Leo Lam © 2010-2011

Signals and Systems

EE235

Leo Lam © 2010-2011

Stanford

• The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake, when it became known as SPLAC.

• SPLAC? Stanford Piecewise Linear Accelerator.

Leo Lam © 2010-2011

Today’s menu

• Today: Fourier Series– 1st topic “Orthogonality”

Leo Lam © 2010-2011

Fourier Series: Introduction

4

• Fourier Series/Transform: Build signals out of complex exponentials– Periodic signals– Extend to more general signals

• Why?– Convolution: hard– Multiplication: easy (frequency domain)

• Some signals are more easily handled in frequency domain

Leo Lam © 2010-2011

Fourier Series: Why Complex Exp?

5

• Complex exponentials are nice signals– Eigenfunctions to LTI– Convolution (in t) Multiplication (in w)

• Frequency: directly related to sensory• Harmonics: Orthogonality (later today)

– Orthogonality simplifies math

Leo Lam © 2010-2011

The beauty of Fourier Series

6

• Recall:

• Write x(t) in terms of est (Fourier/Laplace Transform)

The input is a sum of weighted shifted impulses

The output is a sum of weighted shifted impulses

dthxthtxty )()()()()(

SSpecial input:

Leo Lam © 2010-2011

The beauty of Fourier Series

7

• Write x(t) in terms of est (Fourier/Laplace Transform)

• Make life easier by approximation:

• Output:

( ) js t

jj

x t c e

LTI( ) js t

jj

x t c e ( ) ( ) js t

j jj

y t c H s e

Sum of weighted eigenfunctions

Sum of scaled weighted eigenfunctions

Leo Lam © 2010-2011

Definition: Approximation error

8

• Approximating f(t) by cx(t):• Choose c so f(t) is as close to cx(t) as possible• Minimizing the error energy:

• Which gives:

( ) js t

jj

x t c e

error

2

1

2

1

( ) ( )

( ) ( )

t

t

t

t

f t x t dt

c

x t x t dt

Dot-product

Leo Lam © 2010-2011

Dot product: review

9

• Dot product between two vectors

• Vectors (and signals) are orthogonal if their dot product is zero.

f

x

Angle between the two vectors

Leo Lam © 2010-2011

Vector orthogonality

10

• Vectors (and signals) are orthogonal if their dot product is zero.

• Dot product: length of x projected onto a unit vector f

• Orthogonal: cos(q)=0• Perpendicular vectors=no projection

f

x

f

x

Key idea

Leo Lam © 2010-2011

Visualize dot product

11

• Let ax be the x component of a

• Let ay be the y component of a

• Take dot product of a and b

• In general, for d-dimensional a and b

x-axis

a

y-axis b

Leo Lam © 2010-2011

Visualize dot product

12

• 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?

Leo Lam © 2010-2011

Orthogonal signal (example)

13

• Are x(t) and y(t) orthogonal?

Yes. Orthogonal over any timespan!

Leo Lam © 2010-2011

Orthogonal signal (example 2)

14

• 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 © 2010-2011

Orthogonal signal (example 3)

15

• 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 © 2010-2011

Orthogonal signal (example 3)

16

• 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

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

17

Orthogonal signals

Any special observation here?

Leo Lam © 2010-2011

Summary

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

blocks