주기 [전력] 신호인경우, 기본주기 T0

정현 Fourier Series• 직교함수집합 : 구간 t0 ≤ t ≤ t0 + T0

복소지수함수 Fourier Series• 직교함수집합 : 구간 t0 ≤ t ≤ t0 + T0

Fourier Series

0

2( ) e , 0, 1, 2, ,ojn tn ot n

T

( ) 1, cos , sin , 1,2,3, ,n o ot n t n t n

복소지수함수근저함수 (basis function)집합에의한표현• 구간 t0 ≤ t ≤ t0 + T0에서임의의신호 x(t)는아래선형조합으로표현

• 선형조합의계수 은다음과같이구함

• 만일신호 x(t)가주기 T0를가진주기신호라면 basis함수들도같은주기의주기함수가되므로위의무한급수표현은한주기의구간을넘어서모든시구간 - < t < 로확장할수있음

복소지수함수 Fourier Series

0

2( ) e ,ojn tn o

n

x t CT

0

0

0 020

( ) ( )( ), ( ) 1 ( )( ) 1

o

n oo o

o o o

o o

t T

t Tt jn tnn t T t T t

nt t

x t t dtx t tC x t e dtTt dt dt

nC

• Fourier Series 및 Fourier Series 계수

• FA Coefficient Cn은 n0주파수성분(nth harmonic)의전력량• 일명, 스펙트럼계수(spectrum coefficient)• 복소수 : , 여기서 은 의편각(argument)

0

0

2 /

2 /

0 0

( ) (Fourier Series: Synthesis)

1 1( ) ( ) (FS Coefficients: Analysis)

o

o

o o

jn t j nt Tn n

n n

jn t j nt Tn T T

x t C e C e

C x t e dt x t e dtT T

arg nj Cn nC C e arg nC nC

대칭성• x(t)의값이 real인경우, 즉 인경우

• If x(t) is real & even (i.e. ), then cn is also real & even

• If x(t) is real & odd (i.e. ), then cn is imaginary & odd

Fourier Series 성질

( ) ( )x t x t

(conjugate symmetric)n nc c

(amplitude spectrum is even) arg arg (phase spectrum is odd)

n n

n n

c cc c

( ) ( )x t x t

( ) ( )x t x t

선형성• x(t)와 y(t)가동일한주기를갖는주기신호인경우

• 의푸리에급수는

0

0

( )

( )

jn tn

n

jn tn

n

x t c e

y t d e

( ) ( ) ( )z t x t y t

0( )

with

jn tn

n

n n n

z t g e

g c d

시간천이• x(t)의푸리에계수가 인경우

• x(t-)의푸리에계수 은

• 진폭스펙트럼은변화가없다.

nc

nd

0

0

0

( )

0 0

0

1 1( ) ( )

1 ( )

o o

o o

o

o

jn t jn t jnn T T

jn jn

T

jnn

d x t e dt x t e e dtT T

e x e dT

e c

n nd c

주기신호의평균전력

• Real valued signal의경우

•

Parseval’s Theorem

0

2 2

0

1 ( ) nTn

P x t dt cT

2 20

12 n

nP c c

2 is called power spectrumnc

[proof]• 의푸리에계수

• z(t)의푸리에계수

• z(t)의직류값(k = 0에서의푸리에계수)

• x(t) = y(t)인경우

0 0

0 0

* *

( )* *

( ) ( ) ( ) jn t jm tn m

n m

j n m t jk tn m k m m

n m k m

z t x t y t c e d e

c d e c d e

*( ) ( ) ( )z t x t y t

0

0

* *

0

1 ( ) ( ) jk tk m m T

mc d x t y t e dt

T

0

* *

0

1 ( ) ( )m m Tm

c d x t y t dtT

0

2 2

0

1 ( )n Tn

c x t dtT

크기가 1이고펄스폭이 1인사각펄스(구형파)

• 크기가 A이고펄스폭이 인사각펄스:

Rectangular Pulse

1 1

2 21 for

( ) ( )0 otherwise

1 1 2 2

tt rect t

u t u t

( )t

t12

12

1

( / )A t

t

2

2

A

( / )A t

크기가 1이고펄스폭이 2인삼각펄스

• 크기가 A이고펄스폭이 2인삼각펄스:

Triangular Pulse

1 for 1 1( )

0 otherwiset t

t

( / )A t

( )t

t11

1

( / )A t

t

A

Sa 함수와 sinc함수

Sampling Function

Sa( )t

t

1

03 2 32

sinc( )t

t

1

103 2 1 32

sinSa( ) ttt

sinsinc( ) Sa( )tt tt

[Example] Rectangular Pulse Train

• 주기가 T0이고펄스폭이 인구형펄스열

0( )k

t kTx t

1

0T0T 0 0

2T0

2T

( )x t

t

[Example] Rectangular Pulse Train

i) n = 0: dc value

0 02 / / 2T

0 0

0 0

2 2

2 20 0 0

2

20 0

1 1 2( ) ( ) exp

1 2(1)exp

oT Tjn t

n T T

j ntc x t e dt x t dtT T T

j nt dtT T

2

0 20 0

1 (1)c dtT T

ii) n ≠0

0

0

2 2

2 20 0 0 0

2

0 2

0 0

0 0

0 0

0 00

1 2 1 2( )exp (1)exp

1 2exp2

1 exp exp2

sin / si

sinc

n //

sinc

T

n T

j nt j ntc x t dt dtT T T T

j ntj n T

j n j nj n T T

n T n Tn T n T

nT TT

n

0

0

0

00 2nSa

f

nT T

SaT

[Example] Rectangular Pulse Train• Spectrum shape :

• 의파형은진동하면서감쇠하고, 마다 zero-crossing

• First zero-crossing frequency:

0

00

0 0

0

( )2 2

( )2

nn

nc c n Sa SaT T

c SaT

( )c / ( 0)n n

/

nc

2

0/T

0 02 03

0

( )2

c SaT

16/80

비주기신호의주기적확장

비주기신호의 Fourier Transform

( )x t

t

TT 0

( )px t

t

periodic extensionT

0

2

02

0

( )

1 ( ) ( 2 / )

2 2

o

o

jn tp n

n

T jn tn pT

nn

x t c e

c x t e dt TT

nc Sa SaT T

nc

0T

0

2T

nc

02T

0T

0T T

02T T

0

1. Spectrum shape :

2

2. First zero-crossing : 2

2 2

c SaT

n nT

2

2

Define discrete frequency n as

Define X(n)

Let then

02

n n nT

2

2( ) ( ) o

T jn tn n pT

X Tc x t e dt

T

( ) ( ) ( )T j tn nX Tc x t e dt X

0

0

0

1( ) lim ( ) lim lim

lim ( )2 2

o o

o

jn t jn tp n nT T Tn n

jn t j tnT n

x t x t c e Tc eT

dTc e X e

Fourier Transform

Inverse Fourier Transform

주파수변수를 대신 f를사용하면

F 1( ) ( ) ( )2

j t dx t X X e

( ) ( ) ( ) j tX x t x t e dt

F

2

1 2

( ) ( ) ( )

( ) ( ) ( )

j ft

j ft

X f x t x t e dt

x t X f X f e df

F

F

Continuous Spectrum

• 주기신호의푸리에스펙트럼이이산스펙트럼

• 비주기신호의푸리에스펙트럼은연속스펙트럼

Amplitude spectrum and phase spectrumarg ( )( ) ( )

1) ( ) : amplitude spectrum2) arg ( ) : phase spectrum

j XX X e

XX

[예제 5.1] Rectangular Pulse with pulse width

1 / 2( ) ( / )

0 otherwiset

x t t

2

2

2

2

( ) e

0,

sine 20,

2

2

j t

j t

t dt

If

Ifj

Sa

F

t0

1

( )x t

[예제 5.1] Rectangular Pulse with pulse width

( / )2

t Sa

F

( )X

2

4

6

[예제 5.2] Impulse

( ) 1t F

0( ) ( ) ( ) 1j tt t e dt e t dt

F

t

( ) ( )x t t

0

( ) 1X

0

1　

[예제 5.3] Inverse Fourier transform of ()

1 2 ( ) ( )f F

1 0 1( ) ( ) ( )2 2 2

j t d de e

F

t

( ) 1x t

0

( ) 2 ( )X

0

1　

(2 )

[예제 5.4] Inverse Fourier transform of rectangular spectrum• Rectangular spectrum of bandwidth / 2

21

2

2

2

( ) ( ) e2

0,2

sin1 exp( ) 20,2 2

2

2 2

j t dx t

If t

tj tIf t

tjt

tSa

x F

1 / 2( ) ( / )

0 / 2X

[예제 5.4] Inverse Fourier transform of rectangular spectrum• Rectangular spectrum of bandwidth / 2

12 2

tSa

F

1

1

2

( )x t

2

2

0t

　

( )X

( )a ( )b

[예제 5.5] Exponential function( ) ( ), 0atx t e u t a

( )

0( ) ( )

1 , 0

at j t a j tX e e u t dt e dt

aa j

2 2 2

1

1 1 1( )

1

arg ( ) tan

Xaa

a

Xa

1( ) (0)2

arg ( )4

X a X

X a

[예제 5.5] Exponential function

1( )ate u ta j

F

2

1 1( )

1

Xa

a

1( ) tanXa

( )X

( )X

aa

aa

1a 1

2a

2

2

4

4

t

( ) ( )atx t e u t

1