10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 1 Development of Complex Curricula for Molecular Bionics and InfobionicsPrograms within a consortial* framework** Consortium leader PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has been realisedwith the support of the European Union and has been co-financed by the European Social Fund *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben ***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg. PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY sote_logo.jpg dk_fejlec.gif INFOBLOKK 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 2 Peter Pazmany Catholic University Faculty of Information Technology Digital-and Neural Based Signal Processing & Kiloprocessor Arrays Introduction and Analog to Digital conversion www.itk.ppke.hu Digitális-neurális-, és kiloprocesszoros architektúrákon alapuló jelfeldolgozás Bevezetés és az analóg-digitális átalakítás dr. Oláh András 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 3 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 3 Outline • CourseInformation • Introductionandfocusofthecourse • Whatissignalprocessing?(objectives:algorithms,architecturesandapplications) • Firstlecture:A/Dconversion– Thesamplingtheorem – Uniformquantization – Nonuniformquantization • ADconvertersandmainperformances • AvailableADconvertersonthemarket– SuccessiveApproximationRegisterADC – Delta-SigmaADC Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 4 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 4 Course Information • ClassMailingList:digjel@lists.ppke.hu • Smalltestsinclassesonalltopics; • OnemajortestonDigitalSignalProcessingscheduledinthemiddleofthesemester; • Exam(majorquestionsonNeuralProcessing,smallquestionsonDigitalSignalProcessing); • Grading:– Finalgrade=0.33*av.onSTs+0.33*DSP+0.33*NSP Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 5 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 5 Course Information (Cont’d) • Suggested literature and references:– Lecturenotes(essentialforthetestsandexams) – J.G.Proakis,D.G.Manolakis:„DigitalSignalProcessing”,PrenticeHall,1996,ISBN0-13394338-9 – S.Haykin„Adaptivefilters”,PrenticeHall,1996(recommended) – Haykin,S.:Neuralnetworks-acomprehensivefoundation,MacMillan,2004 – Hassoun,M.:Fundamentalsofartificialneuralnetworks,MITPress,1995 – Chua,L.O.,RoskaT.andVenetianer,P.L.:"TheCNNisasUniversalastheTuringMachine",IEEETrans.onCircuitsandSystems,Vol.40.,March,1993 – J.G.Proakis:Digitalcommunications,McGrawHill,1996. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 6 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 6 Course Syllabus and Scheduling 1. IntroductionandAnalogtoDigitalconversion. 2. Descriptiondigitalsignalsandsystemsintimedomain. 3. Descriptiondigitalsignalsandsystemsintransform(Z,DFT)domain. 4. Efficientcomputationofthetransformdomain(FFT)andfilterdesign. 5. Adaptivesignalprocessing. Midtermexam 6. Introductiontoneuralprocessing(inspiration,historyandapproaches). 7. Signalprocessingbyasingleneuron(linearsetseparation). 8. Hopfieldnetwork,Hopfieldnetasassociativememoryandcombinatorialoptimizer. 9. CellularNeuralNetwork. 10. FeedforwardNeuralNetworks(generalization,representation,learning,appl.). 11. PrincipalComponentAnalysis. 12. Virtualmachines:signalprocessingwithmulticoresystems. Finalexam Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 7 Objective of Digital Signal Processing A/D conversion + computer + SP algorithms Important feature Observed physical process Medical signals, Seismic signals, Vibro analysis, Speech Video Multimedia etc. Predicting epileptic seizure, Predicting earthquake Testing bearings Data compression for transmitting multimedia information linear and nonlinear algorithms Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 8 Seismic signal analysis Seismogram http://en.wikipedia.org/wiki/File:Seismogram.gif Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 9 Epileptic seizure prediction http://www.scholarpedia.org/article/Image:Mormann_SeizurePrediction_Fig1.jpg Signal processing in IT 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 10 Highly complex systems in Information Technologies Optimal operation ? Untractablebyanalyticalmeans(hugeamountoffreeparameters&datatobetakenintoaccount) Some input data desired output Solution Modeling architecture (signal processing elements with free parameters) estimated output + - error signal KNOWLEDGE TRANSFER(LEARNING) Modeling architecture with optimized parameters new input generalized output Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 11 Fundamental issues • Representationcapabilities(isthearchitecturecomplexenoughtomodelthesystem)? • Learning(howtoadjustthefreeparameterstocapturethehiddencharacteristicsofthemodeledsystem)? • Generalization(oncetheknowledgetransferhastakenplace,howtotrusttheoutputgiventoaninputbeingnotpartofthetrainingset)? Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 12 Examples IP network (with routers, switches, buffers …etc.) Input traffic volume (voice, video, multimedia) QoS parameters (average packet loss rate, average packet delay) Financial system (stock market, economical factors) Stock prices, currency exchange rates (financial data series ) Optimal investment for maximizing the return GSM or UMTS, b3G systems Number of users Multiuser interference Endeavour:HOW TO MODEL AND OPTIMIZE THESE SYSTEMS ? Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 13 A simple example –packet delay estimation Inter arrival time x d Complex system (Packet switched network) Input traffic delay Measurements d Modeling architecture y=Ax+B est. delay Future x est. delay learning generalization Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 14 Measurements d Complex system (Packet switched network) Input traffic delay Modeling architecture y=Ax^3+bx^2+Cx+D est. delay Linear approximation is not good A simple example –packet delay estimation (cont’) Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 15 Challenges • Linearornonlinearmodeling(mostofthereal-lifeproblemsareofhighlynonlinearnature)? • Howtodevelopfastlearningalgorithms? • Howtodevelopexactmeasuresexpressingthequalityofgeneralization? A/D conversion + computer + SP algorithms Important feature Observed physical process Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 16 Directions • Fastandreal-timelinearprocessingwithdesignatedHWarchitecture(DSP) • Biologicallyinspirednonlinearprocessing • Emergenceofnovelcomputationalparadigmsbyusingkilo-processorarrays A/D conversion + computer + SP algorithms Important feature Observed physical process Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 17 Biological inspirations Modeling architecture Representation Learning Generalization Robustness Modularity Solution provided by evolution and biology: MAMMAL BRAIN •high representation capability; • large scale adaptation; • far reaching generalizations; • modular structure (nerve cells, neurons); • very robust Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 18 Copying the Brain? Human Brain Neuro- biological model Artificial Neural Network Engineering tools and algorithms solving problems in the field of Information Technologies (IT) Focus of this course: Signal Processing algorithms Feature extraction (Simplification) Technology (VLSI) Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 19 Signal processing introduction -summary • Collectionofalgorithmstosolvehighlycomplexproblemsinreal-time(inthefieldofIT)byusingclassicalmethodsandnovelcomputationalparadigmsroutedinbiology. Complex system (Packet switched network) Input traffic delay Modeling architecture est. delay Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 20 Historical notes • Linear analog filters, 20’s • Artificial neuron model, 40’s (McCulloch-Pitts, J. von Neumann); • Hebbian learning rule, 50’s (Hebb) • Perceptron learning rule, 50’s (Rosenblatt); • Fast Fourier Transformation, 50’s • Nonlinear adaptive filter, 50’s (Gabor) • ADALINE, 60’s (Widrow) • Critical review , 70’s (Minsky) • Adaptive linear signal processing (RM, KW algorithms) , 70’s Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 21 Historical notes (cont’) • DSPs and digital filters, 80’s • Feed forward neural nets, 80’s (Cybenko, Hornik, Stinchcombe) • Back propagation learning, 80’s (Sejnowsky, Grossberg) • Hopfield net, 80’s (Hopfield, Grossberg); • Self organizing feature map, 70’s -80’s (Kohonen) • CNN, 80’s-90’s (Roska, Chua) • PCA networks, 90’s (Oja) • Applications in IT, 90’s -00’s • Kiloprocessor arrays, 2005 Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 22 Analog-to-Digital Conversion Signalanalysisandprocessingisengagedwithstudyingthedifferentphenomenaofnatureanddrawingconclusionsabouthowtheobservedquantitiesarechangingintime.Allapplicationshaveonethingincommon,signalsarestudiedasafunctionoftimeandtheanalysisiscarriedoutbyacomputer.However,computerscanonlyprocessdigitalsequences,thustheanalogsignalmustfirstbeconvertedintoabinarysequence. Book_figure_1 Analog to Digital Conversion analog signal, x(t) binary sequence, cn 00100111101001110111 Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 23 Notations The underlying notation is summarized by the following table: Signal Time Voltage Analog signal x(t) Continuous Continuous Sampled signal x(n) orx(nT) Discrete Continuous Quantized signal Discrete Discrete Coded signal cn Discrete Binary Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 24 ADC x(t) x(nT) . x(n) Sampling Quantization T .T Optimal representation cn Coding Compressing Analog-to-Digital Conversion • ADChasthreemainsteps:– samplingwhensamplethevalueofthesignalx(t)atcertaindiscretetimeinstantsobtainingasequencexk; – quantizationwhenthevaluesofthesamplesxkareroundedtosomealloweddiscretelevels(referredtoasquantizationlevels)andhavingafinitesetoftheselevelstheycantheneasilyberepresentedbybinarycodewords. – codingwhenquantizationsymbolsaremappedintobinarycodewords Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 25 ThechallengeofADC • Question:– Isthereanylossofinformationinthecourseoftheconversion? – Whatistheoptimalrepresentationofsignalsbybinarysequences(intermsoflength…etc.)? • Fundamentalchallengesofsamplingandofquantization:choosingpropersamplingfrequencyandquantizationlevels.ADCisfullycharacterizedby– thesamplingfrequency(denotedbyfs); – thenumberofquantizationlevels(N), – andtheruleofquantization. • OptimizingADCmeansthatweseektheoptimalvaluesoftheseparametersinordertoobtainefficientbinaryrepresentationofsignalswithminimumlossofinformation. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 26 Sampling Samplingiscarriedoutbyaswitchandtemporaryweassumethattheswitchisideal(i.e.theholdingperiodiszero). Book_figure_1 x(t) xs(t) Sampling T .t Analog signal Real sampled signal Book_figure_3 Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 27 Sampling (cont’) ? Book_figure_1 x(t) Reconstructed analog signal Book_figure_1 x(t) x(nT) Sampling T .T Analog signal Sampled signal Book_figure_3 Sampling switch Can analog signals be reconstructed from their samples without any loss? Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 28 Bandwidth of a signal: the concept • Itisdesirabletoclassifysignalsaccordingtotheirfrequency-domaincharacteristics(theirfrequencycontent):– Low-frequencysignal:ifasignalhasitsspectrumconcentratedaboutzerofrequency – High-frequencysignal:ifthesignalspectrumconcentratedathighfrequencies. – Bandpass-signal:asignalhavingspectrumconcentratedsomewhereinthebroadfrequencyrangebetweenlowfrequenciesandhighfrequencies. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 29 Bandwidth of a signal: the concept (cont’) • Thequantativemeasureoftherangeoverwhichthespectrumisconcentratediscalledthebandwidthofsignal. • Weshallsaythatasignalisbandlimitedifitsspectrumiszerooutsidethefrequencyrange|f|.B,whereBistheabsolutebandwidth.Theabsolutebandwidthdilemma:– Band limited signals are not realizable! – Realizable signals have infinite bandwidth! – (No signal can be time-limited and band limited simultaneously.) Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 30 Bandwidth of a signal: the concept (cont’) • Inthecaseofabandpasssignal(fmin.f.fmax),thetermnarrowbandisusedtodescribethesignalifitsbandwidth B= fmax -fmin, ismuchsmallerthanthemedianfrequency (fmax +fmin)/2. Otherwise,thesignaliscalledwideband. • Therearemanybandwidthdefinitionsdependingonapplication:– noise equivalent bandwidth – 3 dB bandwidth – .% energy bandwidth Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 31 The noise equivalent bandwidth Itisdefinedasthebandwidthofasystemwitharectangulartransferfunctionthatreceivesasmuchnoiseasthesystemunderconsideration f White noise PSD B Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 32 The 3 dB bandwidth Isthebandwidthatwhichtheabsolutevalueofthespectrum(energyspectrumorPSD)hasdecreasedtoavaluethatis3dBbelowitsmaximumvalue. f B. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 33 The .% energy bandwidth Isthebandwidththatcontains.%oftotalemitted. f B90% Világos átlós lefelé 90% Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 34 Frequency ranges of a some natural signals Biological Signals Type of Signal Frequency Range [Hz] Electroretinogram 0 -20 Pneumogram 0 -40 Electrocardiogram (ECG) 0 -100 Electroenchephalogram (EEG) 0 -100 Electromyogram 10 -200 Sphygmomanogram 0 -200 Speech 100 -4000 Seicmic signals Seismic exploration signals 10 -100 Eartquake and nuclear explosion signals 0.01-10 Electromagnetic signals Radio bradcast 3x104 -3x106 Common-carrier comm. 3x108 -3x1010 Infrared 3x1011 -3x1014 Visible light 3.7x1014 -7.7x1014 Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 35 The sampling theorem (Shannon –Kotelnikov 1949) Ifabandlimitedsignalx(t)(thebandislimitedtoB)issampledwithsamplingfrequencyfs.2Bthenx(t)canbeuniquelyreconstructedformitssamplesasfollows: where Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 36 Proof of sampling theorem SinceX(f)isbandlimiteditcanbeextendedtoformaperiodicsignal if1/Ts>2Basindicatedbythenextfigure: Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 37 Proof of sampling theorem (cont’) Onemaynotice,thattheconditionfs=1/Ts>2BguaranteethatthereisnooverlappinginXs(f)andasaresult: (furthermoresincefs=1/Ts>2Bthisstatementisalsotrue LetusalsonotethatXs(f)isaperiodicsignal,i.e. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 38 Proof of sampling theorem (cont’) Letusnowexpressasamplex(n)bythemeansofinverseFouriertransform Ontheotherhand,Xs(f)beingaperiodicsignalitcanbeexpandedintoFourierseriesasfollows: where Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 39 Proof of sampling theorem (cont’) FromtheFourierseriesofXs(f)followsthat and Takingintoaccountthatwecanwrite andsubstituting intotheintegral,weobtain whichprovesthetheorem. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 40 Phenomena of aliasing Ifthesamplefrequencyisnotchosentobehighenough(i.e.frequencyfs.2B),thenXs(f)thenthereisanoverlapinthespectrum,whichimpliesthatX(f)cannotberegainedfromXs(f). Aliasing Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 41 Problem 1: Wesamplethefunctionsx1(t)=u(t)e-tandx2(t)=u(t)te-t,respectively. – Whichfunctionhaslargerbandwidth?(Todeterminethebandwidthuseparameter.=0.01) – Whatistheminimumsamplingfrequencytouniquelyrestorethesignalsformtheirsamples? Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 42 Problem 2: Givenathefrequencyresponseofasystemasfollows: . – Determinethebandwidthofthesystemwithparameter.=0.1! – Whatistheimpactonthebandwidthifweset.=0.01? – Whattypeoffilteringdoesthissystemimplement? Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 43 •Inpracticethesamplingiscarriedoutbyaswitchwhichhasafinite(non-zero)holdingtime. • Iftheholdingtime.tissmallenoughthenxs(t)canbeperceivedas • Thesignalxs(t)isoftencalledrealsampledsignal,asxs(t)canbeobtainedfromx(t)byaproperelectroniccircuitry. • Sincethed(t)›.(t)when.t›0,thusifweconstructalow-passfilterwithimpulseresponsefunctionh(t)thentheoutputofthisfiltertotheinputd(t)isapproximatelyh(t)aswell. Sampling in practice Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 44 Summarizing of sampling Inthecaseofpracticalsamplingfirstweobtainxs(t)fromx(t)andthenfromxs(t)theoriginalsignalx(t)canberegainedbylettingxs(t)passthroughalowpassfilter. Filtering Book_figure_1 x(t) Reconstructed analog signal H(f) f Lowpass filter Book_figure_1 x(t) xs(t) Sampling T .T Analog signal Real sampled signal Book_figure_3 Sampling switch Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 45 Oversampling technique AccordingtotheShannontheoremthesamplingfrequencyfsshouldbetwotimeslargerthanthesignalbandwidthB.SuchachoiceofsamplingfrequencycreatesariskthatthesignalsoffrequencyfH>BcangeneratethesignalsfH-fsinthebandwidthaftersampling.Forthatreasonitissafertosetthesamplingfrequencyfstwotimeslargerthanthefrequencywhentheanti-aliasfiltersufficientlyattenuatesthesignals. Analogue anti-alias filter ADC x(t) Kfs Digital anti-alias filter decimal filter :K Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 46 Oversampling technique (cont’) f Kfs Kfs/2 fs/2 analogue filter digital filter X(f) Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 47 Oversampling technique (cont’) Highersamplingfrequencymeanslesscriticalrequirementsofthefilterperformances.Theprofitrelatedtooversampling: – cheaperandlesscomplicatedanti-aliasfilter – noisereductionincreasesthequantizationSNR(seelater) Thismethodiscurrentlyappliedinhighqualitysoundprocessing: – inSACDsystemintroducedbySony(SACD–SuperAudioCompactDisc)thesamplingfrequencyis2.82MHzwhichmeanstheoversamplingfactorK=64. – inDVDAudiosystemintroducedbyTechnicsthesamplingfrequencyis192kHzandtheoversamplingfactorisK=4. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 48 Under sampling technique Letusconsideranothercasewhenweprocessthesignalinthebandwidth30MHz–55MHz.Applyingthesamplingfrequency110MHz(accordingtotheShannontheorem)seemstobeextravagant.InsuchacasewecanmodifytheShannonrule(aliasingfreesampling): where Note:k=1isreturnedtheoriginalShannonsamplingrule Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 49 Under sampling technique (cont’) Inourcaseofthesignalsinbandwidth30MHz–55MHzitissufficienttousesamplingfrequency55MHz.fs.60MHzinsteadof110MHz.Ofcourse,byusingtheundersamplingtechniqueweapplyaband-passanti-aliasfilterinsteadofalow-passfilter. Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 50 Quantization Weassumethatthesignalisalreadysampledandwedealwithsamplesx(n).Sinceeachsamplehascontinuousamplitude,quantizationisconcernedtomappingx(n)intowhichmayhaveonlyafinitenumberofvalues. Quantization Book_figure_2b Book_figure_4 Sampled signal Quantified signal Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 51 Quantization (cont’) • Quantizationalwaysentailslossofinformationduetotheroundingprocess. • Thedesignofaquantizerisconcernedwithtwoparameters:– numberofquantizationlevels; – locationofquantizationlevels(uniformornon-uniform); • ThequalityofquantizationisdescribedbyaSignal-to-QuantizationNoiseRatio(SQNR)wheretheaveragesignalpoweriscomparedtothenoisepowerresultingfromthequantizationerror: Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 52 •Signalvalueisroundedofftopredefinedthresholdscalledasquantizationvalueswhichareequidistantlyplaced. • Notations:– thesamplerangeis[-C,C] – thedistancebetweenthethresholdsis., – thenumberofquantizationlevelisN=2C/.=2n,wherenrepresentsthenumberofbitsbywhichthequantizedsignalcanberepresented. – theerrorsignalisand-./2...-./2. Uniform quantization The quantization characteristics and the quantization error function Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 53 Uniform quantization (cont’) Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 54 Modeling the quantization noise Sincethenatureoferrorsarerandomthespecificvalueof.dependsonthevalueofthecurrentsample,thus.isregardedasarandomvariablesubjecttouniformprobabilitydensityfunction,andtheaveragenoisepoweris Digital-and Neural Based Signal Processing & Kiloprocessor Arrays: Introduction and Analog to Digital conversion 10/6/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 55 SQNR of the uniform quantization • Inthecaseoffull-scalesinewave(withamplitudeC): • Inthecaseofrandominputvariablesubjecttouniformprobabilitydensityfunctionovertheinterval[-C,C]: • InthecaseofsinewavewithamplitudeA(innormaloperationi.e.A