44
urier Series and Fourier Transf

wendi
• Category

## Documents

• view

578

70

description

Fourier Series and Fourier Transform. The Fourier Series. Linear Circuit. I/P. O/P. Sinusoidal Inputs. OK. Nonsinusoidal Inputs. Nonsinusoidal Inputs. Sinusoidal Inputs. Fourier Series. The Fourier Series. Joseph Fourier 1768 to 1830. - PowerPoint PPT Presentation

### Transcript of Fourier Series and Fourier Transform

Fourier Series and Fourier Transform

Linear CircuitI/P O/P

Sinusoidal Inputs OK

Nonsinusoidal Inputs

Nonsinusoidal Inputs Sinusoidal Inputs

The Fourier Series

Fourier Series

The Fourier Series

Joseph Fourier1768 to 1830

Fourier studied the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions.

The Fourier Series Fourier proposed in 1807

A periodic waveform f(t) could be broken down into an infinite series of simple sinusoids which, when added together, would construct the exact form of the originalwaveform.

Consider the periodic function

( ) ( ) ; 1, 2, 3,f t f t nT n

T = Period, the smallest value of T that satisfies the aboveEquation.

The Fourier Series The expression for a Fourier Series is

0 0 01 1

( ) cos sinN N

n nn n

f t a a n t b n t

Fourier Series = a finite sum of harmonically related sinusoids

Or, alternative form

0 01

( ) cos( )N

n nn

f t C C n t

0 , , and are real and are called

Fourier Trigonometric Coefficientsn na a b and 0

2

T

0 0 and are the Complex CoefficientsnC a C

The Fourier Series

0 01

( ) cos( )N

n nn

f t C C n t

0C is the average (or DC) value of f(t)

For n = 1 the corresponding sinusoidis called the fundamental

1 0 1cos( )C t

For n = k the corresponding sinusoidis called the kth harmonic term

0cos( )k kC k t

Similarly, 0 is call the fundamental frequencyk0 is called the kth harmonic frequency

The Fourier Series

A Fourier Series is an accurate representation of a periodic signal and consists of the sum of sinusoids at the fundamental and harmonic frequencies.

Definition N

The waveform f(t) depends on the amplitude and phaseof every harmonic components, and we can generate anynon-sinusoidal waveform by an appropriate combinationof sinusoidal functions.

http://archives.math.utk.edu/topics/fourierAnalysis.html

The Fourier Series To be described by the Fourier Series the waveform f(t)must satisfy the following mathematical properties:

1. f(t) is a single-value function except at possibly a finite number of points.

2. The integral for any t0.

3. f(t) has a finite number of discontinuities within the period T.

4. f(t) has a finite number of maxima and minima within the period T.

0

0

( )t T

tf t dt

In practice, f(t) = v(t) or i(t) so the above 4 conditions are always satisfied.

The Fourier Series Recall from calculus that sinusoids whose frequencies areinteger multiples of some fundamental frequency f0 = 1/Tform an orthogonal set of functions.

0

2 2 2sin cos 0 ; ,

T nt mtdt n m

T T T

and

0 0

2 2 2 2 2 2sin sin cos cos

0 ;

1 ; 0

T Tnt mt nt mtdt dt

T T T T T Tn m

n m

The Fourier Series The Fourier Trigonometric Coefficients can be obtainedfrom

0

0

0

0

0

0

0

0

0

1( )

2( )cos

2( )sin

t T

t

t T

n t

t T

n t

a f t dtT

a f t n t dtT

b f t n t dtT

average value over one period

n > 0

n > 0

The Fourier Series To obtain ak

0 0 00 0

0 0 001

( )cos cos

( cos sin )cos

T T

N T

n nn

f t k t dt a k t dt

a n t b n t k t dt

The only nonzero term is for n = k

00( )cos

2

T

k

Tf t k t dt a

Similar approach can be used to obtain bk

Example 15.3-1 determine Fourier Series and plot for N = 7

0

00

/ 2 / 4

/ 2 / 4

1( )

1 1 1( ) 1

2

t T

t

T T

T T

a f t dtT

f t dt dtT T

0

1

2a

average or DC value

Example 15.3-1(cont.)

An even function exhibits symmetry around the vertical axisat t = 0 so that f(t) = f(-t).

0

00

/ 4

0/ 4

2( )sin

21 sin 0

t T

n t

T

T

b f t n t dtT

n t dtT

/ 4

0/ 4

/ 4

0 / 40

21 cos

2sin |

T

n T

T

T

a n t dtT

n tT n

Determine only an

Example 15.3-1(cont.)

1sin sin

2 2n

n na

n

0 when 2, 4, 6, na n and 2( 1)

when 1, 3, 5, q

na nn

where( 1)

2

nq

01,

1 2( 1)( ) cos

2

qN

n odd

f t n tn

1 3 5 7

2 2 2 2, , ,

3 5 7a a a a

Symmetry of the Function

Four types

1. Even-function symmetry2. Odd-function symmetry3. Half-wave symmetry4. Quarter-wave symmetry

Even function

( ) ( )f t f t All bn = 0

/ 2

00

4( )cos

T

na f t n t dtT

Symmetry of the Function

Odd function

( ) ( )f t f t All an = 0

/ 2

00

4( )sin

T

nb f t n t dtT

Half-wave symmetry

( ) ( )2

Tf t f t

an and bn = 0 for even values of n and a0 = 0

Quarter-wave symmetry

Symmetry of the Function

All an = 0 and bn = 0 for even values of n and a0 = 0

/ 4

00

8( )sin ; for odd

T

nb f t n t dt nT

Odd & Quarter-wave

For Even & Quarter-wave

Symmetry of the Function

All bn = 0 and an = 0 for even values of n and a0 = 0

/ 4

00

8( )cos ; for odd

T

na f t n t dt nT

Table 15.4-1 gives a summary of Fourier coefficients and symmetry.

Example 15.4-1 determine Fourier Series and N = ?

4 and s2mf T

0

2T

T

To obtain the most advantages form of symmetry, we choose t1 = 0 s

0

Odd & Quarter-wave

All an = 0 and bn = 0 for even values of n and a0 = 0

/ 4

00

8( )sin ; for odd

T

nb f t n t dt nT

Example 15.4-1(cont.)

4( ) ; 0 / 4

/ 4m mf f

f t t t t TT T

2

4

32( ) ; 0 / 4f t t t T

/ 4

00

/ 4

0 02 2 2

0 0 0

2 2

8 32sin

512 sin cos

32sin ; for odd

2

T

n

T

b t n t dtT

n t t n t

n n

nn

n

Example 15.4-1(cont.)

The Fourier Series is

021

1( ) 3.24 sin sin ; for odd

2

N

n

nf t n t n

n

2

32

The first 4 terms (upto and including N = 7)

1 1 1( ) 3.24(sin 4 sin12 sin 20 sin 28 )

9 25 49f t t t t t

Next harmonic is for N = 9 which has magnitude 3.24/81 = 0.04 < 2 % of b1 ( = 3.24)

Therefore the first 4 terms (including N = 7) is enough forthe desired approximation

Exponential Form of the Fourier Series

0 01

( ) cos( )N

n nn

f t C C n t

is the average (or DC) value of f(t) and 0C

( )

2n n

n n n

a jbC

C

2 2

2n n

n n

a bC

C

and

where

1

1

tan ; if 0

180 tan ; if 0

nn

n

n

nn

n

ba

a

ba

a

Exponential Form of the Fourier Series or

2 cos and 2 sinn n n n n na C b C Writing in exponential form using Euler’s identity with

0cos( )nn t N

0 00

0

( ) jn t jn tn n

n nn

f t C e e

C C

where the complex coefficients are defined as0

0

0

1( ) n

t T jn t jn nt

f t e dt C eT

C

And ; the coefficients for negative n are the complex conjugates of the coefficients for positive n

*n nC C

Example 15.5-1 determine complex Fourier Series

The average value of f(t) is zero 0 0C

Even function

00

0

1( )

t T jn tn t

f t e dtT

C

We select and define 0 2

Tt 0jn m

0/ 2

/ 2

/ 4 / 4 / 2

/ 2 / 4 / 4

/ 4 / 4 / 2/ 2 / 4 / 4

/ 2 / 2

0

1( )

1 1 1

| | |

2 2

0 4sin 2sin( )

2 2

T jn tn T

T T Tmt mt mt

T T T

mt T mt T mt TT T T

jn jn jn jn

f t e dtT

Ae dt Ae dt Ae dtT T TA

e e emT

Ae e e e

jn T

A nn

n

C

; for even

2sin ; for odd

2sin

where 2

n

An n

nx n

A xx

Example 15.5-1(cont.)

Example 15.5-1(cont.)

Since f(t) is even function, all Cn are real and = 0 for n even

1 1

sin / 2 2

/ 2

A A C C

For n = 1

For n = 2

2 2

sin0A

C C

For n = 3

3 3

sin(3 / 2) 2

3 / 2 3

A A

C C

Example 15.5-1(cont.)

The complex Fourier Series is

0 0 0 0

0 0 0 0

3 3

3 3

0 0

01

2 2 2 2( )

3 32 2

34 4

cos cos33

4 ( 1)cos

j t j t j t j t

j t j t j t j t

q

nn odd

A A A Af t e e e e

A Ae e e e

A At t

An t

n

where1

2

nq

For real f(t) n n C C

2cos

2 sin

jx jx

jx jx

e e x

e e j x

Example 15.5-2 determine complex Fourier Series

Even function

/ 4

/ 4

/ 4/ 4

/ 4 / 4

11

1|

1

T mtn T

mt TT

mT mT

e dtT

emT

e emT

C0jn m Use

/ 2 / 2

( 1) / 2

1

2

0 ; even, 0

( 1) ; odd

jn jnn

n

e ejn

n n

n

C

Example 15.5-2(cont.)

To find C0

0 0

/ 4

/ 4

1( )

1 11

2

T

T

T

C f t dtT

dtT

The Fourier Spectrum The complex Fourier coefficients

n n n C C

nC

Amplitude spectrum

n

Phase spectrum

The Fourier Spectrum The Fourier Spectrum is a graphical display of the amplitude and phase of the complex Fourier coeff.at the fundamental and harmonic frequencies.

Example

A periodic sequence of pulses each of width

The Fourier Spectrum The Fourier coefficients are

0/ 2

/ 2

1 T jn tn T

Ae dtT

C

For 0n

0

0 0

/ 2

/ 2

/ 2 / 2

0

0

0

2sin

2

jn tn

jn jn

Ae dt

TA

e ejn T

A n

n T

C

0

0

sin( / 2)

( / 2)

sin

n

A n

T n

A x

T x

C

The Fourier Spectrum

where 0 / 2x n

For 0n / 2

0 / 2

T T

C

The Fourier Spectrum ˆL'Hopital's rule

sin1 for 0

xx

x

sin( )0 ; 1, 2, 3,

nn

n

05

010

0

The Truncated Fourier Series

0( ) ( )N

jn tn N

n N

f t e S t

C

A practical calculation of the Fourier series requires thatwe truncate the series to a finite number of terms.

The error for N terms is

( ) ( ) ( )Nt f t S t

We use the mean-square error (MSE) defined as

2

0

1( )

TMSE t dt

T

MSE is minimum when Cn = Fourier series’ coefficients

The Truncated Fourier Series

overshoot 10%

Circuits and Fourier Series

An RC circuit excited by a periodic voltage vS(t).

It is often desired to determine the response of a circuitexcited by a periodic signal vS(t).

Example 15.8-1 An RC Circuit vO(t) = ?

1 , 2 F, secR C T

Example 15.3-1

Circuits and Fourier Series

An equivalent circuit.

Each voltage source is a term of the

Fourier series of vs(f).

Using phasors to find

to the sinusoids.

Each input

is a Sinusoid.

Example 15.8-1 (cont.)

01,

1 2( 1)( ) cos

2

qN

sn odd

v t n tn

Example 15.8-1 (cont.)

( 1)

2

nq

where

The first 4 terms of vS(t) is

0 1 3 5

1 2 2 2( ) cos2 cos6 cos10

2 3( ) ) ( ( )

5( )

sv t t

v t v t v t v ts s s s

t t

The steady state response vO(t) can then be found using superposition. 0 1 3 5( ) ( ) ( ) ( ) ( )o o o o ov t v t v t v t v t

Example 15.8-1 (cont.)

The impedance of the capacitor is

0

1 ; for 0,1, 3, 5,C n

jn C Z

0

0

0

1

; for 0,1, 3, 51

4

,

1

on sn

sn

jn Cn

Rjn C

jn CR

V V

V

We can find

Example 15.8-1 (cont.)

The steady-state response can be written as

0

102

( ) cos(

cos( tan 4 )1 16

on on on

snsn

v t n t

n t nn

V V

VV

0

1

22

for 1, 3, 5

0 for 0,1, 3, 5

s

sn

sn

nn

n

V

V

V

In this example we have

Example 15.8-1 (cont.)

0

1

2

1( )

22

( ) cos( 2 tan 4 ) ; for 1,3,51 16

o

on

v t

v t n t n nn n

1

3

5

( ) 0.154cos(2 76 )

( ) 0.018cos(6 85 )

( ) 0.006cos(10 87 )

o

o

o

v t t

v t t

v t t

1( ) 0.154cos(2 76 ) 0.018cos(6 85 )

20.006cos(10 87 )

ov t t t

t

Summary

The Fourier Series

Symmetry of the Function

Exponential Form of the Fourier Series

The Fourier Spectrum

The Truncated Fourier Series

Circuits and Fourier Series