reading assignments - dsp first

DSP First, 2/e

Lecture 7CFourier Series Examples:

Common Periodic SignalsCommon Periodic Signals

READING ASSIGNMENTSREADING ASSIGNMENTS

This Lecture: Appendix C, Section C-2 Various Fourier Series Pulse Waves Triangular Wave Rectified Sinusoids (also in Ch. 3, Sect. 3-5)

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 3

LECTURE OBJECTIVESLECTURE OBJECTIVES

Use the Fo rier Series Integral Use the Fourier Series Integral

0

0 )/2(1 )(T

dtetxa tTkj

D i F i S i ff f

0

0

0

)(1 )( dtetxa jTk

Derive Fourier Series coeffs for common periodic signals

Draw spectrum from the Fourier Series coeffs ak is Complex Amplitude for k-th Harmonic


Harmonic Signal is PeriodicHarmonic Signal is Periodic

tFkj

keatx 02)(

Sums of Harmonic complex exponentials are Periodic signals

k are Periodic signals

PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:

12PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:

0

00

001or22F

TT

F


00

Recall FWRSRecall FWRS1isPeriod)/2sin()( TTTttx 1201 isPeriod)/2sin()( TTTttx

Absolute value flips thenegative lobes of a sine wave


FWRS Fourier Integral {ak}

0

0

0

)/2(

0

)(1 TktTj

k dtetxT

a

Full-Wave Rectified Sine00

)sin( )/2(10

0

00 dteta

TktTj

TTk

Full Wave Rectified Sine

:)/2sin()(

121

0

1

TTPeriodTttx

)/2()/()/(

1

0

0

000

00

dteeeTktTj

tTjtTj

TTk

)/sin()( 0

20

Tttx

2)12)(/()12)(/(

0

00

00

0

ee

dtej

TtkTjTtkTj

T

20

))12)(/((20

))12)(/((2 0000

eekTjTjkTjTj

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 7)14(

22

kak

FWRS Fourier Coeffs: akFWRS Fourier Coeffs: ak

ak is a function of kak is a function of k Complex Amplitude for k-th Harmonic

)14(22

kak

Does not depend on the period, T0

)14( k

DC value is 6336.0/20 a


Spectrum from Fourier SeriesSpectrum from Fourier Series

Plot a for F ll Wa e Rectified Sin soidaPlot a for Full-Wave Rectified Sinusoid

0000 2and/1 FTF 2

ka

)14(22

k

ak


Fourier Series SynthesisFourier Series Synthesis HOW do you APPROXIMATE x(t) ? HOW do you APPROXIMATE x(t) ?

0

0 )/2(1 )(T

tkTj dtt 0

00

)/2(1 )( tkTjTk dtetxa

Use FINITE number of coefficientsN

real is )( when* txaa kk

tFkjN

kkeatx 02)(


Nk

Reconstruct From Finite Number of Harmonic Componentsof Harmonic Components

F ll Wa e Rectified Sin soid )/i ()( TttFull-Wave Rectified Sinusoid

ms100 T

)/sin()( 0Tttx

22

akHz1000 F

6336.0/20 a)14( 2 k

ak

N

tFkjk

tFkjkN eaeaatx 22

000)(

kkkN

10)(

?)/sin()(to)(is closeHow 0TttxtxN


)()()( 0N

Full-Wave Rectified Sine {a }Full-Wave Rectified Sine {ak}22

)(

valuedrealis)14(

2

2

2)14(22 k

a

Ntjk

kk

...

)(

2)116(

22)116(

2)14(

2)14(

22

)14(2

0000

02

eeee

etx

tjtjtjtjNk

tjkkN

)cos()2cos()cos(...

4442

215

2215

232

322

)116()116()14()14(

0000

tNtteeee tjtjtjtj

Plots for N=4 and N=9 are shown next

)cos(...)2cos()cos( 0)14(01503 2 tNttN


Excellent Approximation for N=9

Reconstruct From Finite Number of Spectrum Componentsof Spectrum Components

F ll Wa e Rectified Sin soid )/i ()( TttFull-Wave Rectified Sinusoid )/sin()( 0Tttx

H100ms100

FT

Hz1000 F

6336.0/20 a

2

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 13)14(

22

kak

Fourier Series SynthesisFourier Series Synthesis


PULSE WAVE SIGNALGENERAL FORMGENERAL FORM

Defined over one periodDefined over one period

2/2/0

2/01)(

Tt

tx

2/2/0

)(0Tt

Nonzero DC value


Pulse Wave {ak}{ k}

General PulseWave 2/

))(/2(10

0)(T

tkTjTk dtetxa

dktTj

12/

)/2(1

General PulseWave

2/2/0

2/01)(

0Ttt

tx

2/0

0)(

TTk

dtea

kTjkTjktTj

ktTjTk

1

)2/()/2()2/()/2(2/)/2(

2/

)/2(1

000

0

0

eeekj

kTjkTj

kTj

ktTj

T

)( )2(

)2/()/2()2/()/2(

2/)/2(

)/2(1

00

0

0

0

Tkee kTjkTj )/sin( 0)()/()()/( 00


kkj )2(

Pulse Wave {ak} = sinc{ k}PulseWave

,...2,1,0)/sin( 0 kk

Tkak

2/)0)(/2(1 tTj

0 )/sin(liN Tk

Double check the DC coefficient:

0

0

2/2/

)0)(/2(10 1 tTj

T dtea

0

0

0

)/sin(lim Note,Tk

Tkk

000 22

1

2/

1 1 TTT dt


PULSE WAVE SPECTRAPULSE WAVE SPECTRA

2/T 2/0T

4/0T 4/0T

16/0T


50% duty-cycle (Square) Wave50% duty cyc e (Squa e) a e

)2/sin()/)2/(sin( 00 kTTk

Thus a =0 when k is odd

,...2,1,0)2/sin()/)2/(sin(2/ 000 k

kk

kTTkaT k

Thus, ak=0 when k is odd Phase is zero because x(t) is centered at t=0 different from a previous case different from a previous case

311 k

PulseWave starting at t=0

0

,4,20

,3,1

2)1(1

001

)(

210 k

k

kkj

kja

Ttt

txk

k


02 k

PULSE WAVE SYNTHESISwith first 5 Harmonicswith first 5 Harmonics

2/0T

4/0T


16/0T

Triangular Wave: Time Domaing

D fi d i dDefined over one period

2/2/for/2)( 000 TtTTttx

Nonzero DC value


Triangular Wave {ak}

2/

2/

)/2(

0

0

0

0)(1 T

T

ktTjk dtetx

Ta

Triangular Wave0

0

0

2/

2/

)/2(0

0

|/2|1 T

T

ktTjk dteTt

Ta

Triangular Wave

0/2)( Tttx

02

0

0

02

0

0

02

2/2

2/0

)( ktjT

Tktj

T

T

dtetdtet

T

00 /2 T

220

002

0

0

220

002

0

000

0

2/

122/

0

12

2/0

Tkktjtkj

T

T

kktjtkj

Tk

TTT

eea

220

000022

02

022

022

0

00002

0

00000

12/2/12112/2/2

2/0

kkTjTkj

kTkkkTjTkj

T

TTT

ee

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 22 22

0

000 1 integral indefinite theusekktjtkjktj edtet

Triangular Wave {ak}

12/2/12112/2/222

00002222222

00002 eea kTjTkjkTjTkj

k

12124

220

220

20

220

220

20

eea

kjkjkjkj

kkTkkTk

1212422

02

022

02

022

02

0ee

kkjkj

Tkkjkj

TkT

220

20

1400 2

T

T

)1(4111)1(2122

02

02222

02

022

02

02

022 kTkkT

kjkT

kjTk

kk

00

310,...4,21

)1(1

22

kk

kk

Period

AreaDC


0

,...3,10

21

)(12222

kk

kk 212/

0 0

0 TTa

Triangular Wave {ak}g { k}

S t i 50 H i th Spectrum, assuming 50 Hz is the fundamental frequency

421 k

,...3,10,...4,2

1

122

kk

ak

k

02

1 k


Triangular Wave Synthesis

Hz50ms20

0

0

F

T


Full-Wave Rectified Sine {ak}

0

0 )/2()(1 TktTj

k dtetxT

a

Full Wave Rectified Sine00T0

0

00

)/2(1 )sin(T

ktTjTTk dteta

Full-Wave Rectified Sine

1

1

:)/2sin()(

TTPeriodTttx

0

000

00

)/2()/()/(

1

0T

ktTjtTjtTj

TTk

dteee

121

0: TTPeriod

0

0

0

0

0

)12)(/(1)12)(/(1

0 2T

tkTjT

tkTj

T

dd

dtej

00

00

0

0

0

0

)12)(/()12)(/(

0

)12)(/(21

0

)12)(/(21

TtkTjTtkTj

tkTjTj

tkTjTj dtedte


0

00

0

0))12)(/((2

)12)(/(

0))12)(/((2

)12)(/(

kTjTj

tkTj

kTjTj

tkTj ee

Full-Wave Rectified Sine {ak})12)(/()12)(/( 0

00

0

eea

TtkTjTtkTj

0))12)(/((2

0))12)(/((2 0000

a kTjTjkTjTjk

11 )12)(/()12(2

1)12)(/()12(2

1 0000

ee TkTj

kTkTj

k

11 )12()12(

1)12()12(

1

ee kj

kkj

k

2

)12()12( kk


)14(

21)1( 22

)14()12(12

2

kk

kkk

Half-Wave Rectified SineHalf-Wave Rectified Sine

Signal is positive half cycles of sine wave HWRS = Half-Wave Recitified Sine


Half-Wave Rectified Sine {ak}

)1()(1 00

0

)/2( kdtetxT

aT ktTj

k

Half Wave Rectified Sine00T

2/ )/2(21 00)sin(

T ktTj dteta

Half-Wave Rectified Sine

2/)/2(

)/2()/2(1

0

0

000

00)sin(

TktTj

tTjtTj

T

TTk

dteee

dteta

2/)1)(/2(1

2/)1)(/2(1

0

0

0

0

0

0 2T

tkTjT

tkTj

T

dtedte

dtej

2/)1)(/2(2/)1)(/2(

0

))((21

0

))((21

00

00

0

0

0

0

TtkTjTtkTj

jTj

jTj

ee

dtedte

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 290))1)(/2((2

))((

0))1)(/2((2

))((

00

0

00

0

kTjTj

j

kTjTj

j ee

Half-Wave Rectified Sine {ak}

2/

))1)(/2((2

)1)(/2(2/

))1)(/2((2

)1)(/2( 0

00

00

00

0 eeaT

kTjTj

tkTjT

kTjTj

tkTj

k

00

11 2/)1)(/2(

)1(412/)1)(/2(

)1(41 0000 ee TkTj

kTkTj

k

11 )1(

)1(41)1(

)1(41 ee kj

kkj

k

1?odd 0

1)1()1(4)1(1

2 kk

kk

kk4

1j


even )(

)1(1

)1(4

2

2

kk

k

4j

Half-Wave Rectified Sine {ak}{ k}

S t i 50 H i th Spectrum, assuming 50 Hz is the fundamental frequency

odd0 k

1odd0

14

kkk

a jk

even)1(

12 k

k


HWRS Synthesis

Hz50ms20

0

0

F

T


Fourier Series DemoFourier Series Demo

MATLAB GUI: fseriesdemo Shows the convergence with more terms One of the demos in: http://dspfirst.gatech.edu/matlab/


fseriesdemo GUI


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