離散時(shí)間信號(hào)處理DSP

DigitalSignalProcessingChapter6StructuresforDiscrete-TimeSystems6.0IntroductionSeveralwaystodescribediscrete-timesystems:Impulseresponsesintimedomain.Differenceequationsintimedomain.Theztransformsincomplexfrequencydomain(transferfunctions).Fouriertransformsinfrequencydomain(Frequencyresponse).H(ejω)=|H(ejω)|ejΘ(ω)26.1DescriptionoftheDigitalFilterStructuresIt’sdifferenceequationsintimedomainis將輸入加以延時(shí)，構(gòu)成M節(jié)旳延時(shí)網(wǎng)絡(luò)，把每個(gè)延時(shí)抽頭后加權(quán)，然后把成果相加。將輸出加以延時(shí)，構(gòu)成N節(jié)旳延時(shí)網(wǎng)絡(luò)，把每個(gè)延時(shí)抽頭后加權(quán)，然后把成果相加。所以，網(wǎng)絡(luò)構(gòu)造表達(dá)一定旳運(yùn)算構(gòu)造，不同構(gòu)造所需要旳存儲(chǔ)單元以及運(yùn)算次數(shù)不同，前者影響構(gòu)造復(fù)雜性，后者影響運(yùn)算速度。36.1DescriptionoftheDigitalFilterStructuresThreebasicelementstoimplementdigitalfilters:DelayMultiplierAdderBlockdiagram（方框圖）representationofthreebasicelements.z–1X(z)x(n)x(n–1)z–1X(z)x(n)X(z)kk

X(z)k

x(n)x1(n)X1(z)x2(n)X2(z)x1(n)+x2(n)X1(z)+X2(z)46.1DescriptionoftheDigitalFilterStructuresSignalflowgraph（信號(hào)流圖）representationofthreebasicelements.X(z)x(n)x(n–1)z–1X(z)z–1x1(n)X1(z)x2(n)X2(z)x1(n)+x2(n)X1(z)+X2(z)X(z)x(n)k

x(n)k

X(z)k56.1DescriptionoftheDigitalFilterStructuresTwoclassesofdigitalfilters:Finite-durationimpulseresponsefiltersornonrecursivefilters.Itstransferfunctionsareofthepolynomialform.Infinite-durationimpulseresponsefiltersorrecursivefilters.Itstransferfunctionsareoftherationalpolynomialform.66.3BasicstructuresforIIRdigitalfilters6.3.1DirectformIThetransferfunctionofarecursivefilterisgivenbyAndthedifferenceequationsintimedomainisIngeneral,M

≤N.76.3.1DirectformsIx(n)y(n)z–1z–1b0b1b2z–1bMbM–1z–1z–1z–1–a1–a2–aN–1–aNy(n–1)y(n–2)y(n–N)x(n–1)x(n–2)x(n–M)86.3.1DirectformsIx(n)y(n)b0b1b2bMbM–1z–1z–1z–1–a1–a2–aN–1–aNDirectformsIstructureforIIRdigitalfilters96.3.2DirectformsIIx(n)y(n)z–1z–1b0b1b2z–1bMbM–1z–1z–1z–1–a1–a2–aN–1–aN106.3.2DirectformsIIx(n)y(n)b0b1b2bMbM–1z–1z–1z–1–a1–a2–aN–1–aNDirectformsIIstructureforIIRdigitalfilters11Comparisonofthetwotypesx(n)y(n)b0b1b2bMbM–1z–1z–1z–1–a1–a2–aN–1–aNDirectformsIx(n)y(n)b0b1b2bMbM–1z–1z–1z–1–a1–a2–aN–1–aNDirectformsII12Example1ComputeH(z)fromthefollowingsignalflowgraph.Solution:x(n)y(n)1/4z–1z–11/4-3/82136.3.3Cascadeform WritingthenumeratoranddenominatorpolynomialsofH(z)asproductsofsecond-orderfactors,respectively,wehavethatx(n)y(n)γ11z–1z–1–m11–m21γ21γ1mz–1z–1–m1m–m2mγ2mH0…146.3.4Parallelform H(z)canalsobeexpressedasanadditionofsecond-orderpartial-fractions,suchthatx(n)y(n)β11z–1z–1–m11–m21z–1z–1–m1m–m2mβ0mβ01β1m…15Example2Figurethesignalflowgraphofthefollowingsystembythedirectform(typeIandII),cascadeformandparallelform.Solution:x(n)y(n)1/3z–1z–13/4-1/8TypeIIx(n)y(n)3/4-1/8z–1z–11/3TypeI16Example2x(n)y(n)1/3z–11/4z–11/2Cascadeform17Example2x(n)y(n)z–11/4z–11/210/3-7/3Parallelform18Example3Determinethetransferfunctionofthesystembelow:x(n)y(n)z–11/3z–11/5-15/2-3196.5BasicstructuresforFIRdigitalfiltersThedifferenceequationofFIRfilters

206.5.1Directformx(n)y(n)z–1z–1z–1h(0)h(1)h(2)h(M–1)h(M)216.5.1DirectformTransposeddirectform（直接型構(gòu)造旳轉(zhuǎn)置）x(n)y(n)z–1z–1z–1h(0)h(1)h(2)h(M–1)h(M)22Example4ComputethetransferfunctiongivenbythesignalflowgraphandthedirectformofH(z).x(n)y(n)h(0)h(1)h(2)h(3)h(4)h(5)h(6)h(7)h(8)23Example4x(n)y(n)z–1z–1z–1h(0)h(1)h(2)h(3)h(4)h(5)h(6)h(7)h(8)z–1z–1z–1z–1z–1246.5.2CascadeformWritingH(z)asaproductofsecond-orderfactors,wegetthat

x(n)y(n)z–1z–1γ01γ11γ21z–1z–1γ02γ12γ22z–1z–1γ2Nγ1Nγ0N256.5.3Linear-phaseforms（線性相位型）AnimportantsubclassofFIRdigitalfiltersistheonethatincludeslinear-phasefilters,thatis andthefrequencyresponsehasthefollowingform266.5.3Linear-phaseformswhereb(n)istheinverseFouriertransformofB(ω),andSinceB(ω)isreal,So276.5.3Linear-phaseforms Inthecommoncasewhereallthefiltercoefficientsarereal,so Ifh(n)iscausal,thatish(n)=0,forn<0,thenh(n)isoffinite-length,thatis,h(n)=0,forn>2τ.So,

Thisequationshowsthattheh(n)ofalinear-phasefilterissymmetricorantisymmetricaboutM/2.286.5.3Linear-phaseformsnh(n)210345678nh(n)2103456789nh(n)210345678nh(n)2103456789symmetricantisymmetricMevenModdTypeITypeIITypeIIITypeIV296.5.3Linear-phaseforms:typeI306.5.3Linear-phaseforms:typeIx(n)y(n)z–1z–1z–1h(0)h(1)h(2)h(M/2)z–1z–1z–1h(M/2–1)316.5.3Linear-phaseforms:typeII326.5.3Linear-phaseforms:typeIIx(n)y(n)z–1z–1z–1h(0)h(1)h(2)z–1z–1z–1z–1336.5.3Linear-phaseforms:typeIII346.5.3Linear-phaseforms:typeIIIx(n)y(n)z–1z–1z–1h(0)h(1)h(2)z–1z–1z–1h(M/2–1)–1–1–1–1356.5.3Linear-phaseforms:typeIV366.5.3Linear-phaseforms:typeIVx(n)y(n)z–1z–1z–1h(0)h(1)h(2)z–1z–1z–1z–1–1–1–1–1–137Example5Drawthesignalflow-graghofthedirectformandlinear-phaseformfortheFIRsystem.Solution:directformx(n)y(n)z–1z–1z–11-23-21z–1z–13-4z–138Example5Linear-phaseformx(n)y(n)z–1z–1z–11-23-4z–1z–1z–1396.5.3Linear-phaseformsClearly,thelinear-phaseformstructurerequiresabout50%fewermultiplicationsthanthatofthedirectforms.40DigitalnetworkanalysisTheanalysisofdigitalnetworksisrealized