信息論與編碼第4章信息率失真函數(shù)

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 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4.1.2 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 試驗信道試驗信道: )/(DDabpPijD稱)/(ijabp調(diào)整 使i);(min)()/(YXIDRDijPabp4.1.1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(D)=H(X)連續(xù)：)(lim0DRDDRmaxDD

4、0)(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)/()/(21ijijab、pabp)()( D、RD

5、RnimjjiijiDbadabpapD1111),()/()(nimjjiijiDbadabpapD11 22),()/()()()(log)/()();(111111DRbp)/a(bpabpapYXInimjjjijiiniijjabpapbp11i1)/()()()()()(log)()/();( 112222DRbp/abpapabpYXInimjjijiij定義新試驗信道：)/()1 ()/()/(21ijijijabpabpabpDDDDDbadabpapDnimjjiiji 2111)1 ( )1 ( ),()/()(滿足保真滿足保真度準(zhǔn)則度準(zhǔn)則)1 ()();( DDRDRY

6、XI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)二元信源的率失真函數(shù)DijjiiPabpbadap)/(),(),(

7、nimjjijijibpabpabpapYXI11)()/(ln)/()();(DD niabpmjij, 2 , 1 1)/(1),()/()(11jiijinimjbadabpapDniijijabpapbp1)/()()()()()();(11nimjjiijiD,badabpapSYXI令mjijiabpu1 1)(0)(ij/abp0)()(S)()()()(ln)(ijiiiijijiu,badapapapbpabpap0ln),()()/(lnijjijbaSdbpabpi)(lniiiapu令),()()/(jbiajiijSdebpabp1mjbajijiSdebp1),()

8、(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),(),(nibaSdbaSdjijiiijiiiebadapeapdSd兩邊乘以 對

9、 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驗證 是否大于等于零)(ijabp23456),()(minmin1maxjiniijjjbadapDD70,00D21,1)( 21pppaaXPXj

10、jDDminmaxnijiij,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 ()/(212SSepeppabp)1 ()1 ()/(221SSeppe

11、pabp)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 合格品a1，廢品a2，生產(chǎn)過程抽象成信源：例例01. 099. 021aa 把檢驗過程看作信道，檢驗結(jié)果即為信道輸出。0),(11bad

12、合格品出廠：b1：檢驗合格b2：檢驗不合格廢品報廢：0),(22bad合格品報廢：1),(21bad廢品出廠：100),(12bad 0100102121aabbD 不經(jīng)檢驗全部出廠： 0101P1元1),()/()(2121ijjiijibadabpapD不經(jīng)檢驗全部報廢： 1010P2元99. 0D檢驗完全正確： 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=0.791 元)/(6 .31025. 0791. 0)/(25. 0);(bitbitYXI元元符號符號108. 0)( 892.