Transcript TA Seminar - University of Washington

Signals and Systems
EE235
Leo Lam © 2010-2013
Courtesy of Phillip
Leo Lam © 2010-2013
Today’s menu
• Fourier Series
Leo Lam © 2010-2013
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
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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)
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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)?
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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:
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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
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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:
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Fourier Series: Orthonormal basis
• xn(t) – orthonormal basis:
– Trigonometric functions (sinusoids)
– Exponentials
– Wavelets, Walsh, Bessel, Legendre etc...
Fourier Series functions
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Trigonometric Fourier Series
Note a change in index
• Set of sinusoids: fundamental frequency w0
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Trigonometric Fourier Series
• Orthogonality check:
cos( x) cos( y ) 
1
(cos( x  y )  cos( x  y ))
2
for m,n>0
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Trigonometric Fourier Series
• Similarly:
Also true: prove it to
yourself at home:
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Trigonometric Fourier Series
• Find coefficients:
The average value of f(t) over
one period (DC offset!)
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Trigonometric Fourier Series
• Similarly for:
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Compact Trigonometric Fourier Series
• Compact Trigonometric:
• Instead of having both cos and sin:
• Recall:
Expand and equate
to the LHS
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Compact Trigonometric to est
• In compact trig. form:
• Remember goal: Approx. f(t)Sum of est
• Re-writing:
• And finally:
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Compact Trigonometric to est
• Most common form Fourier Series
• Orthonormal:
,
• Coefficient relationship:
• dn is complex:
• Angle of dn:
• Angle of d-n:
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So for dn
• We want to write periodic signals as a series:
x(t ) 

de
n 
n
jnw0t
T0  2 / w0
• And dn:
1
 jnw0t
d n   f (t )e
dt
TT
• Need T and w0 , the rest is mechanical
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Harmonic Series
• Building periodic signals with complex exp.
x(t ) 

de
n 
n
jnw0t
T0  2 / w0
• Obvious case: sums of sines and cosines
1.
2.
3.
4.
Find fundamental frequency
Expand sinusoids into complex exponentials (“CE’s”)
Write CEs in terms of n times the fundamental frequency
Read off cn or dn
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Harmonic Series
• Example:
x(t )  1  cos(5t  .6) w0  5, T  2 / 5
• Expand:
Fundamental freq.
1 j (w0t .6)  j (w0t .6)
x(t )  1  (e
e
)
2
1 j (.6 ) jw0t 1  j (.6 )  jw0t
0w0t
e  e e  e e
2
2
d0  1
d1  0.5e j 0.6
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d1  0.5e j 0.6
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