EE 424 #1: Sampling and Reconstruction

1 EE424#1: Sampling and ReconstructionJanuary13,2011 ContentsNotation and Definitions2A Review: Signal Manipulations, CT Convolution, CTFT and Its Properties3 Signal manipulations3CT convolution3 CTFT and its properties5 Poisson Sum Formula7 Sampling7 Introduction7 Applications8 Point and impulse sampling8 Sampling theorem11 Reconstruction12 Ideal Reconstruction : Shannon interpolation formula12 Ideal Reconstruction : Summary13A general Reconstruction filter14 Reconstruction with zero-order hold15 Examples of Sampling and reconstruction19 Comments on Lab124 Sampling part of Lab124 Reconstruction part of Lab125 Lowpass Reconstruction filters26DT lowpass Reconstruction filters29 Reading: EE224handouts2,16,18,19, andlctftsummary(review); , , , and the V.

2 Oppenheim and A. S. & Systems. Prentice Hall, UpperSaddle River, NJ,1997ee 424#1: Sampling and Reconstruction 2 Notation and unit rectangle is defined in sinc function is defined assinc(x) =sin( x) x(1)see also indicator function is defined as:1(a,b)(t) ={1,t (a,b)0,otherwise.(2)Figure1: Definition and plot of the (CT impulse).We define the continuous- time (CT) impulse ( )by the property that + x(t) (t)dt=x(0)for all x(t)that are continuous at t= : Plot of the sinc 424#1: Sampling and Reconstruction 3A Review: Signal Manipulations, CT Convolution, CTFT and ItsPropertiesSignal manipulationsPractice examples:Figure3: time shift:y(t) =x(t t0).Where does timet=0 move?Figure4: Scaling:y(t) =x(t/T)whereT> convolutionCTconvolution is defined asx(t)?}

3 H(t) = + x( )h(t )d .Basic CT linear time -invariant (LTI) systems. The time -shift systemy(t) =x(t t0)is LTI with impulse response (t t0):x(t)? (t t0) =x(t t0).(3)Example: Computey(t) = (x?h)(t)forx(t) =21(0,2)(t)andh(t) =1(0,1)(t).First sketchx(t)andh(t):ee 424#1: Sampling and Reconstruction 4 Figure5: Critical time points:t 1=0andt=0 as well ast 1=2 andt=2, , 1, 2, 3, meaning that we have5intervals to consider 424#1: Sampling and Reconstruction 5 CTFT and its propertiesXF( )denotes continuous-timeFourier transform (CTFT)ofx(t):XF( ) = + x(t)e j td t(4a)x(t) =12 + XF( )ej td (4b)where is the frequency in radians per second (rad/s).The textbook usesX(j )to denote theCTFT ofx(t).Review EE224handoutlctftsummaryto solve the practice exam-ples in 424#1: Sampling and Reconstruction 6 Figure6: Examples of CTFT property: Ifx(t)CTFT XF( ), thenx(t)ej 0tCTFT XF( 0)(complex modulation).

4 (5)Generalized modulation property. Find CTFT of a signalx(t)f(t)(6)wheref(t)is periodic with fundamental periodT0and fundamentalfrequency 0=2 /T0. First, expressf(t)using fourier series (FS):f(t) =+ k= akej k 0tand substitute this expansion into (6):x(t)+ k= akej k 0t=+ k= akx(t)ej k 0tCTFT + k= akXF( k 0).(7)To derive the Sampling theorem, we will choosef(t)to be the im-pulse train, defined in the lowpass filter. The frequency response of the ideal lowpassfilter in be written as22 See also ( ) =T1 /T, /T( )(8)and the corresponding impulse responsehLP(t)is33 See (t) =T /T sinc( /T t)=sinc(tT).(9)ee 424#1: Sampling and Reconstruction 7 Figure7: An ideal lowpass Sum FormulaFigure8: The impulse trainpT(t)isdefined aspT(t) =+ n= (t n T)whereTdenotes its sum formula.

5 Consider the fourier -series representationof the impulse trainpT(t)in :pT(t) =+ k= akej k 0twhere 0=2 Tandak=1T TpT(t)e j k 0tdt=1T T/2 T/2 (t)e j k 0tdt= ,pT(t) =+ k= 1 Tej k 0t.(10)SamplingIntroductionSampling: Conversion of a continuous- time signal(usu-ally not quantized)to a discrete - time signal(usuallyquantized).ee 424#1: Sampling and Reconstruction 8 Reconstruction : Conversion of a discrete - time signal(usually quantized)to a continuous- time Sample and Reconstruct? Digital storage (CD, DVD, etc.) Digital transmission (optical fiber, cellular phone, etc.) Digital switching (telephone circuit switch, Internet packet switch,etc.) Digital signal processing (video compression, speech compression,etc.)

6 Digital synthesis (speech, music, etc.).ApplicationsHere is a typical Sampling and Reconstruction system:Quantization causes noise, limiting the signal-to-noise ratio (SNR) to about6dB per bit. We mostlyneglect the quantization effects in this and impulse samplingThere are two waysof looking at the sampled signal: as1. a sequence of numbersx[n] =x(n T),nintegerpoint Sampling of x(t), depicted in (b), oree 424#1: Sampling and Reconstruction 92. a continuous- time signalxP(t) =+ n= x(n T) (t n T)impulse Sampling of x(t), depicted in (c).Figure9: Sampling : (a) CT signalx(t),(b) the point-sampled sequencex[n],and (c) the impulse-sampled signalxP(t).ee 424#1: Sampling and Reconstruction 10 Point Sampling : An actual Sampling system mixes continuous and discrete time .

7 Continuous-timex(t)specified for allt. SpectrumXF( )analyzed by CTFT, frequencyvariable . discrete -timex[n] =x(n T)atn T,ninteger. SpectrumXf( )analyzed by DTFT, frequencyvariable = Sampling : An equivalent all-CTsystem. Continuous- time signalxP(t)specified for allt, but zero except att=n T. SpectrumXFP( )analyzed using CTFT (which is why we use impulse Sampling ), withXFP( ) =Xf( T ).(11)ee 424#1: Sampling and Reconstruction 11 Sampling theoremIn this handout,we focus on impulse Sampling because itrequires only the knowledge of theory ofCTsignals the impulse trainpT(t) = + n= (t n T)and define4 Since this is a course on digital signalprocessing, we will turn to DT signalsand point Sampling starting hand-out #2.

8 Then, (11) will be the bridgebetween the CT Sampling theory devel-oped in this handout and DT results inthe remainder of the (t) =x(t)pT(t) =+ n= x(t) (t n T) =+ n= x(n T) x[n] (t n T)(12)which is formally a CT the Poisson sum formula (10), we5 However, it is clear that the informa-tion it conveys aboutx(t)is limited tothe valuesx(n T), (t) =+ k= 1Tx(t)ej k 0t.(13)Take CTFT of (13):XFP( ) =+ k= 1 TCTFT{x(t)ej k 0t}=1T+ k= XF( k 0)(14)where 0=2 T(rad/s).Forx(t)CTFT XF( )bandlimited to| |< m, we have:Figure10: A bandlimited signal spec-trumXF( )and the spectrumXFP( )ofthe corresponding sampled x(t)CTFT XF( )bandlimited to| |< 424#1: Sampling and Reconstruction 12 If the Sampling frequency satisfies66(15) is known as theNyquist criterion.

9 0>2 m(15)as in , no aliasing occurs and we can perfectly reconstruct x(t)from its samplesx[n] =x(t)|t=n T,n=0, 1, 2, ..or, equivalently, from xP(t). If 062 maliasing occurs and we cannot reconstruct x(t)perfectly from x[n]ingeneral. (In special cases, we can.)ReconstructionAssume that theNyquist requirement 0>2 mis satis-fied. We consider two Reconstruction schemes: ideal Reconstruction (with ideal bandlimited interpolation), Reconstruction with zero-order Reconstruction : Shannon interpolation formulaRecall(14):XP(t) =..+1 TXF( + 0) +1 TXF( ) +1 TXF( 0) +..Figure11: To reconstruct the originalCT signalx(t), apply an ideal lowpassfilter to the impulse-sampled signalxP(t) =x(t)pT(t).

10 Our ideal Reconstruction filter has the frequency response:HF( ) =T1( /T, /T)( )ee 424#1: Sampling and Reconstruction 13and, consequently, the impulse response [see (9)]h(t) =sinc(tT).Figure12: An equivalent all-CT recon-struction , the reconstructed signal isx(t) =xP(t) impulse-sampled signal?h(t) =+ n= x(n T) (t n T)?h(t) h(t n T), see (3)=+ n= x(n T)sinc(t n TT)which is theShannon interpolation ( Reconstruction ) formula. The actualreconstruction system mixes continuous and discrete time . The reconstructed signalxr(t)is a train of sinc pulses scaled by thesamplesx[n]. This system is difficult to implement because each sinc pulse ex-tends over a long (theoretically infinite) time Reconstruction : Summary Easy to analyze.