数字信号处理-第04章-正交变换

2001/10/304.6Hilbert4.766.16.2FIR44.14.2K—L4.3DCT,DST4.4HartleyDHT4.5W24.6Hilbert4.734.1N},......,,{21NspanXϕϕϕ=Nϕϕϕ,......,,21X∑==Nnnnx1ϕαNααα,......,,21xX∈Nϕϕϕ,......,,21Step1:,,,jnnjnnnjjnxϕαϕϕαϕϕα∞=−∞∞=−∞===∑∑Step2:Step2:1ˆϕ2ˆϕ2ϕ1ϕ≠==jijiji01,ϕϕ(biorthogonality)*(),()()()jjjxttxttdtαϕϕ==∫*(),()()()jjjnxnnxnnαϕϕ==∑iiϕϕˆ=12,,,Nϕϕϕ12,,,Nϕϕϕ“”12,,,Nϕϕϕ[(0),(1),,(1)]TxxxN=−xNN×A=yAx,,,〈〉=〈〉=〈〉AxAxxxy,yNN×A1T−=AA(())11.2.N=yAx1T−==xAyAy3.221ϕ2ϕ3ϕ1α2α3αx“”“Frame)”,33““(())””2*||||()(),nxxnxnxx==∑22||||||nnαα==∑Parseval’s41,Nnnnnnxαϕαϕ∗===∑1Lnnnxβϕ∗==∑22(,)||||,xxxxxxxxε=−=−−,1,,nnnLβα==2(,)xxε221(,)NnnLxxεα=+=∑8FIR5CA0111TNλλλ−−==ACAACA¾FSFT,DTFT,DFS,DFT¾DCTDST,DHT¾Walsh-Hadamard,Haar¾SLT(¾¾¾¾¾¾¾¾4.2K—Lˆ(Karhunen--Loeve)[(0),(1),,(1)]TxxxN=−x{}000101101111101111()()TxxxNNNNNNEcccccccccµµ−−−−−−=−−=Cxx(,)(,)xxCijCji=K—LA=yAxyyC1.0xλ−=ICxC011,,,Nλλλ−2.xCN011,,,N−AAA3.011,,,N−AAA,1,0,1,,1;iiiN==−AA4.A011[,,,]N−AAA5.K—L:x=yAx011TyxNλλλ−==CACAyK—L011(0)(1)(1)TNyyyN−==+++−xAyAAA01ˆ(0)(1)()myyymmN=+++xAAAK—Lx{}2ˆ[]Eε=−xx011,,,N−AAAxC11Niimελ−=+=∑=yAxyDFTx()Xk1ˆ−=xAyyK—L¾¾K—LK—LxCA4.3DCT(),0,1,,1xnnN=−∑−==10)(1)0(NncnxNX102(21)()()cos21,2,,1NcnnkXkxnNNkNπ−=+==−∑DCT,2(21)cos2,0,1,,1knknkCgNNnkNπ+==−012;10kggfork==≠DCT0187111135152[coscoscoscos]1161616168721351052[coscoscoscos]16161616ππππππππ==ccCc1,0ijijij=〈〉=≠cc8DCTDCT1TNcNc−==xCXCX1012(21)()(0)()cos2NccknkxnXXnNNNπ−=+=+∑DCTMarkov-1):pdf111100[()(),(),,()]nnnnnnpXtxXtxXtxXtx++−−≤===11[()()],()()nnnnnpXtxXtxXtXn++=≤=:()XtMarkov-1ρ,[],,0,1,,1,1ijxijijNρρ−==−R212231231111NNNxNNNρρρρρρρρρρρρ−−−−−−=RK—LMarkov-1xRxRA1/2,2(1)[]sin(1)(1)22,0,1,,1ijjjNijNijNπωλ+=+−+++=−A22)cos(211ρωρρλ+−−=jjxR,jjλω)cos(2)cos()sin()1()tan(22ωρρωωρω+−−−=Njω1ρ→1ρ→tan()0Nω→1,,1,0,/−==NjNjjπω22(1)(12cos())jjλρρωρ=−−+ρρρρλ−+=−−=11)1(12200,1,1,jjNλ==−∑∑−=−==1010][NjjNjjjxλR0Nλ=0,1,1,jjNλ==−0Nλ=1,,1,0,/−==NjNjjπω1/2,2(1)[]sin(1)(1)22,0,1,,1ijjjNijNijNπωλ+=+−+++=−A,2(21)[]cos2ijijNNπ+=A,01[]iN=ADCTMarkov-1K—LDCTDCTK—LDCTMarkov-1DCTJPEG,MPEG1ρ→8,0.95Nρ==K—LDCTDST(),1,2,,xnnN=DST12()()sin()111,2,,NsnnkXkxnNNkNπ==++=∑12()()sin()111,2,,NsknkxnXkNNnNπ==++=∑,2sin()11,1,2,,knnkSNNnkNπ=++=1,0ijijij==≠ssDST,TNsNs==XSxxSX=)9/64sin()9/16sin()9/8sin()9/16sin()9/4sin()9/2sin()9/8sin()9/2sin()9/sin(928πππππππππSDSTK—L)sin()(mmmmkakθω+=ΦDFT:DCT:DST:K—L:,,,,nknknknkWCSA=yAxxRyR1,12,1(,)(,)NxijijNyijijijijλλ=≠=≠==∑∑RR12/1λλη−=“”80.91Nρ==DCT:DFT:DST:98.05%89.48%84.97%ηηη===NMiiiiNiyMiyE,,2,1),(),(11==∑∑==RRηEηMEη4.4Hartley∫∞∞−Ω−=ΩdtetxjtjF)(21)(πX∫∞∞−ΩΩ=dtejtxtjF)(21)(XπFT:cas()cos()sin()tttΩΩ+Ω∫∞∞−Ω=ΩdttcastxH)()(21)(πX∫∞∞−ΩΩ=dttcastxH)()(21)(XπHT:1()()cas()21[()cos()()sin()]2()()HxttdtxttdtxttdtEOππ∞−∞∞∞−∞−∞Ω=Ω=Ω+Ω=Ω+Ω∫∫∫X()()()()Re[()]Im[()]FHFFXjEjOXjXjXjΩ=Ω−ΩΩ=Ω−ΩFTHT102()()cas()0,1,,1NHnxnknkNkNπ−===−∑XDHT1022cas()cas()0NnNmknkmkmkNNππ−===≠∑HartleyDHTDFTNDHT4.44.5W1022()()sin[()()]40,1,,1NWnXkxnnkNNkNππαβ−==+++=−∑2/1,2/12)2/1)(2/1(4sin22/1,02)2/1(4sin20,2/12)2/1(4sin20,0)24sin(2==+++===++===++===+=βαππβαππβαππβαππNknNNknNNknNNnkNIVNIIINIININ==XWxDFTDWTDWT,/2)(,0)αβ=(,1/2)αβ=1WHartleyDWTDCTDST112342/cos(/),0,1,,2/cos[(1/2)/],0,1,,12/cos[(1/2)/],0,1,,12/cos[(1/2)(1/2)/],0,1,,1NnkNkNnNCNggnkNnkNCNgnkNnkNCNgnkNnkNCNnkNnkNππππ+===+=−=+=−=++=−DCT-DCT-DCT-DCT-112342/sin(/),1,2,,12/sin[(1/2)/],1,2,,2/sin[(1/2)/],1,2,,2/sin[(1/2)(1/2)/],0,1,,1NNkNnNSNnkNnkNSNgnkNnkNSNgnkNnkNSNnkNnkNππππ−==−=−==−==++=−DST-DST-DST-DST-DCTDSTK—LDCT-K—L1:ρ→0:ρ→0.45~0.85:ρ=1:ρ→−:ρDST-K—LDCT-K—LDCT-DCT-DCT-DCT-DCT4.6Hilbert1()ˆ()1()1()xxtdtxtdxttττπτττπτπ∞−∞∞−∞−−==∗∫∫0()sgn()0jHjjj−ΩΩ=−Ω=Ω∵)()()(ΩΩ=ΩϕjejHjH()1Hj∴Ω=ΩΩ−=Ω02/02/)(ππϕ)(ˆ)()(txjtxtz+=ˆ()()()()()()0()()0ZjXjjXjXjjHjXjjXjjXjjΩ=Ω+Ω=Ω+ΩΩ−Ω=Ω+ΩΩΩΩΩ=Ω000)(2)(jXjZ()zt()ztHilbert∫∞∞−−−=∗−=τττππdtxtxttx)(ˆ1)(ˆ1)(0()cos(2)xtAftπ=0ˆ()sin(2)xtAftπ=02ˆ()()()jftztxtjxtAeπ=+=HilbertHilbert−−=00)(ωππωωjjeHj1()()201(1)2jjnnhnHeednnnnπωωπωπππ−=−−==∫∑∞−∞=+−−=∗=mmmnxnhnxnx)12()12(2)()()(ˆπHilbertHilbertStep1.DFT,()xn(),0,,1XkkN=−Step2.−=−===1,,2012,,2,1)(20)()(NNkNkkXkkXkZStep3.DFT,()Zk()znStep4.ˆ()IDFT[(()())]xnjZkXk=−−)]()([)(ˆnxnzjnx−−=Hilbert1.Hilbert2.Hilbert0)(2)(2)](ˆ)[(21)(ˆ)(0202=ΩΩ−ΩΩ−=ΩΩΩ=∫∫∫∫∞−∞−∞∞−∗∞∞−djXjdjXjdjXjXdttxtxπππ0)](ˆ)[(21)(ˆ)(==∫∑−∗∞−∞=ππωωωπdeXeXnxnxjjnHilbert1122ˆ()()ˆ()()ˆ()()xtxtletxtxtxtxt⇔⇔⇔121212:()()()ˆ:()()ˆ()()ifxtxtxtthenxtxtxtxt=∗=∗=∗ˆ()xt3.:()(),()00letxnxnxnforn∗=≡∫−−−=ππθωθθωπdeXeXjRjI)2cot()(21)(∫−+−=ππθθθθωπ)0()2cot()(21)(xdeXeXjIjRHilbert∫−−−=ππθωθθωπdeXeXjj)2cot()(ln21)](arg[4.7))(cos()()(0tttatxϕ+Ω=0Ω()at00hhΩ−ΩΩΩ+Ω-0.0500.05-0.500.51-0.0500.05-101-0.0500.05-101-0.5-0.2500.250.5051015-0.5-0.2500.250.50204060-0.5-0.2500.250.50246))(cos()()(0tttatxϕ+Ω=1.2.()()cos()catattϕ=)sin()()cos()()(00ttattatxscΩ−Ω=)(sin)()(ttatasϕ=3.))(cos()()(0tttatxϕ+Ω=))(sin()()(ˆ0tttatxϕ+Ω=0()ˆ()()()()jtjtztxtjxtateeϕΩ=+