Post on 23-Dec-2015
Leo Lam © 2010-2013
Signals and Systems
EE235
Leo Lam © 2010-2013
So stable
Leo Lam © 2010-2013
Today’s menu
• Chocolates and cookies• Fourier Series (periodic signals)
Leo Lam © 2010-2013
Exponential Fourier Series: formulas
4
• Analysis: Breaking signal down to building blocks:
• Synthesis: Creating signals from building blocks
Leo Lam © 2010-2013
Harmonic Series (example)
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• 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 © 2010-2013
Harmonic Series (example)
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• One period:
• Turn it to: • Fundamental frequency:• Coefficients:
tT
*
All basis function equally weighted and real! No phase shift!
Complex conj.
Leo Lam © 2010-2013
Harmonic Series (example)
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• From:
• To:
• Width between “spikes” is:
tT
Fourier spectra
0
1/T
w
Time domain
Frequency domain
Leo Lam © 2010-2013
Example: Shifted delta-train
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• A shifted “delta-train”
• In this form:• For one period:
• Find dn:
timeT 0 T/2
*
Leo Lam © 2010-2013
Example: Shifted delta-train
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• A shifted “delta-train”
• Find dn:
timeT 0 T/2
Complex coefficient!
Leo Lam © 2010-2013
Example: Shifted delta-train
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• 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 © 2010-2013
Example: Shifted delta-train
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• A shifted “delta-train”• Now as a series in exponentials:
0Phase
0
1/TMagnitude (same as non-shifted)
Leo Lam © 2010-2013
Example: Sped up delta-train
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• 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!
The new T.
Leo Lam © 2010-2013
Lazy ways: re-using Fourier Series
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• 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
Leo Lam © 2010-2013
Graphical: Time scaling: Fourier Series
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• Example: Time scaling up (graphical)
• New signal based on f(t):
• Using the Synthesis equation:
Fourier spectra
0
tjn
nedtg 02)(
Twice as far apart as f(t)’s
Leo Lam © 2010-2013
Graphical: Time scaling: Fourier Series
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• Spectra change (time-scaling up):• f(t)
• g(t)=f(2t)
• Does it make intuitive sense?
0
1
0
1
Leo Lam © 2010-2013
Graphical: Time scaling: Fourier Series
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• Spectra change (time scaling down):• f(t)
• g(t)=f(t/2) 0
1
0
1
Leo Lam © 2010-2013
Fourier Series Table
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0
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
x t
y t ax t b
y t x at
y t x t
y t x t t
0 0
0 0
0 0
0,
,
k
y yk k
y yk k
yk k
jk tyk k
d
d ad k d ad b
d d a
d d
d d e
Added constant only affects DC term
Linear ops
Time scaleSame dk, scale w0reverse
reverse
Shift in time –t0 Add linear phase term –jkw0t0
• Fourier Series Properties:
Leo Lam © 2010-2013
Fourier Series: Fun examples
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• Rectified sinusoids
• Find its exponential Fourier Series:
t0
f(t) =|sin(t)|
Expand as exp., combine, integrate
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n
tjnen
tf 0
)41(
2)(
2
Leo Lam © 2010-2013
Fourier Series: Circuit Application
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• Rectified sinusoids
• Now we know:
• Circuit is an LTI system: • Find y(t)• Remember:
+-sin(t)
fullwaverectifier
y(t)f(t)
Where did this come from?
S
Find H(s)!
Leo Lam © 2010-2013
Fourier Series: Circuit Application
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• Finding H(s) for the LTI system:
• est is an eigenfunction, so• Therefore:• So:
)()( sHety stststst esHesHse )()(3
13
1)(
ssH
Shows how much an exponential gets amplified at different frequency s
Leo Lam © 2010-2013
Fourier Series: Circuit Application
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• Rectified sinusoids
• Now we know:
• LTI system: • Transfer function: • To frequency:
+-sin(t)
fullwaverectifier
y(t)f(t)
Leo Lam © 2010-2013
Fourier Series: Circuit Application
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• Rectified sinusoids
• Now we know:
• LTI system: • Transfer function:• System response:
+-sin(t)
fullwaverectifier
y(t)f(t)
Leo Lam © 2010-2013
Summary
• Fourier Series circuit example