第四章_信息率失真函數(shù)_lgy

1、第第2 2章：信源熵章：信源熵第第3 3章：信道容量章：信道容量第第4 4章：信息率失真函數(shù)章：信息率失真函數(shù)第第5 5章：信源編碼章：信源編碼第第6 6章：信道編碼章：信道編碼第第7 7章：密碼體制的安全性測度章：密碼體制的安全性測度4.1 4.1 信息率失真函數(shù)信息率失真函數(shù)4.2 離散信源的信息率失真函數(shù)離散信源的信息率失真函數(shù)4.3 連續(xù)信息的率失真函數(shù)連續(xù)信息的率失真函數(shù)4.4 保真度準(zhǔn)則下的信源編碼定理保真度準(zhǔn)則下的信源編碼定理實(shí)際通信系統(tǒng)允許一定的失真存在。1打電話；2放電影，視覺暫留性。允許壓縮信源輸出的信息率。研究內(nèi)容：信息率 允許失真4.1.1 失真函數(shù)和平均失真度失真函數(shù)

2、和平均失真度4.1.2 率失真函數(shù)定義率失真函數(shù)定義4.1 .3 率失真函數(shù)性質(zhì)率失真函數(shù)性質(zhì)1naaX1nbbY)/(ijabp對任一),(jiba指定一個非負(fù)數(shù)0),(jibad失真度失真度（函數(shù)）),(.),(),(.),(.),(),(),(.),(),(212221212111mnnnmmbadbadbadbadbadbadbadbadbadDa1 b1a2b2anbmjiajibadji0),(0.0.0aaaaaa11a漢明失真0111011102)(),(ijjiabbad2平方誤差失真函數(shù)平方誤差失真函數(shù)平均失真度平均失真度nimjjiijijibadabpapbadED11

3、),()/()(),(允許失真DD N次擴(kuò)展信道的平均失真度：NNXXXX21Niiiaa 1,11niiaaaaNniiN, 1,1Nni, 2 , 1NNYYYY21Nmj, 2 , 1 ,11mjjbbbbNNjjjbb 1mjjjN1,21NkjijijijjiikkNNNNjibadbadbadbbaadbad1),( ),(),( ),(),(1111由信源和信道的無記憶性)/()/()()( 11kkkijNkijiNkiabppappNjiijnimjiDDdppNDNN111 ),()/()()(DDkDNND)(NDND)(4.1.1失真函數(shù)和平均失真度失真函數(shù)和平均失真度

4、4.1.2 率失真函數(shù)定義率失真函數(shù)定義4.1.3 率失真函數(shù)性質(zhì)率失真函數(shù)性質(zhì)naaax,21信源編碼器信源編碼器mbbby,21XY假想信道假想信道將信源編碼器看作信道將信源編碼器看作信道DD DD D失真許可的 試驗(yàn)信道試驗(yàn)信道: )/(DDabpPijD稱)/(ijabp調(diào)整 使i);(min)()/(YXIDRDijPabpN次無記憶擴(kuò)展信源和信道：)(: )/()(NDNDabpPijND);(min)()()/(YXIDRNDijPpN)()(DNRDRN4.1.1失真函數(shù)和平均失真度失真函數(shù)和平均失真度4.1.2 率失真函數(shù)定義率失真函數(shù)定義定義域：1max, 0 DD=0R(

5、D)=H(X)連續(xù)：)(lim0DRDDRmaxDD0)(DRmaxDD 0)(DR)()/(jijbpabp令mjjjbpnijiimjjbpnimjjijibpDbpbadapbpDbadbpapjjj1)(11)(11max)()(min),()()(min),()()(minjnijiiDbadap1),()(線性分配a1a21021)1 (aammnmnnDDDbadbadbadbadbadbad.211221111),(.),(.),(.),(),(.),(假定所有Dj中，Ds最小，令sjsjbpj01)(jjDDminmax下凸性210)()1 ()()1 ( DRDRDDR)/

6、()/(21ijijab、pabp)()( D、RDRnimjjiijiDbadabpapD1111),()/()(nimjjiijiDbadabpapD11 22),()/()()()(log)/()();(111111DRbp)/a(bpabpapYXInimjjjijiiniijjabpapbp11i1)/()()()()()(log)()/();( 112222DRbp/abpapabpYXInimjjijiij定義新試驗(yàn)信道：)/()1 ()/()/(21ijijijabpabpabpDDDDDbadabpapDnimjjiiji 2111)1 ( )1 ( ),()/()(滿足保真

7、滿足保真度準(zhǔn)則度準(zhǔn)則)1 ()();( DDRDRYXI由 對 的下凸性：);(YXI)(ijabp);()1 ();();(21YXIYXIYXI)()1 ()( DRDR)()1 ()()1 ( DRDRDDR連續(xù)和單調(diào)遞減性3DH(X)R(D)maxDDD 4.1 信息率失真函數(shù)信息率失真函數(shù)4.2 4.2 離散信源的信息率失真函數(shù)離散信源的信息率失真函數(shù)4.3 連續(xù)信息的率失真函數(shù)連續(xù)信息的率失真函數(shù)4.4 保真度準(zhǔn)則下的信源編碼定保真度準(zhǔn)則下的信源編碼定理理4.2.1 離散信源信息率失真函數(shù)離散信源信息率失真函數(shù)的參量表達(dá)式的參量表達(dá)式4.2.2 二元信源的率失真函數(shù)二元信源的率失真

8、函數(shù)DijjiiPabpbadap)/(),(),(nimjjijijibpabpabpapYXI11)()/(ln)/()();(DD niabpmjij, 2 , 1 1)/(1),()/()(11jiijinimjbadabpapDniijijabpapbp1)/()()()()()();(11nimjjiijiD,badabpapSYXI令mjijiabpu1 1)(0)(ij/abp0)()(S)()()()(ln)(ijiiiijijiu,badapapapbpabpap0ln),()()/(lnijjijbaSdbpabpi)(lniiiapu令),()()/(jbiajiijS

9、debpabp1mjbajijiSdebp1),()(12niiijjjbiaSdeapbpbp1),()()()(3i32)(jbp1,i)(jbp)(ijabp),()()()(11),(jijijiibadebpapSDnimjbaSd4)()(ln)()()(),(11),(jjijijijiibpebpebpapSRbaSdnimjbaSdniiiapSSD1ln)()(5SdDdRniiniidDdapdDdSDSdDdRdDdSSRDRdDdRiii11)(SdDdSdSdapDSiiini)(1 負(fù)值在公式 兩邊對S取導(dǎo)數(shù)20),()()(1),(),(nibaSdbaSdji

10、jiiijiiiebadapeapdSd兩邊乘以 對 求和)(jbpj0)(1DdSdapiniii0dDdSDmaxDnapi1)()(DS)(DR4.2.1離散信源信息率失真函數(shù)離散信源信息率失真函數(shù)的參量表達(dá)式的參量表達(dá)式11)(),(1jibaSdniiieapi),(1)(1jibaSdmjjiebp)(jbp),()()(jibaSdijijebpabpnimjbaSdjiijijiebadbpapSD11),(),()()()(iniiapSSDSRln)()()(1)(ijabp驗(yàn)證 是否大于等于零)(ijabp23456),()(minmin1maxjiniijjjbadap

11、DD70,00D21,1)( 21pppaaXPXjjDDminmaxnijiij,badapD1)()(,)11()ijnSd a biiip a e )1 (1pDpD 2pDD2max1)1 (21sepp1)1 (21ppeS)1 (11Sep)1)(1 (12SepijijmjbaSdebp1)(1),()1 ()()(21SSepebpbp)1)(1 ()()(21SSepbpebpSSeeppbp1)1 ()(1SSepepbp1)1 ()(2),()()/(jijiijbaSdebpabp)1 ()1 ()/(211SSepeppabp)1)(1 ()1 ()/(212SSep

12、eppabp)1 ()1 ()/(221SSeppepabp)1)(1 ()1 ()/(222SSeppepabpnimjjbiaSdjijiiebadbpapSD11),(),()()()(SSee1)1ln()1 ()1ln(ln1 ln)()()(1ppeppeeSapSSDSRSSSniii)1ln()1 (ln )1ln()1ln(ln)(ppppDDDDDDR/1/ln21DDS容忍失真允許壓縮的信息率)()(DHpH15 . 02maxDS(D)DD0.50.25P=0.25P=0.5()R D推廣至n元等概率信源，信息率失真函數(shù)為：)1ln()1 (1/lnln)(DDnDDn

13、DR4.1 信息率失真函數(shù)信息率失真函數(shù)4.2 離散信源的信息率失真函數(shù)離散信源的信息率失真函數(shù) 4.3 連續(xù)信源的信息率失真函數(shù)4.4 保真度準(zhǔn)則下的信源編碼定保真度準(zhǔn)則下的信源編碼定理理4.3.1 4.3.1 連續(xù)信源失真函數(shù)的參量表達(dá)式連續(xù)信源失真函數(shù)的參量表達(dá)式)/(),(,xypRyx定義定義0),(yxd dxdyyxdxypD),()(dxdyypxypxypyxIc )()/(log)();(定義PD為滿足保真度準(zhǔn)則的試驗(yàn)信道集合);()()/(YXIInfDRcPxypD下確界:InfDD dxdyexydypxxpSDyxsd),()()()( )()( dxxxpSDSR

14、)( log)()(dDdRS 證明：4.3.2 4.3.2 高斯信源的率失真函數(shù)高斯信源的率失真函數(shù)222)(221)(mxexpdxxxpm)(dxxpmx)()(222)(),( yxyxd 2)(/()( ),()/()(yxxypdxxpdxdyyxdxypxpD dxdyyxdxypD),()( dxyxyxpdyypdxdyyxyxpyp22)(/()()(/()(dxyxyxpyD2)(/()(條件方差)dyyDypD)()()(2log21)/(maxyeDyXH)(2log21)/(yeDyXHcdyyXHypYXHcc)/()()/(maxdyyDype)(log)(21

15、2log21根據(jù)根據(jù)詹森不等式詹森不等式：dyyDyPdyyDyP)()(log)(log)(DlogDeYXHc2log21)/(DD eDYXHc2log21)/(22log21)(eXHceDDe2log212log21 2log21 )/()();(2DeYXHXHYXIcccDDR2log21)(YXN反向加性高斯實(shí)驗(yàn)信道DDR2log21)(1D2 0)( 2DRD2)(DRD2D2)(DS高斯信源的率失真函數(shù)4.34.3連續(xù)信息的率失真函數(shù)連續(xù)信息的率失真函數(shù)連續(xù)率失真函數(shù)的參量表達(dá)式連續(xù)率失真函數(shù)的參量表達(dá)式高斯信源的率失真函數(shù)高斯信源的率失真函數(shù)4.3.3 信息價值信息價值

16、合格品a1，廢品a2，生產(chǎn)過程抽象成信源：例例01. 099. 021aa 把檢驗(yàn)過程看作信道，檢驗(yàn)結(jié)果即為信道輸出。0),(11bad合格品出廠：b1：檢驗(yàn)合格b2：檢驗(yàn)不合格廢品報廢：0),(22bad合格品報廢：1),(21bad廢品出廠：100),(12bad 0100102121aabbD 不經(jīng)檢驗(yàn)全部出廠： 0101P1元1),()/()(2121ijjiijibadabpapD不經(jīng)檢驗(yàn)全部報廢： 1010P2元99. 0D檢驗(yàn)完全正確： 1001P3元0D)/(081. 0 99. 0log99. 001. 0log01. 0 )()(符號符號bitXHDR)/(2 .12081.099.099.0maxbitD元元檢驗(yàn)不十分可靠： 9 . 01 . 01 . 09 . 0P411 . 099. 009 . 099. 0D09 . 001. 01001 . 001. 0元199. 0比最大損失減少了：0.99-0.199=0.