leo lam © 2010-2012 signals and systems ee235. leo lam © 2010-2011 arthur’s knights who was the...
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Leo Lam © 2010-2012
Signals and Systems
EE235
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
• To do– Lab starts soon!
• Dirac Delta Function (cont’)• System properties
– Linearity– Time invariance– Stability– Invertibility– Causality– Memory
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)