数字信号处理英文教学课件PPT.ppt

1 chapter7ltidiscrete timesystemsinthetransformdomain transferfunctionclassificationtypesoflinear phasetransferfunctionssimpledigitalfilters 2 typesoftransferfunctions thetime domainclassificationofadigitaltransferfunctionbasedonthelengthofitsimpulseresponsesequence finiteimpulseresponse fir transferfunction infiniteimpulseresponse iir transferfunction 3 typesoftransferfunctions inthecaseofdigitaltransferfunctionswithfrequency selectivefrequencyresponses therearetwotypesofclassifications 1 classificationbasedontheshapeofthemagnitudefunction h ei 2 classificationbasedontheformofthephasefunction 4 7 1transferfunctionclassificationbasedonmagnitudecharacteristics digitalfilterswithidealmagnituderesponsesboundedrealtransferfunctionallpasstransferfunction 5 7 1 1digitalfilterswithidealmagnituderesponses adigitalfilterdesignedtopasssignalcomponentsofcertainfrequencieswithoutdistortionshouldhaveafrequencyresponseequalto1atthesefrequencies andshouldhaveafrequencyresponseequalto0atallotherfrequencies 6 digitalfilterswithidealmagnituderesponses therangeoffrequencieswherethefrequencyresponsetakesthevalueof1iscalledthepassband therangeoffrequencieswherethefrequencyresponsetakesthevalueof0iscalledthestopband 7 digitalfilterswithidealmagnituderesponses magnituderesponsesofthefourpopulartypesofidealdigitalfilterswithrealimpulseresponsecoefficientsareshownbelow 8 digitalfilterswithidealmagnituderesponses thefrequencies c c1 and c2arecalledthecutofffrequencies anidealfilterhasamagnituderesponseequalto1inthepassbandand0inthestopband andhasa0phaseeverywhere 9 digitalfilterswithidealmagnituderesponses earlierinthecoursewederivedtheinversedtftofthefrequencyresponsehlp ej oftheideallowpassfilter hlp n sin cn n n wehavealsoshownthattheaboveimpulseresponseisnotabsolutelysummable andhence thecorrespondingtransferfunctionisnotbibostable 10 digitalfilterswithidealmagnituderesponses also hlp n isnotcausalandisofdoublyinfinitelength theremainingthreeidealfiltersarealsocharacterizedbydoublyinfinite noncausalimpulseresponsesandarenotabsolutelysummable thus theidealfilterswiththeideal brickwall frequencyresponsescannotberealizedwithfinitedimensionalltifilter 11 digitalfilterswithidealmagnituderesponses todevelopstableandrealizabletransferfunctions theidealfrequencyresponsespecificationsarerelaxedbyincludingatransitionbandbetweenthepassbandandthestopband thispermitsthemagnituderesponsetodecayslowlyfromitsmaximumvalueinthepassbandtothe0valueinthestopband 12 digitalfilterswithidealmagnituderesponses moreover themagnituderesponseisallowedtovarybyasmallamountbothinthepassbandandthestopband typicalmagnituderesponsespecificationsofalowpassfilterareshownas 13 7 1 2boundedrealtransferfunctions acausalstablereal coefficienttransferfunctionh z isdefinedasaboundedreal br transferfunctionif forallvaluesofw letx n andy n denote respectively theinputandoutputofadigitalfiltercharacterizedbyabrtransferfunctionh z withx ej andy ej denotingtheirdtfts 14 boundedrealtransferfunctions integratingtheabovefrom to andapplyingparseval srelationweget thentheconditionimpliesthat 15 boundedrealtransferfunctions thus forallfinite energyinputs theoutputenergyislessthanorequaltotheinputenergy itimpliesthatadigitalfiltercharacterizedbyabrtransferfunctioncanbeviewedasapassivestructure if thentheoutputenergyisequaltotheinputenergy andsuchadigitalfilteristhereforealosslesssystem 16 boundedrealtransferfunctions thebrandlbrtransferfunctionsarethekeystotherealizationofdigitalfilterswithlowcoefficientsensitivity acausalstablereal coefficienttransferfunctionh z withisthuscalledalosslessboundedreal lbr transferfunction 17 boundedrealtransferfunctions example considerthecausalstableiirtransferfunction wherekisarealconstant itssquare magnitudefunctionisgivenby 18 boundedrealtransferfunctions thus for 0 h ej 2 max k2 1 2 0 h ej 2 min k2 1 2 ontheotherhand for 0 2 cos max 2 2 cos min 2 0here h ej 2 max k2 1 2 h ej 2 min k2 1 2 0 19 boundedrealtransferfunctions hence isabrfunctionfork 1 plotsofthemagnitudefunctionfor 0 5withvaluesofkchosentomakeh z abrfunctionareshownonthenextpage 20 boundedrealtransferfunctions lowpassfilter highpassfilter 21 7 1 3allpasstransferfunction themagnituderesponseofallpasssystemsatisfies a ej 2 1 forall theh z ofasimple1th orderallpasssystemis whereaisreal and oraiscomplex theh z shouldbe 22 allpasstransferfunction onerealpole onecomplexpole 23 allpasstransferfunction twoorderallpasstransferfunction ploes zeros 24 allpasstransferfunction generalize themth orderallpasssystemis ifwedenotepolynomial so 25 allpasstransferfunction thenumeratorofareal coefficientallpasstransferfunctionissaidtobethemirror imagepolynomialofthedenominator andviceversa weshallusethenotationtodenotethemirror imagepolynomialofadegree mpolynomialdm z i e 26 allpasstransferfunction theexpression impliesthatthepolesandzerosofareal coefficientallpassfunctionexhibitmirror imagesymmetryinthez plane 27 allpasstransferfunction toshowthat am ej 1weobservethat therefore hence 28 allpasstransferfunction properties acausalstablereal coefficientallpasstransferfunctionisalosslessboundedreal lbr functionor equivalently acausalstableallpassfilterisalosslessstructure themagnitudefunctionofastableallpassfunctiona z satisfies 29 allpasstransferfunction 3 let g denotethegroupdelayfunctionofanallpassfiltera z i e theunwrappedphasefunction c ofastableallpassfunctionisamonotonicallydecreasingfunctionofwsothat g iseverywherepositiveintherange0 w p 30 applicationofallpasssystem anycausalstablesystemcanbedenotedas h z hmin z a z wherehmin z isaminimumphase delaysystem useallpasssystemtohelpdesignstablefilters useallpasssystemtohelpdesignlinearphasesystem asimpleexample p361 fig7 7 31 7 2transferfunctionclassificationbasedonphasecharacteristic 1 thephasedelaywillcausethechangeofsignalwaveform 32 2 thenonlinearityofsystemphasedelaywillcausethesignaldistortion timedelayofsignalisdependedonsystemphasecharacteristic 33 3 ifweignorethephaseinformation then 34 linearphaserequirement 4 thelinearphasefirfilterdesign groupdelay 35 7 2transferfunctionclassificationbasedonphasecharacteristic zero phasetransferfunctionlinear phasetransferfunctionminimum phaseandmaximum phasetransferfunctions 36 7 2 1zero phasetransferfunction onewaytoavoidanyphasedistortionsistomakethefrequencyresponseofthefilterrealandnonnegative todesignthefilterwithazerophasecharacteristic butforacausaldigitalfilteritisimpossible 37 zero phasetransferfunction onlyfornon real timeprocessingofreal valuedinputsignalsoffinitelength thezerophaseconditioncanbemet leth z beareal coefficientrationalz transformwithnopolesontheunitcycle thenf z h z h z 1 hasazerophaseontheunitcycle 38 zero phasetransferfunction pleaselookatbookp362 thefunctionfiltfiltimplementstheabovezero phasefilteringscheme pleaselookatbookp412 p7 5 39 7 2 2linear phasetransferfunction thephasedistortioncanbeavoidedbyensuringthatthetransferfunctionhasaunitymagnitudeandalinear phasecharacteristic thatis h ej e j dhowtoperformthelinear phasefilter y n x n d y ej e j dx ej h ej y ej x ej e j d 40 linear phasetransferfunction example determinetheimpulseresponseofanideallowpassfilterwithalinearphaseresponse applyingthefrequency shiftingpropertyofthedtfttotheimpulseresponseofanidealzero phaselowpassfilterwearriveat 41 linear phasetransferfunction bytruncatingtheimpulseresponsetoafinitenumberofterms arealizablefirapproximationtotheideallowpassfiltercanbedeveloped thetruncatedapproximationmayormaynotexhibitlinearphase dependingonthevalueofn0chosen ifwechoosen0 n 2withnapositiveinteger thetruncatedandshiftedapproximation 42 linear phasetransferfunction figurebelowshowsthefiltercoefficientsobtainedusingthefunctionsincfortwodifferentvaluesofn 43 7 2 3minimum phaseandmaximum phasetransferfunction basedontheexpression thephasefunctionis where kand karezerosandpoles respectively 44 minimum phaseandmaximum phasetransferfunction wedefinetheexpression ej k and ej k aszerovectorsandpolevectors when kand kareinsidetheunitcircle and changefrom0to2 thechangeofphaseofthezero pole vectorsare2 when kand kareoutsidetheunitcircle and changefrom0to2 thechangeofphaseofthezero pole vectorsare0 45 minimum phaseandmaximum phasetransferfunction so onlyzerosorpolesinsidetheunitcirclecanaffectphasefunctionofh ej forcausalstablesystem wecandeduce so it sthephasedelay lag system whenallzerosareallinsidetheunitcircle weget it stheminimumphasedelaysystem 46 minimum phaseandmaximum phasetransferfunction whenallzerosarealloutsidetheunitcircle weget wheremisthenumberofzeros andit sthemaximumphasedelaysystem atransferfunctionwithzerosinsideandoutsidetheunitcircleiscalledamixed phasetransferfunction example7 4 p367 47 7 3typesoflinear phasefirtransferfunctions itisnearlyimpossibletodesignalinear phaseiirtransferfunction itisalwayspossibletodesignanfirtransferfunctionwithanexactlinear phaseresponse wenowdeveloptheformsofthelinear phasefirtransferfunctionh z withrealimpulseresponseh n 48 linear phasefirtransferfunctions ifh z istohavealinear phase itsfrequencyresponsemustbeoftheform wherecand areconstants and calledtheamplituderesponse alsocalledthezero phaseresponse isarealfunctionof consideracausalfirtransferfunctionh z oflengthn 1 i e ofordern 49 linear phasefirtransferfunctions forarealimpulseresponse themagnituderesponse h ej isanevenfunctionof i e h ej h e j since theamplituderesponseistheneitheranevenfunctionoranoddfunctionof i e 50 linear phasefirtransferfunctions thefrequencyresponsesatisfiestherelationh ej h e j orequivalently therelation ifisanevenfunction thentheaboverelationleadstoej e j implyingthateither 0or 51 linear phasefirtransferfunctions from wehave substitutingthevalueof intheaboveweget 52 linear phasefirtransferfunctions replacing with inthepreviousequationweget makingachangeofvariablel n n werewritetheaboveequationas 53 linear phasefirtransferfunctions as wehaveh n e j c n h n n ej c n n theaboveleadstotheconditionwhenc n 2h n h n n 0 n n thus thefirfilterwithanevenamplituderesponsewillhavealinearphaseifithasasymmetricimpulseresponse 54 linear phasefirtransferfunctions ifisanoddfunctionof thenfrom wegetej e j as theaboveissatisfiedif 2or 2 then reducesto 55 linear phasefirtransferfunctions thelastequationcanberewrittenas as fromtheaboveweget 56 linear phasefirtransferfunctions makingachangeofvariablel n n werewritethelastequationas equatingtheabovewith wearriveattheconditionforlinearphaseas 57 linear phasefirtransferfunctions h n h n n 0 n nwithc n 2thereforeafirfilterwithanoddamplituderesponsewillhavelinear phaseresponseifithasanantisymmetricimpulseresponse 58 linear phasefirtransferfunctions sincethelengthoftheimpulseresponsecanbeeitherevenorodd wecandefinefourtypesoflinear phasefirtransferfunctionsforanantisymmetricfirfilterofoddlength i e nevenh n 2 0weexaminenexttheeachofthe4cases 59 linear phasefirtransferfunctions 60 linear phasefirtransferfunctions type1 symmetricimpulseresponsewithoddlengthinthiscase thedegreenisevenassumen 8forsimplicitythetransferfunctionh z isgivenby 61 linear phasefirtransferfunctions becauseofsymmetry wehaveh 0 h 8 h 1 h 7 h 2 h 6 andh 3 h 5 thus wecanwrite 62 linear phasefirtransferfunctions thecorrespondingfrequencyresponseisthengivenby thequantityinsidethebracesisarealfunctionofw andcanassumepositiveornegativevaluesintherange0 63 linear phasefirtransferfunctions wherebiseither0orp andhence itisalinearfunctionofwinthegeneralizedsensethegroupdelayisgivenby indicatingaconstantgroupdelayof4samples thephasefunctionhereisgivenby 64 linear phasefirtransferfunctions inthegeneralcasefortype1firfilters thefrequencyresponseisoftheform 65 linear phasefirtransferfunctions type2 symmetricimpulseresponsewithevenlengthtype3 antisymmetricimpulseresponsewithoddlengthtype4 antisymmetricimpulseresponsewithevenlengthp371 372aboutthesefirtransferfunctions 66 fourtypesoflinearphasefilter 67 fourtypesoflinearphasefilter 68 linear phasefirtransferfunctions whichisseentobeaslightlymodifiedversionofalength 7mo