Signal processing - Fourier representation of signals

1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform a) Relationship to the z – Transform and Periodicity b) Symmetry Properties c) Operational Properties d) Correlation of Signals e) Signals with Poles on the Unit Circle

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Nguyễn Công Phương SIGNAL PROCESSING Fourier Representation of Signals Contents I. Introduction II. Discrete – Time Signals and Systems III. The z – Transform IV.Fourier Representation of Signals V. Transform Analysis of LTI Systems VI. Sampling of Continuous – Time Signals VII.The Discrete Fourier Transform VIII.Structures for Discrete – Time Systems IX. Design of FIR Filters X. Design of IIR Filters XI. Random Signal Processing sites.google.com/site/ncpdhbkhn 2 sites.google.com/site/ncpdhbkhn 3 Fourier Representation of Signals ⋮ Fourier Representation of Signals 1. Sinusoidal Signals and their Properties a) Continuous – Time Sinusoids b) Discrete – Time Sinusoids c) Frequency Variables and Units 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 4 Continuous – Time Sinusoids (1) • A: the amplitude • θ: phase (radians, rad) • F0: frequency (Hertz, Hz) • Ω0: angular frequency (rad/s) • T0: period (s) sites.google.com/site/ncpdhbkhn 5 02( ) cos( ),x t A F t tpi θ= + − ∞ < < ∞ 0 t ( )x t A cosA θ 0T 0 02 FpiΩ = 0 0 0 1 2T F pi = = Ω cos sinje jϕ ϕ ϕ± = ± 0 0 02 2 cos( ) ( ) j t j tj jAA F t e e e eθ θpi θ Ω − Ω−+ = + sites.google.com/site/ncpdhbkhn 6 Continuous – Time Sinusoids (2) 0 t1T 1 12( ) cosx t F tpi= t0 2T 2 22( ) cosx t F tpi= sites.google.com/site/ncpdhbkhn 7 Continuous – Time Sinusoids (3) 0 02 1( ) j t j F t s t e e piΩ= = 0 02 2 2 2( ) j t j F t s t e e piΩ= = ⋮ 0 02( ) jk t j kF t ks t e e piΩ = = The fundamental/first harmonic The second harmonic The kth harmonic 0 0 0 0 0 0 0 0 0 0 t T t T jk t jm t k mt t T k m s t s t dt e e dt k m + + Ω − Ω = = =  ≠ ∫ ∫ * , ( ) ( ) , sites.google.com/site/ncpdhbkhn 8 Continuous – Time Sinusoids (4) 0 t 0 t 1 0 0 1 0 2 02 2 3 2 5( ) cos( ) cos( ) cos( )x t A F t A F t A F tpi pi pi= + + 2 0 0 1 0 2 02 2 2 83 2 7 14( ) cos( ) cos( . ) cos( . )x t B F t B F t B F tpi pi pi= + + Fourier Representation of Signals 1. Sinusoidal Signals and their Properties a) Continuous – Time Sinusoids b) Discrete – Time Sinusoids c) Frequency Variables and Units 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 9 sites.google.com/site/ncpdhbkhn 10 Discrete – Time Sinusoids (1) 0 nT 0 nT 0 nT 0 nT Discrete – Time Sinusoids (5) • The sequence x[n] = Acos(2πf0 + θ) is periodic in ω0 with fundamental period 2π and periodic in f0 with fundamental period one. (periodic in frequency), therefore: 1. Sinusoidal sequences with ω0 separated by k2π are identical 2. Frequencies are always within an interval of 2π radians: –π < ω ≤ π or 0 ≤ ω < 2π (the fundamental frequency range) 3. Acos[ω0(n + n0) + θ] = Acos[ω0n + (ω0n0 + θ)]: a time shift is equivalent to a phase change 4. The rate of oscillation of a discrete – time sine increases if ω0 increases from 0 to π, but becomes slower if ω0 increases from π to 2π. sites.google.com/site/ncpdhbkhn 14 sites.google.com/site/ncpdhbkhn 15 Discrete – Time Sinusoids (6) 2 [ ] , ,k j knj n N k k ks n A e A e k n pi ω = = −∞ < < ∞ periodic in time[ ] [ ], ( )k ks n N s n+ = periodic in frequency[ ] [ ], ( )k N ks n s n+ = 0 0 0 0 2 21 1 0 n n N n n N j kn j mn N N k m n n n n N k m s n s n e e k m pi pi= + − = + − − = = = = =  ≠ ∑ ∑* , [ ] [ ] , Fourier Representation of Signals 1. Sinusoidal Signals and their Properties a) Continuous – Time Sinusoids b) Discrete – Time Sinusoids c) Frequency Variables and Units 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 16 sites.google.com/site/ncpdhbkhn 17 Frequency Variables and Units ,HzF s F− 2 s F − 0 0 0 0 2 s F 1 s F T = rad , sec Ω cycles , samples f radians samples ,ω 2 s Fpi− s Fpi− s Fpi 2 s Fpi 1− 0 5.− 0 5. 1 2pipipi−2pi− 2( ) cos( )x t A Ftpi θ= + ( ) cos( )x t A t θ= Ω + [ ] cos( )x n A nω θ= + 2[ ] cos( )x n A fnpi θ= + Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals a) Fourier Series for Continuous – Time Periodic Signals b) Fourier Transform for Continuous – Time Aperiodic Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 18 sites.google.com/site/ncpdhbkhn 19 Fourier Series for Continuous – Time Periodic Signals (1) ⋮ 0 1 j t c e Ω 02 2 j t c e Ω 03 3 j t c e Ω 04 4 j t c e Ω 05 5 j t c e Ω 0( ) jk t k k x t c e ∞ Ω =−∞ = ∑ Fourier Series for Continuous – Time Periodic Signals (2) • Plot of |ck|: the magnitude spectrum of x(t) • Plot of θk: the phase spectrum of x(t) sites.google.com/site/ncpdhbkhn 20 0( ) jk t k k x t c e ∞ Ω =−∞ = ∑ 0 0 0 00 1 ( ) t T jk t k t c x t e dt T + − Ω = ∫Continuous – Time Fourier Series (CTFS) Fourier Synthesis Fourier Analysis The Fourier series representation of a continuous – time periodic signal k k kc c θ= ∠ sites.google.com/site/ncpdhbkhn 21 Fourier Series for Continuous – Time Periodic Signals (3)Ex. 1 Find the Fourier series? ( )x t A 0 ττ− 0T0T− t 0 0 0 00 1 ( ) t T jk t k t c x t e dt T + − Ω = ∫ 0 0( ) ,x t A kT t kTτ τ= − + ≤ ≤ + 0 0 1 jk t kc Ae dtT τ τ − Ω − → = ∫ 0 0 0 jk tA e T jk τ τ − Ω −   =   − Ω  0 0 0 0 jk jkA e e k T j τ τΩ − Ω − = Ω 0 0 0cos( ) sin( ) jk e k j kτ τ τΩ = Ω + Ω 0 0 0cos( ) sin( ) jk e k j kτ τ τ− Ω = − Ω + − Ω 0 0 0 0 0 0 2 2sin( ) sin( ) k A k kA c k T T k τ ττ τ Ω Ω → = = Ω Ω sites.google.com/site/ncpdhbkhn 22 Fourier Series for Continuous – Time Periodic Signals (4)Ex. 1 Find the Fourier series? ( )x t A 0 ττ− 0T0T− t 0 0 0 0 0 0 0 2 22 2 2 sin( ) sin( ) sin k kA c T k k FA A T k F T ττ τ pi ττ τ φ pi τ φ Ω = Ω = = 0 0 0 2 0 2 0 sinA A c T T τ τ = = 0( ) jk t k k x t c e ∞ Ω =−∞ = ∑ 0 1 0 0 22 2 sin( )FA c T F pi ττ pi τ− − = − 0 1 0 0 22 2 sin( )FA c T F pi ττ pi τ = 0 00 0 0 0 0 0 0 2 22 2 2 2 2 sin( ) sin( ) ( ) ... ... j t j tF FA A A x t e e T F T T F pi τ pi ττ τ τ pi τ pi τ − Ω Ω− = + + + + − sites.google.com/site/ncpdhbkhn 23 Fourier Series for Continuous – Time Periodic Signals (5)Ex. 1 Find the Fourier series? ( )x t A 0 ττ− 0T0T− t 3 3 3 sin( / ) / k A k c k pi pi → = 2 0 33 0 33 0 6 o. . A A Aτ τ = = = ∠0 0 2A c T τ = 0 1 0 0 22 2 sin( )FA c T F pi ττ pi τ− − = − 0 1 0 0 22 2 sin( )FA c T F pi ττ pi τ = 3 0 28 0 28 0 3 3 osin( / ) . . / A A Api pi = = = ∠ 0 6T τ= 3 0 28 0 28 0 3 3 osin( / ) . . / A A Api pi − = = = ∠ − 2 2 3 3 4 4 5 5 6 6 0 14 0 14 0 0 0 069 0 069 0 055 0 055 0 0 . ; . ; ; . ; . ; . ; . ; c A c A c c c A c A c A c A c c − − − − − = = = = = − = − = − = − = = sites.google.com/site/ncpdhbkhn 24 Fourier Series for Continuous – Time Periodic Signals (6)Ex. 1 ( )x t A 0 ττ− 0T0T− t Find the Fourier series? 0 1 1 2 2 3 3 4 4 5 5 6 6 0 33 0 28 0 28 0 14 0 14 0 0 0 069 0 069 0 055 0 055 0 0 . . ; . ; . ; . ; ; . ; . ; . ; . ; ; c A c A c A c A c A c c c A c A c A c A c c − − − − − − = = = = = = = = − = − = − = − = = k 0 1 2 3 4 5 61−2−3−4−5−6− kc sites.google.com/site/ncpdhbkhn 25 Fourier Series for Continuous – Time Periodic Signals (7)Ex. 2 Find the Fourier series? ( )x t A 0 ττ− 0T0T− t A− sites.google.com/site/ncpdhbkhn 26 Fourier Series for Continuous – Time Periodic Signals (8)Ex. 3 Find the Fourier series? ( )x t 0 0T t A 0 0 0 00 1 ( ) t T jk t k t c x t e dt T + − Ω = ∫ 0 0 0 0 0 0 0 0 0 4 2 4 3 2 , / ( ) , / A t A kT t T kT T x t A t A T kT t T kT T  − + < ≤ +  =   − + < ≤ +  0 0 0 0 0 2 0 0 0 2 0 0 1 4 1 4 3 / / T jk t T jk t T A t A e dt T T A t A e dt T T − Ω − Ω   − = + +      + −    ∫ ∫ 0 00 0 0 0 0 0 0 0 2 2 2 0 0 0 0 2 2 2 0 0 4 4 3 / / / / T Tjk t jk t T Tjk t jk t T T A A te dt e dt T T A A te dt e dt T T − Ω − Ω − Ω − Ω = − + + + − ∫ ∫ ∫ ∫ sites.google.com/site/ncpdhbkhn 27 Fourier Series for Continuous – Time Periodic Signals (9)Ex. 3 ( )x t 0 0T t A Find the Fourier series? 0 0 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 2 2 2 0 0 4 4 3 / / / / T Tjk t jk t k T Tjk t jk t T T A A c te dt e dt T T A A te dt e dt T T − Ω − Ω − Ω − Ω = − + + + − ∫ ∫ ∫ ∫ 0 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0 2 4 4 3 / / ( ) ( ) Tjk t jk t jk t Tjk t jk t jk t T A te e A e T jk jk T jk A te e A e T jk jk T jk − Ω − Ω − Ω − Ω − Ω − Ω    = − − + +   − Ω − Ω − Ω      + − −   − Ω − Ω − Ω   ...= sites.google.com/site/ncpdhbkhn 28 Fourier Series for Continuous – Time Periodic Signals (10) Convergence conditions (Dirichlet conditions): 0 0 0 ( ) t T t x t dt + < ∞∫ 0 0 0 0 2 0( ) , lim ( ) ( ) m t Tjk t m k mtmk m x t c e x t x t dt +Ω →∞ =− = − =∑ ∫ 1. The periodic signal x(t) is absolutely integrable over any period, that is, x(t) has a finite area per period: 2. The periodic signal x(t) has a finite number of maxima, minima, and finite discontinuities per period. sites.google.com/site/ncpdhbkhn 29 Fourier Series for Continuous – Time Periodic Signals (11) ( )x t A 0 ττ− 0T0T− t ( )x t 0 0T t A Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals a) Fourier Series for Continuous – Time Periodic Signals b) Fourier Transform for Continuous – Time Aperiodic Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 30 sites.google.com/site/ncpdhbkhn 31 Fourier Transforms for Continuous – Time Aperiodic Signals (1) ( )x t A 0 ττ− 0T0T− t k kc 0 1F T ∆ = 0T →∞ Ex. 1 sites.google.com/site/ncpdhbkhn 32 Fourier Transforms for Continuous – Time Aperiodic Signals (2) 0( ) jk t k k x t c e ∞ Ω =−∞ = ∑ 0 0 0 00 1 ( ) t T jk t k t c x t e dt T + − Ω = ∫Continuous – Time Fourier Series (CTFS) Fourier Synthesis Fourier Analysis The Fourier series representation of a continuous – time periodic signal 22( ) ( ) j Ftx t X j F e dFpipi∞ −∞ = ∫ 22( ) ( ) j FtX j F x t e dtpipi ∞ − −∞ = ∫ Continuous – Time Fourier Transform (CTFT) Fourier Synthesis Fourier Analysis The Fourier transform representation of a continuous – time aperiodic signal sites.google.com/site/ncpdhbkhn 33 Fourier Transforms for Continuous – Time Aperiodic Signals (3) 200 300 400 500 600 700 800 0 0.2 0.4 0.6 0.8 1 200 300 400 500 600 700 800 0 0.5 1 200 300 400 500 600 700 800 0 0.5 1 200 300 400 500 600 700 800 0 0.2 0.4 0.6 0.8 1 sites.google.com/site/ncpdhbkhn 34 Fourier Transforms for Continuous – Time Aperiodic Signals (4) Convergence conditions ( )x t dt ∞ −∞ < ∞∫ 1. The aperiodic signal x(t) is absolutely integrable, that is, x(t) has a finite area. 2. The aperiodic signal x(t) has a finite number of maxima, minima, and finite discontinuities in any finite interval. sites.google.com/site/ncpdhbkhn 35 Fourier Transforms for Continuous – Time Aperiodic Signals (5)Ex. 2 ate− t 0 0 0 0 , ( ) , at e t x t t − > =  < Find the Fourier transform of 22( ) ( ) j FtX j F x t e dtpipi ∞ − −∞ = ∫ 2 0 at j Ft e e dtpi ∞ − − = ∫ 2 0 ( )a j F t e dtpi ∞ − + = ∫ 2 02 ( )a j F te a j F pi pi ∞ − + = − − 1 10 2 2a j F a j Fpi pi= − =− − + 2 2 1 2 2 atan ( ) F aa F pi pi   = ∠ −   + -10 -8 -6 -4 -2 0 2 4 6 8 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 -10 -8 -6 -4 -2 0 2 4 6 8 10 -1.5 -1 -0.5 0 0.5 1 1.5 sites.google.com/site/ncpdhbkhn 36 Fourier Transforms for Continuous – Time Aperiodic Signals (6)Ex. 3 0 , ( ) , A t x t t τ τ  < =  > Find the Fourier transform of 22( ) ( ) j FtX j F x t e dtpipi ∞ − −∞ = ∫ 2j FtAe dt τ pi τ − − = ∫ 2 2 j FtAe j F τpi τ pi − − = − ( )x t A 0 ττ− t 22 2 sin( )FA F pi τ τ pi τ = -5 -4 -3 -2 -1 0 1 2 3 4 5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 sites.google.com/site/ncpdhbkhn 37 Fourier Transforms for Continuous – Time Aperiodic Signals (7) aperiodic( ) :x t periodic( ) :s t ( ) ( ) ( ) s x t x t s t= 02 22( ) ( ) j F kt j Ft s k k X j F x t c e e dtpi pipi ∞ ∞ − −∞ =−∞   → =     ∑∫ 02 ( )( ) j F kF t k k c x t e dtpi ∞ ∞ − − −∞ =−∞   =   ∑ ∫ 02 02 ( ) ( ) [ ( )] j F kF t x t e dt X j F kFpi pi∞ − − −∞ = −∫ 02 2( ) [ ( )]s k k X j F c X j F kFpi pi ∞ =−∞ → = −∑ sites.google.com/site/ncpdhbkhn 38 Fourier Transforms for Continuous – Time Aperiodic Signals (8) 02 2( ) [ ( )]s k k X j F c X j F kFpi pi ∞ =−∞ = −∑ ( )s t 1 0 ττ− 0T0T− t t ( )x t F 2( )X j Fpi F 0 0 0 sin( ) ( )k kF c F kF τ τ τ = 0 0F 02F 03F 04F 05F0F− 02F− 03F− 04F− 05F− F 2( ) s X j Fpi 0 0F 02F 03F 04F 05F 0F− 02F− 03F− 04F− 05F−0 2 2( ) [ ( )] s k k X j F c X j F kFpi pi ∞ =−∞ = −∑ Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals a) Fourier Series for Discrete – Time Periodic Signals b) Fourier Transform for Discrete – Time Aperiodic Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 39 sites.google.com/site/ncpdhbkhn 40 Fourier Series for Discrete – Time Periodic Signals (1) 0( ) jk t k k x t c e ∞ Ω =−∞ = ∑ 0 0 0 00 1 ( ) t T jk t k t c x t e dt T + − Ω = ∫Continuous – Time Fourier Series (CTFS) Fourier Synthesis Fourier Analysis 21 0 [ ] N j kn N k k x n c e pi − = =∑ 21 0 1 [ ] N j kn N k n c x n e N pi − − = = ∑Discrete – Time Fourier Series (DTFS) Fourier Synthesis Fourier Analysis sites.google.com/site/ncpdhbkhn 41 Fourier Series for Discrete – Time Periodic Signals (2)Ex. 1 Find the Fourier coefficients of x[n] = sinω0n = sin2πf0n. 0 02 2 0 1 12 2 2 j j f n j f n j e j x n f n e ej je j θ pi pi θ θ θ pi θ θ − − = +  → = = − = −  cos sin [ ] sin cos sin 0 0 0 02 2 2 22 21 1 1 11 1 2 2 2 2 k k k kj n j n j n j nj n j nN N N Ne x n e e e e ej j j j pi pi pi pipi pi− − = → = − × = −[ ] 0 02 21 1 2 2 k kj n j n N Ne ej j pi pi− = − 0 0 2 21 1 2 2 ( )j k n j N k n N Ne ej j pi pi − = − sites.google.com/site/ncpdhbkhn 42 Fourier Series for Discrete – Time Periodic Signals (3)Ex. 1 Find the Fourier coefficients of x[n] = sinω0n = sin2πf0n. 0 0 2 21 1 2 2 j k n j N k n N Nx n e ej j pi pi − = − ( ) [ ] 0 0 0 0 21 0 2 2 2 2 20 1 1 0 1 1 N j kn N k k j n j n j k n j N k n j N n N N N N N k N k N x n c e c e c e c e c e c e pi pi pi pi pi pi − = × − − − − = = + + + + + + ∑ ( ) ( ) [ ] ... ... ... 0 00 1 2 1 1 10 0 0 0 0 2 2k m N k N N c c c c c c cj j− − −→ = = = = = − = =; ; ... ; ; ...; ; ... ; ; ...; ; 0 0 1 2 3 4 1 12 5 0 0 0 2 2 k N c c c c cj j= = → = = = = − =, ; ; ; ; 0 1 2 3 4 5 6 7 88− 7− 6− 5− 4− 3− 2− 1− 0 5. kc k 9 109−10− sites.google.com/site/ncpdhbkhn 43 Fourier Series for Discrete – Time Periodic Signals (4)Ex. 2 Find the Fourier series of the periodic sequence 1 any interger 0 , , [ ] [ ] , otherwise N l n mN m n n lNδ δ ∞ =−∞ = = − =   ∑ 21 0 1 [ ] N j kn N k n c x n e N pi − − = = ∑ 0 1 2 3 4 5 6 7 88− 7− 6− 5− 4− 3− 2− 1− 1 [ ]N nδ n ⋯⋯21 0 1 [ ] N j kn N n n e N pi δ − − = = ∑ 1 , all k N = 21 0 1 [ ] , all N j kn N N n n e n N pi δ − − = → = ∑ 0 1 2 3 4 5 6 7 88− 7− 6− 5− 4− 3− 2− 1− 1 / N kc k ⋯⋯ sites.google.com/site/ncpdhbkhn 44 Fourier Series for Discrete – Time Periodic Signals (5)Ex. 3 Find the Fourier series of the rectangular pulse sequence. 21 0 1 [ ] N j kn N k n c x n e N pi − − = = ∑ 21 L j knN n L e N pi − =− = ∑ 1 0 1 1 NN n n a a a − = − = − ∑ m n L= + 22 0 1 ( )L j k m LN k m c e N pi − − = = ∑ 2 22 0 1 m Lj kL j k N N m e e N pi pi − =   =     ∑ 2 2 2 1 2 1 1 ( )j kL j k L N N j k N e e N e pi pi pi − + − − = − 2 2 2 21 2 2/ / / /( ) [ sin( / )]j j j j je e e e e jθ θ θ θ θ θ− − − −− = − = 2 1 21 2 1 2 sin sin k k L N c N k N pi pi   +     → =       0 1 2 3 4 5 6 7 88− 7− 6− 5− 4− 3− 2− 1− 1 [ ]x n n ⋯⋯ L N sites.google.com/site/ncpdhbkhn 45 Fourier Series for Discrete – Time Periodic Signals (6)Ex. 3 Find the Fourier series of the rectangular pulse sequence. 2 1 21 2 1 2 sin sin k L N N k N pi pi   +      =       21 L j knN k n L c e N pi − =− = ∑ 2 1 0 2 2 1 21 2 1 2 , , , , ... sin , otherwise sin k L k N N N c k L N N k N pi pi + = ± ±      = +              0 1 2 3 4 5 6 7 88− 7− 6− 5− 4− 3− 2− 1− 1 [ ]x n n ⋯⋯ L N 2 0 0 0 0 2 1 1 1 1 1 1 2 1 terms ... L j kn N n L L c e N e e e N L N pi − =− + =   = + + +     + = ∑  0 5. kc k ⋯⋯ 0 33.0 33. 0 17.−0 17.− 0 Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals a) Fourier Series for Discrete – Time Periodic Signals b) Fourier Transform for Discrete – Time Aperiodic Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 46 sites.google.com/site/ncpdhbkhn 47 Fourier Transform for Discrete – Time Aperiodic Signals (1) 0 1 2 3 4 5 6 7 88− 7− 6− 5− 4− 3− 2− 1− 1 [ ]x n n ⋯⋯ L N 0 5. kc k ⋯⋯ 0 33.0 33. 0 17.−0 17.− 0 sites.google.com/site/ncpdhbkhn 48 Fourier Transform for Discrete – Time Aperiodic Signals (2) 21 0 [ ] N j kn N k k x n c e pi − = =∑ 21 0 1 [ ] N j kn N k n c x n e N pi − − = = ∑ Discrete – Time Fourier Series (DTFS) Fourier Synthesis Fourier Analysis 0 0 21 2 [ ] ( ) t j j n t x n X e e d pi ω ω ω pi + = ∫ ( ) [ ] j j n n X e x n eω ω ∞ − =−∞ = ∑ Discrete – Time Fourier Transform (DTFT) Fourier Synthesis Fourier Analysis 21 ( ) j k N kc X eN pi = sites.google.com/site/ncpdhbkhn 49 Fourier Transform for Discrete – Time Aperiodic Signals (3)Ex. Find the DTFT of the sequence x[n] = δ[n + 1] + δ[n] + δ[n – 1]. ( ) [ ] j j n n X e x n eω ω ∞ − =−∞ = ∑ 1 1 [ ] j n n x n e ω− =− = ∑ 1j je eω ω−= + + 1 2cosω= + 0 1 2 3 44− 3− 2− 1− 1 [ ]x n n -6 -4 -2 0 2 4 6 0 1 2 3 ω M a g n i t u d e s p e c t r u m -6 -4 -2 0 2 4 6 -2 0 2 ω P h a s e s p e c t r u m Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 50 Continuous – time signals Discrete – time signals Time – domain Frequency – domain Time – domain Frequency – domain P e r i o d i c s i g n a l s A p e r i o d i c s i g n a l s F o u r i e r s e r i e s F o u r i e r t r a n s f o r m s Continuous & periodic Discrete & aperiodic Discrete & periodic Discrete & periodic Continuous & aperiodic Continuous & aperiodic Discrete & aperiodic Continuous & periodic ( )x t A 0 ττ− 0T0T− t k 0 1 2 3 4 5 61−2−3−4−5−6− kc 0 0 2 T piΩ = 0( ) jk t k k x t c e ∞ Ω =−∞ =∑ 0 0 0 00 1 ( ) t T jk t k t c x t e dt T + − Ω = ∫ ICTFS CTFS 2 0 [ ] N j kn N k k x n c e pi = =∑ 21 0 1 [ ] N j kn N k n c x n e N pi − − = = ∑ IDTFS DTFS 1 2 ( ) ( ) j tx t X j e d pi ∞ Ω −∞ = Ω Ω∫ ( ) ( ) jk tX j x t e dt∞ − Ω −∞ Ω = ∫ ICTFT CTFT 0 0 0 1 2 [ ] ( ) t T j j n t x n X e e dω ω ω pi + = ∫ ( ) [ ] j j n n X e x n eω ω ∞ − =−∞ =∑ IDTFT DTFT ( )x t A 0 ττ− t -5 -4 -3 -2 -1 0 1 2 3 4 5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ( )X jΩ 0 [ ]x n n ⋯⋯ N kc k ⋯⋯ 0 0 1 2 3 44− 3− 2− 1− [ ]x n n -6 -4 -2 0 2 4 60 0.5 1 1.5 2 2.5 3 ( )jX e ω Ω ω sites.google.com/site/ncpdhbkhn 51 Summary of Fourier Series and Fourier Transforms Periodic ty w th “period” α in one domain implies discret zation with “spacing” of 1/α in the other domain, and vice versa Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform a) Relationship to the z – Transform and Periodicity b) Symmetry Properties c) Operational Properties d) Correlation of Signals e) Signals with Poles on the Unit Circle sites.google.com/site/ncpdhbkhn 52 sites.google.com/site/ncpdhbkhn 53 Relationship to the z – Transform and Periodicity (1) ( ) [ ] ( )j j j n j z e n z e X z x n e X eωω ω ω ∞ − = =−∞ = → = =∑ ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ ( ) ( [ ] ) j j n j n n z re X re x n r eω ω ω ∞ − − =−∞ = → = ∑ sites.google.com/site/ncpdhbkhn 54 Relationship to the z – Transform and Periodicity (2)Ex. Draw the magnitude spectrum of 0 1 2 2 01 2 ( ) [ cos( ) ] K bH z r z r zω − − = − + Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform a) Relationship to the z – Transform and Periodicity b) Symmetry Properties c) Operational Properties d) Correlation of Signals e) Signals with Poles on the Unit Circle sites.google.com/site/ncpdhbkhn 55 sites.google.com/site/ncpdhbkhn 56 Symmetry Properties (1) ( ) [ ] j j n n X e x n eω ω ∞ − =−∞ = ∑ [ ] [ ] [ ] ( ) ( ) ( ) cos( ) sin( ) R I j j j R I j n x n x n jx n X e X e jX e e n j n ω ω ω ω ω ω− = + = + = − ( ) { [ ] [ ]}{cos( ) sin( )} j R I n X e x n jx n n j nω ω ω ∞ =−∞ → = + −∑ { [ ](cos ) [ ]cos( ) [ ]sin( ) [ ]sin( )}R I R I n x n n jx n n jx n n x n nω ω ω ω ∞ =−∞ = + − +∑ { [ ]cos( ) [ ]sin( )} { [ ]cos( ) [ ]sin( )}R I I R n n x n n x n n j x n n x n nω ω ω ω ∞ ∞ =−∞ =−∞ = + + −∑ ∑ ( ) { [ ]cos( ) [ ]sin( ) ( ) { [ ]sin( ) [ ]cos( ) j R R I n j I R I n X e x n n x n n X e x n n x n n ω ω ω ω ω ω ∞ =−∞ ∞ =−∞  = +  →   = − −  ∑ ∑ sites.google.com/site/ncpdhbkhn 57 Symmetry Properties (2) 2 0 1 2 [ ] ( ) j j nx n X e e d pi ω ω ω pi = ∫ [ ] [ ] [ ] ( ) ( ) ( ) cos( ) sin( ) R I j j j R I j n x n x n jx n X e X e jX e e n j n ω ω ω ω ω ω = + = + = + 0 0 1 2 1 2 [ ] ( ) cos( ) ( )sin( ) [ ] ( )sin( ) ( )cos( ) j j R R I j j I R I x n X e n X e n d x n X e n X e n d pi ω ω pi ω ω ω ω ω pi ω ω ω pi 2 2   = −  →    = +  ∫ ∫ sites.google.com/site/ncpdhbkhn 58 Symmetry Properties (3) 0 0 1 2 1 2 ( ) { [ ]cos( ) [ ]sin( ) [ ] ( ) cos( ) ( )sin( ) [ ] ( )sin( ) ( ) cos( )( ) { [ ]sin( ) [ ]cos( ) j j j R R I R R I n j jj I R II R I n X e x n n x n n x n X e n X e n d x n X e n X e n dX e x n n x n n piω ω ω pi ω ωω ω ω ω ω ω pi ω ω ωω ω pi ∞ 2 =−∞ ∞ 2 =−∞   = +  = −    ↔     = += − −    ∑ ∫ ∑ ∫ If x[n] is real 0[ ] [ ]; [ ]R Ix n x n x n= = ( ) [ ]cos( ) ( ) [ ]sin( ) j R n j I n X e x n n X e x n n ω ω ω ω ∞ =−∞ ∞ =−∞  =  →   = −  ∑ ∑ cos( ) cos( ); sin( ) sin( )α α α α− = − = − even symmetry( ) [ ]cos( ) [ ]cos( ) ( ) ( ) ( ) [ ]sin( ) [ ]sin( ) ( ) (odd symmetry) j j R R n n j j I I n n X e x n n x n n X e X e x n n x n n X e ω ω ω ω ω ω ω ω ∞ ∞ − =−∞ =−∞ ∞ ∞ − =−∞ =−∞  = − = =  →   = − − = = −  ∑ ∑ ∑ ∑ sites.google.com/site/ncpdhbkhn 59 Symmetry Properties (4) If x[n] is real 0[ ] [ ]; [ ]R Ix n x n x n= = 2 2 atan ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j j R I j j j jR I j I j R X e X e X e X e X e jX e X eX e X e ω ω ω ω ω ω ω ω ω  = +  = + →  ∠ =  ( ) ( ); ( ) ( ) j j j j R R I IX e X e X e X e ω ω ω ω− − = = − 2 2 even symmetry atan ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (odd symmetry) ( ) j j j j R I j j jI j R X e X e X e X e X eX e X e X e ω ω ω ω ω ω ω ω − − − − − −  = + =  →  ∠ = = −∠  0 0 1 2 1 2 ( ) { [ ]cos( ) [ ]sin( ) [ ] ( ) cos( ) ( )sin( ) [ ] ( )sin( ) ( ) cos( )( ) { [ ]sin( ) [ ]cos( ) j j j R R I R R I n j jj I R II R I n X e x n n x n n x n X e n X e n d x n X e n X e n dX e x n n x n n piω ω ω pi ω ωω ω ω ω ω ω pi ω ω ωω ω pi ∞ 2 =−∞ ∞ 2 =−∞   = +  = −    ↔     = += − −    ∑ ∫ ∑ ∫ sites.google.com/site/ncpdhbkhn 60 Symmetry Properties (5) If x[n] is real 0[ ] [ ]; [ ]R Ix n x n x n= = 0 1 2 [ ] ( ) cos( ) ( )sin( ) j j R Ix n X e n X e n d pi ω ωω ω ω pi 2  = − ∫ ( ) ( ) ( ) cos( ) ( )cos( ) cos( ) cos( ) ( ) ( ) ( )sin( ) ( )sin( ) sin( ) sin( ) j j j jR R R R j j j jI I I I X e X e X e n X e n n n X e X e X e n X e n n n ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω − − − − = → = − =  = − → = − =  0 1 [ ] ( )cos( ) ( )sin( )j jR Ix n X e n X e n d pi ω ωω ω ω pi  → = − ∫ 0 0 1 2 1 2 ( ) { [ ]cos( ) [ ]sin( ) [ ] ( ) cos( ) ( )sin( ) [ ] ( )sin( ) ( ) cos( )( ) { [ ]sin( ) [ ]cos( ) j j j R R I R R I n j jj I R II R I n X e x n n x n n x n X e n X e n d x n X e n X e n dX e x n n x n n piω ω ω pi ω ωω ω ω ω ω ω pi ω ω ωω ω pi ∞ 2 =−∞ ∞ 2 =−∞   = +  = −    ↔     = += − −    ∑ ∫ ∑ ∫ sites.google.com/site/ncpdhbkhn 61 Symmetry Properties (6) 0 0 1 2 1 2 ( ) { [ ]cos( ) [ ]sin( ) [ ] ( ) cos( ) ( )sin( ) [ ] ( )sin( ) ( ) cos( )( ) { [ ]sin( ) [ ]cos( ) j j j R R I R R I n j jj I R II R I n X e x n n x n n x n X e n X e n d x n X e n X e n dX e x n n x n n piω ω ω pi ω ωω ω ω ω ω ω pi ω ω ωω ω pi ∞ 2 =−∞ ∞ 2 =−∞   = +  = −    ↔     = += − −    ∑ ∫ ∑ ∫ If x[n] is real and even [ ] [ ]x n x n− = ( ) [ ]cos( ); ( ) [ ]sin( ) j j R I n n X e x n n X e x n nω ωω ω ∞ ∞ =−∞ =−∞ = = −∑ ∑ 0 1 [ ] ( ) cos( ) ( )sin( ) j j R Ix n X e n X e n d pi ω ωω ω ω pi  = − ∫ 1 0 0 2 even symmetry 0 1 ( ) [ ] [ ]cos( ) ( ) ( ) [ ] ( )cos( ) (odd symmetry) j R n j I j R X e x x n n X e x n X e n d ω ω pi ω ω ω ω pi ∞ =  = +  → =   =  ∑ ∫ sites.google.com/site/ncpdhbkhn 62 Symmetry Properties (7) 0 0 1 2 1 2 ( ) { [ ]cos( ) [ ]sin( ) [ ] ( ) cos( ) ( )sin( ) [ ] ( )sin( ) ( ) cos( )( ) { [ ]sin( ) [ ]cos( ) j j j R R I R R I n j jj I R II R I n X e x n n x n n x n X e n X e n d x n X e n X e n dX e x n n x n n piω ω ω pi ω ωω ω ω ω ω ω pi ω ω ωω ω pi ∞ 2 =−∞ ∞ 2 =−∞   = +  = −    ↔     = += − −    ∑ ∫ ∑ ∫ If x[n] is real and odd [ ] [ ]x n x n− = − ( ) [ ]cos( ); ( ) [ ]sin( ) j j R I n n X e x n n X e x n nω ωω ω ∞ ∞ =−∞ =−∞ = = −∑ ∑ 0 1 [ ] ( ) cos( ) ( )sin( ) j j R Ix n X e n X e n d pi ω ωω ω ω pi  = − ∫ 0 0 2 odd symmetry 1 ( ) ( ) [ ]sin( ) ( ) [ ] ( )sin( ) (odd symmetry) j R j I n j I X e X e x n n x n X e n d ω ω pi ω ω ω ω pi ∞ =−∞   =   → = −   =  ∑ ∫ sites.google.com/site/ncpdhbkhn 63 Symmetry Properties (8) Ex. 1 Draw the spectra of the sequence x[n] = anu[n]. ( ) [ ] j j n n X e x n eω ω ∞ − =−∞ = ∑ 0 n j n n a e ω ∞ − = =∑ 1 1If jae aω− < → < 1 1 ( )j jX e ae ω ω− → = − 1 1 (cos sin )a jω ω= − − 1 1 1 cos sin ( cos sin )( cos sin ) a ja a ja a ja ω ω ω ω ω ω − − = − + − − 2 2 1 1 cos sin ( cos ) ( sin ) a ja a ja ω ω ω ω − − = − − 2 2 1 1 2 1 2 cos sin cos cos a aj a a a a ω ω ω ω − = − − + − + 2 2 22 1 1 2 1 2 1 atan 1 21 2 cos sin ( ) ( ); ( ) ( ) cos cos sin ( ) ( ) ; ( ) ( ) coscos j j j j R R I I j j j j a aX e X e X e X e a a a a aX e X e X e X e a aa a ω ω ω ω ω ω ω ω ω ω ω ω ω ωω − − − − = = = = − − + − + →  −  = = ∠ = = −∠ − + − + sites.google.com/site/ncpdhbkhn 64 Symmetry Properties (9) Ex. 1 Draw the spectra of the sequence x[n] = anu[n]. -3 -2 -1 0 1 2 3 0.8 1 1.2 1.4 1.6 1.8 2 omega (rad) R e a l p a r t a = 0.5 a = -0.5 -3 -2 -1 0 1 2 3 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 omega (rad) I m a g i n a r y p a r t a = 0.5 a = -0.5 -3 -2 -1 0 1 2 3 0.8 1 1.2 1.4 1.6 1.8 2 omega (rad) M a g n i t u d e a = 0.5 a = -0.5 -3 -2 -1 0 1 2 3 -0.5 0 0.5 omega (rad) A n g l e a = 0.5 a = -0.5 sites.google.com/site/ncpdhbkhn 65 Symmetry Properties (10) Ex. 2 Find x[n] from 1 0 , ( ) , cj c X e ω ω ω ω ω pi  < =  < < 0 0 21 2 [ ] ( ) t j j n t x n X e e d pi ω ω ω pi + = ∫ 1 2 c c j n e d ω ω ω ω pi − = ∫ 1 2 c c j nejn ω ω ω pi − = 2 cos( ) sin( ) c c n j n jn ω ω ω ω pi − + = 0sin( ) ,cn n n ω pi = ≠ 010 2 [ ] c c jx e d ω ω ω ω pi − = ∫ 1 2 c c c ω ω ω ω pi pi− = = k [ ]x n ( ) jX e ω 1 0 cωcω− 0T0T− ω sites.google.com/site/ncpdhbkhn 66 Symmetry Properties (11) Ex. 3 Find spectra of 0 1 0 , [ ] , otherwise A n L x n ≤ ≤ − =   ( ) [ ] j j n n X e x n eω ω ∞ − =−∞ = ∑ 1 0 L j n n Ae ω − − = =∑ 1 1 j L j eA e ω ω − − − = − 2 2 2 2 2 2 / / / / / / ( ) ( ) j L j L j L j j j e e eA e e e ω ω ω ω ω ω − − − − − = − 1 2 2 2 ( ) / sin( / ) sin( / ) j L LAe ω ω ω − − = 2 2 21 2 2 sin( / ) ( ) sin( / ) sin( / ) ( ) ( ) sin( / ) j j LX e A LX e A L ω ω ω ω ω ω ω  =  →  ∠ = ∠ − − +∠  0 1 2 3 44− 3− 2− 1− A [ ]x n n -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 omega (rad) M a g n i t u d e - 3 -2 -1 0 1 2 3 -15 -10 -5 0 5 10 15 omega (rad) A n g l e Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform a) Relationship to the z – Transform and Periodicity b) Symmetry Properties c) Operational Properties d) Correlation of Signals e) Signals with Poles on the Unit Circle sites.google.com/site/ncpdhbkhn 67 sites.google.com/site/ncpdhbkhn 68 Operational Properties (1) sites.google.com/site/ncpdhbkhn 69 Operational Properties (2) [ ]c j n e x n ω [ ]( )c jX e ω ω− DTFT Frequency shifting ⊗[ ]x n [ ]y n [ ] c j n c n e ω = 0 ( ) jX e ω ω 2pi− 2pimω− mω ⋯ ⋯ 0 ( ) [ ]c jY X e ω ω−= ω 2pi− 2pi 2 cpi ω+cω ⋯ ⋯ c m ω ω− c m ω ω+ sites.google.com/site/ncpdhbkhn 70 Operational Properties (3) sites.google.com/site/ncpdhbkhn 71 Operational Properties (4) [ ]c j n e x n ω 1 1 2 2 [ ] [ ] ( ) ( )c c j jX e X eω ω ω ω+ −+ DTFT Modulation ⊗[ ]x n [ ]y n [ ] cos( ) c c n nω= 0 ( ) jX e ω ω 2pi− 2pimω− mω ⋯ ⋯ 0 ( ) jY e ω ω 2pi− 2pi ⋯ ⋯ c ω c ω− 0 5 ( ). [ ]cjX e ω ω−0 5 ( ). [ ]cjX e ω ω+ sites.google.com/site/ncpdhbkhn 72 Operational Properties (5) Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform a) Relationship to the z – Transform and Periodicity b) Symmetry Properties c) Operational Properties d) Correlation of Signals e) Signals with Poles on the Unit Circle sites.google.com/site/ncpdhbkhn 73 sites.google.com/site/ncpdhbkhn 74 Correlation of Signals (1) [ ] [ ] [ ],xy n r l x n y n l l ∞ =−∞ = − −∞ < < ∞∑ [ ] [ ] xy yxr l r l= − 1 1 [ ] [ ] xy xy x y r l l E E ρ− ≤ = ≤ [ ] [ ]* [ ] xyr l x l y l= − [ ] [ ] [ ] [ ]* [ ], k y n x k h n k x n h n n ∞ =−∞ = − = −∞ < < ∞∑ [ ] [ ]* [ ] xyr l x l y l= − ( ) ( ) ( ) j j xyR X e Y e ω ωω −= DTFT [ ] [ ]* [ ] x r l x l x l= − 2( ) ( )j x R X e ωω = DTFT sites.google.com/site/ncpdhbkhn 75 Correlation of Signals (2) Ex. 1 Find the autocorrelation of the sequence x[n] = anu[n]. [ ] [ ] [ ]x n r l x n x n l ∞ =−∞ = −∑ [ ] [ ] n l x n x n l ∞ = = −∑ n n l n l a a ∞ − = =∑ 1 1 2 2( ) ( )l l l l l l l l la a a a a a− + + − + + −= + + 2 41( ...)la a a= + + + 21 la a = − Fourier Representation of Signals 1. Sinusoidal Signals and their Properties 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform a) Relationship to the z – Transform and Periodicity b) Symmetry Properties c) Operational Properties d) Correlation of Signals e) Signals with Poles on the Unit Circle sites.google.com/site/ncpdhbkhn 76 sites.google.com/site/ncpdhbkhn 77 Signals with Poles on the Unit Circle 1 1 ROC 1 1 [ ] ( ) , :u n X z z z− → = > − 1 0 0 1 2 0 1 ROC 1 1 2 (cos ) cos( ) [ ] ( ) , : (cos ) z n u n X z z z z ω ω ω − − − − → = > − +

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