leo lam © 2010-2012 signals and systems ee235. leo lam © 2010-2012 pet q: has the biomedical...
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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
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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
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!