Leo Lam © 2010-2012

Signals and Systems

EE235Lecture 19

Leo Lam © 2010-2012

Today’s menu

• Fourier Series

Leo Lam © 2010-2012

Visualize dot product

3

• 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

Fancy word: What does it mean physically?

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Orthogonal signal (example)

4

• Are x(t) and y(t) orthogonal?

Yes. Orthogonal over any timespan!

Leo Lam © 2010-2012

Orthogonal signal (example 2)

5

• Are a(t) and b(t) orthogonal in [0,2p]?• a(t)=cos(2t) and b(t)=cos(3t)• Do it…(2 minutes)

0)sin()5sin(5

1

2

1)cos()5cos(

2

1

))cos()(cos(2

1)cos()cos(

2

0

2

0

ttdttt

yxyxyx

Leo Lam © 2010-2012

Orthogonal signal (example 3)

6

• 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

Orthogonal signal (example 3)

7

• 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

1-1

Leo Lam © 2010-2012

x1(t)

t

x2(t)

t

x3(t)

t

T

T

T

T/2

1 2

0

( ) ( ) 0T

x t x t dt

x1(t)x2(t)

tT

2 3

0

( ) ( ) 0T

x t x t dt

x2(t)x3(t)

tT

Orthogonal signals

Any special observation here?

Leo Lam © 2010-2012

Summary

• Intro to Fourier Series/Transform• Orthogonality• Periodic signals are orthogonal=building

blocks

Leo Lam © 2010-2012

Fourier Series

10

• 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

Fourier Series: Orthogonality

11

• 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 © 2010-2012

Fourier Series: Orthogonality in signals

12

• 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

Fourier Series: Signal representation

13

• 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 © 2010-2012

Fourier Series: Signal representation

14

• 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 © 2010-2012

Fourier Series: Parseval’s Theorem

15

• 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 © 2010-2012

Fourier Series: Orthonormal basis

16

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

Fourier Series functions