Leo Lam © 2010-2013

Signals and Systems

EE235

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Courtesy of Phillip

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Today’s menu

• Fourier Series

Leo Lam © 2010-2013

Fourier Series

4

• 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 © 2010-2013

Fourier Series: Orthogonality

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• Vectors as a sum of orthogonal unit vectors• Signals as a sum of orthogonal unit signals

• How much of x and of y to add?

• x and y are orthonormal (orthogonal and normalized with unit of 1)

x

y a = 2x + y

of x

of ya

Leo Lam © 2010-2013

Fourier Series: Orthogonality in signals

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• 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 © 2010-2013

Fourier Series: Signal representation

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• For a signal f(t) from t1 to t2

• Orthonormal set of signals x1(t), x2(t), x3(t) … xN(t)

• Let

• Error:

of

of

of

Leo Lam © 2010-2013

Fourier Series: Signal representation

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• For a signal f(t) from t1 to t2

• Error:

• Let {xn} be a complete orthonormal basis • Then:

• Summation series is an approximation• Depends on the completeness of basis

Does it equal f(t)?

of

of

of Kind of!

Leo Lam © 2010-2013

Fourier Series: Parseval’s Theorem

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• Compare to Pythagoras Theorem

• Parseval’s Theorem

• Generally:

c

a

b

Energy of vector Energy of

each oforthogonalbasis vectors

All xn are orthonormal vectors with energy = 1

Leo Lam © 2010-2013

Fourier Series: Orthonormal basis

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• xn(t) – orthonormal basis:– Trigonometric functions (sinusoids)– Exponentials– Wavelets, Walsh, Bessel, Legendre etc...

Fourier Series functions

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Trigonometric Fourier Series

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• Set of sinusoids: fundamental frequency w0

Note a change in index

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Trigonometric Fourier Series

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• Orthogonality check:

for m,n>0

))cos()(cos(2

1)cos()cos( yxyxyx

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Trigonometric Fourier Series

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• Similarly:

Also true: prove it to yourself at home:

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Trigonometric Fourier Series

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• Find coefficients:

The average value of f(t) over one period (DC offset!)

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Trigonometric Fourier Series

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• Similarly for:

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Compact Trigonometric Fourier Series

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• Compact Trigonometric:• Instead of having both cos and sin:• Recall:

Expand and equate to the LHS

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Compact Trigonometric to est

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• In compact trig. form:

• Remember goal: Approx. f(t)Sum of est

• Re-writing:

• And finally:

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Compact Trigonometric to est

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• 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

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• 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 © 2010-2013

Harmonic Series

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• Building periodic signals with complex exp.

• Obvious case: sums of sines and cosines1. Find fundamental frequency2. Expand sinusoids into complex exponentials (“CE’s”)3. Write CEs in terms of n times the fundamental frequency4. Read off cn or dn

00 0( ) 2 /jn t

nn

x t d e T

Leo Lam © 2010-2013

Harmonic Series

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• Example:

• Expand:

0( ) 1 cos(5 .6) 5, 2 / 5x t t T

Fundamental freq.

0 0

(.6) (.6)0 0 0

( .6) ( .6)

0

1( ) 1 ( )

21 1

2 2

j j

j t j t

t j t j t

x t e e

e e e e e

0 1d 0.61 0.5 jd e

0.61 0.5 jd e