Article信州大学工学部紀要 37: 1-10 (1974)departmental bulletin pape

NAKAMURA,  Yatsuka

English

NAKAMURA  Yatsuka

Shinshu University Institutional Repository

        Jeurnal of the Faculty of Engineering, Shinshu University, No.37,1974 1                         Gti･ls[Jfitl}EIIIZ*ut*aee gRi37fy'                       On Markov Channels                           Yatsuka NAKAMURA*                         (Receivecl October 29, 1974)                            1. IRtroduetjon   The inforrnation theory originated by C. E. Shannon was extended to thecase of finite rnemory chaBnels by Hinckin3). Moreover the case of infinitememory channels was also studied by Adleri) and others. However, amongthese generalized channels, tkere are not many which are useful for concretemodels. In this paper, we define a Markov channel which is a natural genera-lization of the Markov chain and useful to make examples of generalized chan-nels. We skow that it is determined by a pair of vector valued mappings, andgive two representative examples of Markov channeis and simulate how errorsoccur in these chaltnels. We also show that a stationary stochastic statemachine is a Markov channel and we can compute the capacity of this ckannel.                              2. NotatioRs   The inPat sPace and the outPut sPace are represented by the alphabet spacesX=Ai and Y=-LBi respectively, where A== {ai,a2,･･-･･･,ale} and B:= {bi,b2,･･････,bi}are sets of finite letters (or states), and J={O, tl, ±2, ･･･} is a set of all lntegerswhich represents the time, i. e. , X and Y are the infinite dimensional productspaces of the countable copies of finite measurable spaces A and B respectively.Let ,,,,t21`7and Wbe Borel fields generated by all cylinder sets of X and Y res-pectively. For the element x in X, xn represents the nt th letter in A, and foryin Z yn represents the ni th letter in B. In the followings, we omit theexplanation for y as it is same as x. The notation [xg,xg+i,･･･,x?] represents athin cylinder on the time interval (s,s+1,･･･,t>, i.e.,            [xg, x,O.i, ･･･, x?] == £x E X: x. :=-L x9,, n = s, s + 1, ･･･, t},where s,tEf <s<t). A a-sub'field of .;Y generated by all thin cylinders of theform [xs,xs.x, ･-･,xt] on the time interva! (s,s+1, ･･･,t) is denoted by (Eg) i.L, ,,sz"-i,and a a-subfield generated by all thin cylinders of the form [xs] is deRoted by.,'2e"'s and is called the time-s Partition, as it is also seen as a finite partitien ofX. Let 2x be a trivial Partition of X', i. e. , 2x x {X}. A shij)5 transformation S't: Assistant Professor, Department of Information Engineering2 Yatsuka NAKAMuRA(or T) on X (er Y) represents time advance and is defined by the formula (Sx)n==: xn+i ((Ty)n == yn+i). The shift transformation S <or T) is a measurable trans-formation on the rrieasurable space X (or Y). The inPut source is an S-invarariantprobability measure P on the rneasurable space X, i.e. , which satisfies P(S-iE) = P(E) for all E E .2e. The outPut source is defined similarly. We denote a setof all input sources by U.    A channel from an input space X to an output space Y is a two variablenumerical function vx (F) of x E X and Fe Ziand satisfies the followings:    I. If we fix x Eff X, then vx(･) is a probability measure on <Y, fZY),    II. if we fix FG cZi, then v. (F) is a measurable function on X,and III. for all xEXandFEii Zi, vsx (F)=vx (SmiF>.    An input source P E ll and a channel v determine an output source q(･);                           q(F) =l.vx(.t7)p(dx) (FE czi ).Let (Xop }i] ff (2} 7) be the direct product measurable space of (X, .;2lf) and(Y; CZi), then we call it a comPound sPace, which is also an alphabet space(A (8) B)t. A shift transformation S (E9 T on the compound space is defined by (S (E])･T) (x, y) = (Sx, Ty). A comPound source is an S (g) T-invariant probability measureon the compound space. An input source P and a channel v also determine acompound source r;                           r(G) == I.vx(Gx)P(dx) (G E xo op cztt ),where Gx is an x-section of the set G. The sources q(･) and r(･) deterrnined byan input source P and a chanRel v are sometlrnes denoted by q(･;P, v) and r(･;p, v), respectively. We sometimes abbreviate vx([ys, ys+i, ･･･, yt]) to vx<ptsys+i･･･yt).   If there exists an lnteger m such that f< ･> = v. (yiy2･ ny ･yn) is a op i-rm" i ffi-measurable function for all positive integer n and for all letters Ni,y2,･･･,Nn,and if fl･) is constant for n smaller than 1-m,then the channel v is called afinite memory channel or an m-memory channel.   For any finite partitions pai and Y2 of X, we write              Y,V £f', = {A, fi A,: A, Ei LSY ,, A, E pt2},which is a new partition of X. Then the entroPy hp(S) of the transformation Srelative to an inpat source P E ll is defined by                  hp(s) = lim -IL H<2ea vSmi 2ov ･･･vST""+i ge),                          nnwhere .2:;7e is the time-O partition, and H(..gvf) is the sum -X P <A) log P (A).                                                        AE vftif                            On Markov channels 3Ifqandran output source and a compound source respective!y, then wecan also define the entropy hq(T> and hr (S(g>T> of the transformation T relativeto an output source q and of the transformation S(g>T relative to a cornpoundsource r. Then the transmission rate Rp(v) of a channel v relative to an inputsource P is given by                     Rp(v) = hp(S) + ha(T) - hr(S(El)T)where both q(･) =q(･;P,v) and r(･)= r(･;P,v) are determined by P and v. The(stationary) capacity Cs (v) of a channel v is defined by                       Cs (v) =: sup Rp (v).                              pErr                          3. MarkovChannels   We begin with the definiton of a Markev channel.    Definition. A channelvfrorn the input space X to the output space Y iscalied an m-fold Marfeov channel if there exists a positive number m such that    IV. vx(.Yn j JYe'''Yn-i) = Vx(Nn j ptn-m''' Yn-i)for all xe X, for all yi,･･･,yn Ei B and for all positive integer n (n:}lm), where           ..(,,...y,,, i yt･･･ys-i) : ("x(Y`UYS'i)i"Y`l.'2''ilMy)):111xs,[:l :.'.i 8:The errors which occur in the m-fold Marlcov channel depend on the only msteps in the past.    Next let us see that the Markov channel is characterized byapair ofmeasurable mappings. Now for every e!ement x in X, we consider a stochasticvector on Bm                             Q(x)=(qb- (x)) (bEBM)and we also consider a stochastic matrix                             (li<x) = (ijb t-b(x)),whose row is indexed byb in B and whose column is indexed by bin BM.Moreover we assume that ab-(x) and abl-fi(x) are measurable functions of x on(X, .fi}a), and assume that          qyn--m+!･･･yn (SX) = X ayniyn-n;･･･yn-i(X)gyn-m･･･yn-i(X). (1)                         Yn-m emBThe pair (Q(x), Q"U(x)  is called a stochastic Pair.4 Yatsul<a NAKAMuRA   Theorem 1. Let (Q(x),QN(x>) be a stochastic Pair. Then it dojines an m-foldMarfeov channel. Conversely every m-fold Markov channel defines a stochasticpair.   Proof. For a stochas£ic pair (Q(x>, Q-"(x)>, we can construct an m-£old Mar-kov channel by tke foliowing proceedure:We put     v.(y,･･･yn) : (z;}. ;;..,ptj.{,li/i//' ,4y-+ii yi'"y-"'-i (S`-iX' 111 l.i Iillllll i.l glAssumming n-m>O, we see          y.l!lilBVx (Yi'''Y") = y.IIIillBqyi'''ym(x) llYJIn4ynt+i : yi'''ym+i-i (x)                      "= qy,･･･y. (v)ll1.i;l'imm1 4y.+iT yi･･･y.+i-, (Si'ix)                      == vx(Yi'''Yn-i)･As a similar equaiity also holds in case of n-m f{ O, we conclude that vx(･) canbe extended to a probabllity measure on op ico..i cZii by the Kolmogorov extenslontheorem. Next, for any thin cylinder [yi,･･･,yn] in Y (n 2}})l2}i i, n>m),     vx(yl･･･yn)= Z vx(Y,･･･Y.)               Yl"'Yl-1             = Z qyii･･y.,(x) fl",I;'l ay.,.Myi･･･y..i.,(Si-ix>              J,1-i･3'lfit            =y,.II.ily -,qy2･･･ynt+i (Sx) ll1･i:.L" Aq'y..il yi.･.y.,i", (si-ix>             = qyl･･-ym÷l., (S l x> IIZItrl"Z 4y.." yi...y.,i-., (si-lx) ･`･if n-m l}) l,                  Z qy.m.･･･y.-,(S"-MX) ･･･if n-m<l.                Yn-･m""Yl-.1In both cases of n>m and num<m, we get                   vx (Yl'''Yn) :vslx(Tl [Yl'''Yn]). <2>Replacinglin (2) by l-1, we get              vsx (T [Yl'''Yn]) == vsl-1(sx)(Tl-iT [Yt, ''', yn])                            On Markov channeis S                              u' vslx (Tl [Yl,''', Yn])as T[yt,･･･,yn] is a thin cylinder on the time interval (l-1,･･･,n-1). Hence                       vx (Yl'''Yn) = vsx (T [Nl'''Yn])holds. We extend vx(･) from (Ebioo..i SIxi to :'U -- opi--corm.. Yi, by the formula                    vx (Y-l'''Yn) == vs"lx (T-l IY-l, ''', Yn]).Tken v satisfies the condition III of the channeL For any [yi,･･･,yn], v.(yi･-･yn)is .c20-measurable on X, as thls is defined by the product of the measurablefunctions q5(x) and ab,s(x). And for any measurable set F <E Y), v.(F) is alsomeasurab!e, as F can be approximated'by a countable sequence of cylinder setswith the measure yx(･). Therefore the condition II is a!so clear.   Converse!y let vbe an m-fold Markov channel. We define a stochastic pair(Q(x),-Q" (x)> by the following;                   qyi (x) == vx (Y,>and                          ijy.., 1 Zl(x) = vx(y..A Zi)where S = [),i,y2,･･･ym]. Clearly the pair is well defined by the condition III oftke channel.        X aym+i[yi･･･yn; (X)qyi･･･ynt (X) ": £ vx (Yi'''Y,n) vx (Mn+dYi'''Ynt)       )'i EB yi EB             := ]Z] vx(yi'''ytn+,t> =vx(y2'''ym+i)=vsx(T(y2'''ym+i))              yi EB             == qy2･･-y.+, (Sx).The above chaiR of formulae shows that the pair satisfies the condition of thestochastic pair. Q. E. D.   This theorerri tells us that any Markov channeHs determined by a suit ofmappings, which resembles to the fact that a memoryless channel (cf. 2) p. 154)is determined by a stochastic matrix Let (cab) be a stochastic matrix whichdetermines a memoryless channel, then taking qb(x) = cxib aRd 'q'bis(x) =r cx,b,we see that the memoryless channel is a kind of Markov channels.                              4. Examples   We put A =xx B == {O, 1}, i. e. , we consider a binary case. For a stochasticpair, Iet us choose6 Yatsuka NAKAMliRA            Q(x) wr (1 il p)= o<o) ･･･ if x, .. o,                    (i5ab)== Q(') ''' if xe -= i,     -Q- (X) == (1Sq 1;")=. Q-" <OO) ''' if xo == O, xi=o,                             -. ･･･ if xo =O, xl=1,             (i1q i1,)=:e(oi)             (1 -;r 1ilq)=" Qi (10) ''' if xe= 1, xt == o,             (i 11 r i li- q)= Zl> (ii) ''' if xe = i, xi = i,and assume that Q(OO)Q(O) =Q(O) i.e.,P=(r-1)/(r+q-2). Then we findthat Q(11)Q(i) =- Q(1), Q(Ol)Q(O) == Q(1) and Q(10)Q(1) :: Q(O). Hence (Q(x), Q(x))is a stochastic pair, so that we can define a Markov channei, which we callan additive noise channel. Errors in this channel occur as shown in Fig.1.Fig.2isaresuit ofa simple simulation in the case ofq=9/10, r=1/2 andP == 5/7.                   q 1--q r                          CE                                1-r                            C: correct                            E: erroneous                                Fig. 1         input sequence IOIIOOIOIIOOIOIIIOOIIOOOOIOIIIOIOOIOI                ･         output sequence OIIIOOIOIIOOIOIIIIIOIIIIIOOIIIOIOOIOI                                Fig. 2                            On Markev channels 7   Next we give another example;                    Q(x> =(;/ )=:Q for anyxc! X,                   Q(X) =- (, [ll p i 1] P) -= Q(O)                            lf x, = o,                           (, iZ, iII` q) =Q<i) if xi := i.Then we can also define a Markov channel as shown in Fig. 3, which we caila stationarN stochastic state machine channel (SSSM . channel). Fig. 4 is a resultof simulation when p == 3/4 and q= 1/4. In Fig. 4, the output letters seem tohave little correlation wlth the input letters, but we can show later that thecapacity of this channel is not zero, therefore, the transmission of the infor-mation is possible through this noisy channel by the Shannon-Hinchin theorem.                  ×,-ob,trPO                                       1-p                  x,-ibot"qO                                       1--q                                 Fig 3         input sequence 10110010110010111001100001011aOIOOIOI                 e         output sequence 1101110010110010001011000010110111001                                 Fig. 4                      5. Capacity of SSSM Ckannels   The second example of the previous section can be extended to a moregeneral case, i.e., the letters need not be binary. Let Q(x) be independent ofxeX. And we assume that Q"V(x) is only dependent on x, hence we can writeQ(x) == Q(xi), and assume tlaat Q(xi)Q--Q for all xEA, where Q==: Q(x). Wecail this generalized one an SSSM channel also. We can cornpute the capacity8 Yatsul<a NAKAMuRACs(v) of an SSSM ckannel v. First, we give a proposition in a more generalsituation.   Proposition 1. Let v be a finite memory Markov channel, then           Rp == hq(T) + lx y,li.IIII.y..q,yi･･･ym(x) rp(ay,.+iiyi･･･y. (x))p (dx)where rp<t) == -tlog t.   Proof. By the definkion of Rp, Rp = hp(S) + hq(T> - hr<S & T)    = hq(T) + hp(S> - lim -IL U( ;Z' 'o X fZi oVS-i,,2"o (El) T-i fZ/ eV                   n71                                     ･･･ V S-n+i ,,sZe"e (Ei) T-n+i :Z/ Q)    = hq(T) + hp(S)    - lim -IL H(sm-i2oe (g) 2y V Sm,s2oe (g> 2y V      nn                 '''VS.,S" eV2yV.;2lfe (El) WoVS"i,ff'o (E9 T-i f?iieV                   '･ '''VS'7i'i,..20e (Eg) T-"Mi C}l/e)     :hq(T)÷hp(S)+ li,m, i. I!iEl... ,;E.).., rp( I vx(yi'''y")P(dx>)･                        X-flt n 1 ll                                  X ""X             1- tn ilHere we assume thatvis m-foid Markovian and of m-memory with the com-mon integer m, which does not lose generality.Rp =hq(T>+hp(S)+<iiml Z Z rp(P(xi-"t'''Xn) vx<Yi'''Yn))                  u nx -･x .v --y                      IH ?)z n l n  =:h,(T)+hp(S)+lim-l- Z] Z q(P(xi-m'''xn)qyi･･･ym(X)                 n ?2x -･pt ", -.>i                      1-･71t n 1 n                                      ' llla)klTlit Nqyi+tn 1 yi･･･yi÷nt-i(X))   = hq(T) + hp(S)  -i- lim -!L X Z p(xit..'''x.>qy,･･y.(x> ll'i-M'q-y,+m l y,-･yi+tn-i(x>×     n nx -･x )T ･-.v it-1     l-m nl n                                 n-m.  ×(log P(xi-jn･･･xn) + log qy,･･･y.(x> + X. Iog ayj+. i yj･･･yj"nt-i<X))                                 J==1                        On Marl<ov Channels 9  ==h,(T>+lim-l- lli]i Z] p(xi--m'''xn)qyi'''ym(x)           n VZ x -･x ), --J,                1-m nl n                                        n-m                  ･ll'it=-1'i ijyi+,. I yi･J･yi÷mmi(x)) (Xj..i log "q'yi+m I yj'･･･yi+,n-i(X))                 }t-f･11   n iI g'rm- x -x y･-y  =: hq<T)÷lim- £ X X P(Xi"m'''Xn) vx(Yi'''Nn) 10g ay"'+mfyi･･yn,÷j'--i<X)l                    1-flt n 1 11                 lt-?n umnv hq(T) + ii,m -h ( ,Z･..,l.,.mli, ],....,..., ,l.ll],..,,...e[Xi-"z'''X""m)                  . yx<yt･･yl'+m-1) log 'q"yj+. 1 yi･･･ynt÷J'--i(X) l                 il.-'tll. == hq(T) + iipa -h I ,]Z.. l H'(si"m2x (g) Ti+m c}eit,                 ¥,.-l-mui. si,･/z･--･, op Ti 2.v v'k/L.iliwws'fe.,sz., (g} Tk :y,)]  -= hq(T) + I y ¥..,, qyi･･Jy.(x) rp(qy.., i y,･･ty.(x))p(dx),             1 M÷I    X   which completes the proof, where H(.orl S') means ff(,.9fV,.KZ. ")-ff<.(Z, i                                                       Q.E.D.   Next we give another proposition:   Proposition 2. Let v be an SSSM channel defated by Q== (qb> and Q(xi)=(ablbt (xi)>. if Pe is a Bernoulli izzPntt source, i.e., time independent probabilitymeasure on X=: Ar, then the outPztt source qo(･)= q(･;Po,v) is a Marfeov measureon Y= Bi.   Proof. Let us prove that                qe([Nn]1[Yr''Nn-i]) == qo([Yn]1[Yn-i])･ (3>We abride the brackets [] in the following formulae. Then, qe(Yn l Yi'''Y"-･i) ='nt qG(Yi'''Yn--iMi)/ae(Yi'''Y"-i) = q( Yi>'q"y2iyi(X2)'' "q- yn l y..i(Xn)P(XD'''P(Xn) / (q(Yi)'q'y, I y,( C2)                         ' '''ij3'n--iiJ'n-2(Xn-1)P(Xl)'''P(X"-1))  := i7yn1yn-i(Xn)P(Xn) = ae(YnIYn-･i),which proves the equality (3). Q. E. D.   Tlteorem 2. Let v be an SSSM channel, then the capacity of v is given by10 Yatsuka NAKAMuRA                                                ' CS(") = SS,P( .?}t],b,qbiab2[bi (a2)Po(a2) 10g (ab,lb, (a2))/<III,;, 4b,1b,(a2i)Po (a2t))],where Pe moves on all Probability measures on A.    Proof. Let P be any iRput source, then by Proposition 1,                    Rp == hq(T) + Z ay,rp(ay,Iyi(X2))P(X2)･                                YlThe following inequality and equality are well known (cf. 2) p. 82 and p. 78) :            hq(T)=limH(S :ZiieIViml'oS-i CZ/o);:i{H(S CZie1 CZ;io)･                     nThus,                          '            hq(T> -< Z q(ye) rp (q<yi 1 yo>> == £ qy, ty (:Ii ) ayi i y, <xi)P(xi))･                    ye yo ,xlNow Iet Pe be a Bernoulii source given by                       Pe(Xi'''Xn) =: P(Xi>P(X2)'''P(Xn),then by Proposition 2, denoting qe(･>-- q(･;Po,v),                         H<S-i ZieI 7o) " hae(T)･Therefore         Rp =: hq(T) + X qyt ny(4y2 i yi(X2))P(X2)                      J'1           E{: hq,(T) + X qy, rp<ay, i y, <x2))Pe(x2) < = Rp,)                     Yl            =: Z qb,4b,lb, (a2)Pe(a2) 10g (4b,lb,(a2))/<E4b,ib,(a2t)Po(a2i)),              a2bib2 a2'which proves the result. Q. E. D.   Therefore we can easily compute the capacity of the channel shown inFig.2 of the previous section, which is about O. 1886 bit/step.                               Reference1) ADLER, R.L., Ergodicand mixing properties of infinite memory channels..Proc. Amer. Ii4izth. Soc. I2, 924-930 (1961).2) BILLINGsLEy, P., Ergodic theory and information. John Wiley and Sons, (1965).3) HiNcmN, A. I., Mathematical foundations of information theory. Dover, <1958). (English translation)

On Markov Channels

        Jeurnal of the Faculty of Engineering, Shinshu University, No.37,1974 1                         Gti･ls[Jfitl}EIIIZ*ut*aee gRi37fy'                       On Markov Channels                           Yatsuka NAKAMURA*                         (Receivecl October 29, 1974)                            1. IRtroduetjon   The inforrnation theory originated by C. E. Shannon was extended to thecase of finite rnemory chaBnels by Hinckin3). Moreover the case of infinitememory channels was also studied by Adleri) and others. However, amongthese generalized channels, tkere are not many which are useful for concretemodels. In this paper, we define a Markov channel which is a natural genera-lization of the Markov chain and useful to make examples of generalized chan-nels. We skow that it is determined by a pair of vector valued mappings, andgive two representative examples of Markov channeis and simulate how errorsoccur in these chaltnels. We also show that a stationary stochastic statemachine is a Markov channel and we can compute the capacity of this ckannel.                              2. NotatioRs   The inPat sPace and the outPut sPace are represented by the alphabet spacesX=Ai and Y=-LBi respectively, where A== {ai,a2,･･-･･･,ale} and B:= {bi,b2,･･････,bi}are sets of finite letters (or states), and J={O, tl, ±2, ･･･} is a set of all lntegerswhich represents the time, i. e. , X and Y are the infinite dimensional productspaces of the countable copies of finite measurable spaces A and B respectively.Let ,,,,t21`7and Wbe Borel fields generated by all cylinder sets of X and Y res-pectively. For the element x in X, xn represents the nt th letter in A, and foryin Z yn represents the ni th letter in B. In the followings, we omit theexplanation for y as it is same as x. The notation [xg,xg+i,･･･,x?] represents athin cylinder on the time interval (s,s+1,･･･,t>, i.e.,            [xg, x,O.i, ･･･, x?] == £x E X: x. :=-L x9,, n = s, s + 1, ･･･, t},where s,tEf <s<t). A a-sub'field of .;Y generated by all thin cylinders of theform [xs,xs.x, ･-･,xt] on the time interva! (s,s+1, ･･･,t) is denoted by (Eg) i.L, ,,sz"-i,and a a-subfield generated by all thin cylinders of the form [xs] is deRoted by.,'2e"'s and is called the time-s Partition, as it is also seen as a finite partitien ofX. Let 2x be a trivial Partition of X' , i. e. , 2x x {X}. A shij)5 transformation S't: Assistant Professor, Department of Information Engineering2 Yatsuka NAKAMuRA(or T) on X (er Y) represents time advance and is defined by the formula (Sx)n==: xn+i ((Ty)n == yn+i). The shift transformation S <or T) is a measurable trans-formation on the rrieasurable space X (or Y). The inPut source is an S-invarariantprobability measure P on the rneasurable space X, i.e. , which satisfies P(S-iE) = P(E) for all E E .2e. The outPut source is defined similarly. We denote a setof all input sources by U.    A channel from an input space X to an output space Y is a two variablenumerical function vx (F) of x E X and Fe Ziand satisfies the followings:    I. If we fix x Eff X, then vx(･) is a probability measure on <Y, fZY),    II. if we fix FG cZi, then v. (F) is a measurable function on X,and III. for all xEXandFEii Zi, vsx (F)=vx (SmiF>.    An input source P E ll and a channel v determine an output source q(･);                           q(F) =l.vx(.t7)p(dx) (FE czi ).Let (Xop }i] ff (2} 7) be the direct product measurable space of (X, .;2lf) and(Y; CZi), then we call it a comPound sPace, which is also an alphabet space(A (8) B)t. A shift transformation S (E9 T on the compound space is defined by (S (E])･T) (x, y) = (Sx, Ty). A comPound source is an S (g) T-invariant probability measureon the compound space. An input source P and a channel v also determine acompound source r;                           r(G) == I.vx(Gx)P(dx) (G E xo op cztt ),where Gx is an x-section of the set G. The sources q(･) and r(･) deterrnined byan input source P and a chanRel v are sometlrnes denoted by q(･;P, v) and r(･;p, v), respectively. We sometimes abbreviate vx([ys, ys+i, ･･･, yt]) to vx<ptsys+i･･･yt).   If there exists an lnteger m such that f< ･> = v. (yiy2･ ny ･yn) is a op i-rm" i ffi-measurable function for all positive integer n and for all letters Ni,y2,･･･,Nn,and if fl･) is constant for n smaller than 1-m,then the channel v is called afinite memory channel or an m-memory channel.   For any finite partitions pai and Y2 of X, we write              Y,V £f', = {A, fi A,: A, Ei LSY ,, A, E pt2},which is a new partition of X. Then the entroPy hp(S) of the transformation Srelative to an inpat source P E ll is defined by                  hp(s) = lim -IL H<2ea vSmi 2ov ･･･vST""+i ge),                          nnwhere .2:;7e is the time-O partition, and H(..gvf) is the sum -X P <A) log P (A).                                                        AE vftif                            On Markov channels 3Ifqandran output source and a compound source respective!y, then wecan also define the entropy hq(T> and hr (S(g>T> of the transformation T relativeto an output source q and of the transformation S(g>T relative to a cornpoundsource r. Then the transmission rate Rp(v) of a channel v relative to an inputsource P is given by                     Rp(v) = hp(S) + ha(T) - hr(S(El)T)where both q(･) =q(･;P,v) and r(･)= r(･;P,v) are determined by P and v. The(stationary) capacity Cs (v) of a channel v is defined by                       Cs (v) =: sup Rp (v).                              pErr                          3. MarkovChannels   We begin with the definiton of a Markev channel.    Definition. A channelvfrorn the input space X to the output space Y iscalied an m-fold Marfeov channel if there exists a positive number m such that    IV. vx(.Yn j JYe'''Yn-i) = Vx(Nn j ptn-m''' Yn-i)for all xe X, for all yi,･･･,yn Ei B and for all positive integer n (n:}lm), where           ..(,,...y,,, i yt･･･ys-i) : ("x(Y`UYS'i)i"Y`l.'2''ilMy)):111xs,[:l :.'.i 8:The errors which occur in the m-fold Marlcov channel depend on the only msteps in the past.    Next let us see that the Markov channel is characterized byapair ofmeasurable mappings. Now for every e!ement x in X, we consider a stochasticvector on Bm                             Q(x)=(qb- (x)) (bEBM)and we also consider a stochastic matrix                             (li<x) = (ijb t-b(x)),whose row is indexed byb in B and whose column is indexed by bin BM.Moreover we assume that ab-(x) and abl-fi(x) are measurable functions of x on(X, .fi}a), and assume that          qyn--m+!･･･yn (SX) = X ayniyn-n;･･･yn-i(X)gyn-m･･･yn-i(X). (1)                         Yn-m emBThe pair (Q(x), Q"U (x)) is called a stochastic Pair.4 Yatsul<a NAKAMuRA   Theorem 1. Let (Q(x),QN(x>) be a stochastic Pair. Then it dojines an m-foldMarfeov channel. Conversely every m-fold Markov channel defines a stochasticpair.   Proof. For a stochas£ic pair (Q(x>, Q-"(x)>, we can construct an m-£old Mar-kov channel by tke foliowing proceedure:We put     v.(y,･･･yn) : (z;}. ;;..,ptj.{,li/i//' ,4y-+ii yi'"y-"'-i (S`-iX' 111 l.i Iillllll i.l glAssumming n-m>O, we see          y.l!lilBVx (Yi'''Y") = y.IIIillBqyi'''ym(x) llYJIn4ynt+i : yi'''ym+i-i (x)                      "= qy,･･･y. (v)ll1.i;l'imm1 4y.+iT yi･･･y.+i-, (Si'ix)                      == vx(Yi'''Yn-i)･As a similar equaiity also holds in case of n-m f{ O, we conclude that vx(･) canbe extended to a probabllity measure on op ico..i cZii by the Kolmogorov extenslontheorem. Next, for any thin cylinder [yi,･･･,yn] in Y (n 2}})l2}i i, n>m),     vx(yl･･･yn)= Z vx(Y,･･･Y.)               Yl"'Yl-1             = Z qyii･･y.,(x) fl",I;'l ay.,.Myi･･･y..i.,(Si-ix>              J,1-i･3'lfit            =y,.II.ily -,qy2･･･ynt+i (Sx) ll1･i:.L" Aq'y..il yi.･.y.,i", (si-ix>             = qyl･･-ym÷l., (S l x> IIZItrl"Z 4y.." yi...y.,i-., (si-lx) ･`･if n-m l}) l,                  Z qy.m.･･･y.-,(S"-MX) ･･･if n-m<l.                Yn-･m""Yl-.1In both cases of n>m and num<m, we get                   vx (Yl'''Yn) :vslx(Tl [Yl'''Yn]). <2>Replacinglin (2) by l-1, we get              vsx (T [Yl'''Yn]) == vsl-1(sx)(Tl-iT [Yt, ''', yn])                            On Markov channeis S                              u' vslx (Tl [Yl,''', Yn])as T[yt,･･･,yn] is a thin cylinder on the time interval (l-1,･･･,n-1). Hence                       vx (Yl'''Yn) = vsx (T [Nl'''Yn])holds. We extend vx(･) from (Ebioo..i SIxi to :'U -- opi--corm.. Yi, by the formula                    vx (Y-l'''Yn) == vs"lx (T-l IY-l, ''', Yn]).Tken v satisfies the condition III of the channeL For any [yi,･･･,yn], v.(yi･-･yn)is .c20-measurable on X, as thls is defined by the product of the measurablefunctions q5(x) and ab,s(x). And for any measurable set F <E Y), v.(F) is alsomeasurab!e, as F can be approximated'by a countable sequence of cylinder setswith the measure yx(･). Therefore the condition II is a!so clear.   Converse!y let vbe an m-fold Markov channel. We define a stochastic pair(Q(x),-Q" (x)> by the following;                   qyi (x) == vx (Y,>and                          ijy.., 1 Zl(x) = vx(y..A Zi)where S = [),i,y2,･･･ym]. Clearly the pair is well defined by the condition III oftke channel.        X aym+i[yi･･･yn; (X)qyi･･･ynt (X) ": £ vx (Yi'''Y,n) vx (Mn+dYi'''Ynt)       )'i EB yi EB             := ]Z] vx(yi'''ytn+,t> =vx(y2'''ym+i)=vsx(T(y2'''ym+i))              yi EB             == qy2･･-y.+, (Sx).The above chaiR of formulae shows that the pair satisfies the condition of thestochastic pair. Q. E. D.   This theorerri tells us that any Markov channeHs determined by a suit ofmappings, which resembles to the fact that a memoryless channel (cf. 2) p. 154)is determined by a stochastic matrix Let (cab) be a stochastic matrix whichdetermines a memoryless channel, then taking qb(x) = cxib aRd 'q'bis(x) =r cx,b,we see that the memoryless channel is a kind of Markov channels.                              4. Examples   We put A =xx B == {O, 1}, i. e. , we consider a binary case. For a stochasticpair, Iet us choose6 Yatsuka NAKAMliRA            Q(x) wr (1 il p)= o<o) ･･･ if x, .. o,                    (i5ab)== Q(') ''' if xe -= i,     -Q- (X) == (1Sq 1;")=. Q-" <OO) ''' if xo == O, xi=o,                             -. ･･･ if xo =O, xl=1,             (i1q i1,)=:e(oi)             (1 -;r 1ilq)=" Qi (10) ''' if xe= 1, xt == o,             (i 11 r i li- q)= Zl> (ii) ''' if xe = i, xi = i,and assume that Q(OO)Q(O) =Q(O) i.e.,P=(r-1)/(r+q-2). Then we findthat Q(11)Q(i) =- Q(1), Q(Ol)Q(O) == Q(1) and Q(10)Q(1) :: Q(O). Hence (Q(x), Q(x))is a stochastic pair, so that we can define a Markov channei, which we callan additive noise channel. Errors in this channel occur as shown in Fig.1.Fig.2isaresuit ofa simple simulation in the case ofq=9/10, r=1/2 andP == 5/7.                   q 1--q r                          CE                                1-r                            C: correct                            E: erroneous                                Fig. 1         input sequence IOIIOOIOIIOOIOIIIOOIIOOOOIOIIIOIOOIOI                ･         output sequence OIIIOOIOIIOOIOIIIIIOIIIIIOOIIIOIOOIOI                                Fig. 2                            On Markev channels 7   Next we give another example;                    Q(x> =(;/ )=:Q for anyxc! X,                   Q(X) =- (, [ll p i 1] P) -= Q(O)                            lf x, = o,                           (, iZ, iII` q) =Q<i) if xi := i.Then we can also define a Markov channel as shown in Fig. 3, which we caila stationarN stochastic state machine channel (SSSM . channel). Fig. 4 is a resultof simulation when p == 3/4 and q= 1/4. In Fig. 4, the output letters seem tohave little correlation wlth the input letters, but we can show later that thecapacity of this channel is not zero, therefore, the transmission of the infor-mation is possible through this noisy channel by the Shannon-Hinchin theorem.                  ×,-ob,trPO                                       1-p                  x,-ibot"qO                                       1--q                                 Fig 3         input sequence 10110010110010111001100001011aOIOOIOI                 e         output sequence 1101110010110010001011000010110111001                                 Fig. 4                      5. Capacity of SSSM Ckannels   The second example of the previous section can be extended to a moregeneral case, i.e., the letters need not be binary. Let Q(x) be independent ofxeX. And we assume that Q"V (x) is only dependent on x, hence we can writeQ(x) == Q(xi), and assume tlaat Q(xi)Q--Q for all xEA, where Q==: Q(x). Wecail this generalized one an SSSM channel also. We can cornpute the capacity8 Yatsul<a NAKAMuRACs(v) of an SSSM ckannel v. First, we give a proposition in a more generalsituation.   Proposition 1. Let v be a finite memory Markov channel, then           Rp == hq(T) + lx y,li.IIII.y..q,yi･･･ym(x) rp(ay,.+iiyi･･･y. (x))p (dx)where rp<t) == -tlog t.   Proof. By the definkion of Rp, Rp = hp(S) + hq(T> - hr<S & T)    = hq(T) + hp(S> - lim -IL U( ;Z' 'o X fZi oVS-i,,2"o (El) T-i fZ/ eV                   n71                                     ･･･ V S-n+i ,,sZe"e (Ei) T-n+i :Z/ Q)    = hq(T) + hp(S)    - lim -IL H(sm-i2oe (g) 2y V Sm,s2oe (g> 2y V      nn                 '''VS.,S" eV2yV.;2lfe (El) WoVS"i,ff'o (E9 T-i f?iieV                   '･ '''VS'7i'i,..20e (Eg) T-"Mi C}l/e)     :hq(T)÷hp(S)+ li,m, i. I!iEl... ,;E.).., rp( I vx(yi'''y")P(dx>)･                        X-flt n 1 ll                                  X ""X                                   1- tn ilHere we assume thatvis m-foid Markovian and of m-memory with the com-mon integer m, which does not lose generality.Rp =hq(T>+hp(S)+<iiml Z Z rp(P(xi-"t'''Xn) vx<Yi'''Yn))                  u nx -･x .v --y                      IH ?)z n l n  =:h,(T)+hp(S)+lim-l- Z] Z q(P(xi-m'''xn)qyi･･･ym(X)                 n ?2x -･pt ", -.>i                      1-･71t n 1 n                                      ' llla)klTlit Nqyi+tn 1 yi･･･yi÷nt-i(X))   = hq(T) + hp(S)  -i- lim -!L X Z p(xit..'''x.>qy,･･y.(x> ll'i-M'q-y,+m l y,-･yi+tn-i(x>×     n nx -･x )T ･-.v it-1         l-m nl n                                 n-m.  ×(log P(xi-jn･･･xn) + log qy,･･･y.(x> + X. Iog ayj+. i yj･･･yj"nt-i<X))                                 J==1                        On Marl<ov Channels 9  ==h,(T>+lim-l- lli]i Z] p(xi--m'''xn)qyi'''ym(x)           n VZ x -･x ), --J,                1-m nl n                                        n-m                  ･ll'it=-1'i ijyi+,. I yi･J･yi÷mmi(x)) (Xj..i log "q'yi+m I yj'･･･yi+,n-i(X))                 }t-f･11           n iI g'rm-1x -x y･-y  =: hq<T)÷lim- £ X X P(Xi"m'''Xn) vx(Yi'''Nn) 10g ay"'+mfyi･･yn,÷j'--i<X)l                    1-flt n 1 11                 lt-?n umnv hq(T) + ii,m -h ( ,Z･..,l.,.mli, ],....,..., ,l.ll],..,,...e[Xi-"z'''X""m)                  . yx<yt･･yl'+m-1) log 'q"yj+. 1 yi･･･ynt÷J'--i(X) l                 il.-'tll.   == hq(T) + iipa -h I ,]Z...l H'(si"m2x (g) Ti+m c}eit,                 ¥,.-l-mui. si,･/z･--･, op Ti 2.v v'k/L.iliwws'fe.,sz., (g} Tk :y,)]  -= hq(T) + I y ¥..,, qyi･･Jy.(x) rp(qy.., i y,･･ty.(x))p(dx),             1 M÷I    X   which completes the proof, where H(.orl S') means ff(,.9fV,.KZ. ")-ff<.(Z, i                                                       Q.E.D.   Next we give another proposition:   Proposition 2. Let v be an SSSM channel defated by Q== (qb> and Q(xi)=(ablbt (xi)>. if Pe is a Bernoulli izzPntt source, i.e., time independent probabilitymeasure on X=: Ar, then the outPztt source qo(･)= q(･;Po,v) is a Marfeov measureon Y= Bi.   Proof. Let us prove that                qe([Nn]1[Yr''Nn-i]) == qo([Yn]1[Yn-i])･ (3>We abride the brackets [] in the following formulae. Then, qe(Yn l Yi'''Y"-･i) ='nt qG(Yi'''Yn--iMi)/ae(Yi'''Y"-i) = q( Yi>'q"y2iyi(X2)'' "q- yn l y..i(Xn)P(XD'''P(Xn) / (q(Yi)'q'y, I y,( C2)                         ' '''ij3'n--iiJ'n-2(Xn-1)P(Xl)'''P(X"-1))  := i7yn1yn-i(Xn)P(Xn) = ae(YnIYn-･i),which proves the equality (3). Q. E. D.   Tlteorem 2. Let v be an SSSM channel, then the capacity of v is given by10 Yatsuka NAKAMuRA                                                ' CS(") = SS,P( .?}t],b,qbiab2[bi (a2)Po(a2) 10g (ab,lb, (a2))/<III,;, 4b,1b,(a2i)Po (a2t))],where Pe moves on all Probability measures on A.    Proof. Let P be any iRput source, then by Proposition 1,                    Rp == hq(T) + Z ay,rp(ay,Iyi(X2))P(X2)･                                YlThe following inequality and equality are well known (cf. 2) p. 82 and p. 78) :            hq(T)=limH(S :ZiieIViml'oS-i CZ/o);:i{H(S CZie1 CZ;io)･                     nThus,                          '            hq(T> -< Z q(ye) rp (q<yi 1 yo>> == £ qy, ty (:Ii ) ayi i y, <xi)P(xi))･                    ye yo ,xlNow Iet Pe be a Bernoulii source given by                       Pe(Xi'''Xn) =: P(Xi>P(X2)'''P(Xn),then by Proposition 2, denoting qe(･>-- q(･;Po,v),                         H<S-i ZieI 7o) " hae(T)･Therefore         Rp =: hq(T) + X qyt ny(4y2 i yi(X2))P(X2)                      J'1           E{: hq,(T) + X qy, rp<ay, i y, <x2))Pe(x2) < = Rp,)                     Yl            =: Z qb,4b,lb, (a2)Pe(a2) 10g (4b,lb,(a2))/<E4b,ib,(a2t)Po(a2i)),              a2bib2 a2'which proves the result. Q. E. D.   Therefore we can easily compute the capacity of the channel shown inFig.2 of the previous section, which is about O. 1886 bit/step.                               Reference1) ADLER, R.L., Ergodicand mixing properties of infinite memory channels..Proc. Amer. Ii4izth. Soc. I2, 924-930 (1961).2) BILLINGsLEy, P., Ergodic theory and information. John Wiley and Sons, (1965).3) HiNcmN, A. I., Mathematical foundations of information theory. Dover, <1958). (English translation)

https://soar-ir.repo.nii.ac.jp/?action=repository_action_common_download&item_id=12935&item_no=1&attribute_id=65&file_no=1

On Markov Channels

Abstract

Similar works

Full text

Available Versions

Shinshu University Institutional Repository

Shinshu University Institutional Repository