Leo Lam © 2010-2013 Signals and Systems EE235 Leo Lam.
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Transcript of Leo Lam © 2010-2013 Signals and Systems EE235 Leo Lam.
Leo Lam © 2010-2013
Signals and Systems
EE235Leo Lam
Leo Lam © 2010-2013
Today’s menu
• From Wednesday: Manipulating signals• How was Lab 1?• To Do: Really memorize u(t), r(t), p(t)• Today: More of that!• Even and odd signals• Dirac Delta function
Leo Lam © 2010-2013
People types
There are 10 types of people in the world:
Those who know binary and those who don’t.
Leo Lam © 2010-2013
Playing with time
t
1
What does look like?
2
1
-2
Time reverse of speech:Also a form of time scaling, only with a negative number
Leo Lam © 2010-2013
Playing with time
t
1
2
Describe z(t) in terms of w(t)
1
-2 1 3 t
Leo Lam © 2010-2013
Playing with time
time reverse it: x(t) = w(-t)delay it by 3: z(t) = x(t-3)so z(t) = w(-(t-3)) = w(-t + 3)
t1
2
1
-2 1 3
x(t)
you replaced the t in x(t) by t-3. so replace the t in w(t)by t-3: x(t-3) = w(-(t-3))
Leo Lam © 2010-2013
Playing with time
z(t) = w(-t + 3)
t1
2
1
-2 1 3
x(t)
Doublecheck:w(t) starts at 0so -t+3 = 0 givest= 3, this is the start (tip) of thetriangle z(t).
w(t) ends at 2So -t+3=2 givest=1, z(t) endsthere
Leo Lam © 2010-2013
Summary:
• Arithmetic: Add, subtract, multiple• Time: delay, scaling, shift, mirror/reverse• And combination of those
Leo Lam © 2010-2013
Even and odd signals
An even signal is such that:
tSymmetrical across
the t=0 axis
tAsymmetrical across
the t=0 axis
An odd signal is such that:
( ) ( )e ex t x t
( ) ( )o ox t x t 0
( ) 2 ( )L L
e e
L
x t dt x t dt
( ) 0L
o
L
x t dt
Leo Lam © 2010-2013
Even and odd signals
1 1( ) ( ( ) ( )) ( ( ) ( ))
2 2x t x t x t x t x t
Every signal sum of an odd and even signal.
( ) ( )e ex t x t
Even signal is such that:
The even and odd parts of a signal
Odd signal is such that:
( ) ( )o ox t x t
1( ) ( ( ) ( ))
21
( ) ( ( ) ( ))2
e
o
x t x t x t
x t x t x t
Leo Lam © 2010-2013
Even and odd signals
1( ) ( ( ) ( ))
21
( ) ( ( ) ( ))2
e
o
x t x t x t
x t x t x t
Euler’s relation:
j te What are the even and odd parts of
)sin()(2
1
)cos()(2
1
)sin()cos(
tjee
tee
tjte
tjtj
tjtj
tj
Even part
Odd part
Leo Lam © 2010-2013
Summary:
• Even and odd signals• Breakdown of any signals to the even and odd
components
Leo Lam © 2010-2013
Delta function δ(t)
“a spike of signal at time 0”
0
The Dirac delta is: • The unit impulse or impulse• Very useful• Not a function, but a “generalized function”)
Leo Lam © 2010-2013
Delta function δ(t)
0lim
Each rectangle has area 1, shrinking width, growing height ---limit is (t)
1
1
Leo Lam © 2010-2013
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
0 t0
Shifted to time instant t0:
Leo Lam © 2010-2013
Dirac Delta example
• Evaluate
10
2
)( dtt
= 0. Because δ(t)=0 for all t≠0
Leo Lam © 2010-2013
Dirac Delta – Your turn
• Evaluate
= 1. Why?/ 4
sin( / 2) ( / 2)t dt
Change of variable: / 2t ( 1 )d
d dtdt
/2
/4 /4sin( / 2) sin( / 2)( / 2) ( )t dt d
1
1
Leo Lam © 2010-2013
Dirac Delta – Another one
• Evaluate
Leo Lam © 2010-2013
• 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
