(푸리에 변환)
제3장 신호의 주파수 영역 분석 30 / 120
Fourier Transform Fourier Transform
주기함수는 Fourier Series로 표현되며 기본주파수
f0(=1/T0)의 정수배 주파수 성분들만 존재함 (선스펙트럼)
앞의 구형파 펄스열 예제에서
• 주기 신호의 기본주기가 증가할수록 푸리에 급수의 기본주파수는 감소하고, 선 스펙트럼의 간격이 줄어든다.
• 주기가 무한대로 접근할수록 선 스펙트럼의 간격이 무한소로 줄어들어 스펙트럼은 연속 스펙트럼으로 되고 결과적으로 i 줄어들어 스펙트럼은 연속 스펙트럼으로 되고, 결과적으로 sinc 함수 모양이 된다.
비주기 신호는 푸리에 변환을 이용하여 주파수 성분분석
제3장 신호의 주파수 영역 분석 31 / 120
Fourier Transform Fourier Transform
비주기 신호의 푸리에 변환
Fourier Transform Fourier Transform
비주기 신호의 푸리에 변환
Fourier Transform Fourier Transform
푸리에 변환(Fourier Transform)
2 j f
∞∫
푸리에 역변환 (Inverse Fourier Transform)
( ) [ ( )] ( ) j2 ft
X f F x t x t e− π dt
−∞
= = ∫
∞
푸리에 변환쌍 (Fourier Transform Fair)
1 2
amplitude spectrum X f
( ) ( ) phase spectrum = ∠X f =θ f
( ) amplitude spectrum = X f
Fourier Transform Fourier Transform
1
Fourier Transform Fourier Transform
1 / 2
( ) 0 otherwise
t t
x t τ
τ
⎧ ≤
⎛ ⎞ ⎪
= Π⎜ ⎟ = ⎨
⎝ ⎠ ⎪⎩
( ) X f
τ
0 otherwise
⎝ ⎠ ⎪⎩τ
1 τ
( ) X f
1 τ
( ) sinc( ) X f =τ fτ
제3장 신호의 주파수 영역 분석 36 / 120
Fourier Transform Fourier Transform
Fourier transform pair
( ) 1 ( ) ( ) ( )
Fourier transform pair
2 2
Fourier Transform Fourier Transform
예제 3.6: rectangular pulse
( )
• A smaller τ produces a wider main lobe (broad bandwidth)
38 / 120
Fourier Transform Fourier Transform
예제 3.7: impulse
( ) ( ) ( ) 1
x t = δ t ⇔ X f =
2
0
[ ( )]δ t ∞ δ( )t e− j π ftdt
−∞
∞
=
∫
∫
F
0 ( )
1
e δ t dt
= −∞
=
∫
( ) ( )
x t =δ t X f( ) =1
1
0 t f
0
←⎯→F 1
제3장 신호의 주파수 영역 분석 39 / 120
Fourier Transform Fourier Transform
예제 3.8: constant
Fourier Transform Fourier Transform
예제 3.9: exponential functionp
( ) ( ), 0 ( ) 1
Fourier Transform Fourier Transform
( )
제3장 신호의 주파수 영역 분석 43 / 120
Properties of Fourier Transform Properties of Fourier Transform
Linearityy
1
Time shifting; shifting in time changes only the phase
2
Properties of Fourier Transform Properties of Fourier Transform
Time scaling g ⎛ ⎞
( ) 1
1
x at X
a f a
⎛ ⎞ω
⇔ ⎜ ⎟⎝ ⎠
⎛ ⎞
• expanding in time (|a|<1) leads to slow variation
1 f
a X a
⎛ ⎞⎜ ⎟
⎝ ⎠
(deemphasizing high frequency components)
1 x (t)
0 t
τ
제3장 신호의 주파수 영역 분석 45 / 120
Properties of Fourier Transform Properties of Fourier Transform
Time scalingg
1
Properties of Fourier Transform Properties of Fourier Transform
Frequency shifting and modulation
0
Properties of Fourier Transform Properties of Fourier Transform
Frequency shifting and modulation EX3-14)
Properties of Fourier Transform Properties of Fourier Transform
Dualityy
Example:
pf) 2π x t( ) ∞ X ( )ω ej tω dω t↔ω 2π x( ω) ∞ X t e( ) −j tω dt
Properties of Fourier Transform Properties of Fourier Transform
Dualityy
Properties of Fourier Transform Properties of Fourier Transform
Differentiation
( ) ( ) (obtained by differentiating both sides w.r.t. )
2 ( )
Example:
2
Integration
( jω) Y( )ω + 2( jω) ( )Y ω +Y( )ω = ( jω) ( )X ω + X( )ω
Properties of Fourier Transform Properties of Fourier Transform
Example : triangular pulse
( )
Properties of Fourier Transform Properties of Fourier Transform
Parseval’s Theorem
2 1 2 2
Energy Spectral Density (ESD) :
{ }
Energy Spectral Density (ESD) : X f
• the frequency distribution of total energy
( ) X f
제3장 신호의 주파수 영역 분석 53 / 120
Properties of Fourier Transform Properties of Fourier Transform
Symmetry
given ( )x t ⇔ X( ),ω
Properties of Fourier Transform Properties of Fourier Transform
Symmetry (continued)y y ( )
Properties of Fourier Transform Properties of Fourier Transform
Convolution (in time)( )
Example 1
( )f ( )f ( f )
( ) dx t( ) ( ) ( ) ( )
y t Y j X H j
dt ω ω ω ω ω
= ⇔ = ∴ =
Example 2
dt
Properties of Fourier Transform Properties of Fourier Transform
Convolution (in time)( )
Properties of Fourier Transform Properties of Fourier Transform
Multiplication (in time) : dual of convolution propertyp ( ) p p y
( ) ( ) 1 ( ) ( ) x t m t 2 X ω M ω
⇔ π ∗
)
( ( )
X f ∗M f
제3장 신호의 주파수 영역 분석 58 / 120
Application of Fourier Transform Application of Fourier Transform
Communication system : modulation and demodulation Modulation
Application of Fourier Transform Application of Fourier Transform
Communication system : modulation and demodulation Demodulation
Application of Fourier Transform Application of Fourier Transform
Multiplexing (frequency division multiplexing)
제3장 신호의 주파수 영역 분석 61 / 120
Useful Fourier Transform Pairs Useful Fourier Transform Pairs
Unit impulse x t( )=δ( )t X f( )=1
Constant
1 ⇔ 2πδ ω( ), δ ( f )
( ) 1
x t = X f( )=δ( )f
1 ←⎯→F
Unit step function
( ) ( ) 1 1 1
( )
u t ⇔ πδ ω + δ f +
0 t f
0
Exponential
( ) ( ) , ( )
Useful Fourier Transform Pairs Useful Fourier Transform Pairs
Complex exponential
0
Sinusoidal functions
{
0 0}
{ }Useful Fourier Transform Pairs Useful Fourier Transform Pairs
Rectangular pulse
( ) t ( ) sinc( )
x t X f T fT
T
= Π⎛ ⎞⎜ ⎟ ⇔ =
⎝ ⎠
Triangular pulse
( ) t ( ) sinc (2 ) x t = Λ⎛ ⎞⎜ ⎟ ⇔ X f =T fT
Impulse train
( ) ( ) sinc ( )
x t X f T fT
= Λ⎜ ⎟⎝ ⎠T ⇔ =
p
0 0 0
( ) ( ) ( ) ( ) 1
n n
x t t nT X f f f nf f
δ δ T
∞ ∞
=−∞ =−∞
= ∑ − ⇔ = ∑ − =
n=−∞ n=−∞
제3장 신호의 주파수 영역 분석 64 / 120
Fourier Transform of Periodic Signals Fourier Transform of Periodic Signals
Arbitrary periodic signal x(t) with period Ty p g ( ) p 00
Fourier series
2 0
Take Fourier transform
( ) F j2 nf t
Fourier Transform of Periodic Signals Fourier Transform of Periodic Signals
Arbitrary periodic signal x(t) with period Ty p g ( ) p 00 (continued)( )
Define truncated signal
0 0
( ) T T
x t t
⎧ − ≤ ≤
⎪⎨
Then the Fourier coefficients are
( ) ( ) 2 2
0, otherwise
T
Fourier Transform of Periodic Signals Fourier Transform of Periodic Signals
Arbitrary periodic signal x(t) with period Ty p g ( ) p 00 (continued)( )
Fourier Transform of Periodic Signals Fourier Transform of Periodic Signals
예제3.16) Impulse train
0 0
i) Fourier series
0 T0 2T0 3T0
ii) Fourier transform to get
0
ii) Fourier transform to get
0 0 0
Fourier Transform of Periodic Signals Fourier Transform of Periodic Signals
예제
( ) 0 k
t kT
x t A
τ
∞
= ∞
⎛ − ⎞
= ∑ Π ⎜⎝ ⎟⎠
k=−∞ ⎝ τ ⎠
( ) x t
L L
( ) x t A
0
t T0
T0
−
제3장 신호의 주파수 영역 분석 69 / 120
Fourier Transform of Periodic Signals Fourier Transform of Periodic Signals
풀이
Truncated signal
( ) ( / )