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Leo Lam © 2010-2012
Signals and Systems
EE235Lecture 22
Leo Lam © 2010-2012
Today’s menu
• Fourier Series (periodic signals)
Leo Lam © 2010-2012
It’s here!
Solve
Given
Solve
Leo Lam © 2010-2012
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 © 2010-2012
Harmonic Series
5
• 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 © 2010-2012
Harmonic Series
6
• Graphically:
(zoomed out in time)
One period: t1 to t2
All time
Leo Lam © 2010-2012
Harmonic Series (example)
7
• 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 © 2010-2012
Harmonic Series (example)
8
• One period:
• Turn it to: • Fundamental frequency:• Coefficients:
tT
*
All basis function equally weighted and real! No phase shift!
Complex conj.
Leo Lam © 2010-2012
Harmonic Series (example)
9
• From:
• To:
• Width between “spikes” is:
tT
Fourier spectra
0
1/T
w
Time domain
Frequency domain
Leo Lam © 2010-2012
Exponential Fourier Series: formulas
10
• Analysis: Breaking signal down to building blocks:
• Synthesis: Creating signals from building blocks
Leo Lam © 2010-2011
Example: Shifted delta-train
11
• A shifted “delta-train”
• In this form:• For one period:
• Find dn:
timeT 0 T/2
*
Leo Lam © 2010-2011
Example: Shifted delta-train
12
• A shifted “delta-train”
• Find dn:
timeT 0 T/2
Complex coefficient!
Leo Lam © 2010-2011
Example: Shifted delta-train
13
• A shifted “delta-train”
• Now as a series in exponentials:
timeT 0 T/2
0
Same magnitude; add phase!
Phase of Fourier spectraw
Leo Lam © 2010-2011
Example: Shifted delta-train
14
• A shifted “delta-train”• Now as a series in exponentials:
0Phase
0
1/TMagnitude (same as non-shifted)
Leo Lam © 2010-2011
Example: Sped up delta-train
15
• Sped-up by 2, what does it do?
• Fundamental frequency doubled
• dn remains the same (why?)• For one period:
timeT/2 0 m=1 2 3
Tdtet
Td
T
Ttjn
n
1)(
14
4
0
Great news: we can be lazy!
Leo Lam © 2010-2011
Lazy ways: re-using Fourier Series
16
• Standard notation: “ ” means “a given periodic signal has Fourier series coefficients ”
• Given , find where is a new signal based on
• Addition, time-scaling, shift, reversal etc.• Direct correlation: Look up table!• Textbook Ch. 3.1 & everywhere online:
http://saturn.ece.ndsu.nodak.edu/ecewiki/images/3/3d/Ece343_Fourier_series.pdf
kdtx )(
kd)(tx
kdtx )( kdtx )( )(tx)(tx
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