Lecture 11: Introduction to Fourier Series Sections 2.2.3, 2.3.

Lecture 11:Introduction to Fourier Series

Sections 2.2.3, 2.3

Trigonometric Fourier Series

•Outline▫Introduction▫Visualization▫Theoretical Concepts▫Qualitative Analysis▫Example▫Class Exercise

Introduction• What is Fourier Series?

▫ Representation of a periodic function with a weighted, infinite sum of sinusoids.

• Why Fourier Series?▫ Any arbitrary periodic signal, can be approximated by

using some of the computed weights

▫ These weights are generally easier to manipulate and analyze than the original signal

Periodic Function

• What is a periodic Function?▫ A function which remains unchanged when time-shifted

by one period f(t) = f(t + To) for all values of t

• What is To

To To

Properties of a periodic function 1

• A periodic function must be everlasting▫ From –∞ to ∞

• Why?

• Periodic or Aperiodic?

Properties of a periodic function

• You only need one period of the signal to generate the entire signal▫ Why?

• A periodic signal cam be expressed as a sum of sinusoids of frequency F0 = 1/T0 and all its harmonics

VisualizationCan you represent this simple function using sinusoids?

Single sinusoid representation

Visualization

To obtain the exact signal, an infinite number of sinusoids are required

)cos( 01 ta amplitude Fundamental

frequency

)3cos( 03 ta

New amplitude 2nd Harmonic

)5cos( 05 ta amplitude 4th Harmonic

Theoretical Concepts                                                                   

(6)

,...3,2,1....)sin()(2

,...3,2,1....)cos()(2

2

)sin()cos()(

01

1

01

1

00

00

00

01

01

0

ndttntfT

b

ndttntfT

a

T

tnbtnaatf

Tt

t

n

Tt

t

n

nn

nn

PeriodCosine terms

Sine terms

Theoretical Concepts                                                                   

(6)

n

nn

nnn

nn

n

a

b

bac

ac

tncctf

1

22

00

01

0

tan

)cos()(

DC Offset

What is the difference between these two functions?

A

0 1 2-1-2

-A

A

0 1 2-1-2

Average Value = 0

Average Value ?

DC OffsetIf the function has a DC value:

01

1

)(1

)sin()cos(2

1)(

00

01

01

0

Tt

t

nn

nn

dttfT

a

tnbtnaatf

Qualitative Analysis• Is it possible to have an idea of what your solution

should be before actually computing it?

For Sure

Properties – DC Value• If the function has no DC value, then a0 = ?

-1 1 2

-A

A

DC?

A

0 1 2-1-2

DC?

Properties – Symmetry

A

A

0 π/2 π 3π/2

f(-t) = -f(t)

• Even function

• Odd function

0

-A

A

π/2 π 3π/2

f(-t) = f(t)

Properties – Symmetry• Note that the integral over a period of an odd function is?

,...3,2,1....)sin()(2 01

1

00

ndttntfT

bTt

tn

If f(t) is even:

EvenOddX = Odd

,...3,2,1....)cos()(2 01

1

00

ndttntfT

aTt

tn

EvenEvenX = Even

Properties – Symmetry• Note that the integral over a period of an odd function is

zero.

,...3,2,1....)cos()(2 01

1

00

ndttntfT

aTt

tn

If f(t) is odd:

OddEvenX = Odd

,...3,2,1....)sin()(2 01

1

00

ndttntfT

bTt

tn

OddOddX = Even

Properties – Symmetry•If the function has:

▫even symmetry: only the cosine and associated coefficients exist

▫odd symmetry: only the sine and associated coefficients exist

▫even and odd: both terms exist

Properties – Symmetry

• If the function is half-wave symmetric, then only odd harmonics exist

Half wave symmetry: f(t-T0/2) = -f(t)

-1 1 2

-A

A

Properties – Discontinuities•If the function has

▫ Discontinuities: the coefficients will be proportional to 1/n

▫ No discontinuities: the coefficients will be proportional to 1/n2

• Rationale:

-1 1 2

-A

A

A

0 1 2-1-2

Which is closer to a sinusoid?

Which function has discontinuities?

Example

• Without any calculations, predict the general form of the Fourier series of:

-1 1 2

-A

A

DC? No, a0 = 0;

Symmetry? Even, bn = 0;

Half wave symmetry?

Yes, only odd harmonics

Discontinuities?No, falls of as

1/n2Prediction an 1/n2 for n = 1, 3, 5, …;

Example• Now perform the calculation

2/

00

00

0

001

1

)cos()(4

)cos()(2

TTt

tn dttntf

Tdttntf

Ta

;20 T 2

20

...5,3,1...8

22 n

n

Aan

)cos(14

)cos(2222

1

0

nn

AdttnAtan

zero for n even

Example• Now compare your calculated answer with your

predicted form

DC? No, a0 = 0;

Symmetry?Even, bn = 0;

Half wave symmetry?

Yes, only odd harmonics

Discontinuities?No, falls of as

1/n2

Class exercise

• Discuss the general form of the solution of the function below and write it down

• Compute the Fourier series representation of the function

• With your partners, compare your calculations with your predictions and comment on your solution

A

0 1 2-1-2

Spectral Lines

,...3,2,1....)sin()(2

,...3,2,1....)cos()(2

2

)sin()cos()(

01

1

01

1

00

00

00

01

01

ndttntfT

b

ndttntfT

a

T

tnbtnatf

Tt

tn

Tt

tn

nn

nn

n

nn

nnn

nn

n

a

b

bac

ac

tncctf

1

22

00

01

0

tan

)cos()(

Spectral Lines

•Gives the frequency composition of the function▫Amplitude, phase of sinusoidal components

•Could provide information not found in time signal▫E.g. Pitch, noise components

•May help distinguish between signals ▫E.g speech/speaker recognition

Spectral Lines Example

-3 -2 -1 0 1 2 30.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/pi

y(t

)

exp(-t/2)

QUESTIONS --DC Yes____ ao = ? No_____ ao = 0Symmetry

Even____ an = ? bn = 0Odd____ an = 0 bn = ?Nether even nor odd ____ an = ? bn = ?

Halfwave symmetryYes_____ only odd harmonicsNo______ all harmonics

DiscontinuitiesYes_____ proportional to1/nNo______ proportional to1/n2

Note ? means find that variable.Comment on the general form of the Fourier Series coefficients [an and/or bn.]

X

X

X

X

Spectral Lines Example

,3,2,1....0

)2sin(2/

,...3,2,1....01

1

)0sin()(

0

2

,3,2,1....0

)2cos(2/

,3,2,1....01

1

)0cos()(

0

2

22

0

20

5042.02/01

1

)(

00

0

11

ndttnt

e

n

Tt

tdttntf

Tnb

ndtntt

e

n

Tt

tdttntf

Tna

T

dtt

e

Tt

tdttf

Ta

-3 -2 -1 0 1 2 30.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/pi

y(t

)

exp(-t/2)

Spectral Lines Example

,....3,2,1.....1

216

4.0342n

12

16

)2/

1(2/

82

12

16

)2/

44(2/

22

,...3,2,1.....1

216

2/4)2sin()2cos(4(

2/22

,...3,2,1.....1

216

008.1

12

16

)2/

1(2/

22

,...3,2,1.....1

216

2/)2cos()2sin(4(

2/22

nnn

ene

n

nene

nn

nennnenb

nnn

ee

nn

ennnena

Spectral Lines Example

-3 -2 -1 0 1 2 30

0.5

1

t/pi

y(t)

exp(-t/2)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

a n

n

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

b n

n

0 1 2 3 4 5 6 7 8 9 100

0.5

1C

n

n

0 1 2 3 4 5 6 7 8 9 10-2

-1

0

n [ra

d]

n

n

nn

nnn

nn

n

a

b

bac

ac

tncctf

1

22

00

01

0

tan

)cos()(