leo lam © 2010-2011 signals and systems ee235. leo lam © 2010-2011 speed of light
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Leo Lam © 2010-2011
Signals and SystemsEE235
Leo Lam © 2010-2011
Speed of light
Leo Lam © 2010-2011
Today’s menu
• Fourier Series (Exponential form)• Fourier Transform!
Leo Lam © 2010-2011
Fourier Series: Circuit Application
4
• 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 © 2010-2011
Fourier Series: Circuit Application
5
• Finding H(s) for the LTI system:
• est is an eigenfunction, so• Therefore:• So:
)()( sHety stststst esHesHse )()(3
13
1)(
s
sHShows how much an exponential gets amplified at different frequency s
Leo Lam © 2010-2011
Fourier Series: Circuit Application
6
• Rectified sinusoids
• Now we know:
• LTI system: • Transfer function: • To frequency:
+-sin(t)
fullwaverectifier
y(t)f(t)
Leo Lam © 2010-2011
Fourier Series: Circuit Application
7
• Rectified sinusoids
• Now we know:
• LTI system: • Transfer function:• System response:
+-sin(t)
fullwaverectifier
y(t)f(t)
Leo Lam © 2010-2011
Fourier Series: Dirichlet Conditon
8
• Condition for periodic signal f(t) to exist has exponential series:
• Weak Dirichlet:
• Strong Dirichlet (converging series): f(t) must have finite maxima, minima, and discontinuities in a period
• All physical periodic signals converge
Leo Lam © 2010-2011
End of Fourier Series
9
• We have accomplished:– Introduced signal orthogonality– Fourier Series derivation– Approx. periodic signals:– Fourier Series Properties
• Next: Fourier Transform
Leo Lam © 2010-2011
Fourier Transform: Introduction
10
• Fourier Series: Periodic Signal• Fourier Transform: extends to all signals• Recall time-scaling:
Leo Lam © 2010-2011
Fourier Transform:
11
• Recall time-scaling:
0
Fourier Spectrafor T,
Fourier Spectrafor 2T,
Leo Lam © 2010-2011
Fourier Transform:
12
• Non-periodic signal: infinite period T
0
Fourier Spectrafor T,
Fourier Spectrafor 2T,
0 0" / "
T
Leo Lam © 2010-2011
Fourier Transform:
13
• Fourier Formulas:• For any arbitrary practical signal
• And its “coefficients” (Fourier Transform):
• F(w) is complex• Rigorous math derivation in Ch. 4 (not required
reading, but recommended)
Time domain toFrequency domain
Leo Lam © 2010-2011
Fourier Transform:
14
• Fourier Formulas compared:
Fourier transform coefficients:
Fourier transform(arbitrary signals)
Fourier series (Periodic signals):
Fourier series coefficients:
and
0( ) jn tn
n
f t d e
Leo Lam © 2010-2011
Fourier Transform (example):
15
• Find the Fourier Transform of
• What does it look like?
)()( tuetf at
If a <0, blows up
magnitudevaries with
phasevaries with
Leo Lam © 2010-2011
Fourier Transform (example):
16
• Fourier Transform of
• Real-time signals magnitude: even phase: odd
)()( tuetf at
-100 -80 -60 -40 -20 0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-100 -80 -60 -40 -20 0 20 40 60 80 100-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
magnitude phase
Leo Lam © 2010-2011
Fourier Transform (Symmetry):
17
• Real-time signals magnitude: even – why?
-100 -80 -60 -40 -20 0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
magnitude
*
*
*
*
( ) real-valued ( ) ( )
( ) ( )
(
(
))
)
(
j t j t
j t
f t f t f t
f t e dt f t e dt
f t e dt
F
F
( ) * ( )
*
*
| ( ) | | ( ) | | (
( ) ( ) ( ) ( )
( (
|
( )
)
) )
j j
F F F
F r e F r e
F F F
Even magnitude
Odd phase
Useful for checking answers
Leo Lam © 2010-2011
Fourier Transform/Series (Symmetry):
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• Works for Fourier Series, too!
Fourier transform(arbitrary practical signal)
Fourier series (periodic functions)
0( ) jn tn
n
f t d e
Fourier coefficients*
n nd d Fourier transform coefficients
*( ) ( )F F
magnitude: even & phase: odd
Leo Lam © 2010-2011
Fourier Transform (example):
19
• Fourier Transform of
• F(w) is purely real
| |( ) a tf t e
dteeF tjta
)(
dteedteeF tjattjat
0
0)(
22
2
11
a
a
jaja
F(w) for a=1
Leo Lam © 2010-2011
Summary
• Fourier Transform intro• Inverse etc.