Signals and Systems
EE235
Lecture 19
Leo Lam © 2010-2012
Today’s menu
• Fourier Series
Leo Lam © 2010-2012
Visualize dot product
• In general, for d-dimensional a and b
• For signals f(t) and x(t)
• For signals f(t) and x(t) to be orthogonal from
t1 to t2
• For complex signals
Leo Lam © 2010-2012
Fancy word: What does it
mean physically?
3
Orthogonal signal (example)
• Are x(t) and y(t) orthogonal?
Yes. Orthogonal over any timespan!
Leo Lam © 2010-2012
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Orthogonal signal (example 2)
• Are a(t) and b(t) orthogonal in [0,2p]?
• a(t)=cos(2t) and b(t)=cos(3t)
• Do it…(2 minutes)
1
cos(x) cos(y )  (cos(x  y )  cos(x  y))
2
2p
1 2p
1 1

  cos(5t )  cos(t )dt   sin(5t )  sin(t )  0
2 0
2 5
0
Leo Lam © 2010-2012
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Orthogonal signal (example 3)
•
•
•
•
x(t) is some even function
y(t) is some odd function
Show a(t) and b(t) are orthogonal in [-1,1]?
Need to show:
• Equivalently:
• We know the property of odd function:
• And then?
Leo Lam © 2010-2012
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Orthogonal signal (example 3)
• x(t) is some even function
• y(t) is some odd function
• Show x(t) and y(t) are orthogonal in [-1,1]?
• Change in variable v=-t
• Then flip and negate:
Same, QED
-1
Leo Lam © 2010-2012
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7
Orthogonal signals
x1(t)x2(t)
x1(t)
T
t
 x (t ) x (t )dt  0
1
t
2
0
T/2
x2(t)x3(t)
T
x3(t)
T
t
T
 x (t ) x (t )dt  0
2
0
Leo Lam © 2010-2012
t
Any special observation
T here?
x2(t)
T
T
3
t
Summary
• Intro to Fourier Series/Transform
• Orthogonality
• Periodic signals are orthogonal=building
blocks
Leo Lam © 2010-2012
Fourier Series
• Fourier Series/Transform: Build signals out of
complex exponentials
• Established “orthogonality”
• x(t) to X(jw)
• Oppenheim Ch. 3.1-3.5
• Schaum’s Ch. 5
Leo Lam © 2010-2012
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Fourier Series: Orthogonality
• Vectors as a sum of orthogonal unit vectors
• Signals as a sum of orthogonal unit signals
y
a = 2x + y
x
• How much of x and of y to add?
of x
of y
a
• x and y are orthonormal (orthogonal and
normalized with unit of 1)
Leo Lam © 2010-2012
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Fourier Series: Orthogonality in signals
• Signals as a sum of orthogonal unit signals
• For a signal f(t) from t1 to t2
• Orthonormal set of signals x1(t), x2(t), x3(t) …
xN(t)
of
of
of
Does it equal f(t)?
Leo Lam © 2010-2012
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Fourier Series: Signal representation
• For a signal f(t) from t1 to t2
• Orthonormal set of signals x1(t), x2(t), x3(t) …
xN(t)
of
of
of
• Let
• Error:
Leo Lam © 2010-2012
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Fourier Series: Signal representation
• For a signal f(t) from t1 to t2
• Error:
• Let {xn} be a complete orthonormal basis
• Then:
of
of
Does it equal f(t)?
of
Kind of!
• Summation series is an approximation
• Depends on the completeness of basis
Leo Lam © 2010-2012
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Fourier Series: Parseval’s Theorem
• Compare to Pythagoras Theorem
c
Energy of
vector
Energy of
b
a
• Parseval’s
Theorem
each of
orthogonal
basis vectors
All xn are orthonormal
vectors with energy = 1
• Generally:
Leo Lam © 2010-2012
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Fourier Series: Orthonormal basis
• xn(t) – orthonormal basis:
– Trigonometric functions (sinusoids)
– Exponentials
– Wavelets, Walsh, Bessel, Legendre etc...
Fourier Series functions
Leo Lam © 2010-2012
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