δ =0.586 Average number of nearest signal points (NN): 0.5 NN: 1.5 δ =2.000

Transcription

1 IEICE TRANS. FUNDAMENTALS, VOL. E{A, NO. JANUARY 999 PAPER Regula pape Tadeos between Eo Pefomance and Decoding Complexity in Multilevel 8-PSK Codes with UEP Capabilities and Multistage Decoding Motohiko ISAKA y, Robet H. MORELOS-ARAGOA yy, Membes, Mac P.C. FOSSORIER yyy, Shu LIN yyy, Nonmembes, and Hideki IMAI y, Membe SUMMARY In this pape, we investigate multilevel coding and multistage decoding fo satellite boadcasting with modeate decoding complexity. An unconventional signal set patitioning is used to achieve unequal eo capabilities. Two possibilities ae shown and analyzed fo pactical systems: (i) linea block component codes with nea optimum decoding, (ii) punctued convolutional component codes with a common tellis stuctue. key wods: multilevel codes, multistage decoding, tadeos, decoding complexity, eo pefomance. Intoduction Unequal eo potection (UEP) coding techniques ae known to povide eective tansmission systems especially ove boadcasting channels with dieent channel condition fo each eceive. In many applications with analog data like high-denition television (HDTV), hieachical souce coding enables itself to match UEP coding by poviding moe poweful eo coecting capabilities fo infomation bits which ae sensitive to noise and distotions added on the channels. This efes to a system in which the bit steam caying the basic esolution of the TV signal is eceived always at a low bit eo ate (BER), even unde degaded channel conditions. As the channel condition impoves, the bit steams caying pogessively highe esolutions ae eceived as well, thus achieving so-called \gaceful degadation". Fo highe eciency, UEP capabilities should be given in the context of some combined coding and modulation stuctues. In pevious woks, the authos have investigated multilevel coded 8-PSK modulation in conjunction with multistage decoding [] based on unconventional patitionings in []{[4]. A simila appoach can be found in [5]. Howeve, these schemes have been designed pimaily with linea block component codes, so that maximum likelihood decoding (MLD) at each Manuscipt eceived Mach 3, 999. Manuscipt evised Mach 3, 999. y The authos ae with the Institute of Industial Science, The Univesity of Tokyo, Tokyo, Japan. yy The autho is with Advanced Telecommunications Laboatoy, SONY Compute Science Laboatoies, Inc., Tokyo 4 Japan. yyy The authos ae with the Dept. of Electical Engineeing, Univesity of Hawaii, Honolulu, HI, 968 USA. stage becomes quite complex if poweful linea block codes of length N > = 64 ae chosen to potect the most impotant class(es) of bit steams stongly. This motivates the concept of pactical multilevel coding and multistage decoding whee elatively good eo pefomance is achieved with modeate decoding complexity. In this pape we investigate two such possibilities. Fist, linea block component codes and suboptimal decoding with educed decoding complexity ae applied at each decode stage. The odeed statistics algoithm of [6] is used to decode poweful block component codes with few epocessing phases only, so that nea optimum decoding is achieved. Theoetical bounds ae also deived to detemine the ode of epocessing equied to achieve a desied eo pefomance. We note that Wachsmann et al. analyzed the use of had decision decoding in multistage decoding fom infomation theoetic point of view fo some paticula cases [5], but thee has not been any epot o analysis on educed-complexity soft decision decoding at each decode stage. Second, the use of punctued convolutional codes geneated fom a single mothe code togethe with a Hamming code of length 63 ae chosen as component codes to allow decoding with a common tellis stuctue. Guidelines fo choosing the code ates associated with each level of the unconventional patitioning ae given based on the theoetical esults of []. It is shown that good unequal eo potection capabilities ae achieved by this simple stuctue. Both appoaches achieve a good tade-o between eo pefomance and decoding complexity, and most impotantly, the st system allows the use of long poweful codes in applications of pactical inteest while the second scheme allows to achieve poweful UEP capabilities with \o-the-shelf" Vitebi decodes. The est of this pape is oganized as follows: unconventional patitioning fo UEP is eviewed in Section. Multilevel coding with block component codes and nea optimum multistage decoding is studied in Section 3. In Section 4, we conside the use of punctued convolutional codes geneated fom a mothe code togethe with a Hamming code of length 63 to allow decoding with a common tellis stuctue. Finally, conclusions on this wok ae dawn in Section 5.

2 IEICE TRANS. FUNDAMENTALS, VOL. E{A, NO. JANUARY 999. Unconventional (hybid) patitioning fo multilevel codes with multistage decoding and UEP Fo digital video boadcasting, UEP coding with twolevel eo coecting capability and elatively small popotion of most impotant bits (MIB) is desied. In [], multilevel coded modulation and multistage decoding based on unconventional patitioning is poposed to povide UEP capabilities eciently. The labeling of hybid patitioning fo an 8-PSK signal constellation is depicted in Fig.. The signal points ae patitioned by colo (black and white) at the st patitioning level and by symbols (squae and cicle) at the second patitioning level. The coesponding patitioning chain is shown in Fig.. Fig. Aveage numbe of neaest signal points (NN):.5 NN:.5 Hybid patitioning δ =.586 Fig. Patitioning chain δ =.586 δ =. 3 Dene N i (i = ; ; 3) and i (i = ; ; 3) as the aveage numbe of neaest neighbos (neaest signal points) and the intaset distance (minimum squaed Euclidean distance between signal points in disjoint subsets) at level-i of the patitioning, espectively. The paametes at the st level of hybid patitioning can be easily found as = 4 sin (=8) and N = =. A vey low bit eo pobability can be obtained by applying poweful component codes at this level []. At the second level of the patitioning, Ungeboeck's set patitioning ule [7] is intoduced, esulting in = 4 sin (=8), N = 3= fo the second level, and 3 = :, N 3 = fo the thid level. Theefoe, the eo pefomances of these two levels can be balanced by choosing a less poweful code at the thid level. As a esult, two levels of potection ae achieved. 3. Multilevel codes with block component codes and nea optimum multistage decoding With hybid patitioning, the choice of a poweful block component code at the st level ts UEP applications due to the lage minimum Hamming distance d H and the esulting low multiplicity which allow to eectively potect the most impotant class of infomation bits. Howeve, it is also dicult to achieve pactically optimal eo pefomance fo poweful block codes even with the computationally ecient soft-decision algoithm of [6]. As a esult, suboptimum decoding with a educed decoding complexity becomes vey attactive as it enables a good balance between decoding complexity and eo pefomance. Soft decision decoding algoithm based on odeed statistics was oiginally poposed and analyzed fo binay tansmission [6]. In this section, we extend this algoithm to multilevel signaling and analyze its eo pefomance. 3. Channel model and decoding algoithm with multilevel signaling Let us descibe the channel model and soft decision decoding algoithm based on odeed statistics [6] fo the multilevel signaling used in this pape. Suppose an (N; K i ; d i ) binay linea code C i is used at level i (i = ; ; 3). Let c i = (c i ; c i ; ; c in ) be a codewod in C i (c ij f; g; j = ; ; N). The codewods c; c; c3 ae mapped to the 8-PSK symbol sequence s = (s; s; ; s N ) such that a signal point s j with a label (c j ; c j ; c 3j ) in Fig. is selected as each symbol. The eceived sequence at the output of the eceive is = (; ; ; N ) with j = s j + w j, whee w j is a Gaussian andom vecto with zeo mean and covaiance matix N = I. With multistage decoding, the component code at level i is decoded with the associated decode based on the had decisions at lowe index stage decodes and abitay codewods at highe index levels. De- ne x;j as the pojection of j onto the -coodinate axis. Since at the st stage decode, the had decision decoding fo position-j is made based on the sign of x;j only [], j x;j j epesents the eliability associated with this decision. Assume that the decoding algoithm begins by eodeing the components of the pojected eceived sequence x = ( x; ; x; ; ; x;n ) based on thei eliability values in a deceasing ode. Dene p = (p ; p ; ; p N ) as the eodeed sequence at the st stage decode, so that jp j > jp j > > jp N j; ()

3 ISAKA et al: TRADEOFFS BETWEEN ERROR PERFORMANCE AND DECODING COMPLEITY 3 Fom this odeing, the most eliable basis (MRB) of the code is detemined, so that the st K -odeed eceived symbols occupy the most eliable independent (MRI) positions of the pojected eceived block [6]. A had-decision decoding of this infomation set coesponds to a codewod c of the equivalent code constucted afte a pope pemutation. This decoding is efeed as ode- epocessing [6]. The codewod is then epocessed in successive stages until eithe pactically optimum o a desied level of eo pefomance is achieved. Fo each phase of epocessing l < = m, we conside all possible K l codewod candidates obtained by ipping l MRI positions out of the positions of the MRB. This pocedue is efeed as phase-l of ode-m epocessing [6]. The same decoding algoithm is applied to stage- and stage-3, based on the two dimensional eceived symbols deliveed fom stage-. Since pactically optimum eo pefomance is consideed at these stages in the following, the eade is efeed to []-[4] fo a detailed desciption of the algoithm. 3. Pefomance analysis 3.. Uppe bounds on BER with MLD at each stage decode Uppe bounds on BER with MLD at each decoding stage have been deived in [], based on union bound aguments. The esults ae biey eviewed in the following fo the late discussions. Fo simplicity binay linea block codes ae consideed as component codes. Extensions to convolutional component codes ae staightfowad. Fo i = ; ; 3, let A (i) w denote the numbe of codewods of weight w in C i. Dene = sin(=8) and = cos(=8) as the absolute value of the pojection of 8-PSK signal points into (Y ) coodinates. The pobability of a bit eo P b, with hybid patitioning, at level, is uppe bounded by union bound aguments as N w w w P b < = N A() w w k w=d k= Q REb N d P (k)! ; whee R = (K + K + K 3 )=N is the ate of the oveall multilevel code in bits/symbol, and d P(k) = w (k + (w k) ) ; (3) Q(x) = p e n = dn: (4) x () as The BER at level i (i = ; 3) can be uppe bounded P bi < = P? b i = N w=d i w N A(i) w N w i Q! REb wi ; (5) N with N = :5 and N 3 =. With UEP capabilities achieved by the patitioning consideed, the eo popagation fom stage to can be vitually ignoed. Taking eo popagation fom stage to 3 into account, a good appoximated uppe bound at decoding stage 3 is given by [], P b3 < = P? b 3 + P? b : (6) Based on the bounds shown in this and the subsequent subsections, it tuns out that the multilevel codes based on hybid patitioning can potect MIB moe powefully at the cost of some pefomance degadation fo least impotant bits, as in othe efeences on coded modulation fo UEP [8]{[]. 3.. Odeed statistics associated with the multilevel signaling In this section, some odeed statistics associated with multilevel signaling consideed ae analyzed. We deive, as an example, the pobability P e(i; N) (espectively P e(i; j; N)) coesponding to the event that the had decision of symbol i (and symbol j) of the odeed sequence p of length N is (ae) in eo. Extensions to moe than two positions ae also possible with the technique descibed below. Fo i = ; ; N, dene f Wi j i (w i j i = x i ) as the density function of the ith noise value w i in the odeed sequence p, conditioned on the fact that i = x i was tansmitted, whee x i = 6 ; 6. The odeed eceived signals should satisfy jx i + w i j > jx k + w k j with new labeling i < k accoding to (). Conside the case of i =, and jw k + x k j < = jw i j; (7) should hold fo i < k. This implies that fo x k = W k [ + w i ; + w i ]; if w i > = ; W k [ + w i ; + w i ]; if w i < = ; and fo x k = W k [ + w i ; + w i ]; W k [ + w i ; + w i ]; if w i > = ; if w i < = : Assuming the tansmitted signals ae independent, equipobable, and independent of the zeo-mean white

4 4 IEICE TRANS. FUNDAMENTALS, VOL. E{A, NO. JANUARY 999 Gaussian noise, we deive the following elation fo both w i > = and w i < = P (W k [ + w i ; + w i j i = ]) = P(W k [ + w i ; + w i j i = ]) = P (W [ + w i ; + w i ]): (8) The same deivations hold fo i = ; 6. It follows with P (jw k + x k j < = jw i + x i jj i = x i ; k = x k ) = P(W [z (w i ; x i ; jx k j); (w i ; x i ; jx k j)] j i = x i ; k = x k );(9) z (w i ; x i ; jx k j) = min(jx k j + jx i j + sgn(x i ) w i ; jx k j jx i j sgn(x i ) w i ); () (w i ; x i ; jx k j) = max(jx k j + jx i j + sgn(x i ) w i ; jx k j jx i j sgn(x i ) w i ): () We dene the thee following disjoint events with espect to the value w i [3], with E = fjw + xj > jw i + x i jg; E = fw i < = W < = w i + dw i g; E 3 = fjw + xj < jw i + dw i + x i jg: () Also, each event has the associated pobabilities, P(E ; x i ; jx k j) = Q (z(w i ; x i ; jx k j) + Q ((w i ; x i ; jx k j) ; P(E ) = REb N e w i RE b =N dw i ; P(E 3 ; x i ; jx k j) = Q (z(w i ; x i ; jx k j)) z(w i ; x i ; jx k j) = (w i ; x i ; jx k j) = Q ((w i ; x i ; jx k j)) ;(3) R E b N z (w i ; x i ; jx k j); (4) R E b N (w i ; x i ; jx k j): (5) Let us conside the case in which E occus (i ) times, E once and E 3 (N i) times to evaluate the pobability that the had decision of the ith symbol afte odeing is in eo. Note that thee ae N possibilities in specifying this ith symbol in the whole sequence. Assume among the emaining N symbols, l ( < = l < = N ) symbols have pojection k = 6 (k j= i) on -coodinate (efeed to as \symbols A") and the est of the N l symbols have the value k = 6 (efeed to as \symbol B"). Finally if m symbols out of the l \symbols A" ae coupted by noise values that poduce event E, events in () occus as follows: m symbols with jx k j = ae associated with event E, (i m) symbols with jx k j = ae associated with event E, (l m) symbols with jx k j = ae associated with event E 3, (N i l + m) symbols with jx k j = ae associated with event E 3, Only one symbol (odeed i-th symbol) is associated with event E. As the numbe of each event should not be negative, m anges between m min and m max with m min = max(; (N i l)) and m max = min(l; i ). We obtain f Wi j i (w i j i = x i ) dened by (6) by taking all the combinations of events into consideation. Finally, the pobability that the had decision of the ith symbols of the sequence of length N is in eo is given by P e(i; N) = f Wi j i (w i j i = )dw i + f Wiji (w i j i = )dw i :(7) It should be noted that BPSK signaling is the special case in which x i = 6 and x k = 6. Also, these esults ae useful fo extending the appoach of [6] to 4-PAM signaling. The joint conditional density function of W i and W j (i < j) in the odeed sequences p, denoted by f Wi ;W j j i ; j (w i ; w j j i = x i ; j = x j ), can be deived afte dening the following ve events: E = fjw + xj > jw i + x i jg; E = fw i < = W < = w i + dw i g; E 3 = fjw j + x j j < jw + xj < jw i + dw i + x i jg; E 4 = fw j < = W < = w j + dw j g; E 5 = fjw + xj < jw j + dw j + x j jg: (8) Each event E m (m = ; ; 5) has the associated pobability given by P (E ; x i ; jx k j) = Q (z(w i ; x i ; jx k j)) + Q ((w i ; x i ; jx k j)) ;

5 ISAKA et al: TRADEOFFS BETWEEN ERROR PERFORMANCE AND DECODING COMPLEITY 5 f Wi j k (w i j i = x i ) = (N) N N l= N l m max m=m min l m N l i m fp (E ; x i ; )g m P(E ) fp (E ; x i ; )g im fp (E 3 ; x i ; )g lm fp(e 3 ; x i ; )g Nil+m ; (6) P (E ) = REb N e w i RE b=n dw i ; P(E 3 ; x i ; x j ; jx k j) = Q ((w j ; x j ; jx k j)) Q ((w i ; x i ; jx k j)) + Q (z(w i ; x i ; jx k j)) Q (z(w j ; x j ; jx k j)) ; REb P (E 4 ) = e w j RE b =N dw j ; N P (E 5 ; x j ; jx k j) = Q (z(w j ; x j ; jx k j)) Q ((w j ; x j ; jx k j)) : (9) Fo xed i and j, assume that among the emaining N symbols, l ( < = l < = N ) symbols have pojection k = (k j= i) on -coodinate (efeed to as \symbols A") and the est of the N l symbols have the value k = (efeed to as \symbol B"). In addition, assume m symbols out of the l \symbols A" ae coupted by noise values which poduce event E, and n symbols out of the emaining (l m) symbols poduce event E 3 in (). Again, since the numbe of each event should be nonnegative, we have m fm min ; m max g and n fn min ; n max g whee m min = maxf; (N i l)g; m max = minfl; i g; n min = maxf; (N l + m j)g; n max = minfl m; j ig; () espectively. By following the same pocedue fo f Wiji (w i j i = x i ), f Wi;Wjj i; j (w i ; w j j i = x i ; j = x j ) is given by (), whee A (x) = if x A, and A (x) =, othewise. The pobability P e(i; j; N) can be calculated fom (). The same aguments can be easily extended when moe than two odeed noise value ae consideed. In Fig. 3, P e(3; 64) and P e(5; 35; 64) deived by (7) and (), espectively, have been compaed with the coesponding simulation esults. Fo P e(3; 64), the numeical esults ovelap at all E b =N and thus veies the validity of (7). Simila behavio can be also obseved fo P e(5; 35; 64) Uppe bounds on BER fo suboptimum ode-i epocessing An uppe bound on ode-i epocessing can be deived by using P (u ; u ; ; u i ; N) in which two cases of i = ; has been deived in (7) and (), espectively. log(pb) "Sim-Pe(3;64)" "Pe(3;64)" "Sim-Pe(5,35;64)" "Pe(5,35;64)" Eb/No (db) Fig. 3 Simulation and analytical esults: Pe(3;64) and Pe(5,35;64) Based on the aguments in [6], an uppe bound on ode-i epocessing can be given by the union of () and (3) whee P j epesents the pobability that j dependent columns have been encounteed in nding the K MRI positions of the eceived sequence, and N ave (i) denotes the aveage numbe of columns depending on i dimensions fo a given code C whose geneato matix is in systematic fom [6]. 3.3 Simulation esults Simulation esults and some uppe bounds on the optimum bit eo ate (BER) and that of educed complexity decoding ae depicted in Fig. 4 (fo st decoding stage) and in Fig. 5 (fo all the decoding stages togethe with uncoded QPSK) fo the constellation of Fig. ; the component codes used at the st, second and thid levels ae the (64,8,), (64,45,8) and (64,63,) extended BCH (ebch) codes, espectively, so that the oveall ate is (bits/symbol). We can clealy see that two levels of UEP capability ae achieved. Also shown in Fig. 5 fo compaison puposes ae the simulation esults and uppe bounds of multilevel codes fo equal eo potection (EEP) based on Ungeboeck's set patitioning [7] with.9375 (bits/symbol); (64,6,4), (64,5,6) and (64,57,4) extended BCH (ebch) codes ae used as component codes.

6 6 IEICE TRANS. FUNDAMENTALS, VOL. E{A, NO. JANUARY 999 f Wi ;W j j i ; j (w i ; w j j i = x i ; j = x j ) = (N) N(N ) P(E ) P(E 4 ) fp (E ; x i ; )g m fp(e ; x i ; )g im n max l m n n=n min N l= N l mmax N i l + m j i n m=m min l m N l i m fp (E 3 ; x i ; x j ; )g n fp (E 3 ; x i ; x j ; )g jin fp (E 5 ; x j ; )g lmn fp(e 5 ; x j ; )g Njl+m+n [z (w i ;x i ;jx j j); (w i ;x i ;jx j j)](w j ); () Pe(i; j; N) = 4 +wi f Wi ;W j j i ; j (wi; wj ji = ; j = )dwidwj +wi + f Wi ;W j j i ; j (w i ; w j j i = ; j = )dw i dw j + f Wi ;W j j i ; j (w i ; w j j i = ; j = )dw i dw j + ; () wi wi f Wi ;W j j i ; j (w i ; w j j i = ; j = )dw i dw j 9 P (i) b Nave() K(i+) j = K j i+ =j i + NK P e(k j i+ ; ; K j ; N) + K+j j i+ =j i + j = j! P j j= K+j i j = P e(k + j j i+ ; ; K + j j ; K + j ; N) A A ; (3) At the st stage decode, fou simulation cases coesponding to decoding with ode-i epocessing with i = ; ; 3; 4 have been caied out. It can be obseved that ode-3 epocessing achieves pactically optimum pefomance without any signicant impovement due to ode-4 epocessing, while ode- epocessing sues only a slight degadation of about :5 db at the BER P b 5 with educed computational cost. A substantial coding loss is obseved fo ode- epocessing. Howeve, it is impotant to note that ove boadcast channels, not all the eceives in geneal sue fom low E b =N values, and some eceives would have sucient eliability infomation even with ode- epocessing. Fom Fig. 5, we also obseve little pefomance degadation between ode- and ode- epocessings at the second stage decoding. Fo the thid stage decoding, only ode- epocessing is consideed since it achieves pactically optimum pefomance fo the component code used [6]. In summay, ode- epocessing of the (64,8) ebch at st stage, ode- epocessing of the (64,45) ebch at second stage and ode- epocessing of the (64,63) ebch at thid stage decodes achieves an excellent tade-o between eo pefomance and decoding complexity. Also, even lowe epocessing odes can be chosen at some eceives. Finally we notice fom Fig. 5 that the union bound obtained fom (3) fo i = is vey tight, and that fo i = is elatively tight. 4. Multilevel codes with punctued convolutional component codes and multistage decoding In [4], Vitebi et al. investigated the application of \industy standad" convolutional codes to tellis coded modulation. By analogy to this pagmatic appoach, we popose a pactical multilevel UEP coding scheme whee component codes ae punctued convolutional codes geneated fom a single mothe code. The decoding at each stage can be pefomed with the same tellis stuctue.

7 ISAKA et al: TRADEOFFS BETWEEN ERROR PERFORMANCE AND DECODING COMPLEITY 7 log(pb) UB-MLD "Sim(64,8)-ode4" "Sim(64,8)-ode3" "Sim(64,8)-ode" "Sim(64,8)-ode" "Sim(64,8)-ode" "UB(64,8)-MLD" "UB(64,8)-ode" "UB(64,8)-ode" UB-ode UB-ode Eb/No (db) Fig. 4 Results: Block component codes with nea optimum decoding (st decoding stage) log(pb) stage EEP "uncodedqpsk" "Sim-stage" "UB-stage-MLD" "Sim-stage" "UB-stage-MLD" "Sim-stage3" "UB-stage3-MLD" "Sim-EEP" "UB-EEP" stage,3 uncodedqpsk 5 5 Eb/No (db) Fig. 7 Simulation esults: ate /3 convolutional code fo the st level, ate /3 punctued convolutional code fo the second level, and (63,57) Hamming code fo the thid level, all with a common tellis stuctue log(pb) stage- MLD EEP stage- ode "uncodedqpsk" "UB(64,8)-MLD" "UB(64,8)-ode" "UB(64,8)-ode" "Sim(64,45)-ode" "Sim(64,45)-ode" "UB(64,45)-MLD" "Sim(64,63)" "UB(64,63)-MLD" "Sim-EEP" "UB-EEP" stage,3 uncodedqpsk stage- ode Eb/No (db) Fig. 5 Results: Block component codes with nea optimum decoding (all decoding stages) LIB MIB ate-/3 CC (m=6) [ ] ate-/ CC (m=6) [7 33] Hamming code (63,57,3) punctuing b b b3 8PSK signal mapping (Hybid I patition) Fig. 6 Multilevel coding with a common tellis stuctue using convolutional code (CC) and Hamming code of length 63 As an example, we conside the 64-state ate-= convolutional code with geneato polynomials 7,33 (in octal fom) as mothe code, as depicted in Fig. 6. Simulation esults and uppe bounds of multilevel coding and multistage decoding with these punctued convolutional codes and MLD at all decoding stages ae shown in Fig. 7, with uncoded QPSK and multilevel codes fo EEP. It can be obseved that UEP capability is again attained. In these simulations, the st level component code is modied to potect most impotant bits (MIB) stongly by epeating the second output bit of the encode twice, which esults in a ate-=3 convolutional code with geneato polynomials 7; 33; 33 and fee distance d H = 3. Note that the optimal convolutional code of the same ate and constaint length with espect to the fee distance has d H = 6, which means an asymptotic coding gain of :9dB is saci- ced to obtain this coding stuctue. A ate-=3 punctued convolutional code with d H = 6 is chosen at the second stage, and the (63,57,3) Hamming code is used at the thid stage since this code can also be decoded with the same tellis stuctue [5]. The oveall ate of this multilevel code is.9476 (bits/symbol). Decoding based on a common tellis stuctue is possible with thee output bits on a banch, and insetion of dummy signals as thid symbols on evey banch at the second and thid stage decodes. This scheme is inteesting fo UEP applications due not only to the eo pefomances achieved, but also to its ealization with \o-the-shelf" component codes. 5. Conclusion In this pape, we investigated pactical multilevel coding and multistage decoding fo satellite boadcasting

8 8 IEICE TRANS. FUNDAMENTALS, VOL. E{A, NO. JANUARY 999 with modeate decoding complexity. An unconventional patitioning is used in ode to achieve UEP capabilities at pactical ange of bit eo pobabilities. Two possibilities ae consideed to tade o elatively good eo pobabilities with easonable computational costs. Fist, we consideed linea block component codes with nea optimum multistage decoding. It was shown that computational cost can be eectively educed at the expense of elatively small pefomance degadation. A theoetical analysis on bit eo pobability has been deived fo this suboptimal appoach. Second, punctued convolutional codes geneated by a single mothe code ae used as component codes, by analogy to Vitebi et al.'s pagmatic TCM appoach. This coding scheme allows multistage decoding with a common tellis stuctue, and using a single \o-theshelf" Vitebi decodes. Desiable featues of UEP channel coding ae exible numbe of levels, coding gains and popotions of infomation bits fo each impotance level. This e- nement is easy to obtain, due to the lage numbe of degees of feedom in ou design by intoducing asymmety in the signal constellations [3], o some simple extensions like the use of binay linea UEP codes at some of the patitioning levels []. In both cases, the tadeos between eo pefomance and decoding complexity obseved in this pape apply, and povide additional exibility to multilevel coding fo UEP. acknowledgement The authos ae gateful to anonymous eviewes whose comments wee helpful in impoving the pesentation of this pape. Refeences [] H. Imai and S. Hiakawa, \A new multilevel coding method using eo-coecting codes," IEEE Tans. Infom. Theoy, vol. IT-3, no.3, pp , May 977. [] R.H. Moelos-aagoza, M.P.C. Fossoie, S. Lin and H. Imai \Multilevel coded modulation with unequal eo potection and multistage decoding pat-i: symmetic constellations," accepted fo publication in IEEE Tans. Commun. [3] M. Isaka, R.H. Moelos-aagoza, M.P.C. Fossoie, S. Lin and H. Imai \Coded modulation fo satellite boadcasting based on unconventional patitionings," IEICE Tans. Fundamentals, vol., pp.55{63, Oct., 998. [4] M. Isaka, M.P.C. Fossoie, R.H. Moelos-aagoza, S. Lin and H. Imai \Multilevel coded modulation with unequal eo potection and multistage decoding pat-ii: asymmetic constellations," accepted fo publication in IEEE Tans. Commun. [5] U. Wachsmann, R.F.H. Fische and J.B. Hube,\Multilevel codes: theoetical concepts and pactical design ules, " IEEE Tans. Infom. Theoy, vol.45, no.5, pp.36-39, July 999. [6] M.P.C. Fossoie and S. Lin, \Soft-decision decoding of linea block codes based on odeed statistics," IEEE Tans. Infom. Theoy, vol.4, pp , Sept [7] G. Ungeboeck, \Channel coding with multilevel/phase signals, " IEEE Tans. Infom. Theoy, vol.8, pp.55-67, Jan. 98. [8] A.R. Caldebank and N. Seshadi, \Multilevel codes fo unequal eo potection," IEEE Tans. Infom. Theoy, vol. 39, no. 4, pp , July 993. [9] L.F. Wei, \Coded modulation with unequal eo potection," IEEE Tans. Commun., vol. 4, no., pp , Oct [] G. Taicco and E. Bigliei,\Pagmatic unequal eo potection coded schemes fo satellite communications," in Poc. of the Fifth Communication Theoy Mini-Confeence, GLOBECOM'96, London, U.K., pp. -5, Nov [] A. Seege, \Hieachical channel coding fo Rayleigh and Rice fading," in Poc. of the Sixth Communication Theoy Mini-Confeence, GLOBECOM'97, Phoenix, A, pp. 8-, Nov [] R.H. Moelos-aagoza, N. Uetsuki, T. Takata, T. Kasami and S. Lin,\Eo Pefomance of Multilevel Block Coded 8- PSK Modulations Using Unequal Eo Potection Codes fo the Rayleigh Fading Channel,"IEICE Tans. Fundamentals, vol. E8-A, no. 6, pp , June 997. [3] A. Papoulis, Pobability, Random Vaiables, and Stochastic Pocesses, 3d ed., New Yok: McGaw-Hill, 99. [4] A.J. Vitebi, J.K. Wolf, E. ehavi and R. Padovani,\A pagmatic appoach to tellis-coded modulation", IEEE Commun. Magazine, July 989. [5] A.M. Michelson and D.F. Feeman, \Vitebi decoding of the (63,57) Hamming code{implementation and pefomance esults ", IEEE Tans. Commun., vol. 43, no.,pp , Nov Motohiko Isaka eceived the B.E. and M.E. degees in electonic engineeing, and Ph.D degee in infomation and communication engineeing, all fom the Univesity of Tokyo, Japan, in 994, 996 and 999, espectively. He is cuently with the Institute of Industial Science, the Univesity of Tokyo as a post doctoal fellow. His cuent inteests include coding theoy, coded modulation, communication theoy and cyptogaphy. He is a membe of the IEEE. Robet H. Moelos-aagoza eceived the B.S. and M.S. degees in Electical Engineeing fom the National Autonomous Univesity of Mexico, Mexico city, Mexico, in 985 and 987, espectively, and the Ph.D. degee in Electical Engineeing fom the Univesity of Hawaii, Honolulu, Hawaii, in 99. Fom Mach 995-June 997, he was a eseach associate at the Institute of Industial Science, the Univesity of Tokyo, Tokyo. Fom June 997 to July 999, he was with the Channel Coding Goup of LSI Logic Copoation, Milpitas, CA. Since August 999, he is a eseache at the Advanced Telecommunications Laboatoy, SONY Compute Science Laboatoies, Inc., Tokyo,

9 ISAKA et al: TRADEOFFS BETWEEN ERROR PERFORMANCE AND DECODING COMPLEITY 9 Japan. His eseach inteests ae in eo contol coding, coded modulation and digital communications. D. Moelos-aagoza is a senio membe of IEEE and a membe of Eta Kappa Nu. Mac P.C. Fossoie was bon in Annemasse, Fance, on Mach 8, 964. He eceived the B.S. degee fom the National Institute of Applied Sciences (I.N.S.A.), Lyon, Fance, in 987, and the M.S. and Ph.D. degees in 99 and 994 fom the Univesity of Hawaii at Manoa. In 996, he joined the Faculty of the Univesity of Hawaii, Honolulu, as an Assistant Pofesso of Electical Engineeing. He was pomoted to Associate Pofesso in 999. His eseach inteests include decoding techniques fo linea codes, code constuctions, communication algoithms, and statistics. He is a membe of IEEE. Shu Lin eceived the B.S.E.E. degee fom the National Taiwan Univesity, Taipei, Taiwan, in 959, and the M.S. and Ph.D. degees in electical engineeing fom Rice Univesity, Houston, T, in 964 and 965, espectively. In 965 he joined the faculty of the Univesity of Hawaii, Honolulu, as an Assistant Pofesso of Electical Engineeing. He was pomoted to Associate Pofesso in 969, and to Pofesso in 973. In 986 he joined Texas A&M Univesity, College Station, T, as the Ima Runyon Chai Pofesso of Electical Engineeing. In 987 he etuned to the Univesity of Hawaii. His cuent eseach aeas include: algebaic coding theoy, coded modulation, eo contol systems, and satellite communications. D. Lin is an IEEE Fellow. Hideki Imai was bon in Shimane, Japan, on May 3, 943. He eceived the B.E., M.E. and Ph.D. degees in electical engineeing fom the Univesity of Tokyo, in 966, 968, 97, espectively. Fom 97 to 99 he was on the faculty of Yokohama National Univesity. In 99, he joined the faculty of the Univesity of Tokyo, whee he is cuently a Full Pofesso in the Institute of Industial Science. His cuent eseach inteests include infomation theoy, coding theoy, cyptogaphy, spead spectum systems and thei applications. D. Imai is an IEEE Follow.