

y t x t

y t x t

y t tx t

y t x t t

y t x t

Stable

NOT Stable

NOT Stable

Stable

Stable

Leo Lam © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012


• 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 © 2010-2012


• 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!


Leo Lam © 2010-2012


• 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!


Leo Lam © 2010-2012


• 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?


Shift then scale

Leo Lam © 2010-2012


• 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 © 2010-2012


• 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

Leo Lam © 2010-2011

System properties

• Linearity: A System is Linear if it meets the following two criteria:

• Together…superposition

1 1{ ( )} ( )T x t y t 2 2{ ( )} ( )T x t y t

1 2 1 2{ ( ) ( )} { ( )} { ( )}T x t x t T x t T x t

If and

Then

{ ( )} ( )T x t y tIf { ( )} { ( )}T ax t aT x tThen

“System Response to a linear combination of inputs is the linear

combination of the outputs.”

Additivity

Scaling

1 2 1 2{ ( ) ( )} ( ) ( )T ax t bx t ay t by t

Leo Lam © 2010-2011

Linearity

• Order of addition and multiplication doesn’t matter.

=

SystemT

SystemT

Linearcombination

System 1st

Combo 1st

1 2( ), ( )x t x t

1 2( ), ( )y t y t

1 2( ) ( )ax t bx t

1 2( ) ( )ay t by t

1 2{ ( ) ( )}T ax t bx tLinear

combination

Leo Lam © 2010-2011

Linearity

• Positive proof– Prove both scaling & additivity separately– Prove them together with combined formula

• Negative proof– Show either scaling OR additivity fail

(mathematically, or with a counter example)– Show combined formula doesn’t hold

Leo Lam © 2010-2011

Linearity Proof

• Combo ProofStep 1: find yi(t)Step 2: find y_combo

Step 3: find T{x_combo}Step 4: If y_combo = T{x_combo}Linear

SystemT

SystemT

Linearcombination

System 1st

Combo 1st

1 2( ), ( )x t x t

1 2( ), ( )y t y t

1 2( ) ( )ax t bx t

1 2( ) ( )ay t by t

1 2{ ( ) ( )}T ax t bx tLinear

combination

Leo Lam © 2010-2011

Linearity Example

• Is T linear?

Tx(t) y(t)=cx(t)

1 1 2 2

1 2 1 2 1 2

1 2 1 2

( ) ( ); ( ) ( )

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

{ ( ) ( )} ( ( ) ( ))

y t cx t y t cx t

ay t by t acx t bcx t c ax t bx t

T ax t bx t c ax t bx t

Equal Linear

Leo Lam © 2010-2011

2

2

2 2 2

( ) ( ( ))

( ) ( ( ))

{ ( )} ( ( )) ( ( ))

y t x t

ay t a x t

T ax t ax t a x t

Linearity Example

• Is T linear?

Not equal non-linear

Tx(t) y(t)=(x(t))2

Leo Lam © 2010-2011

Linearity Example

• Is T linear?

( ) ( ) 5

( ) ( ( ) 5) ( ) 5

{ ( )} ( ) 5

y t x t

ay t a x t ax t a

T ax t ax t

Not equal non-linear

Tx(t) y(t)=x(t)+5

Leo Lam © 2010-2011

2

2 2

2

2

2 2

2

2

1 1

2

2

1 1

2

2

1

2

2

1 2 1

2

2

2

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ( ) )

{ ( ) ( )} ( ( ))

( )

) (

y t x t d

ay t a x t d

ax t

y t x t

d

T ax t bx t ax t bx t

d

by t b x t d

bx t

d

Linearity Example

• Is T linear? 2

2

( ) ( )y t x t d

=

Leo Lam © 2010-2011

Linearity unique case

• How about scaling with 0?

• If T{x(t)} is a linear system, then zero input must give a zero output

• A great “negative test”

( ) { ( )}

( ) { ( )} 0 if 0

{ ( )} ( ) 0 if linear

y t T x t

ay t aT x t a

T ax t ay t

Leo Lam © 2010-2011

Spotting non-linearity

• multiplying x(t) by another x()• y(t)=g[x(t)] where g() is nonlinear• piecewise definition of y(t) in terms of values

of x, e.g.

( ) ( ) 0( ) | ( ) |

( ) ( ) 0

x t x ty t x t

x t x t

(although sometimes ok)NOT Formal

Proofs!

Leo Lam © 2010-2012 Signals and Systems EE235. Leo Lam © 2010-2012 Pet Q: Has the biomedical...

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