ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Trigonometric Fourier Series Outline –Introduction –Visualization –Theoretical

ES 240: Scientific and Engineering Computation. Introduction to Fourier Series

Trigonometric Fourier SeriesTrigonometric Fourier Series

Outline– Introduction– Visualization– Theoretical Concepts– Qualitative Analysis– Example– Class Exercise


IntroductionIntroduction

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 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

TToo TToo


Properties of a periodic function 1Properties of a periodic function 1

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

Why?

Periodic or Aperiodic?


Properties of a periodic functionProperties 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


VisualizationVisualization

Can you represent this simple function using sinusoids?

Single sinusoid Single sinusoid representationrepresentation


VisualizationVisualization

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

)cos( 01 ta amplitudeamplitude Fundamental Fundamental

frequencyfrequency)3cos( 03 ta

New amplitudeNew amplitude 22ndnd Harmonic Harmonic

)5cos( 05 ta amplitudeamplitude 44thth Harmonic Harmonic


Theoretical ConceptsTheoretical 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

PeriodPeriodCosine termsCosine terms

Sine termsSine terms


Theoretical ConceptsTheoretical Concepts

(6)

n

nn

nnn

nn

n

a

b

bac

ac

tncctf

1

22

00

01

0

tan

)cos()(


DC OffsetDC OffsetWhat is the difference between these two functions?

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-A-A

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Average Average Value = 0Value = 0

Average Average Value ?Value ?


DC OffsetDC Offset

If 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 AnalysisQualitative Analysis

Is it possible to have an idea of what your solution should be before actually computing it?

For SureFor Sure


Properties – DC ValueProperties – DC Value

If the function has no DC value, then a0 = ?

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DC?DC?

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DC?DC?


Properties – SymmetryProperties – Symmetry

AA

AA

00 ππ/2/2 ππ 33ππ/2/2

f(-t) = -f(t)f(-t) = -f(t)

Even function

Odd function00

-A-A

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ππ/2/2 ππ 33ππ/2/2

f(-t) = f(t)f(-t) = f(t)



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:If f(t) is even:

EvenEvenOddOddXX == OddOdd

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

1

00

ndttntfT

aTt

tn

EvenEvenEvenEvenXX == EvenEven



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:If f(t) is odd:

OddOddEvenEvenXX == OddOdd

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

1

00

ndttntfT

bTt

tn

OddOddOddOddXX == EvenEven



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



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

Half wave symmetry: f(t-THalf wave symmetry: f(t-T00/2) = -f(t)/2) = -f(t)

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Properties – DiscontinuitiesProperties – 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:

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Which is closer Which is closer to a sinusoid?to a sinusoid?

Which function has Which function has discontinuities?discontinuities?


ExampleExample

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

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DC?DC? No, aNo, a00 = 0; = 0;

Symmetry?Symmetry? Even, bEven, bnn = 0; = 0;

Half wave Half wave symmetry?symmetry?

Yes, only odd Yes, only odd harmonicsharmonics

Discontinuities?Discontinuities? No, falls of asNo, falls of as

1/n1/n22

Prediction aPrediction ann 1/n 1/n22

for n = 1, 3, 5, …;for n = 1, 3, 5, …;


ExampleExample

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 zero for n evenn even


ExampleExample

Now compare your calculated answer with your predicted form

DC?DC? No, aNo, a00 = 0; = 0;

Symmetry?Symmetry? Even, bEven, bnn = 0; = 0;

Half wave Half wave symmetry?symmetry?

Yes, only odd Yes, only odd harmonicsharmonics

Discontinuities?Discontinuities? No, falls of asNo, falls of as

1/n1/n22


Class exerciseClass 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

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Documents

ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Trigonometric Fourier Series Outline –Introduction –Visualization –Theoretical