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Leo Lam © 2010-2011
Signals and Systems
EE235
Today’s Cultural Education:Liszt: Von der Wiege bis zum Grabe, Symphonic
Poem No. 13
Leo Lam © 2010-2011
Arthur’s knights
Who was the largest knight at King Arthur’s round table?
Sir Cumfrence, he got his size from eating too much pie.
Leo Lam © 2010-2011
Today’s menu
• Homework 1 due today!• Lab starts tomorrow• Dirac Delta Function (cont’)• System properties
– Linearity– Time invariance– Stability– Invertibility– Causality– Memory
Leo Lam © 2010-2011
Recap: Dirac Delta function δ(t)
“a spike of signal at time 0”
0
It has height = , width = 0, and area = 1
• δ(t) Rules1. δ(t)=0 for t≠02. Area:
3. If x(t) is continuous at t0, otherwise undefined
1)( dtt
)()()()()( 0000 txtttxtttx
Leo Lam © 2010-2011
Scaling the Dirac Delta
• Proof:
• Suppose a>0
• a<0
( )at dt
d dat a dt
dt a
/
/
1 1( ) ( ) ( )
t a
t a
dat dt d
a a a
/
/
1 1( ) ( ) ( )
a
a
d dd
a a a a
Leo Lam © 2010-2011
Scaling the Dirac Delta
• Proof:
• Generalizing the last result
( )at dt
1 1( ) ( )
t
tat dt d
a a
Leo Lam © 2010-2011
• Multiplication of a function that is continuous at t0 by δ(t) gives a scaled impulse.
• Sifting Properties
• Relation with u(t)
Summary: Dirac Delta Function
0 0 0( ) ( ) ( ) ( )x t t t x t t t
0 0( ) ( ) ( )x t t t dt x t
( ) ( )t
u t d
( ) ( )d
t u tdt
Leo Lam © 2010-2011
• Evaluate
Dirac Delta – Another one
Leo Lam © 2010-2011
• Is this function periodic? If so, what is the period? (Sketch to prove your answer)
Slightly harder
k
kt
tx )24()( 2
Not periodic – delta function spreads with k2 for t>0And x(t) = 0 for t<0
Leo Lam © 2010-2011
Energy and power
• The energy of a signal
• Definition: An energy signal is any signal such that:
• Physically: this signal has finite energy
2( )E x t dt
2( )x t dt
Leo Lam © 2010-2011
Power
• The power of a signal
• Definition: A power signal is any signal such that:
• Physically: this signal has finite average power
/ 2 2
/ 2
1lim ( )
T
TTP x t dt
T
/ 2 2
/ 2
1lim ( )
T
TTx t dt
T
Leo Lam © 2010-2011
Signal power and energy
• What is the energy of u(t)
2( )E u t dt
00
21 dt t
Why?
Leo Lam © 2010-2011
Signal power and energy
• What is the power of u(t)
/ 2 2
/ 2
1lim ( )
T
TTP u t dt
T
/ 2 2
0
/ 2
0
1
2
1 1lim 1 lim
2
T T
T Tdt t
T
Leo Lam © 2010-2011
Summary: Signal energy/power
• Defined Energy and Power of signals• Defined Energy signal/Power signal
Leo Lam © 2010-2011
System
( ) ( )y t Ax t
0( ) ( )y t x t t
( ) ( )t
y t x d
0
0
( ) ( ) ( )
( )
y t x t d
x t
delay
amplifier
integrator
sifter
x(t)
x(t)
x(t)
x(t)
Leo Lam © 2010-2011
System properties
• Linearity: A System is Linear if it meets the following two criteria:
• Time-invariance: A System is Time-Invariant if it meets this criterion
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
{ ( )} ( )T x t y t 0 0{ ( )} ( )T x t t y t t If Then
“System Response to a linear combination of inputs is the linear
combination of the outputs.”
“System Response is the same no matter when you run the system.”
Leo Lam © 2010-2011
System properties
• Stability: A System is BIBO Stable if it meets this criterion
• Invertibility: A System is Invertible if it meets this criterion:
“If you know the output signal, then you know exactly what the input signal was.”
BIBO = “Bounded input, bounded output”
| ( ) |x t M t | { ( )} | | ( ) |T x t y t L t If Then
“The system doesn’t blow up if given reasonable inputs.”
You can undo the effects of the system.
{ ( )} ( ) . . { ( )} { { ( )}} ( )i i iT x t y t T s t T y t T T x t x t If
Leo Lam © 2010-2011
System properties
• Causality: A System is Causal if it meets this criterion
• Memory: A System is Memoryless if it meets this criterion
“The output depends only on the current value of the input.”“The system does not anticipate the input.”
(It does not laugh before it’s tickled!)
The output depends only on current or past values of the input.
If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a
If T{x(t)}=y(t) then y(t+a) depends only on x(t+a)
(If a system is memoryless, it is also causal.)
Leo Lam © 2010-2011
Summary:
• System properties
