Signals and Systems Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair • Examples of Fourier Transforms Topic three • Fourier Transforms of Periodic Signals • Properties of the CT Fourier Transform • The Convolution Property of the CTFT • Frequency Response and LTI Systems Revisited • Multiplication Property and Parseval’s Relation • The DT Fourier Transform •
Book Chapter4: Section1 Fourier’s Derivation of the CT Fourier Transform x ( t ) - an aperiodic signal view it as the limit of a periodic signal as T → ∞ For a periodic signal, the harmonic components are spaced ω 0 = 2π/T apart ... As T → ∞, ω 0 → 0, and harmonic components are spaced closer and closer in frequency Fourier series Fourierintegral Computer Engineering Department, Signals and Systems 2
Book Chapter4: Section1 Motivating Example: Square wave increases kept fixed 2sin( k T ) Discrete 0 1 a k k T frequency 0 points become denser in 2sin( T ) ω as T 1 Ta increases k k 0 Computer Engineering Department, Signals and Systems 3
Book Chapter4: Section1 So, on with the derivation ... For simplicity, assume x ( t ) has a finite duration. T T x t ( ), t 2 2 x t ( ) T periodic , t 2 As T , x t ( ) x t ( ) for all t Computer Engineering Department, Signals and Systems 4
Book Chapter4: Section1 Derivation (continued) 2 jk t x t ( ) a e ( ) 0 k 0 T k T T 2 2 1 1 jk t jk t a x t e ( ) dt x t e ( ) dt 0 0 k T T T T 2 2 x t ( ) x t ( )in this interval 1 jk t x t e ( ) dt (1) 0 T If we define j t X ( j ) x t e ( ) dt then Eq.(1) X ( jk ) 0 a k T Computer Engineering Department, Signals and Systems 5
Book Chapter4: Section1 Derivation (continued) T T Thus, for t 2 2 1 jk t x t ( ) x t ( ) X ( jk ) e 0 0 T k a k 1 jk t X ( jk ) e 0 0 0 2 k As T , d ,we get the CT Fourier Transform pair 0 1 j t x t ( ) X ( j ) e d Synthesis equation 2 j t X ( j ) x t e ( ) dt A nalysis equation Computer Engineering Department, Signals and Systems 6
Book Chapter4: Section1 For what kinds of signals can we do this? It works also even if x ( t ) is infinite duration, but satisfies: (1) 2 Finite energy x t ( ) dt a) In this case, there is zero energy in the error 1 2 j t e t ( ) x t ( ) X j ( ) e d Then e t ( ) dt 0 2 Dirichlet conditions b) 1 j t (i) X j ( ) e d x t ( ) at points of continuity 2 1 j t (ii) X j ( ) e d midpoint at discontinuity 2 (iii) Gibb's phenomenon By allowing impulses in x(t )or in X(j ω ), we can represent even more Signals c) E.g. It allows us to consider FT for periodic signals Computer Engineering Department, Signals and Systems 7
Book Chapter4: Section1 Example #1 ( ) a x t ( ) ( ) t j t X j ( ) ( ) t e dt 1 1 j t ( ) t e d Synthesis equation for ( ) t 2 ( ) b x t ( ) ( t t ) 0 j t X j ( ) ( t t e ) dt 0 j t e 0 Computer Engineering Department, Signals and Systems 8
Book Chapter4: Section1 Example #2: Exponential function at x t ( ) e u t a ( ), 0 j t at j t X j ( ) x t e ( ) dt e e dt 0 ( a j ) t e 1 1 ( a j ) t ( ) e 0 a j a j Even symmetry Odd symmetry Computer Engineering Department, Signals and Systems 9
Book Chapter4: Section1 Example #3: A square pulse in the time-domain 2sin T T 1 j t 1 X j ( ) e dt T 1 Note the inverse relation between the two widths ⇒ Uncertainty principle Computer Engineering Department, Signals and Systems 10
Book Chapter4: Section1 Useful facts about CTFT’s X (0) x t dt ( ) 1 x (0) X j ( ) d 2 Example above: x t dt ( ) 2 T X (0) 1 1 Ex. above: (0) x X j ( ) d 2 1 (Area of the triangle) 2 Computer Engineering Department, Signals and Systems 11
Book Chapter4: Section1 Example #4: e 2 at x t ( ) A Gaussian,important in probability, optics, etc. 2 at j t X j ( ) e e dt j j 2 2 2 a t [ j t ( ) ] a ( ) a 2 a 2 a e dt 2 j 2 a t ( ) [ e 2 a dt e ]. 4 a a 2 (Pulse width in t )•(Pulse width in ω) e 4 a ⇒ ∆ t•∆ ω ~ (1/a 1/2 )•(a 1/2 ) = 1 a Also a Gaussian! Uncertainty Principle! Cannot make both ∆ t and ∆ω arbitrarily small. Computer Engineering Department, Signals and Systems 12
Book Chapter4: Section1 CT Fourier Transforms of Periodic Signals Suppose ( X j ) ( ) 0 1 1 j t j t x ( ) t ( ) e d e periodic in with freq t ue ncy 0 0 0 2 2 That i s j t e 2 ( ) All the energy is concentrated in one fr equency 0 0 0 More generall y jk t x ( ) t a e X j ( ) 2 a ( k ) 0 k k 0 k k Computer Engineering Department, Signals and Systems 13
Book Chapter4: Section1 Example #5: 1 1 j t j t x t ( ) cos t e e 0 0 0 2 2 X j ( ) ( ) ( ) 0 0 “Line Spectrum” Computer Engineering Department, Signals and Systems 14
Book Chapter4: Section1 Example #6: x t ( ) ( t nT ) Sampling function n 1 1 T 2 x(t) jk t x t ( ) a x t e ( ) dt 0 k T T T 2 2 k 2 X j ( ) ( ) T T n 2 a k k 0 Same function in the frequency-domain! Note: (period in t ) T ⇔ (period in ω) 2π/ T Inverse relationship again! Computer Engineering Department, Signals and Systems 15
Book Chapter4: Section1 Properties of the CT Fourier Transform 1) Linearity ax t ( ) by t ( ) aX j ( ) bY j ( ) j t 2) Time Shifting ( x t t ) e X j ( ) 0 0 j t j t j t Proof: x t ( t e ) dt e x t e ( ) dt 0 0 t X ( j ) FT magnitude unchanged j t e X j ( ) X j ( ) 0 Linear change in FT phase j t ( e X j ( )) X j ( ) t 0 0 Computer Engineering Department, Signals and Systems 16
Book Chapter4: Section1 Properties (continued) Conjugate Symmetry * x t ( ) real X ( j ) X ( j ) X ( j ) X j ( ) Even X ( j ) X j ( ) Odd Re X { ( j )} Re X j { ( )} Eve n Im X { ( j )} Im X j { ( )} Od d Computer Engineering Department, Signals and Systems 17
Book Chapter4: Section1 The Properties Keep on Coming ... 1 Time-Scaling ( x at ) X j ( ) a a a 1 E.g. a 1 at t x ( t ) X ( j ) compressed in time stretched in frequency a) ( )real and even x t ( ) x t x ( t ) * X j ( ) X ( j ) X ( j ) Real & even b) ( )real and odd ( ) x t x t x ( t ) −𝑌 ∗ (𝑘𝜕) * X j ( ) X ( j ) X ( j ) Purely imaginary &: c) X j ( ) Re X j { ( )}+ jIm X j { ( )} For real ( ) x t Ev x t { ( )} Od x t { ( )} Computer Engineering Department, Signals and Systems 18
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