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. 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 (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)?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).

3 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).(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).

4 (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. 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.)

5 Digital signal processing (video compression, speech compression,etc.) 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 .

6 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. 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.

7 (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. 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) =.

8 +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).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.

9 Hard to implement. Based on bandlimited sinc 424#1: Sampling and Reconstruction 14 Figure13: The interpolated signal isa sum of shifted sincs, weighted bythe samplesx(n T). The sinc functionh(t) =sinc(t/T)shifted ton T, (t T), is equal to one atn Tand zeroat all other samplesl T,l6=n. The sumof the weighted shifted sincs will agreewith all samplesx(n T), general Reconstruction filterFor the development of the theory, it is handy to consider theimpulse-sampled signalxP(t)and its ( )in notbe a fre-quency response of an ideal lowpassfilter, in contrast withHF( )in : Reconstruction in the fre-quency domain is lowpass 424#1: Sampling and Reconstruction 15 Here, the reconstructed signal isxr(t), with CTFTXFr( ) =HFLP( )XFP( ) Sampling ( )1T+ k= XF( 2 kT k 0).Note: As sketched in ,hLP(t)CTFT HFLP( )can be made moreflexible than the ideal sinc/boxcar pair; yet, we can still achieve per-fect Reconstruction .

10 The more we sample above the Nyquist rate, themore flexibility we gain in terms of designing this filter. An exampleof a more flexible filter is given in : Frequency response of aflexible lowpass Reconstruction m= 0/2, then this frequencyresponse reduces to the standard boxcarfrequency with zero-order hold Many practical Reconstruction systemsuse zero-orderhold circuits for Reconstruction . Why? Rectangular pulses are (much) easier to generate than (ap-proximate) sinc pulses. Replace the ideal sinc with a rectangular pulse77 See (t) =rect(t )yieldingxZOH(t) =+ n= x[n]hZOH(t n T).Frequency response of the zero-order hold:HFZOH( ) = T0e j tdt=1 e j Tj =Tsinc( T2 )e T=Tsinc( 0)e j 0(16)recall 0=2 /Tand (1).ee 424#1: Sampling and Reconstruction 16 Reconstruction system(mixes continuous and discrete time ).