Leo Lam © 2010-2011 Signals and Systems EE235 Lecture 25.
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Transcript of Leo Lam © 2010-2011 Signals and Systems EE235 Lecture 25.
Leo Lam © 2010-2011
Signals and Systems
EE235Lecture 25
Leo Lam © 2010-2011
Today’s menu
• Fourier Series (periodic signals)
Leo Lam © 2010-2011
It’s here!
Solve
Given
Solve
Leo Lam © 2010-2011
Reminder from last week
4
• We want to write periodic signals as a series:
• And dn:
• Need T and w0 , the rest is mechanical
00 0( ) 2 /jn t
nn
x t d e T
T
tjnn dtetfT
d 0)(1
Leo Lam © 2010-2011
Harmonic Series
5
• Example:
• Fundamental frequency:– w0=GCD(1,2,5)=1 or
• Re-writing:
( ) cos( ) 3sin(2 ) cos(5 .6)x t t t t
(2 , , 2 / 5) 2T LCM
0 0
0 0
0 0
2 2
(5 .6) (5 .6)
1( ) ( )
23( )
2
1( )2
j t j t
j t j t
j t j t
x t e e
e ej
e e
1 1 0.5d d
(.6) (0.6) *5 5 5
1 1;
2 2j jd e d e d
*2 2 2
3 3;
2 2d d d
j j
dn = 0 for all other n
Leo Lam © 2010-2011
Harmonic Series
6
• Example (your turn):
• Write it in an exponential series:
• d0=-5, d2=d-2=1, d3=1/2j, d-3=-1/2j, d4=1
4( ) 5 2cos(2 ) sin(3 ) j tx t t t e
(0)( ) (2)( ) ( 2)( )
(3)( ) ( 3)( ) (4)( )
1( ) 5 2
2
1(1)
2
j t j t j t
j t j t j t
x t e e e
e e ej
0
Leo Lam © 2010-2011
Harmonic Series
7
• Graphically:
(zoomed out in time)
One period: t1 to t2
All time
Leo Lam © 2010-2011
Harmonic Series (example)
8
• Example with d(t) (a “delta train”):
• Write it in an exponential series:
• Signal is periodic: only need to do one period• The rest just repeats in time
t
T
Leo Lam © 2010-2011
Harmonic Series (example)
9
• One period:
• Turn it to: • Fundamental frequency:• Coefficients:
tT
*
All basis function equally weighted and real! No phase shift!
Complex conj.
Leo Lam © 2010-2011
Harmonic Series (example)
10
• From:
• To:
• Width between “spikes” is:
tT
Fourier spectra
0
1/T
w
Time domain
Frequency domain
Leo Lam © 2010-2011
Summary
• Fourier series• Examples