Leo Lam © 2010-2012

Signals and Systems

EE235Lecture 21

Leo Lam © 2010-2012

Today’s menu

• Fourier Series (periodic signals)

Leo Lam © 2010-2012

It’s here!

Solve

Given

Solve

Leo Lam © 2010-2012

Reminder from last week

4

• 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 © 2010-2012

Harmonic Series

5

• Example:

• Fundamental frequency:– w0=GCF(1,2,5)=1 or

• Re-writing:

( ) cos( ) 3sin(2 ) cos(5 .6)x t t t t

(2 , , 2 / 5) 2T LCM

0 0

0 0

0 0

2 2

(5 .6) (5 .6)

1( ) ( )

23( )

2

1( )2

j t j t

j t j t

j t j t

x t e e

e ej

e e

1 1 0.5d d

(.6) (0.6) *5 5 5

1 1;

2 2j jd e d e d

*2 2 2

3 3;

2 2d d d

j j

dn = 0 for all other n

Leo Lam © 2010-2012

Harmonic Series

6

• Example (your turn):

• Write it in an exponential series:

• d0=-5, d2=d-2=1, d3=1/2j, d-3=-1/2j, d4=1

4( ) 5 2cos(2 ) sin(3 ) j tx t t t e

(0)( ) (2)( ) ( 2)( )

(3)( ) ( 3)( ) (4)( )

1( ) 5 2

2

1(1)

2

j t j t j t

j t j t j t

x t e e e

e e ej

0

Leo Lam © 2010-2012

Harmonic Series

7

• Graphically:

(zoomed out in time)

One period: t1 to t2

All time

Leo Lam © 2010-2012

Harmonic Series (example)

8

• Example with d(t) (a “delta train”):

• Write it in an exponential series:

• Signal is periodic: only need to do one period• The rest just repeats in time

t

T

Leo Lam © 2010-2012

Harmonic Series (example)

9

• One period:

• Turn it to: • Fundamental frequency:• Coefficients:

tT

*

All basis function equally weighted and real! No phase shift!

Complex conj.

Leo Lam © 2010-2012

Harmonic Series (example)

10

• From:

• To:

• Width between “spikes” is:

tT

Fourier spectra

0

1/T

w

Time domain

Frequency domain

Leo Lam © 2010-2012

Exponential Fourier Series: formulas

11

• Analysis: Breaking signal down to building blocks:

• Synthesis: Creating signals from building blocks

Leo Lam © 2010-2011

Example: Shifted delta-train

12

• A shifted “delta-train”

• In this form:• For one period:

• Find dn:

timeT 0 T/2

*