Lesson05QuantV50.docx
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Lesson05QuantV50.docx
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Lesson05QuantV50
Topic:
Quantization
LessonNumber:
[05]UF[05]IEEE
What’sitallabout?
Whatisaquantizationerror?
什么是量化误差?
Howdoyoucomputequantizationerrors?
怎样计算量化误差?
Howdoyouinterpretquantizationerrors?
怎样解释量化误差?
FilterRepresentation滤波器的表述
Digitalfiltersareamultiply-accumulate(MAC)intensive.DSPfilterandtransformalgorithmsareclassifiedasbeingSAXPY(S=AX+Y)innature,andoftenarerequiredtoruninreal-timewithaminimalamountofarithmetichardwaresupport.Thiscancreateamajorarithmeticbottleneckthatcaninvalidateorcompromiseacandidatesolution.Toachievehighperformanceandprecision,withinrealisticpackage,power,andcostconstraints,engineersmustcarefullyaccesstheirdesignchoiceswhichcollectivelywillinfluencetheoutcome.Oneofthesedecisionsrelatedtothechoiceofnumbersystemtobeemployed.Inpractice,thearithmeticoptionsare:
数字滤波器就是乘加器,DSP滤波器和变换自然的被分类,通常要求用最少的硬件实运算。
这可能引起主要算法的瓶颈效应。
这可能抛弃和折中一个候选方案。
为取得高的性能和精度,以及现实的可实现性、电源和价格限制。
工程师必须小心地在他们的设计选择数字类型。
1.fixedpoint定点数
2.floatingpoint浮点数
3.blockfloatingpoint块浮点数
Signedfixedpointsystemsappearinintegerorfractionalform.Whileanintegerrepresentationisthenumberrepresentationchoiceforthosedevelopingcode,theDSPengineersprefertoworkwithrealnumbers(coefficientsanddata).Toillustrate,supposearealdigitalfilterparameterhasarealvalueofα=4.2344.Totheprogrammer,asigned10-bitintegerversionofαisgivenby:
有符号定点数可以表示整数或分数,当开发代码使用整数时,工程师更愿意用实数表示系数和数据。
例如,假定一个正式的数字滤波器,参数是4.2344。
用10位二进制数表示:
[1signbit:
9integerbits]=[0:
100001111]2=27110
符号位整数二进制数十进制整数
TheDSPengineer,however,wouldpreferuseα=4.2344interpretedasa10-bitfractionalnumber:
解释为4.2344
[1signbit:
3integerbits6fractionalbits]=[0:
100001111]2=4.234410
符号位整数位小数点小数部分=(+)(22+0+0).(0+0+2-3+2-4+2-5+2-6)=4.234410
代表小数点,定点小数用[N:
F]表示;上面的例子表达为[10:
6],N—总位数,F—小数位数
wheredenotedthebinarypointlocation.Historically,thefractionalfixedpointdataformathasaformatwhichisgiventhenotationalrepresentation[N:
F]which,inthe10-bitexampleunderconsiderationwouldbeexpressedas[10:
6].Forthecasewhere16-bitTexasInstrument’sprocessorsareused,N=16whichresultsinthe“Q”notationwhere[16:
F]=Q(F).Fixed-pointrepresentations,inanyoftheseformsproduceanapproximationtoarealnumberα.Thedifferencebetweenαanditsfixed-pointrepresentationiscalledthequantizationerror.
德克公司16位仪器的处理器则16用Q表示,则[16:
F]=Q(F)。
定点数表述的是实数α的近似数
Foroff-linefiltering(e.g.,MATLAB)andsomereal-timeapplicationswhereprecisionisanissue,floatingpointispreferred.Thefloatingpointrepresentationofasignednumberx=mxrex,wheremxiscalledthemantissa(normallynormalized),ristheradix,andexistheexponent.Floatingformatshavebeenstandardized(e.g.,IEEE).Theiruseindigitalfilteringisjustifiedonthebasisofhighprecisionbuttheyintroduceahostofproblemsaswell.Notableisthefactthatfloatingpointisslow,resourceintense(hardware)andhasdatadependentlatencieswhichcreaterealtimecontrolandoperationalproblems.
对于非实时滤波器精度更重要,所以就使用浮点数。
浮点数的符号为x=mxrex,4.2344=42344*10-4二进制[0:
100001111]2=0100001111*2-6
浮点数精度高,但在滤波器里带来处理的问题,就是速度太慢
Avariationonthefloating-pointthemeisblockfloatingpoint.SomeDSPapplicationspecificstandardparts(ASSP),suchasFFTs,useblockfloatingpoint.Ablockfloatingpointsystemisactuallynothingmorethanascaledfixed-pointsystemthanitisafloatingpointnumbersystem.Specifically,givenanarrayofdata{xi},withmaximalelementxmaxM,theninblockfloatingpointform,{xi}equals{yi}={xi/K}KwherethescalefactorKcanbeappliedattheendofacomputationalcycleor,insomecaseignoredaltogether.
为提高速度,引入了块浮点
Ofthesechoices,theoverwhelmingchoicefordigitalfilteringisfractionalfixed-pointforhardware-baseddesignsandeitherfixed-pointorfloating-pointforsoftwareenabledsolutions.Thefixed-pointhardwareadvantagesaremanifold,includinghighspeedalongwithreducedcomplexity,powerdissipation,andcost.Thefixed-pointsolutionoftenbeginswithanADCwhichdefines,intotalorinpart,thedataformatandquantizationerrorforthefixed-pointsolution.Statisticallyquantifyingthiserrorisnecessarytobeabletorigorouslyanalyzeadigitalfilter.
Figure2:
Digitizingananalogsignalwitha3bitADC(needaMATLABversion)
Quantization图二讲错了
Digitalsignalscanbegeneratedbydigitaldevices,suchasananalogtodigitalconverter(ADC)asshowninFigure1.AnidealimpulsesamplerinstantaneouslycapturesasamplevaluexS[k]oftheanalogsignalx(t)att=kTs.ThesampledvalueisthenpassedtoaquantizerthatconvertsxS[k]intoadigitalwordxD[k],wherexD[k]isaquantizedapproximationofxS[k].Thedifferencebetweenthediscrete-timeandquantizedsamplevalueiscalledthequantizationerrorandisformallydefinedtobe:
e[k]=xD[k]–xS[k].1.
Toillustrate,ananalogsignalx(t)havinga1voltswing,showninFigure2,issenttoanideal3-bitADC.ThequantizedADCoutcomeshas3-bitresolutioncoveringthe1voltrangewith23=8possibleoutcomes.Thevalueoftheleastsignificantbit(LSB)isLSB=1/4voltsperbit.InpracticeitwouldbeusefultoexpresstheADCerrorsandpropertiesintermsofcomputablestatistics.FortunatelyforlinearADCapplications,thereisastraight-forwardstatisticalanalysisprocedurethatachievesthisgoal.Itisbasedontheproductionofthequantizationerrorwhichcanbeexpressedintermsofthequantizationstep-size,denoted∆(alsocommonlydenotedQ).Iftheinputsignalx(t)isdouble-ended,rangingover–Ax[k] (double-ended)2
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