Kĩ thuật viễn thông - Sampling of continuous – time signals

Nguyễn Công Phương SIGNAL PROCESSING Sampling of Continuous – Time 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 Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction 6. Sampling of Bandpass Signals sites.google.com/site/ncpdhbkhn 3 sites.google.com/site/ncpdhbkhn 4 Ideal Periodic Sampling of Continuous – Time Signals (1) Ideal analog – digital converter Fs = 1/T ( ) c x t [ ] ( ) c x n x nT= [ ] ( ) ( ), c ct nT x n x t x nT n = = = −∞ < < ∞ 0 1 2 3 4 5 6 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 sites.google.com/site/ncpdhbkhn 5 Ideal Periodic Sampling of Continuous – Time Signals (2) CTFT 1 2 ( ) ( ) ( ) ( ) j t j t c c c c X j x t e dt x t X j e d pi ∞ ∞ − Ω Ω −∞ −∞   Ω = ←→ = Ω Ω     ∫ ∫ DTFT 1 2 ( ) [ ] [ ] ( ) j j n j j n n X e x n e x n X e e d pi ω ω ω ω pi ω pi ∞ − − =−∞     = ←→ =      ∑ ∫ 2 2 2 s FT FT f F ω pi pi pi= Ω = = = 1 2 ( ) j T c k X e X j j k T T pi∞Ω =−∞   = Ω −    ∑ 1 2 ( ) j c k X e X j j k T T T ω ω pi ∞ =−∞   = −    ∑ 1 2( ) [ ( )]j FT c s k X e X j F kF T pi pi ∞ 2 =−∞ = −∑ sites.google.com/site/ncpdhbkhn 6 Ideal Periodic Sampling of Continuous – Time Signals (3) 1 2 ( ) j T c k X e X j j k T T pi∞Ω =−∞   = Ω −    ∑ 0 ( )cX jΩ 2 FpiΩ = 1 H−Ω HΩ 2H HFpiΩ = 0 ( ) j TX e Ω 2 FpiΩ =H−Ω HΩ sΩs−Ω 1 T s HΩ −Ω Guard bandGuard band 2s sFpiΩ =2s HΩ > Ω 0 ( ) j TX e Ω 2 FpiΩ =HΩ 2 sΩ2 s− Ω s−Ω s Ω s HΩ −Ω 1 T 2 s s FpiΩ =2s HΩ < Ω sites.google.com/site/ncpdhbkhn 7 Ideal Periodic Sampling of Continuous – Time Signals (4) 1 2 ( ) j T c k X e X j j k T T pi∞Ω =−∞   = Ω −    ∑ 0 ( ) j TX e Ω [rad/s]ΩH−Ω HΩ 2 T pi s −Ω 1 T T pi Nyquist rate s F 2 s HΩ > Ω 2 sF HF 2 HΩ Sampling frequency Folding frequency Nyquist frequency sites.google.com/site/ncpdhbkhn 8 Ideal Periodic Sampling of Continuous – Time Signals (5) 0 ( )cX jΩ 2 FpiΩ = 1 H−Ω HΩ 2H HFpiΩ = 0 ( ) j TX e Ω 2 FpiΩ =H−Ω HΩ sΩs−Ω 1 T s HΩ −Ω Guard bandGuard band 2s sFpiΩ =2s HΩ > Ω 2 0 2 ( ), / ( ) , / j T s c s TX e X j Ω Ω ≤ ΩΩ =  Ω > Ω Let xc(t) be a continuous – time bandlimited signal with Fourier transform 0 for( ) c HX jΩ = Ω > Ω Then xc(t) can be uniquely determined by its samples x[n] = xc(nT), where n = 0, ±1, ±2, , if the sampling frequency Ωs satisfies the condition 2 2 s HT piΩ = ≥ Ω Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction 6. Sampling of Bandpass Signals sites.google.com/site/ncpdhbkhn 9 sites.google.com/site/ncpdhbkhn 10 Reconstruction of a Bandlimited Signal from its Samples (1) ( ) [ ] ( ) r r n x t x n g t nT ∞ =−∞ = −∑ t nT 1( )n T+ 2( )n T+1( )n T−2( )n T− ( ) r g t [ ] ( ) r x n g t nT− ( ) r x t ( ) [ ] ( ) j nT r r n X j x n G j e ∞ − Ω =−∞ Ω = Ω∑ ( ) [ ] j nT r n G j x n e ∞ − Ω =−∞ = Ω ∑ ( ) [ ] j j n n X e x n eω ω ∞ − =−∞ = ∑ ( ) ( ) ( ) j T r r X j G j X e Ω→ Ω = Ω sites.google.com/site/ncpdhbkhn 11 Reconstruction of a Bandlimited Signal from its Samples (2) ( ) ( ) ( ) j T r r X j G j X e ΩΩ = Ω 0 ( )cX jΩ 2 FpiΩ = 1 H−Ω HΩ 2H HFpiΩ = 0 ( ) j TX e Ω 2 FpiΩ =H−Ω HΩ sΩs−Ω 1 T s HΩ −Ω Guard bandGuard band 2s sFpiΩ =2s HΩ > Ω 2 0 2 ( ), / ( ) , / j T s c s TX e X j Ω Ω ≤ ΩΩ =  Ω > Ω 2 If 0 2 , / ( ) ( ) , / s r Band Limited s T G j G j  Ω ≤ ΩΩ = Ω =  Ω > Ω ( ) ( ) r cX j X j→ Ω = Ω ( ) ( )r cx t x t→ = sites.google.com/site/ncpdhbkhn 12 Reconstruction of a Bandlimited Signal from its Samples (3) 2 0 2 , / ( ) ( ) , / s r BL s T G j G j  Ω ≤ ΩΩ = Ω =  Ω > Ω sin( / ) ( ) ( ) / r BL t Tg t g t t T pi pi → = = sin[ ( ) / ] ( ) [ ] ( ) / r n t nT T x t x n t nT T pi pi ∞ =−∞ − → = − ∑ ( ) [ ] ( ) r r n x t x n g t nT ∞ =−∞ = −∑ ( )BLG jΩ T 0 T pi T pi − Ω -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 ( )BLg t t T 2T 3T 4T 5T5T− 4T− 3T− 2T− T− Ideal digital – analog converter Fs = 1/T [ ]x n ( )rx t sites.google.com/site/ncpdhbkhn 13 Reconstruction of a Bandlimited Signal from its Samples (4) sin[ ( ) / ] ( ) [ ] ( ) / r n t nT T x t x n t nT T pi pi ∞ =−∞ − = − ∑ 0 t T 2T 0[ ]x 1[ ]x ( ) c x t 1[ ] ( ) r x g t T− 0[ ] ( ) r x g t sites.google.com/site/ncpdhbkhn 14 Reconstruction of a Bandlimited Signal from its Samples (5) ( )cx t ( )cX jΩ ( ) j TX e Ω[ ]x n S a m p l i n g R e c o n s t r u c t i o n A liasing L o w pass – Filtering Continuous – time Fourier Transform Pairs Discrete – time Fourier Transform Pairs ( ) ( ) j t c c X j x t e dt∞ − Ω −∞ Ω = ∫ 1 2 ( ) ( ) j t c c x t X j e d pi ∞ Ω −∞ = Ω Ω∫ ( ) [ ] j T j Tn n X e x n e ∞ Ω − Ω =−∞ = ∑ 1 2 / / [ ] ( ) T j T j Tn T x n TX e e d pi pipi Ω Ω − = Ω∫ D i s c r e t e – T i m e C o n t i n u o u s – T i m e F requ ency N o rm alized F requ ency Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction 6. Sampling of Bandpass Signals sites.google.com/site/ncpdhbkhn 15 sites.google.com/site/ncpdhbkhn 16 The Effect of Undersampling: Aliasing (1) Ideal digital – analog converter Fs = 1/T [ ]x n ( )ry tIdeal analog – digital converter Fs = 1/T ( ) c x t 2( ) c X j Fpi 2( )j FTX e pi 2( )cY j Fpi 0 1 2 3 4 5 6 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 sites.google.com/site/ncpdhbkhn 17 The Effect of Undersampling: Aliasing (2)Ex. 1 02( ) cos( )cx t F tpi= 0 0 2 ( ) j F t j F t c e e x t pi pi2 − 2+ = Spectrum of xc(t) 0 0 1 2 s F F< 0F− 0F sFsF− 1 2 F 00F− 0F sFsF− 1 2T F 0 0 1 2 ( ) [ ( )] j F T c s k X e X j F kF T pi pi 2 ∞ =−∞ = −∑ 2 2 0 2 ( ) , / , / BL s s G j F T F F F F pi  ≤ =  > 2 sF − 2 sF T 2( )BLG j Fpi Spectrum of xr(t) 0 0 1 2 s F F< 0F− 0F sFsF− 1 2 F No aliasing 02 02 2( ) ( ) ( ) j F T r r X j F G j F X e pipi pi= sites.google.com/site/ncpdhbkhn 18 The Effect of Undersampling: Aliasing (3)Ex. 1 02( ) cos( )cx t F tpi= Spectrum of xc(t) 0 0 1 2 s s F F F< < 0F− 0F sFsF− 1 2 F 00F− 0F sFsF− 1 2T F 2 sF − 2 sF T 2( )BLG j Fpi Spectrum of xr(t) 0 0 1 2 s F F< 0( )sF F− − 0sF F− sFsF− 1 2 F Aliasing 0 0 2 ( ) j F t j F t c e e x t pi pi2 − 2+ = 0 0 1 2 ( ) [ ( )] j F T c s k X e X j F kF T pi pi 2 ∞ =−∞ = −∑ 2 2 0 2 ( ) , / , / BL s s G j F T F F F F pi  ≤ =  > 02 02 2( ) ( ) ( ) j F T r r X j F G j F X e pipi pi= 02( ) cos[ ( ) ] ( )r s cx t F F t x tpi= − ≠ sites.google.com/site/ncpdhbkhn 19 The Effect of Undersampling: Aliasing (4)Ex. 1 02( ) cos( )cx t F tpi= 0 2 2/ , /s sF F F F F= + ∆ ∆ ≤ 2 2( ) cos[ ( / ) ]c sx t F F tpi→ = + ∆ 02( ) cos[ ( ) ]r sx t F F tpi= − 0 2/apparent s sF F F F F= − = − ∆ 2 2 2( ) cos ( ) cos[ ( / ) ] r a s x t F t F F tpi pi→ = = − ∆ sites.google.com/site/ncpdhbkhn 20 The Effect of Undersampling: Aliasing (5)Ex. 2 ( ) At cx t e − = 2 2 2 0 ( )c AX j A A Ω = +Ω > [ ] ( ) ( ) , AnT c n nAT AT x n x nT e e a a e − − − = = = = = 2 2 1 1 2 ( ) [ ] , cos( ) j j n n s X e x n e a a a F ω ω ω ω ∞ − =−∞ = − = − + Ω = ∑ -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.5 1 t x c ( t ) -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.2 0.4 Ω X c ( j Ω ) -10 -8 -6 -4 -2 0 2 4 6 8 10 0 2 4 6 Ω X ( e j ω ) -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.5 1 t y r ( t ) Sum X c (jΩ) shifted 2pi to left & scaled 1/T X c (jΩ) scaled 1/T X c (jΩ) shifted 2pi to right & scaled 1/T Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction 6. Sampling of Bandpass Signals sites.google.com/site/ncpdhbkhn 21 sites.google.com/site/ncpdhbkhn 22 Discrete – Time Processing of Continuous – Time Signals (1) Ideal DAC Fs = 1/T [ ]x n ( )ry tIdeal ADC Fs = 1/T ( )cx t ( )cX jΩ ( )jX e ω ( )rY jΩ LTI Discrete – Time System h[n] H(ejω) ( ) jY e ω [ ]y n 0 ( )cX jΩ Ω 1 2 HFpi− 2 HFpi 0 1 2 ( )j T c k X e X j k T T pi∞Ω =−∞    = Ω−      ∑ 2 T pi2 T pi − 1 T Ω2 HFpi− 2 HFpi 0 t ( ) c x t 0 T [ ] ( ) ( )c ct nTx n x t x nT== = 2T t3T sites.google.com/site/ncpdhbkhn 23 Discrete – Time Processing of Continuous – Time Signals (2) Ideal DAC Fs = 1/T [ ]x n ( )ry tIdeal ADC Fs = 1/T ( )cx t ( )cX jΩ ( )jX e ω ( )rY jΩ LTI Discrete – Time System h[n] H(ejω) ( ) jY e ω [ ]y n 0 1 [ ] [ ]* [ ]y n h n x n= 2 n3 0 1 [ ]x n 2 n3 ⋯⋯ ⋯⋯ 0 ( ) jX e ω 2pi2pi− 1 T ω Hω− Hω ( ) jH e ω c ω− cω 0 ( ) ( ) ( ) j j jY e H e X eω ω ω= 2pi2pi− 1 T ωcω− cω sites.google.com/site/ncpdhbkhn 24 Discrete – Time Processing of Continuous – Time Signals (3) Ideal DAC Fs = 1/T [ ]x n ( )ry tIdeal ADC Fs = 1/T ( )cx t ( )cX jΩ ( )jX e ω ( )rY jΩ LTI Discrete – Time System h[n] H(ejω) ( ) jY e ω [ ]y n 0 ( ) ( ) ( ) j T r r Y j G j Y e ΩΩ = Ω 1 Ωc−Ω cΩ 0 T ( ) [ ] r y nT y n= 2T t3T 0 ( ) r G jΩ 2 T pi2 T pi − 1 T Ωc−Ω cΩ T pi − T pi T ( )j TY e Ω 0 ( ) r y t t sites.google.com/site/ncpdhbkhn 25 Discrete – Time Processing of Continuous – Time Signals (4) Ideal DAC Fs = 1/T [ ]x n ( )ry tIdeal ADC Fs = 1/T ( )cx t ( )cX jΩ ( )jX e ω ( )rY jΩ LTI Discrete – Time System h[n] H(ejω) ( ) jY e ω [ ]y n 1 2 ( ) j T c k X e X j k T T pi∞Ω =−∞    = Ω −      ∑ ( ) ( ) ( ) j j jY e H e X eω ω ω= ( ) ( ) ( ) j T r r Y j G j Y e ΩΩ = Ω 1 2 ( ) ( ) ( ) j T r BL c k Y j G j H e X j k T T pi∞Ω =−∞    → Ω = Ω Ω −      ∑ 0 ( ) r G jΩ 2 T pi2 T pi − 1 T Ωc−Ω cΩ T pi − T pi T ( )j TY e Ω ( ) jH e ω 0 ( ) ( ), / , / j T cH e X j T T pi pi Ω Ω Ω ≤ =  Ω > sites.google.com/site/ncpdhbkhn 26 Discrete – Time Processing of Continuous – Time Signals (5) Ideal DAC Fs = 1/T [ ]x n ( )ry tIdeal ADC Fs = 1/T ( )cx t ( )cX jΩ ( )jX e ω ( )rY jΩ LTI Discrete – Time System h[n] H(ejω) ( ) jY e ω [ ]y n 0 ( ) ( ), / ( ) , / j T c r H e X j T Y j T pi pi Ω Ω Ω ≤Ω =  Ω > 0 ( ), / ( ) , / j T effective H e T H j T pi pi Ω Ω ≤Ω =  Ω > ( ) ( ) ( ) r effective cY j H j X j→ Ω = Ω Ω sites.google.com/site/ncpdhbkhn 27 Discrete – Time Processing of Continuous – Time Signals (6)Ex. 1 ( ) ( ) cc dx ty t dt = ( ) ( ) c Y j j X j→ Ω = Ω Ω ( )( ) ( ) c Y jH j j X j Ω → Ω = = Ω Ω 0 otherwise , ( ) , H c j H j  Ω Ω ≤ ΩΩ =   2 ( ) [ ] ( )s Hc ch t h n h nT Ω = Ω → = 1 2 ( ) , j cT k H H e H j j k T T T ω ω pi pi∞ =Ω =−∞   → = Ω− =  Ω ∑ 2 1 ( ) ( / ) , j c jH e H j T T T ω ωω ω pi→ = = ≤ 2 2 0 01 2 0 , [ ] cos( ) , j n njh n e d nT n nT pi ω pi ω ω pi pi − =   → = =   ≠   ∫ sites.google.com/site/ncpdhbkhn 28 Discrete – Time Processing of Continuous – Time Signals (7)Ex. 2 2 2 22 ( ) ( ) ( ) c n c c n n Y sH s X s s sζ Ω = = + Ω +Ω ( )22If 0 1 11( ) sin ( )ntnc nh t e t u tζζ ζζ − ΩΩ  < < → = Ω −  − ( )22 11[ ] ( ) sin ( )nnTnc nh n h nT e nT u nζ ζζ − ΩΩ  → = = Ω −  − ( ) ( )22 11 sin ( )n nTn ne T n u nζ ζζ − ΩΩ  = Ω −  − ( ) ( )22 0 1 1 ( ) sinn nT nn n n H z e T n zζ ζζ ∞ − Ω − = Ω  → = Ω −   − ∑ ( ) ( ) 2 1 2 22 1 2 1 1 1 2 1 sin cos n n n T n n T T n e T z e T z e z ζ ζ ζ ζ ζ ζ − Ω − − Ω − Ω− − Ω −Ω = × − − Ω − + sites.google.com/site/ncpdhbkhn 29 Discrete – Time Processing of Continuous – Time Signals (8)Ex. 2 2 2 22 ( ) ( ) ( ) c n c c n n Y sH s X s s sζ Ω = = + Ω +Ω ( )22If 0 1 11( ) sin ( )ntnc nh t e t u tζζ ζζ − ΩΩ  < < → = Ω −  − ( ) ( )22 11[ ] sin ( )n nTn nh n e T n u nζ ζζ − ΩΩ  = Ω −  − ( ) ( ) 2 1 2 22 1 2 1 1 1 2 1 sin ( ) cos n n n T n n T T n e T z H z e T z e z ζ ζ ζ ζ ζ ζ − Ω − − Ω − Ω− − Ω −Ω = × − − Ω − + ( ) ( ) 2 2 22 1 1 1 2 1 1 2 [ ] sin [ ] cos [ ] [ ] n n n Tn n T T n y n e T x n e T y n e y n ζ ζ ζ ζζ ζ − Ω − Ω − Ω Ω → = Ω − − − + Ω − − − − Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction a) Analog – to – Digital Conversion b) Digital – to – Analog Conversion 6. Sampling of Bandpass Signals sites.google.com/site/ncpdhbkhn 30 sites.google.com/site/ncpdhbkhn 31 Practical Sampling and Reconstruction A/D converter Fs = 1/T ( )ax t [ ]qx nAntialiasing filter Ha(jΩ) ( )cx t Sample and hold Fs = 1/T Discrete – time system D/A converter Fs = 1/T( )SHy t [ ]y n Reconstruction filter Ha(jΩ)( )ry t Sample and hold Fs = 1/T Practical approximation of ideal A/D converter Practical approximation of ideal D/A converter sites.google.com/site/ncpdhbkhn 32 Analog – to – Digital Conversion (1) A/D converter Fs = 1/T ( )ax t [ ]qx nAntialiasing filter Ha(jΩ) ( )cx t Sample and hold Fs = 1/T Practical approximation of ideal A/D converter ( )inx t ( )outx tR C Hold Sample 0 t ( )inx t ( ) outx t sites.google.com/site/ncpdhbkhn 33 Analog – to – Digital Conversion (2) A/D converter Fs = 1/T ( )ax t [ ]qx nAntialiasing filter Ha(jΩ) ( )cx t Sample and hold Fs = 1/T Practical approximation of ideal A/D converter 0 t ( )inx t ( ) outx t ∆ 011 010 001 000 111 110 101 100 Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction a) Analog – to – Digital Conversion b) Digital – to – Analog Conversion 6. Sampling of Bandpass Signals sites.google.com/site/ncpdhbkhn 34 sites.google.com/site/ncpdhbkhn 35 Digital – to – Analog Conversion (1) ( ) [ ] ( )SH q SH n x t x n g t nT ∞ =−∞ = −∑ ( ) [ ] ( ) r r n x t x n g t nT ∞ =−∞ = −∑ D/A converter Fs = 1/T( )SHy t [ ]y n Reconstruction filter Ha(jΩ)( )ry t Sample and hold Fs = 1/T Practical approximation of ideal D/A converter 21 0 2 2 0 otherwise / , sin( / ) ( ) ( ) , CTFT j T SH SH t T Tg t G j e− Ω≤ ≤ Ω= ←→ Ω = Ω sin( / ) ( ) ( ) / r BL t Tg t g t t T pi pi = = sites.google.com/site/ncpdhbkhn 36 Digital – to – Analog Conversion (2) 21 0 2 2 0 otherwise / , sin( / ) ( ) ( ) , CTFT j T SH SH t T Tg t G j e− Ω≤ ≤ Ω= ←→ Ω = Ω -5 0 5 0 0.5 1 1.5 2 Ω | G S H ( j Ω ) | -5 0 5 -4 -2 0 2 4 Ω ∠ G S H ( j Ω ) 2 T pi2 T pi − 2 T pi − 2 T pi Ideal bandlimited interpolator GBL(jΩ) T pi T pi − pi pi− T ( ) ( ) ( )SH r BLG F H F G F= sites.google.com/site/ncpdhbkhn 37 Digital – to – Analog Conversion (3) 21 0 2 2 0 otherwise / , sin( / ) ( ) ( ) , CTFT j T SH SH t T Tg t G j e− Ω≤ ≤ Ω= ←→ Ω = Ω ( ) ( ) ( )SH r BLG F H F G F= 22 2 0 otherwise // , / ( ) sin( / ) , T r T e T H j T pi ΩΩ Ω <Ω = Ω   -5 0 5 0 1 2 Ω | G S H ( j Ω ) | -5 0 5 0 1 2 Ω | H r ( j Ω ) | -5 0 5 0 2 4 Ω | G S H ( j Ω ) H r ( j Ω ) | Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction 6. Sampling of Bandpass Signals a) Integer Band Positioning b) Arbitrary Band Positioning sites.google.com/site/ncpdhbkhn 38 sites.google.com/site/ncpdhbkhn 39 Integer Band Positioning (1) 0 2( )cX j Fpi1 HF− HF 3( )H H LF F F= − FLFLF− PN 0 2 0 2 , ( ) , L L c H H F X j F pi pi  Ω ≤Ω =Ω =  Ω ≥Ω = 2 H L H LB F F pi Ω −Ω = − = 0 2( )j FTX e pi1 T 2sF B= FLF−HF− LF HF N P 1P1N 2N 3N 0 2( ) r G j Fpi T FLF−HF− LF HF ( )H H LF K F F KB= − = 2 1 2( ) [ ( )]j FT c s k X e X j F kF T pi pi ∞ =−∞ = −∑ 2 2 sin( ) ( ) cos( ), H L r c c Bt F Fg t F t F Bt pi pi pi − = = ( ) ( ) ( )c c r n x t x nT g t nT ∞ =−∞ = −∑ sites.google.com/site/ncpdhbkhn 40 Integer Band Positioning (2) 0 2( )cX j Fpi1 HF− HF 4( )H H LF F F= − FLFLF− PN 0 2( ) r G j Fpi T FLF−HF− LF HF 0 2( )j FTX e pi1 T 2sF B= FLF−HF− LF HF N P 1N 2N 3N 4N 1P Sampling of Continous – Time Signals 1. Ideal Periodic Sampling of Continous – Time Signals 2. Reconstruction of a Bandlimited Signal from its Samples 3. The Effect of Undersampling: Aliasing 4. Discrete – Time Processing of Continuous – Time Signals 5. Practical Sampling and Reconstruction 6. Sampling of Bandpass Signals a) Integer Band Positioning b) Arbitrary Band Positioning sites.google.com/site/ncpdhbkhn 41 sites.google.com/site/ncpdhbkhn 42 Arbitrary Band Positioning (1) 0 2( )cX j Fpi1 HF− FLF− HFLF PN 0 2( )j FTX e pi1 T 2sF B≥ FHFLF P thk 1PN N 1( )thk − 1( ) s Lk F F− − s HkF F− 2 2 1 H L s F FF k k → ≤ ≤ − 2 12H Hs F FF B k B k → ≥ = × 2 1 Hs FF B k B  ≥ → ≤ ≤    2 2 1 2 2 1 ; min / / sH H H H H s H FF F F F FF B k B B k B B B F B     × ≤ ≤ − = =    −        1( ) s L L s H H k F F F kF F F − − ≤  − ≥ sites.google.com/site/ncpdhbkhn 43 Arbitrary Band Positioning (2) 2 2 1 1 sH HFF F k B B k B   × ≤ ≤ −  −   1 2 3 4 5 6 7 8 9 10 2 2.5 3 3.5 4 4.5 5 5.5 6 FH/B F s / B k = 1 k = 2 k = 3 k = 4 sites.google.com/site/ncpdhbkhn 44 Arbitrary Band Positioning (3) Ex. Given a bandpass signal with FL = 1.5kHz and FH = 2.5kHz, find the appropriate Fs? 2 5 1 5 1 . . kHz H LB F F= − = − = 2 2 52 2 5 1 2 52 2 2 5 min / . . / . . kHz H s H FF F B =    =    = = 2 5 2 5 1 . .H F B = = 2 5 2 5 1 . .S F B = = 1 2 3 4 5 6 7 8 9 10 2 2.5 3 3.5 4 4.5 5 5.5 6 FH/B F s / B k = 1 k = 2 k = 3 k = 4 sites.google.com/site/ncpdhbkhn 45 Arbitrary Band Positioning (4) Ex. Given a bandpass signal with FL = 1.5kHz and FH = 2.5kHz, find the appropriate Fs? 2 5 2 5. ; .sH FF B B = = 1 2 3 4 5 6 7 8 9 10 2 2.5 3 3.5 4 4.5 5 5.5 6 FH/B F s / B k = 1 k = 2 k = 3 k = 42k = 2 2 1 H L s F FF k k ≤ ≤ − 2 2 5 2 1 5 2 2 1s F× ×≤ ≤ − . . 2 5 3 s F≤ ≤. kHz kHz