The Capacity Region of the Degraded Finite-State Broadcast Channel

Transcription

1 The Capacity Regio of the Degraded Fiite-State Broadcast Chael Ro Dabora ad Adrea Goldsmith Dept. of Electrical Egieerig, Staford Uiversity, Staford, CA Abstract We cosider the discrete, time-varyig broadcast chael with memory uder the assumptio that the chael states belog to a set of ite cardiality. We rst dee the physically degraded ite-state broadcast chael for which we derive the capacity regio. We the dee the stochastically degraded ite-state broadcast chael ad derive the capacity regio for this sceario as well. I both scearios we cosider the oidecomposable ite-state chael as well as the idecomposable oe. I. INTRODUCTION The broadcast chael (BC) was itroduced by Cover i 972. I this sceario a sigle seder trasmits three messages, oe commo ad two private, to two receivers over a chael deed by X, p(y, z x), Y Z. Here, X is the chael iput from the trasmitter, Y is the chael output at Rx ad Z is the chael output at Rx 2. I the years followig its itroductio the study of the BC focused o memoryless scearios, i.e., whe the probability of a block of trasmissios is give by p(y, z x ) p(y i, z i x i ). I recet years, models of time-varyig broadcast chaels with memory have attracted a lot of attetio, especially i Gaussia setups. This was motivated by the proliferatio of mobile commuicatios, for which the chael is subject to time-varyig correlated fadig. The correlatio of the fadig process itroduces memory to the BC. The fadig BC is oe istace of the geeral BC with chael states. While fadig BCs received cosiderable attetio, discrete BCs with chael states received oly little attetio. A otable exceptio is the degraded arbitrarily varyig BC (DAVBC) cosidered i [2] ad [3]. I [2] DAVBCs with causal ad o-causal side iformatio at the trasmitter were cosidered. The states are assumed i.i.d. ad the chael is memoryless: p(y, z x, s ) p(y i, z i x i, s i ). I [3], the capacity regio for DAVBCs with causal side iformatio at the trasmitter ad o-causal side iformatio at the good receiver was derived. I [3] the state distributio is geeral ad is ot subject to the i.i.d. restrictio, but the chael outputs, give the states ad the chael iputs are agai memoryless. The arbitrarily varyig chael (AVC) is oe model for a time-varyig chael with states. It models a memoryless chael whose law varies i time i a arbitrary maer. The state trasitios are idepedet of the chael iputs ad outputs. I this work we study the discrete time-varyig BC with memory i the framework of ite-state chaels (FSCs). I cotrast to the AVC, i the FSC both the chael output ad the curret state deped o both the chael iput ad the previous state. The authors are with the Wireless Systems Lab, Departmet of Electrical Egieerig, Staford Uiversity, Staford, CA ro,adrea@wsl.staford.edu. This work was supported i part by the DARPA ITMANET program uder grat TFIND. The ite-state chael model was cosidered for the poit-to-poit sceario as early as 953 []. This chael is characterized by the distributio p(y, s x, s ) where S is the curret state ad S is the previous state. For a block of trasmissios, the p.m.f. at the i'th symbol time satises p(y i, s i x i, s i, y i, s 0 ) p(y i, s i x i, s i ), () where s 0 was the state of the chael whe trasmissio bega. Equatio () implies that S i cotais all the history iformatio for time i. Recetly, the ite-state multipleaccess chael was studied i [5]. This sceario is characterized by the chael distributio p(y, s x, x 2, s ), ad the work i [5] also cosidered the effect of feedback o the rates. I the preset work we study the ite-state broadcast chael (FSBC). Here, the chael from the trasmitter to the receivers is govered by a state sequece that depeds o the chael iputs, outputs ad previous states. The way these symbols iteract with each other is captured by the trasitio fuctio p(y, z, s x, s ). Mai Cotributios ad Orgaizatio I this paper we cosider for the rst time the capacity of the FSBC. Here, there is a uique aspect ot ecoutered i the poit-to-poit ad the MAC couterparts, amely the applicatio of superpositio codig to the FSC. We iitially dee the physically degraded FSBC ad d the capacity regio of this sceario. We the dee the stochastically degraded FSBC ad give examples of commuicatio scearios represeted by this model. We derive the capacity regio for this chael as well. The rest of this paper is orgaized as follows: Sectio II itroduces the chael model. Sectio III presets a summary of the results together with a discussio. Lastly, Sectio IV outlies the proof of the capacity regio for the physically degraded FSBC. II. CHANNEL MODEL AND DEFINITIONS First, a word about otatio. I the followig we deote radom variables with upper case letters, e.g. X, Y, ad their realizatios with lower case letters x, y. A radom variable (RV) X takes values i a set X. We use X to deote the cardiality of a ite, discrete set X, X to deote the -fold cartesia product of X, ad p X (x) to deote the probability mass fuctio (p.m.f.) of a discrete RV X o X. For brevity we may omit the subscript X whe it is obvious from the cotext. We use p X Y (x y) to deote the coditioal p.m.f. of X give Y. We deote vectors with boldface letters, e.g. x, y; the i'th elemet of a vector x is deoted with x i ad we use x j i where i < j to deote the vector (x i, x i+,..., x j, x j );

2 is short form otatio for x j, ad x x. A vector of radom variables is deoted by X, ad similarly we dee X j i (X i, X i+,..., X j, X j ) for i < j. We use H( ) to deote the etropy of a discrete radom variable ad I( ; ) to deote the mutual iformatio betwee two radom variables, as deed i [6, Chapter 2]. I( ; ) q deotes the mutual iformatio evaluated with a p.m.f. q o the radom variables. Fially, co R deotes the covex hull of the set R. Deitio : The discrete, ite-state broadcast chael is deed by the triplet X S, p(y, z, s x, s ), Y Z S where X is the iput symbol, Y ad Z are the output symbols, S is the state of the chael at the ed of the previous symbol trasmissio ad S is the state of the chael at the ed of the curret symbol trasmissio. S, X, Y ad Z are discrete alphabets of ite cardialities. The p.m.f of a block of trasmissios is x j p(y, z, s, x s 0 ) p(y i, z i, s i, x i y i, z i, s i, x i, s 0 ) p(x i x i )p(y i, z i, s i y i, z i, s i, x i, s 0 ) (a) p(x ) p(y i, z i, s i x i, s i ), (2) where s 0 is the iitial chael state. Here (a) captures the fact that give S i, the symbols at time i are idepedet of the past. Deitio 2: The FSBC is called physically degraded if its p.m.f. satises p(y i x i, y i, z i, s 0 ) p(y i x i, y i, s 0 ), p(z i x i, y i, z i, s 0 ) p(z i y i, z i, s 0 ). (3a) (3b) Coditio (3a) captures the ituitive otio of degradedess, amely that Z i is a degraded versio of Y i, thus it does ot add iformatio whe Y i is give. Note that i the memoryless case this coditio is ot ecessary as, give X i, Y i is idepedet of the history. Coditio (3b) follows from the stadard otio of degradedess. Usig coditios (3a) ad (3b) we obtai (whe p(y, x s 0 ) > 0) p(z y, x, s 0 ) p(z, y, x s 0 ) p(y, x s 0 ) p(z i, y i, x i z i, y i, x i, s 0 ) p(y i, x i y i, x i, s 0 ) p(x i z i, y i, x i ) p(z i, y i z i,y i,x i,s 0 ) p(x i y i, x i ) p(y i y i, x i, s 0 ) (a) p(x i x i ) p(z i, y i z i, y i, x i, s 0 ) p(x i x i ) p(y i y i, x i, s 0 ) (b) p(y i y i, x i, s 0 ) p(z i z i, y i, x i, s 0 ) p(y i y i, x i, s 0 ) (c) p(z i z i, y i, s 0 ), (4) where (a) is because there is o feedback, (b) follows from (3a) ad (c) follows from (3b). We coclude that whe (3) holds, p(z y, x, s 0 ) p(z y, s 0 ). Hece, p(y, z x, s 0 ) p(y x, s 0 )p(z y, s 0 ). (5) Note that (4) shows how to obtai p(z y, x, s 0 ) i a causal maer. Also ote that Z is a degraded versio of Y but still depeds o the state sequece (i.e. degradedess does ot elimiate the memory). A special case of the physically degraded FSBC occurs whe i (3b) it holds that p(z i x i, y i, z i, s 0 ) p(z i y i ). Hece, p(z y, x, s 0 ) p(z y ) p(z i y i ). (6) Equatio (6) is similar to the deitio of degradedess for the DAVBC used i [2]. Deitio 3: The FSBC is called stochastically degraded if there exists a p.m.f. p(z y) such that p(z, s x, s ) Y p(y, s x, s )p(z y, s, x, s ) p(y, s x, s ) p(z y). (7) Y Note that whe (7) holds the p(z x, s 0 ) p(z, s x, s 0 ) (a) p(z i, s i x i, s i ) p(y i, s i x i, s i ) p(z i y i ) y i Y Y (b) p(y i, s i x i, s i ) p(z i y i ) Y p(y, s x, s 0 ) Y p(y x, s 0 ) p(z i y i ) p(z i y i ), (8) where (a) ad (b) follow from (2). Deitio 3 does ot costitute oly a mathematical coveiece, but represets a physical sceario. For example, cosider a sceario i which a base statio trasmits to two mobile uits, located approximately o the same lie-of-sight from the base statio (BS), as idicated by the dashed lie i Figure. Let the BS trasmit a BPSK sigal ad let the received sigals be subject to additive Gaussia thermal oise due to the receivers' frot-eds. Whe decodig at the receivers takes place after a hard threshold at zero, the resultig sceario is the biary symmetric broadcast chael (BSBC). Deote the situatio where there is o trafc o the road betwee the BS ad the mobiles as state A. Let the chael BS Rx have a crossover probability ɛ (A) 0. ad the chael BS Rx 2 have a crossover probability ɛ 2 (A) 0.5. This ca be represeted as a stochastically degraded BC with a degradig chael whose crossover probability is ɛ 2 (A) ɛ 2 ɛ 2ɛ

3 Assume that o occasios, a car passes o the road betwee the BS ad the mobiles. This causes atteuatio i both chaels simultaeously. Call this state B ad let ɛ (B) 0.8 ad ɛ 2 (B) Agai we have ɛ 2 (B) Hece, the degradig chael is the same for both states, irrespective of the state sequece (i this example the state sequece represets the trafc patter, ad is ot a idepedet sequece). This satises coditio (8). Note that we assume o kowledge of the states at the trasmitter ad receivers. Deitio 6: The average probability of error of a code for the FSBC is give by P () e max s0 S P () e (s 0 ), where, P e () (s 0 )Pr ( g y (Y ) (M 0, M ) or g z (Z ) ) (M 0, M 2 ) s 0, where each of the messages M 0 M 0, M M ad M 2 M 2 is selected idepedetly ad uiformly. Deitio 7: A rate triplet (R 0, R, R 2 ) is called achievable for the FSBC if for every ɛ > 0 ad δ > 0 there exists a (ɛ, δ) such that for all > (ɛ, δ) a (R 0 δ, R δ, R 2 δ, ) code with P e () ɛ ca be costructed. Deitio 8: The capacity regio of the FSBC is the covex hull of all achievable rate triplets. Base Statio Mobile Mobile 2 Fig.. A degraded FSBC sceario: the mobile uits are located o the same lie-of-sight from the base-statio (idicated by the dashed lie). Passig cars affect the chaels to both mobile uits simultaeously. More geerally, we ca dee a set of states for this sceario, e.g. S, 2,..., K, with Y Z 0, ad p(z i, s i y i, s i ) p(s i s i )p(z i y i, s i ) ɛ2 (k), z y p(z y, s k), ɛ 2 (k), z y ɛ 2 (k) (0, 0.5), k S. This results i a collectio of physically degraded BSBCs that ca give more exibility i modelig the sceario of Figure, as the degradig chael may deped o the state. However, for this reaso, this model does ot satisfy our deitio of stochastic degradedess i Deitio 3. Deitio 4: (see [4, Sectio 4.6]) The FSBC is called idecomposable if for every ɛ > 0 there exists N 0 (ɛ) such that for all > N 0 (ɛ), p(s x, s 0 ) p(s x, s 0) < ɛ, for all s, x, ad iitial states s 0 ad s 0. Deitio 5: A (R 0, R, R 2, ) determiistic code for the FSBC cosists of three message sets, M 0, 2,..., 2 R0, M, 2,..., 2 R ad M2, 2,..., 2 R2, ad three mappigs (f, g y, g z ) such that f : M 0 M M 2 X (9) is the ecoder ad g y : Y M 0 M, g z : Z M 0 M 2, are the decoders. Here, M 0 is the set of commo messages ad M ad M 2 are the sets of private messages to Rx ad Rx 2 respectively. The sceario parameters assumed i this example are: Two-ray propagatio model, Rx decodig scheme is maximum-likelihood, Base statio Tx power 30 dbm, Base statio atea gai 0 dbi, Rx atea gai 0 dbi, Rx oise oor 90 dbm, Base statio atea height 0 m, Rx atea height.5 m, BS Rx distace 7.2 Km ad BS Rx 2 distace 8 Km. We also assume a passig car icreases the path atteuatio by 3 db. Dee rst III. MAIN RESULTS AND DISCUSSION R, (p, s 0 ) I(X ; Y U, s 0 ) p log 2 S R 2, (p, s 0 ) I(U ; Z s 0 ) p log 2 S. The mai result is stated i the followig theorem, whose proof is outlied i Sectio IV: Theorem : Let Q be the set of all joit distributios o ( U i, X ) such that the cardiality of the radom vector U is bouded by U i mi X, Y, Z. For the physically degraded FSBC of Deitio 2, dee the regio R (s 0 ) as R (s 0 ) co q Q (R 0, R, R 2 ) : R 0 0, R 0, R 2 0, R R, (q, s 0 ), R 0 + R 2 R 2, (q, s 0 ).(0) The capacity regio of the physically degraded FSBC is give by C pd lim R (s 0 ), () s 0 S ad the limit exists. Sice the capacity of the broadcast chael depeds oly o the coditioal margials p(y x, s 0 ) ad p(z x, s 0 ) (see [6, Chapter 4.6]) the the capacity regio of the stochastically degraded FSBC is the same as the correspodig physically degraded FSBC: Corollary : For the stochastically degraded FSBC of Defiitio 3, the capacity regio is give by Theorem where p(z s, y, x, s ) is replaced by p(z y) that satises equatio (7). Whe the FSBC is idecomposable, the the effect of the iitial state fades away as icreases. Therefore we have the followig corollary: Corollary 2: For the idecomposable physically degraded FSBC, the capacity regio is give by Theorem. For the idecomposable stochastically degraded FSBC, the capacity regio is obtaied from Corollary. I both cases the parameter s 0 i R, (q, s 0 ) ad R 2, (q, s 0 ) ad the itersectio over S i the expressio for C pd are omitted.

4 Proof outlie: Loosely speakig, the corollary is true sice for large eough the effect of the iitial state fades away. Therefore, for asymptotically large the maximum over all iitial states s 0 S equals the miimum. Discussio First, ote that if lim R (s 0 ) exists for all s 0 S the the capacity regio () ca be writte as C pd lim R (s 0 ) (a) lim R (s 0 ). Here, (a) is permitted because S is ite. Thus, the capacity regio ca be viewed as the itersectio of all the capacity regios obtaied whe the iitial state is kow at the receivers (but ot at the trasmitter). We also ote the followig coclusios: ) Sice the limit of the regio exists, the as icreases, optimizig the code will result i better performace (which is ot guarateed whe the limits caot be show to exist, cosider for example a o-statioary chael with oise that oscillates with time). 2) The codebook structure that achieves capacity is a superpositio codebook. This itroduces a structural costrait whe optimizig the codebook for achievig the maximum rate triplets. 3) The auxiliary RV U itroduces difculties maily i places where we eed to rely o the its cardiality. This is because we caot traslate the boud o the cardiality of U ito a boud o the cardiality of a subset of U. I particular, we caot use the cardiality of U whe derivig the capacity regio for the idecomposable FSBC. Moreover, lettig m + m 2, the from Equatio () we have that p(z m, y m, s m x, s 0 ) p(z m, y m, s m x m, s 0 ). But because p(x m u ) p(x m u m ) the p(z m, y m, s m u, s 0 ) p(z m, y m, s m u m, s 0 ). This is a major differece from the poit-to-poit ad the MAC chaels. Cosider, for example, the expressio max max p(u,x ) s 0 I(U ; Z s 0 )+λ max s 0 I(X ; Y U, s 0). (2) While i the MAC ad the poit-to-poit chaels the correspodig expressios coverge for all chaels, for the FSBC (2) ca be show to coverge oly for the idecomposable case. Therefore, usig superpositio codig, the chael betwee U ad (Y, Z ) is fudametally differet from the chael betwee X ad (Y, Z ). This is i cotrast also to the discrete, memoryless BC. IV. PROOF OUTLINE I the derivatio we focus o the physically degraded FSBC. The derivatio requires oly that coditio (5) holds. I the derivatio we shall cosider oly the two private messages case as the commo message ca be icorporated by splittig the rate to Rx 2 ito private ad commo rates, as i [6, Theorem 4.6.4]. R 2 0 s A Achievable Regio R Fig. 2. Lies boudig the achievable regios for the FSBC for iitial states s A ad s B, ad the resultig regio of positive error expoets. A. Achievability Theorem Due to space limitatios we omit the details of the achievability proof ad give oly the coclusio. For complete details see [7]. Dee rst F (λ) max p(u,x ) 0 s B mi R 2,(p, s 0 ) + λ mi R,(p, s 0) Followig [8, Sectio 2], the boudary of the regio of positive error expoets for a give ca be writte as R2 (R ) if F (λ) λr. (3) 0 λ This characterizatio is illustrated i Figure 2. I the achievability proof we show that for a give p(u (, x ), whe trasmittig at the positive rate pair mis 0 S R, (p, s 0), mi s0 S R 2, (p, s 0 ) ), the the error expoet is positive ad bouded away from zero. Hece, the probability of error ca be made less tha ay arbitrary ɛ > 0 by takig a block legth K with a large eough iteger K. Furthermore, i sectio IV-D we show that the largest regio is obtaied by takig the limit R 2 (R ) if 0 λ lim F (λ) λr., (4) ad that this limit exists ad is ite. The fact that the limit exists ad is ite implies that we ca approach the rates of Theorem arbitrarily close by takig large eough, thus by Deitio 7 these rates are achievable. Before cosiderig the coverse we discuss the cardiality of the auxiliary RV U, as the evaluatio of R 2 (R ) of (3) depeds o the existece of such a boud. B. Cardiality Bouds From the derivatio i [8], it follows that maximizig the regio R (s 0 ) of Equatio (0) over all joit distributios p(u, x ), ca be carried out while the cardiality of the auxiliary radom variable U is bouded by U i mi X, Y, Z. (5) Now ote that from (), the achievable regio for a xed is give by the itersectio s 0 S R (s 0 ). As for each R (s 0 ), s 0 S we have the same cardiality boud, the this boud also holds for maximizig the itersectio of the regios R (s 0 ), s 0 S.

5 C. Coverse Lemma : If for some ɛ > 0, λ 0, R 2 + λr > lim F (λ) + ɛ, the there exists a pair of iitial states s 0 ad s 0 such that P () e2 (s 0)R 2 +λ ( P () e (s 0)R ) > ɛ (+λ)( + log 2 S ). The implicatio of this iequality, as explaied i [8, Sectio 3], is that for large eough the probability of error P e () caot be made arbitrarily small outside the regio (4). Proof: From Fao's iequality we have that H(M 2 Z, s 0 ) P () e2 (s 0)R 2 + (6a) H(M Y, s 0 ) P () e (s 0)R +. (6b) Next write mi I(M 2; Z s 0 ) R 2 max H(M 2 Z, s 0 ) (7) s 0 S s 0 S mi I(M ; Y M 2, s 0) R max H(M Y, M 2, s 0) R max H(M Y, s 0).(8) Now ote that I(M 2 ; Z s 0 ) H(Z s 0 ) H(Z M 2, s 0 ) I(U ; Z s 0 ), (9) where U i M 2, i, 2,...,. We also have I(M ; Y M 2, s 0) H(Y M 2, s 0) H(Y M, M 2, s 0) H(Y U, s 0) H(Y X, U, s 0) I(X ; Y U, s 0), (20) where the deitio of U satises the Markov relatioship U s 0 X s 0 Y s 0. Combiig (9) ad (20) we have that for this choice of U : mi I(M 2; Z s 0 ) + λ mi I(M ; Y M 2, s 0) mi I(U ; Z s 0 ) + λ mi I(X ; Y U, s 0) F (λ) + ( + λ) log 2 S, (2) sice F (λ) is obtaied by maximizig over all joit distributios p(u, x ) subject to the cardiality costrait (5), which is also satised by our choice of U. Let s 0, ad s 0, be the maximizig states for H(M 2 Z, s 0 ) ad H(M Y, s 0) respectively. Pluggig (7) ad (8) ito (2) yields R 2 H(M 2 Z, s 0, ) + λ(r H(M Y, s 0,)) ( + λ) log 2 S F (λ). Thus, H(M 2 Z, s 0, ) + λh(m Y, s 0,) + ( + λ) log 2 S (R 2 + λr F (λ)) (R 2 + λr lim F (λ)) > ɛ. Combied with (6), this completes the proof of the lemma. D. Covergece I this subsectio we show that lim F (λ) exists ad is ite for the chael uder cosideratio, whe λ [0, ]. The proof of covergece exteds the argumets i [4, Appedix 4A] to the FSBC. The mai difculty here is the itroductio of the auxiliary RV U ad its iteractio with the other RVs,, X, Y ad Z. We actually show that lim F (λ) sup F (λ) which implies that the limit exists. Due to its legth, the full proof is omitted ad oly the mai poits are highlighted. Let s 0 s z 0(l) miimize l I(U l ; Z l s 0 ) ad ( s 0 s y 0 (l) miimize l I(Xl ; Y l U l, s 0), for the triplet q (u l, x l ), s z 0(l), s y 0 (l)) that achieves the max-mi solutio for F l (λ), ad let (q 2 (u m, x m ), s z 0(m), s y 0 (m)) achieve the max-mi solutio F m (λ). Fially, let s z 0() ad s y 0 () be the states that achieve the max-mi solutio for F (λ). We show that F (λ) is sup-additive, i.e., for every iteger m, l [0, ] with m + l we have F (λ) lf l (λ) + mf m (λ). Sup-additivity is veried by breakig the legth expressios ito expressios of legth l ad expressios of legth m. The critical part here is to cosider the legth m sequece from l+ to. Here we use the fact that give the iitial state the chael is statioary, so p(zl+, Y l+ x l+, s l s 0 ) p(z m, Y m x m x l+, s 0). This, combied with the fact the cardiality boud depeds oly o the legth of the sequece, leads to the coclusio that the joit distributio q 2 (u m, x m ) that maximizes F m (λ) will maximize the segmet from l + to (i.e. is the maximizig distributio for (Ul+, X l+ ), with the same iitial state). Additioally, both I(U ; Z s 0 ) ad I(X ; Y U, s 0) are bouded from above, idepedet of : I(U ; Z s 0 ) log 2 Z, sice all the Z i 's are deed over the same alphabet Z i Z, ad similarly I(X ; Y U, s 0) log 2 X. Thus, F (λ) log 2 Z + λ log 2 X < for ay λ [0, ]. The fact that F (λ) is bouded from above idepedet of ad is also sup-additive implies that lim F (λ) exists ad is ite. Combiig the fact that the limit exists with sectios IV-A, IV-B ad IV-C gives the capacity of the FSBC of Theorem. REFERENCES [] B. McMilla. The Basic Theorems of Iformatio Theory. The Aals of Mathematical Statistics, Vol-24(2):96 29, 953. [2] Y. Steiberg. Codig for the Degraded Broadcast Chael With Radom Parameters, With Causal ad Nocausal Side Iformatio. IEEE Tras. Iform. Theory, IT-5(8): , [3] A. Wishtok ad Y. Steiberg. The Arbitrarily Varyig Degraded Broadcast Chael with States Kow at the Ecoder. Iteratioal Symposium o Iformatio Theory (ISIT) 2006, Seattle, WA, pp [4] R. G. Gallager. Iformatio Theory ad Reliable Commuicatio. Joh Wiley ad Sos Ic., 968. [5] H. Permuter ad T. Weissma. Capacity Regio of the Fiite-State Multiple Access Chael with ad without Feedback. Submitted to the IEEE Tras. Iform. Theory, [6] T. M. Cover ad J. Thomas. Elemets of Iformatio Theory. Joh Wiley ad Sos Ic., 99. [7] R. Dabora ad A. Goldsmith. The Capacity Regio of the Degraded Fiite-State Broadcast Chael. Submitted to the IEEE Tras. Iform. Theory, [8] R. G. Gallager. Capacity ad Codig for Degraded Broadcast Chaels. Problemy Peredachi Iformatsii, vol. 0(3):3 4, 974.