Properties of Continuous-Time Fourier Transform and Functions
Discover the properties and transformations of the continuous-time Fourier transform, including linearity, time scaling, time reversal, time shifting, frequency shifting, symmetry properties, and Fourier transform for real functions. Explore the Fourier transform for real and even/odd functions, understanding their characteristics and spectral representations.
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Continuous-Time Fourier Transform Properties of Fourier Transform
Notation = [ ( )] ( ) f t F j F = 1 - [ ( )] ( ) F j f t F ( ) ( ) F f t F j
Linearity + + ( ) ( ) ( ) ( ) F a f t a f t a F j a F j 1 1 2 2 1 1 2 2
Time Scaling j 1 ( ) F f at F | | a a
Time Reversal ( ) ( ) F f t F j Pf) = t = = j t j t [ ( )] ( ) ( ) f t f t e dt f t e dt F = = = t = t = t = j t ( ) ( ) f t e d t j t ( ) ( ) f t e d t = t = t = = t t = = j t j t ( ) ( ) f t e dt f t e dt = t t = = ( ) F j j t ( ) f t e dt
Time Shifting ( ) j t ( ) F f t t F j e 0 0 Pf) = t = = j t j t [ ( )] ( ) ( ) f t t f t t e dt f t t e dt F 0 0 0 = t + = t t = + + ( ) j t t 0 ( ) ( ) f t e d t t 0 0 + = t t 0 = t j = t j t ( ) e f t e dt 0 = t j = j t = t ( ) j t F j e ( ) e f t e dt 0 0
Frequency Shifting (Modulation) ) 0 j ( ) ( F f t e F j 0 Pf) = j t j t j t [ ( ) ] ( ) f t e f t e e dt F 0 0 = ( ) j t ( ) f t e dt 0 ) 0 = ( F j
Symmetry Property = [ ( )] 2 ( ) F jt f F Pf) d = j t 2 ( ) ( ) f t F j e = j t 2 ( ) ( ) f t F j e d Interchange symbols and t = [ ( )] F jt = j t F 2 ( ) ( ) f F jt e dt
Fourier Transform for Real Functions If f(t) is a real function, and F(j ) = FR(j ) + jFI(j ) F( j ) = F*(j ) = j t ( ) ( ) F j f t e dt = = j t * ( ) ( ) ( ) F j f t e dt F j
Fourier Transform for Fourier Transform for Real Functions Real Functions If f(t) is a real function, and F(j ) = FR(j ) + jFI(j ) F( j ) = F*(j ) FR(j ) is even, and FI(j ) is odd. FR( j ) = FR(j ) FI( j ) = FI(j ) Magnitude spectrum |F(j )| is even, and phase spectrum ( )is odd.
Fourier Transform for Fourier Transform for Real Functions Real Functions If f(t) is real and evenF(j ) is real If f(t) is real and odd F(j ) is pure imaginary Pf) Pf) = F = ) = Even Odd ( ( ) j ( ) ( ( ) j ( F ) f t f = t ( f t f t ( ) ) ) F j F j j = j = Real Real ( ) * ( ) ( ) * ( ) F j F j F F = = ( ) * ( ) ( ) * ( ) F j F j F j F j
Example: t = = [ ( ) cos ] ? f t [ ( )] ( ) f t F j F F 0 Sol) 1 + = j t j t ( ) cos ( )( ) f t t f t e e 0 0 0 2 1 1 = + j t j t [ ( ) cos ] [ ( ) ] [ ( ) ] f t t f t e f t e F F F 0 0 0 2 1 2 1 = + + [ ( )] [ ( )] F j F j 0 0 2 2
Example: f(t)=wd(t)cos 0t wd(t) 1 t t d/2 d/2 d/2 d/2 2 d / 2 d = = = j t ( ) [ ( )] sin W j w t e dt F d d 2 / 2 d d d + sin ( ) sin ( ) 0 0 2 2 = + = ( ) [ ( ) cos ] F j w t 0t F + d 0 0
1.5 d=2 0=5 1 Example: F(j ) 0.5 0 -0.5 -60 -40 -20 0 20 40 60 f(t)=wd(t)cos 0t wd(t) 1 t t d/2 d/2 d/2 d/2 2 d / 2 d j j = = = t ( ) [ ( )] sin W w t e dt F d d 2 / 2 d d d + sin ( ) sin ( ) 0 0 2 2 = + = ( ) [ ( ) cos ] F j w t 0t F + d 0 0
wd(t) 1 Example: t d/2 d/2 =sin at = ( ) f t ( ) ? F j t Sol) 2 d = ( ) sin Wd j 2 2 t sin td = = [ ( )] sin 2 ( ) W jt w F F d d 2 0 | | a at = = = [ ( )] ( ) f t w F F 2 a t 1 | | a
Fourier Transform of f(t) ( ) = ( ) and lim t ( ) 0 F f t F j f t ( ' ) ( ) F f t j F j Pf) = j t [ ( ' )] ( ' ) f t f t e dt F = + j t j t ( ) ( ) f t e j f t e dt = ( j ) j F
Fourier Transform of f (n)(t) ( ) j = ( ) and lim t ( ) 0 F f t F f t ( ) n n ( ) ( ) ( ) F f t j F j
Fourier Transform of f (n)(t) ( ) j = ( ) and lim t ( ) 0 F f t F f t j j ( ) n n ( ) ( ) ( ) F f t F
Fourier Transform of Integral ( ) ( ) 0 = = ( ) and ( ) 0 F f t F j f t dt F 1 ( ) t = ( dx ) f x F j F j t F = = ( ) ( dx ) t f x lim t j ( ) 0 t Let ( ' = = = [ )] [ 1 j ( )] ( ) ( ) t f t F j j F = ( ) ( ) j F j
The Derivative of Fourier Transform ( ) dF j [ ( )] F jtf t F d Pf) = j t ( ) ( ) F j f t e dt d ( ) dF j = = j t j t ( ) ( ) f t e dt f t e dt d d =F = [ ( )] jtf t j t [ ( )] jtf t e dt
Continuous-Time Fourier Transform Convolution
Basic Concept fo(t)=L[fi(t)] fi(t) Linear System fi(t)=a1fi1(t) + a2fi2(t) fo(t)=L[a1fi1(t) + a2fi2(t)] fo(t) = a1L[fi1(t)] + a2L[fi2(t)] = a1fo1(t) + a2fo2(t) A linear system satisfies
Basic Concept fo(t) fi(t) Time Invariant System fi(t +t0) fi(t t0) fo(t + t0) fo(t t0) fi(t) fo(t) t t fi(t+t0) fo(t+t0) t t fi(t t0) fo(t t0) t t
Basic Concept fo(t) fi(t) Causal System A causal system satisfies fi(t) = 0 for t < t0 fo(t) = 0 for t < t0
Which of the following systems are causal? Basic Concept fo(t) fi(t) Causal System fi(t) fo(t) t t t0 t0 fi(t) fo(t) t t t0 t0 fi(t) fo(t) t t t0 t0
Unit Impulse Response h(t)=L[ (t)] (t) LTI System f(t) L[f(t)]=? Facts: = = ( ) ( ) ( ) ( ) ( ) f t d f t d f t = = ( ) [ ( )] f L t d [ ( )] ( ) ( ) L f t L f t d Convolution = ( ) ( ) f h t d
Unit Impulse Response h(t)=L[ (t)] (t) LTI System f(t) [ L L[f(t)]=? * ) ( Facts: ( = f )] = h = f ( f ( )] ) f f t t ) t ( ) ( ) ( ) ( ) ( t d t d t = = ( ) [ ( f L t d [ ( )] ) ( ) L f t L f t d Convolution = ( ) ( ) f h t d
Unit Impulse Response Impulse Response LTI System h(t) f(t) f(t)*h(t)
Convolution Definition The convolution of two functions f1(t) and f2(t) is defined as: = ( ) ( ) ( ) f t f f t d 1 2 = ( ) * ( ) f t f t 1 2
Properties of Convolution = ( ) * ( ) ( ) * ( ) f t f t f t f t 1 2 2 1 = = = ( ) * ( ) ( ) ( ) ( ) ( ) f t f t f f t d f f t d 1 2 1 2 1 2 = = t = ] ) ( ) [ ( ( ) f t f t t d t 1 2 = t = = = ( ) ( ) f t f d 1 2 = f = ( ) * ( ) f t f t ( ) ( ) t f d 2 1 1 2
Properties of Convolution = ( ) * ( ) ( ) * ( ) f t f t f t f t 1 2 2 1 f(t) f(t)*h(t) Impulse Response LTI System h(t) h(t)*f(t) h(t) Impulse Response LTI System f(t)
Properties of Convolution = [ ( ) * ( )] * ( ) ( [ * ) ( ) * ( )] f t f t f t f t f t f t 1 2 3 1 2 3
The following two systems are identical Properties of Convolution = [ ( ) * ( )] * ( ) ( [ * ) ( ) * ( )] f t f t f t f t f t f t 1 2 3 1 2 3 h1(t) h2(t) h3(t) h2(t) h3(t) h1(t)
Properties of Convolution = ( ) * ( ) ( ) f t t f t f(t) f(t) (t) = ( ) * ( ) ( ) ( ) f t t f t d = = ( ) ( ) f t d (t ) f
Properties of Convolution = ( ) * ( ) ( ) f t t f t f(t) f(t) (t) = ( ) * ( ) ( ) f t t T f t T = ( ) * ( ) ( ) ( ) f t t T f t T d = = ( ) ( ) f t T d ( ) f t T
Properties of Convolution = ( ) * ( ) ( ) f t t T f t T (t T) f(t) f(t T) 0 T f(t) f(t) t t 0 0 T
System function (tT) serves as an ideal delay or a copier. Properties of Convolution = ( ) * ( ) ( ) f t t T f t T (t T) f(t) f(t T) 0 T f(t) f(t) t t 0 0 T
Properties of Convolution ( ) * ( ) ( ) ( ) F f t f t F j F j 1 2 1 2 j = t [ ( ) * ( )] ( ) ( ) F f t f t f f t d e dt 1 2 1 2 = j t ( ) ( ) f f t e dt d 1 2 = j ( ) ( ) f F j e d 1 2 = = j ( ) ( ) F j F j ( ) ( ) F j f e d 1 2 2 1
Time Domain Frequency Domain convolution multiplication Properties of Convolution ( ) * ( ) ( ) ( ) F f t f t F j F j 1 2 1 2 j = t [ ( ) * ( )] ( ) ( ) F f t f t f f t d e dt 1 2 1 2 = j t ( ) ( ) f f t e dt d 1 2 = j ( ) ( ) f F j e d 1 2 = j j = j ( ) ( ) ( ) ( ) F F F j f e d 2 1 1 2
Time Domain Frequency Domain convolution multiplication Properties of Convolution j j ( ) * ( ) ( ) ( ) F f t f t F F 1 2 1 2 Impulse Response LTI System h(t) f(t) f(t)*h(t) Impulse Response LTI System H(j ) F(j ) F(j )H(j )
Time Domain Frequency Domain convolution multiplication Properties of Convolution j j ( ) * ( ) ( ) ( ) F f t f t F F 1 2 1 2 F(j )H1(j )H2(j )H3(j ) F(j )H1(j ) H1(j ) H2(j ) H3(j ) F(j ) F(j )H1(j )H2(j )
Properties of Convolution j j ( ) * ( ) ( ) ( ) F f t f t F F 1 2 1 2 Fo(j ) H(j ) Fi(j ) 1 p p 0 0 0 An Ideal Low-Pass Filter
Properties of Convolution j j ( ) * ( ) ( ) ( ) F f t f t F F 1 2 1 2 Fo(j ) H(j ) Fi(j ) 1 p p 0 0 0 An Ideal High-Pass Filter
Properties of Convolution 1 ( ) ( ) ( ) [ ( )] F f t f t F j F j d 1 2 1 2 2 1 ( ) ( ) ( ) * ( ) F f t f t F j F j 1 2 1 2 2
Time Domain Frequency Domain multiplication convolution Properties of Convolution 1 ( ) ( ) ( ) [ ( )] F f t f t F j F j d 1 2 1 2 2 1 j j ( ) ( ) ( ) * ( ) F f t f t F F 1 2 1 2 2
