Signals and Systems Chapter 4: The Continuous Time Fourier Transform - PowerPoint PPT Presentation

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