Fourier Series.pdf

8/14/2019 Fourier Series.pdf

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Fourier Series


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Periodic Functions

A function f(x)is said to be periodic if its

function values repeat at regular intervals

of the independent variable. The regular

interval between repetitions is theperiod

of the oscillations.


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Periodic Waveforms

Consider the given sine wave function:

where:

tAtf

tAtf

sin)(

)sin()(

frequency

frequencangular

amplitudeA

2

_

ntdisplacemetime

anglephase

periodT

_

_

2


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Periodic Waveforms

A

T


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Fundamental Frequency

Consider the given linear combinationofsinusoids:

where:

1= fundamental angular frequency

2sin 1t = fundamental component

0.8sin2 1t = first harmonic

0.7sin41t = component w/ lowestfrequency

ttttf 111 4sin7.02sin8.0sin2)(


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Harmonics

A function f(x)is sometimes expressed asa series of a number of different sinecomponents. The component with the

largest period is the first harmonic, orfundamentalof f(x).

y = A1sint is the first harmonic orfundamental

y = A2sint is the second harmonic

y = A3sint is the third harmonic, etc.


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Example 1

Find the fundamental frequency of the

following Fourier series:

ttttfbtttfa

80cos40cos220cos5)(:)(80cos40cos5)(:)(


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Example 2

Find the amplitude and phase of the

fundamental component of the function:

tttttf

1

111

3cos3...................2sin5.3cos5.1sin5.0)(


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Recall:

ab

baR

where

tbtatR

1

22

tan

:

sincos)cos(


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Example 3

Sketch the graph of the periodic function

defined by

1)(.......10..........)( Tperiodtttf


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Example 4

Write down a mathematical expression of

the function whose graph is:

-2 -1 0 1 2 3 4 t

1

f(t)


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Example 5:

Sketch the graph of the following periodic

functions:

2;11,)(:)( 2 Ttttfa

tt

Tt

tfb

2,sin

;2

0,0

)(:)(

10,

3;02,)(:)(

tt

Ttttfc


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ODD & EVEN FUNCTIONS:

A. Even Function if:

B. Odd Function if:

)()( tftf

)()( tftf


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Example 6

Show that f(t) is even

a)

t

f(t)

4

4


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Example 6


b)

t

f(t)


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Example 6


c)

t

f(t)

3


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Example 7

Show that f(t) is odd

t

f(t)

4

4


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Produtcs of ODD-EVEN Functions:

even x even = even

odd x odd = even

even x odd = odd


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Example 8

State the product of the following

functions:

(a) f(t) = t3sin wt

(b) f(t) = t cos 2t

(c) f(t) = t + t2


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Fourier Series

Decompose periodic functions or periodic

signals into the sum of a (possibly infinite)

set of simple oscillating functions, namely

sines and cosines (or complexexponentials).

The study of Fourier series is a branch of

Fourier analysis.


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Fourier Analysis

A set of mathematical tools to break down

a wave into its various frequency

components.


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Dirichlet Conditions

A function f(t) satisfying the followingconditions:

1. f(t) is periodic with period 2; i.e.

f(t+2) = f(t)2. f(t) is a single-valued and finite in each

period.

3. f(t) has a finite number of finitediscontinuities in each period.

4. f(t) has a number of finite maxima andminima in each period.


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Fourier Series

Is a series used to represent a periodic

wave in either exponential or trigonometric

form.

The series has the form of;

1

0 sincos

2

)(

n

nn tnbtnaa

tf

tnbtbtbtb

tnatatataa

tf

n

n

sin...3sin2sinsin

cos...3cos2coscos2

)(

321

3210


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Fourier series representation of f(t)

over the interval t .

Let the function f(t) be defined on the

interval t . Then the Fourier

coefficients anand bnin the Fourier series

representation of f(t)

10

sincos)(n nn

ntbntaatf


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over the interval t .

where:

dttfa )(

2

10

ntdtfan cos)(

1

ntdtfbn sin)(

1for n = 1,2,.


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over the intervalL t L.

Let the function f(t) be defined on the

intervalL x L. Then the Fourier

coefficients anand bnin the Fourier series

representation of f(t)

1

0 sincos)(

n

nn

L

tnb

L

tnaatf


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over the intervalL t L.

where:

L

L

dttf

L

a )(

2

10

L

Ln dt

L

tntf

La

cos)(

1

L

Ln dt

Ltntf

Lb sin)(1 for n = 1,2,.


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Fourier Series of Even Functions

(Half Range Cosine Series)

If f(t) is an even periodic function which

satisfies the Dirichlet condition, the

coefficients in the Fourier series of f(t) are

given by the formula;

1

0 cos)(n

nL

tnaatf

where:

L

dttfL

a0

0 )(1

L

n dtL

tntf

La

0cos)(

2 0nb;;


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Fourier Series of Odd Functions

(Half Range Sine Series)

If f(t) is an odd periodic function which

satisfies the Dirichlet condition, the

coefficients in the Fourier series of f(t) are

given by the formula;

1

sin)(n

nL

tnbtf

where:

;0;00 naa L

n dtL

tntf

Lb

0sin)(

2


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Example

Find the Fourier series of the function

tonttf )(

1

0

sincos2

)(

cos2

0

0

n

n

n

ntnn

tf

nnb

a

a

Answer:


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Example

Expand the given function into a Fourier

series on the indicated interval.

05,4

50,4

)( t

t

tf

Answer:

1

0

5sin)cos1(

8

)(

)cos1(8

0

0

n

n

n

tn

nntf

nn

b

a

a


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Example

Find the Fourier series of the function

,)( 2 onttf

1

2

2

2

2

0

coscos4

3)(

0

cos4

3

n

n

n

ntnn

tf

b

n

n

a

a

Answer:


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Example

Write the sine series of f(x) = 1 on [0,5]

1

0

5sin)cos1(

2)(

)cos1(2

00

n

n

n

xnn

ntf

n

n

b

aa

Answer:


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Convergence of Fourier Series:


First Convergence Theorem:

Let f be piecewise smooth on [-L,L].

Then the Fourier series of f(x)

converges at each point x of (-L,L) to

(f(x+) + f(x-))


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Convergence of Fourier Series:


This means that at each x betweenL

and L, the Fourier series converges to

the average of the left and right limits of

f(x) at x.

If f is continuous at x, then the left and

right limits are both equal to f(x), and the

Fourier series converges to f(x) itself. If fhas a jump discontinuity at x, then the

series converges to the point midway in

the gap in the graph at this point.


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Convergence of Fourier Integrals

Let f be piecewise continuous on everyinterval [-L,L] and suppose that converges.Then the Fourier integral f converges to

[f(x+)+f(x-)] At each x at which f has a left and right

derivative. In particular, if f is continuous atx and has a left and right derivative there,then the Fourier integral at x converges tof(x).


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Fourier Transform

Let f be piecewise continuous on [-L,L] for

every positive number L and suppose

converges. The Fourier transform of f is

defined to be

The fourier transform of f is therefore a

function F{f(t)} of the new variable . This

function, evaluated at , is F{f(t)} ().

)()())}(({ FdtetftfF ti


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The Heaviside Function (Unit Step

Function)

The Unit Step function is defined by

0,1

0,0)(

t

ttH


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Example

Express the function in terms of H(t) and find its

Fourier transform

0,0,0)(

tettf

at


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Properties of Fourier Transform

1. Amplitude Spectrum

2. Linearity

3. Time Shifting Theorem

4.Frequency Shifting Theorem

5. Scaling

6. Time Reversal

7.Symmetry

8.Modulation

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