第7章信息率失真函數(shù)

1、第第7 7章：信息率失真函數(shù)章：信息率失真函數(shù)7.1 7.1 基本概念基本概念7.2 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)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7.1.1 失真函數(shù)和平均失真度失真函數(shù)和平均失真度7.1.2 率失真函數(shù)定義率失真函數(shù)定義7.1 .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指定一個非負數(shù)0),(jibad失真度失真度（函數(shù)）),(.),(),(.),(.),(),(),(.),(),(212221212111mnnnmmbadba

2、dbadbadbadbadbadbadbadDa1 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),()/()(),(允許失真DD 7.1.1失真函數(shù)和平均失真度失真函數(shù)和平均失真度7.1.2 率失真函數(shù)定義率失真函數(shù)定義7.1.3 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

3、道試驗信道: )/(DDabpPijD稱)/(ijabp調(diào)整 使i);(min)()/(YXIDRDijPabp7.1.1失真函數(shù)和平均失真度失真函數(shù)和平均失真度7.1.2 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)：)(lim0DRDDRmaxDD0)(DRmaxDD 0)(DR)()/(jijbpabp令mjjjbpnijiimjjbpnimjjijibpDbpbadapbpDbadbpapjjj1)(11)(11max)()(min),()()(min),()()(minjnijiiDbadap1),()(線性分配a1a21021)1 (aammn

4、mnnDDDbadbadbadbadbadbad.211221111),(.),(.),(.),(),(.),(假定所有Dj中，Ds最小，令sjsjbpj01)(jjDDminmax下凸性210)()1 ()()1 ( DRDRDDR)/()/(21ijijab、pabp)()( D、RDRnimjjiijiDbadabpapD1111),()/()(nimjjiijiDbadabpapD11 22),()/()()()(log)/()();(111111DRbp)/a(bpabpapYXInimjjjijiiniijjabpapbp11i1)/()()()()()(log)()/();( 1

5、12222DRbp/abpapabpYXInimjjijiij定義新試驗信道：)/()1 ()/()/(21ijijijabpabpabpDDDDDbadabpapDnimjjiiji 2111)1 ( )1 ( ),()/()(滿足保真滿足保真度準則度準則)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 7.2 7.2 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)7.2.1 離散信源信息率失真函數(shù)

6、離散信源信息率失真函數(shù)的參量表達式的參量表達式7.2.2 二元信源的率失真函數(shù)二元信源的率失真函數(shù)DijjiiPabpbadap)/(),(),(nimjjijijibpabpabpapYXI11)()/(ln)/()();(DD niabpmjij, 2 , 1 1)/(1),()/()(11jiijinimjbadabpapDniijijabpapbp1)/()()()()()();(11nimjjiijiD,badabpapSYXI令mjijiabpu1 1)(0)(ij/abp0)()(S)()()()(ln)(ijiiiijijiu,badapapapbpabpap0ln),()()

7、/(lnijjijbaSdbpabpi)(lniiiapu令),()()/(jbiajiijSdebpabp1mjbajijiSdebp1),()(12niiijjjbiaSdeapbpbp1),()()()(3i32)(jbp1,i)(jbp)(ijabp),()()()(11),(jijijiibadebpapSDnimjbaSd4)()(ln)()()(),(11),(jjijijijiibpebpebpapSRbaSdnimjbaSdniiiapSSD1ln)()(5SdDdRniiniidDdapdDdSDSdDdRdDdSSRDRdDdRiii11)(SdDdSdSdapDSiii

8、ni)(1 負值在公式 兩邊對S取導(dǎo)數(shù)20),()()(1),(),(nibaSdbaSdjijiiijiiiebadapeapdSd兩邊乘以 對 求和)(jbpj0)(1DdSdapiniii0dDdSDmaxDnapi1)()(DS)(DR7.2.1離散信源信息率失真函數(shù)離散信源信息率失真函數(shù)的參量表達式的參量表達式11)(),(1jibaSdniiieapi),(1)(1jibaSdmjjiebp)(jbp),()()(jibaSdijijebpabpnimjbaSdjiijijiebadbpapSD11),(),()()()(iniiapSSDSRln)()()(1)(ijabp驗證

9、是否大于等于零)(ijabp23456),()(minmin1maxjiniijjjbadapDD70,00D21,1)( 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 ()/(212SSepeppabp)1 ()1 ()/(221SSeppepabp)1)(1 ()1 ()/(222SSeppepabpnimjjbiaSdjijiiebadbpapSD11),(),()()()(SSee1)1ln()1 ()1ln(ln1 ln)()()(1ppeppeeSapSSDSRSSSniii)1ln()1 (ln