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essge essge essge Inftn se Inftn Tey tnstte (ene) ntn nnel eeve (ee) essge sgnl essge nse se estntn Tnstn Mtx Te fst nbe s ne f beng, f beng, n f beng Sttng f, te next nbe ll be (), (8), () Sttng f, te next nbe ll be (), ( ), (6) Sttng f, te next nbe ll be (), (), () OR sybl sybl sybl AND essge=sybl, sybl essge=sybl, sybl Inftn se selets ese essge f set f pssble essges OR selets seqene f sybls f set f sybls t epesent essge. Destntn ees essge ng set f (gee) pssble essges, te nftn se sent. F T..8..6.. v = [...] "..8 P =..6.. & F Dsete Mv Cn T..8..6...8....6 As gntng n Stte As gntng n Stte As gntng n Stte. p()...!! Dg pbbltes Wt e te eltve feqenes f te bntn f sybls j=,, (g) fte ne te step? yvnestep. p(,j) p(,j)=p()p (j) p (j) Afte ne step j..6.6!!..!! j..8..6...!!! Wt bt n stey stte? 6 6

Wt s te pbblty stbtn n te stey stte? V ss P= V ss v x + v y + v z +...= v ss =[v x v y v z ] n+ eqtns n nnns Stey Stte Dg pbbltes Wt e te stey stte eltve feqenes f te bntn f sybls j=,,, (g)? p(,j)=p()p (j) V ss P= V ss "..8 [ v x v y v z ]..6.. & = [ v x v y v z ]... p()... p (j) j..8..6.. yvss..v x +.v y +.v z = v x.8v x + v y +.v z = v y v x +.6v y + v z = v z v x + v y + v z = [ v x v y v z ] = [...]... yv., yvss. p(,j) Step n stey stte j.78.8.66... 7 8 6 In te e, te ntl n stey stte pbblty stbtns e te se. H=-(.78)lg(.78)-(.8)*lg(.8)--(.66)lg(.66)- (.)lg(.)- (.)lg(.) =.6 bts/(sybl b) H(X,Y)=H(X)+H(Y) Entpy Avege nftn/sybl lle Entpy H H = " p lg( p ) F nf n, p H =p, p T =(-p) H ls beys tve ppety f events e nepenent. Inftn se H se (p) Inftn Tey sybls tnstte (ene) ntn nnel / s eeve (ee) sybls estntn H s te vege nbe f bts/sybl t epesent te nftn. If y ene sng e bts/sybl, ten te ext bts e ennt. Cpessn eves te enny. Y nt pess e fte te enny s gne; ll y e left t s nftn. Te entpy gves s le lt n te nbe f bts pe sybl e n eve. Qnn s ntepettn Wen senng stgt bny e, f se lettes e e n tn tes, s n s nt be eqlly pbble n te (/) ste. Usng enng, e n ty t e te (/) ste ve eqpbble s n s. Snnn-Fn ng splts te sybl feqeny t /, ten epets f e bn. E (/) ste ll be nseng "s te sybl y nt t sen n te ppe bn le bn?" n tees (lse t) eqpbble ne t ll be ete. Te vege nftn pe sybl s getest en te sybls eqpbble. S, te nt f nftn n te (/) ste s nese sng te pessn ng ve te sple bny ng. Usng e plte pessn ng, y n e lse t te (/) ste vng eqpbble s n s.

Snnn Fn ZIP plsn lgt ses ts Cpessn Splt sybls s pbbltes lve Ex. H l Feqeny t.9.6.9.9.9.97.6.. l. H = " tes ASCII 7bts/te, s 7 bts p lg( p ) H=-(.9 lg.9 +.6 lg.6 + ) H=.76 bts/sybl, s 8.7 bts f sentene s -lg.9=.7 bts f nftn l s -lg.=.97 bts f nftn Te e lettes y e nftn Cpessn Snnn Fn Splt sybls s pbbltes lve ( s lse s pssble) Ex. H l tes Feqeny t.9.6.9.9.9.97.6.. l..8.6.9.9.8.8.6.9.9.9.97.6.6.96..6.. Cpessn Cpessn Snnn Fn Splt sybls s pbbltes lve Snnn Fn Splt sybls s pbbltes lve Ex. H l tes Ex. H l tes Feqeny t.9.6.9.9.9.97.6.. l. Pefx fee - ne e s neve te stt f nte e Feqeny t.9.6.9.9.9.97.6.. l.

Snnn Fn Cpessn Splt sybls s pbbltes lve Ex. H l Enng t p.9 6.6.9.9.9.97.6.. l. tes 6()+()+()+()+()+ ()+()+()+(6)+(6) =97 bts 97bts/ tes =.9 bts/te H=.76 bts/sybl.9.6.9.9.9.97.6.. l. Hffn Cng A t lest pbbltes gp sybls Rest Repet Ex. H l Feqeny t.9.6.9.9.9.97.6 l.6..9.6.9.9.9.97 l.96.6.9.6 l.6.9.9.9.97 Hffn Cng Ex. H l Hffn Cng Ex. H l.9.6.9.9.9.97.6.. l..9.6.9.9.9.97.6 l.6..9.6.9.9.9.97 l.96.6.9.6 l.6.9.9.9.97.6.9.6 l.6.9.9 l l l l l.8.6.9.6 l.6 l..8.6.9. l..8 l.8. l l l l l l

l l Hffn Cng Ex. H l l l l l l l l l Hffn l 6 Hffn Cng Ex. H l Nte s bts;,l ve 6 bts 6()+()+()+()+()+ ()+()+()+(6)+(6) =97 bts 97bts/ tes =.9 bts/te H=.76 bts/sybl Snn Fn l 97bts/ tes =.9 bts/te l Hffns lgt s et f blng n extene bny tee f t n egte pt lengt f set f gven egts. Hffn 6 l 8 8 6 Feqenes*(eges t t)= egte pt lengt 7 l Hffns lgt s et f blng n extene bny tee f t n egte pt lengt f set f gven egts. Hffn 6 l 6 E bn s bt. Mnze (bnes * feqeny) Lest feqent sybl fte y. Me feqeny, lse. 8 8 7

Hffn Hffn ng s se n te fnl step f etng n MP fle.! Te MP ft ses fes f sple vles.! If te sple te s.hz, te te tt e fe epesents s ~6s.! Te spet f ts -sple fe s spetlly nlyze n te feqenes e gpe n nnels (tl bns).! Te sng effets tn bn e nlyze bse n psy-stl el.! Ts el etenes te tne-le nse-le nte f te sng n e nnel n ten ees te effet f e nnel n ts negbng bns.! Te sng nftn f ll f te nnels n te fe s ebne nt te vyng sgnl.! Ts sgnl s nelly ffeent f te gnl sgnl bt te ffeene s ly nteble lly.! Te sgnl f te fe s ten Hffn e.! A seqene f tese fes es p n MP fle. ttp://ebpyss.vsn.e/flty/b/py/ffn_ng.t Inftn se H(p) sybl, p() sybl, p() essge sen+nse sent nse eeve flppe? n n yes yes Inftn Tey tnstte (ene) sybl -> sybl -> ntn nnel, pty C sgnl nse nse se H y (x) eeve (ee) H y (x) = - p( nt flp ) lg(p( nt flp )) - p( flp ) lg(p( flp )) essge eqvtn: nfsn n eeve sgnl sps nftn n nse entpy f nse se. estntn + nse=?? -> sybl? + nse=?? -> sybl? -> n flppe bt p( nt flp ) -> flppe bt p( flp ) = - p( nt flp ) H y (x) x en p( nt flp ) = p( flp )=/ H y (x) n en p( nt flp ) =, p( flp )= OR p( nt flp ) =, p( flp )= R= H(x) - H y (x) Atl nf tnsssn te s te nf se entpy ns te nse se entpy C = x{r} ve ll pssble nftn ses C >H(x) ten y n ene t get! es (e etn es -> se enny) A bny ntn syste ntns p f e-pne eless nnels, s sn bel. sene Nsy Cnnel /6 / eeve/ sene e te e te Cpte te f pbbltes: sent eeve sent eeve sent eeve sent eeve eeve Asse tt n e nnel t s eqlly lely tt ll be tne nt tt nt. Asse ls tt n te fst nnel te pbblty f n e n ny ptl bt s /6, n n te sen nnel t s /. sene Cpte te f pbbltes: sent eeve:, sent eeve:, sent eeve:, sent eeve:, /6 / eeve/ sene e te e te /6 /6 / / / / / / / / eeve Cnnel syet S p(->)= p(->), et.

sene /6 eeve/ / e te sene e te eeve sene /6 eeve/ / e te sene e te eeve Cnnel syet S p(->)= p(->), et. Cpte te f pbbltes: sent eeve: p()+p() sent eeve: p()+p() sent eeve: p()+p() sent eeve: p()+p() /6 /6 / / / / / / / / p() = p(->)*p(->) p() = p(->)*p(->) p() = p(->)*p(->) p() = p(->)*p(->) p() = p(->)*p(->) p() = p(->)*p(->) p() = p(->)*p(->) p() = p(->)*p(->) /6 / Cnnel syet S p(->)= p(->), et. / p(): (/6)*(/)= /7 /6 p(): (/6)*(/) = /7 / / p(): ()*(/)= /7 / p(): ()*(/) = /7 Cpte te f pbbltes: sent eeve: p()+p()= /7 +/7= 6/7!.778 sent eeve: p()+p()= /7 + /7 = 6/7!. sent eeve: p()+p()= /7 + /7 = 6/7!. sent eeve: p()+p()=/7 + /7 = 6/7!.778 yflps. / / / p(): ()*(/)= /7 p(): ()*(/) = /7 p(): (/6)*(/)= /7 p(): (/6)*(/) = /7 A bny ntn syste ntns p f e-pne eless nnels, s sn bel. sene E Cetn: Repet Ce /6 / eeve/ sene e te e te eeve Repet e: s tnstte s tee sessve s n s tee sessve s. At te ee, jty esn le s se: f gp f tee bts s e s tn s (e.g.,,, ), t s sse tt s ent, n f e s tn s tt s ent. If te gnl se essge s n eql lel f s n s, t s te pbblty tt ee bt ll be net? sene E Cetn: Repet Ce A bny ntn syste ntns p f e-pne eless nnels, s sn bel. /6 eeve/ / e te sene e te eeve Repet e: s tnstte s tee sessve s n s tee sessve s. At te ee, jty esn le s se. Wt s te pbblty tt ee bt ll be net? sen sene eeve esn bts flppe p sent eeve:!.778 sent eeve:!. sent eeve:!. sent eeve:!.778 bts flppe.778*.778 *.778 =.7 bts flppe.*. 778 *. 778 =. bts flppe. *. *. 778 =.8 bts flppe. *. *. =. yflps.

A bny ntn syste ntns p f e-pne eless nnels, s sn bel. sene E Cetn: Repet Ce /6 eeve/ / e te sene e te eeve Repet e: s tnstte s tee sessve s n s tee sessve s. At te ee, jty esn le s se. Wt s te pbblty tt ee bt ll be net? sen sene eeve esn bts flppe p.7...8..8.8. sent eeve:!.778 sent eeve:!. sent eeve:!. sent eeve:!.778 ee bt net,,,.8+.8+.8+.=.6.6 ne ee bt ll be net Y e gven t fle tt s been pesse t lengt f, bts, n tl tt t s eslt f nnng n el entpy e n seqene f t. Y e ls tl tt te gnl t e sples f ntns vef, qntze t t bts pe sple. Te pbbltes f te npesse vles e s p(s) s p(s) / /6 /8 /6 Wt (ppxtely) s te lengt f te npesse fle, n bts? pesse fle Cpessn npesse fle H=-(. lg. )- = bts/sple bts/sple, bts x bts x bts bts / sple = bts bts / sple Ce sng Mtlb sltn Te nbe f (t-bt) sples n te npesse fle s lf te vle y pte n pt ). Y e tl tt te ntns vef s sple t te n pssble te s tt te vef l be enstte extly f te sples (t lest befe tey ee qntze), n y e tl tt te fle epesents sens f t. Wt s te gest feqeny pesent n te ntns sgnl? npesse fle bts/sple x bts Cpte nbe f sples Cpessn s p(s) s p(s) / /6 /8 /6 y sple = x bts / ( bts/sple) Cpte splng te splng te = y sples / t sens Use Snnn Splng Tee n " & " 6 & E Cetn: Hng Ce (7,) ( -, --), = H " & lns e bny gts f t - H tests f s vl e vl es stsfy H = " " & 6 = & +++6= +++6= +++6= Mx feqeny (Nyqst te) = splng te/ se l- t slve syste f eqtns f enng bts,, 6 t e vl e

" n 6 & " = + + 6 = + + 6 = + + 6 & 6 ese t slve f enng bts t e vl e f sng ts lyt. E Cetn: Hng Ce (7,) =++6 =++6 =++6 vl es stsfy H = " " & 6 = & +++6= +++6= +++6= se l- t n " & " & E Cetn: Hng Ce (7,) vl e f =++6 =++ =++6 =++ =++6 =++ e vl e sen " & sene nsy v " & eeve E Cetn: Hng Ce (7,) sen " nsy & sene v " & eeve H " s tee s n e " " " " = = & & & & se s ln, teefe e n te pstn Hng (7,) n et f bt es 6 t " = & se s n. Bt flp ete

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