8th Internatonal Symposum on Wreless Communcaton Systems, Aachen Herarchcal Generalzed Cantor Set Modulaton Smon Görtzen, Lars Schefler, Anke Schmenk Informaton Theory and Systematc Desgn of Communcaton Systems UMIC Research Centre, RWTH Aachen Unversty, 6 Aachen, Germany Emal: {goertzen, schefler, schmenk}@umc.rwth-aachen.de Abstract In ths paper, we show that arbtrary herarchcal pulse ampltude modulaton PAM) schemes can be fully descrbed by generalzed Cantor sets. Generalzed Cantor sets are modfed versons of the Cantor ternary set, a famous mathematcal construct known for ts set-theoretcal propertes. The fractal nature of generalzed Cantor sets allow for a natural renterpretaton as a modulaton scheme. The resultng Cantor set descrpton of one-dmensonal herarchcal modulaton schemes covers the constellaton ponts as well as the boundary ponts of the decson regons. Furthermore, we derve smple formulas for the average sgnal power as well as for teratve demodulaton. All results can be extended to two dmensons and herarchcal quadrature ampltude modulaton QAM) schemes. As such, ths paper offers a novel perspectve on the classfcaton and parametrzaton of practcal herarchcal modulaton schemes. I. INTRODUCTION In hs nformaton-theoretc work on broadcast channels, Cover [] shows that t s possble to outperform tme-sharng strateges by supermposng hgh-rate nformaton on lowrate nformaton. The hgh-rate, low-prorty nformaton s recovered by recevers wth a hgh sgnal-to-nose rato SNR), but appears as nose to recevers n low SNR envronments. Despte ths, those recevers stll recover the low-rate, hghprorty nformaton. These nsghts have motvated further research n the area of unequal error protecton [], [3] wth applcatons, for example, n the feld of dgtal vdeo broadcastng [4] [6]. To acheve unequal error protecton n practce, herarchcal sgnal constellatons are employed, specfcally herarchcal quadrature ampltude modulaton QAM) [7] []. By applyng nonunform sgnal spacng, the most sgnfcant bts experence much better error protecton than others. To evaluate the performance of herarchcal QAM, most research focuses on the computaton of the bt error rate BER) of certan herarchcal sgnal constellatons. These computatons are usually performed for systems wth a small number of nformaton layers and parametrzed by dstances obtaned from the constellaton dagram. In ths paper, we present a constructve approach to obtan arbtrary herarchcal QAM constellatons. We show that the constellaton ponts of herarchcal QAM concde wth the elements of sets that arse naturally n the constructon of generalzed Cantor sets. These are modfed versons of the famous Cantor ternary set, a mathematcal construct known manly for ts set-theoretcal propertes. We show that by explotng ther fractal nature, Cantor sets can be used to Ths work was supported by the UMIC Research Centre at RWTH Aachen Unversty n Germany. descrbe generalzed Gray-coded herarchcal pulse ampltude modulaton PAM) schemes. Ths ncludes a Cantor-based descrpton of the decson regons of herarchcal PAM. The extenson to two dmensons then yelds correspondng results for square and rectangular herarchcal QAM constellatons. The remander of ths paper s organzed as follows. In Secton II, we defne the real-valued channel model and present the bascs of PAM and herarchcal PAM. In Secton III, we present generalzed Cantor sets and ther propertes. Secton IV ntroduces the Cantor set modulaton scheme. The extenson to two dmensons s presented n Secton V. Followng ths secton, we present exemplary symbol error rate computatons and analyze how path loss effects naturally enable herarchcal nformaton transmsson n Secton VI. Secton VII concludes the paper. II. THE AWGN CHANNEL AND PAM Consder the recepton of a broadcasted real-valued symbol X through addtve whte Gaussan nose AWGN) wth power spectral densty N /. If we assume perfect channel state nformaton at the recever, the receved sgnal Y s Y = X + W, ) wth W N, N /). The average symbol energy s defned as E s = EX ). Assumng X to be unformly dstrbuted between symbols x, =,..., M, we obtan E s = M M x. ) = Wth ths, the average SNR γ at the recever can be expressed n terms of the sgnal energy per symbol E s dvded by N, γ = E s N. 3) The constellaton for PAM s desgned to mnmze the symbol error rate when transmttng over the real-valued AWGN channel. Ths s acheved by an equdstant dstrbuton of the constellaton ponts, centered at the orgn. The average symbol error rate P s s a functon of γ and can be expressed as P s γ) = M ) M Q ) 6γ M, 4) where Qx) = Φx) descrbes the tal probablty of the standard normal dstrbuton. A detaled descrpton can be 978--684-4-//$6. IEEE 6
Fg.. d d Herarchcal /4-PAM. The larger crcles denote the vrtual symbols. found n []. We denote an equdstant PAM scheme wth M symbols by M-PAM. Next, we gve a short descrpton of herarchcal PAM. To acheve unequal error protecton, the decson regons for low-rate nformaton are enlarged, whle the decson regons for hgh-rate nformaton are reduced n sze. For users wth low SNR, the symbols carryng hgh-rate nformaton are ndstngushable from each other and the modulaton appears to be an N-PAM scheme wth N < M. Note however, that these N vrtual symbols are not part of the constellaton dagram and only appear at the recever sde when nterpretng the hgh-rate nformaton as nose. We denote herarchcal PAM wth k layers of nformaton and N, =,..., k, vrtual) symbols n each layer by N /... /N k -PAM. See Fgure for an exemplary /4-PAM scheme. To descrbe a general N /... /N k -PAM scheme, parameters d,..., d k are requred. They are computed from the constellaton dagram as follows: For =,..., k, a recever recoverng -th layer nformaton experences an N /... /N - PAM system. Except for = k, the symbols N are vrtual ones. The mnmum dstance between two symbols n N s defned to be d. Refer to Fgure for the sgnfcance of d and d n the /4-PAM case. Clearly, d d has to hold to acheve reasonable error protecton. For d = d, the system s equvalent to 4-PAM, and for d = t degrades to -PAM. III. THE GENERALIZED CANTOR SET The Cantor set s a mathematcal construct named after Georg Cantor and famous for ts set-theoretcal and topologcal propertes. We gve a short defnton taken from []. The Cantor set conssts of all ponts n the closed unt nterval whch can be expressed to the base 3 wthout usng the dgt. Ths ternary representaton s the reason why t s also called Cantor ternary set. Geometrcally, the Cantor set s obtaned by deletng a sequence of mddle thrds from the closed unt nterval [, ]. Frst, /3, /3) s removed, leavng [, /3] [/3, ]. In the next step, the ntervals /9, /9) and 7/9, 8/9) are deleted. Ths process of removng the mddle thrd of each nterval s contnued ad nfntum and the remanng ponts make up the Cantor set. See Fgure for a vsualzaton of ths teratve process. Motvated by the Cantor ternary set, we now defne generalzed Cantor sets. In a prelmnary step, we center the set around the orgn. Therefore, we start wth the nterval [, ] nstead of the unt nterval. The next step s to allow not only the mddle thrd of each nterval to be removed, but arbtrary ratos. We ntroduce a scalng factor f such that from each nterval I, the mddle porton of sze I f )/f s deleted. Iteraton stage 3 4 Iteratve constructon of the Cantor ternary set f=3) /9 /9 /3 4/9 /9 /3 7/9 8/9 Fg.. Ths fgure shows the teratve constructon of the Cantor set for the specal case f = 3, N. Shown are the Cantor sets of stage to. Clearly, the choce f = 3 yelds the Cantor ternary set. Fnally, we allow the use of dfferent factors f at every stage of the teraton. We now formally defne the generalzed Cantor set C on the nterval [, ] subect to scalng factors f ) N wth f, N. Let C n denote the Cantor set of stage n, n N. Startng wth C = [, ], the Cantor set of stage s obtaned by removng a porton from the mddle of each nterval I n C, such that two ntervals of sze I /f reman. For the specal case f =, t holds that C = C as an empty nterval s removed, but we stll regard ths step as a further dvson nto two equdstant ntervals. We obtan the teratve defnton C = [, ], ) C = f C f )) f C + f )) 6) for all, n whch we use the notaton a S + b = {as + b s S} 7) for a set S and a, b R. Clearly, C n s a unon of n ntervals of length / = f. The generalzed Cantor set tself s defned as C = lm n C n, 8) however, for obvous practcal reasons, we focus on Cantor sets of fnte stage n ths paper. Note that the generalzed Cantor set can be extended to multple dmensons. In partcular, C C denotes the two-dmensonal Cantor set, whch s generally referred to as Cantor dust. Consequently, C n C n descrbes the Cantor dust of stage n. Followng ths defnton, we analyze partcular dscrete subsets of stage-n Cantor sets. Let P n denote the set of centers of all ntervals of C n. Clearly, P n = n holds. Furthermore, the famly P n ) n N can be represented through a bnary tree, n whch the nodes at depth n correspond to P n. Wth the descrpton P = P = {}, 9) ) ) P f = f P + f = f ) 7
each node x at depth n has offsprng x±f n )/ = f and all levels of the tree are ordered from left to rght. Ths bnary tree representaton allows to map each element of P n to a bt sequence of length n. We derve a closed-form expresson for the average power of a real-valued symbol X n takng on the values P n. Proposton. For n N, let X n be a unformly dstrbuted random varable wth support P n,.e., P X n = p) = for n all p P n. Then, the average power of X n computes to ) n EXn) f = = f. ) = Proof: It holds that EX ) = and EX n) = n = n p P n = n p + p P n = EX n ) + p ) p P n p f ) n = f + p + f ) n = f 3) fn = f ) 4) fn = f ) = ) n f = f. ) = Ths result concludes the secton on the generalzed Cantor set. In the followng secton we show how the fractal nature of generalzed Cantor sets naturally motvates herarchcal modulaton schemes. IV. CANTOR SET MODULATION Let n N denote the number of bts to be transmtted, wth correspondng scalng factors f,..., f n. The transmsson of an arbtrary bt sequence b,..., b n ) {, } n s realzed through the followng steps. To reduce the overall bt error rate, the sequences correspondng to two adacent symbols are only allowed to dffer by one bt. Ths s easly accomplshed wth Gray codng [3]. The Gray coded sequence s computed wth an exclusve-or operaton ) and gven by b,..., b n) as b = b, 6) b + = b b +. 7) Through the bnary tree representaton of P n, we obtan a one-to-one mappng {, } n P n, 8) n f b,..., b n ) ) b = f. 9) = Therefore, the constellaton ponts of the n-bt Cantor set modulaton are the elements of P n. At the recever sde, teratve decodng of the receved value Y = y s performed as follows: s = sgny ), ) y + = f y s f ), ) c = + s ), ) c = c, 3) c + = c c +. 4) Thus, c,..., c n ) denotes the receved bt sequence. In the case of an error-free recepton, b = c, =,..., n, holds. The teratve decodng n ) s made possble by the fractal nature of the generalzed Cantor set. We traverse the nodes of the bnary tree contanng the elements of P n at depth n and at each step decde whether to go left c = ) or rght c = ) based on the sgn of y. Despte the complexty of the Cantor set, t s possble to exactly descrbe the decson regons of ths modulaton scheme. Those are the ntervals bounded by the ponts D = n = P {± }. ) Furthermore, the elements of P are the boundary ponts of the decson regons whch are relevant for the decodng of b, the -th most sgnfcant bt. We call a modulaton of ths type Cantor set pulse ampltude modulaton, or CPAM for short. The notaton CPAMf,..., f n ) s used to descrbe the choce of scalng factors. Clearly, ths modulaton allows for almost arbtrary sgnal constellatons whle mantanng a smple teratve demodulaton procedure. In fact, CPAM can be used to holstcally descrbe arbtrary herarchcal PAM schemes as ntroduced n Secton II. Proposton. For n N, the constellaton ponts of PAM wth M = n concde wth the constellaton ponts of CPAMf,..., f n ) wth f = = f n =. The same s true for the decson regons. Thus, PAM s a specal case of CPAM wth PAM = CPAM,..., ). Ths result extends to herarchcal PAM schemes of arbtrary sze. Let d,..., d k descrbe the dstance parameters of a full N /... /N k -PAM scheme wth N =, =,..., k. Then, wth for =,..., k. N /... /N k -PAM = CPAMf,..., f k ) 6) f k =, 7) f = d f + +, d + f + 8) Proof: From ) we conclude that the dstances d have to concde wth F = f = f 9) 8
. Constellaton ponts of CPAMf,) Constellaton ponts of CQAMf,) Ampltude norm.).. Imagnary part. f= f=3 f=4....4.6.8 3 3. 3.4 3.6 3.8 4 Fg. 3. Sample of constellaton ponts of CPAMf, ). Note the equdstant 4-PAM spacng at f = and the slow convergence towards BPSK -PAM)... Real part Fg. 4. Sample of constellaton ponts of CQAMf, ). Note the 6-QAM spacng for f = and the slow convergence towards QPSK 4-QAM). up to a constant factor. Thus, 6) s equvalent to F = d 3) F + d + for =,..., k. Ths s easly verfed wth 8): F f + = f ) F + f + 3) = d f + f + d + f + f + = d. d + 3) For an example of ths equvalence, refer to Fgure 3, whch depcts the constellaton ponts of CPAMf, ) for varyng values of the scalng factor f. Here, a -bt transmsson s modulated n whch the sgn of the symbol determnes the value b of the frst bt of nformaton. For f =, the CPAM constellaton concdes wth regular PAM. For ncreasng f, the dstance between ponts carryng dfferent values of b ncreases, whle the dstance between ponts wth the same frst bt decreases. Thus, the recepton of the frst bt s smplfed at the cost of a more complcated recepton of the whole symbol. V. CANTOR DUST QAM To extend PAM wth constellaton sze M to QAM, the real-valued symbols are placed nto the complex plane n a square fashon. The resultng modulaton scheme has M dstnct symbols wth an average energy per symbol whch s twce as hgh as for PAM. Ths s equvalent to a halvng of the SNR γ computed for PAM. The extenson of CPAM to Cantor dust QAM CQAM) follows the same rules and results n a smlar formula for the average symbol error probablty. Let Ps CPAM γ) denote the average symbol error rate of CPAM at SNR γ. Then, the average symbol error rate n two dmensons s computed as Ps CQAM γ) = P CPAM s γ )). 33) See Fgure 4 for a sample of the CQAMf, ) constellaton ponts for dfferent values of f. Here, the quadrant of the Fg.. SNR [db] Second layer recepton bts) Frst layer recepton bt) BPSK recepton bt). 3 3. 4 Requred SNR for CPAMf, ) transmsson at P CPAM s = 3. symbol carres the nformaton of the frst two bts. For ncreasng f, ths nformaton s protected at the cost of a more complcated recepton of the whole symbol. For non-square constellatons, the extenson to twodmensonal rectangular CQAM s performed analogously. We omt the detals. VI. HIERARCHICAL INFORMATION TRANSMISSION In ths secton, we nvestgate the symbol error rates of a -bt transmsson wth two layers of nformaton,.e., /4- PAM. Correct recepton s assumed as long as the average symbol error rate does not exceed a target P s. We show that by applyng a CPAMf, )-modulaton, the requred SNR for the frst layer can be decreased whle the requred SNR for the second layer ncreases compared to a regular 4-PAM scheme. See Fgure for the numercal results for varyng scalng factors f. Next, we cover the extenson of ths example to two dmensons,.e., 4/6-QAM. As the number of symbols s squared, the number of transmtted bts n each layer doubles. Ths means that the frst layer of CQAM transmsson ncludes two bts of nformaton, whle the second layer transmsson ncludes four. From 33), t s possble to compute the requred SNR for a successful CQAMf, )-transmsson. See Fgure 6 for the results. Evdently, the extenson to two dmensons 9
Fg. 6. SNR [db] Second layer recepton 4 bts) Frst layer recepton bts) QPSK recepton bts). 3 3. 4 Requred SNR for CQAMf, ) transmsson at P CQAM s = 3. has no sgnfcant nfluence on the general behavor of the modulaton scheme. In a fnal step we analyze how path loss naturally enables the broadcastng of herarchcal nformaton. We assume a smple path loss model wth a path loss exponent α such that the SNR γ at the recever s proportonal to D α, wth D denotng the dstance between the transmtter and the recever. We denote the maxmal dstance for layer one and layer two recepton by D and D, respectvely. Note that ths mples D D. Thus, the quotent ρ = D /D denotes the porton of the maxmal dstance for second layer recepton normalzed by the maxmal dstance for frst layer recepton. As a result, the SNR gap between dstances D and D s computed as γ = ρ α. If we requre the frst layer recepton to nclude two bts of nformaton, the SNR at the recever has to be at least equal to that of QPSK, or 4-QAM, recepton see Fgure 6). If we utlze a mult-layered CQAMf, ) scheme wth 6 symbols nstead, the addtonal energy requred to guarantee the same frst layer recepton qualty can be calculated as the dfference between the dashed red lne and the dotted black lne n Fgure 6. From the same fgure, we obtan the SNR gap γ between second layer and frst layer recepton as the dfference between the blue lne and the dashed red lne. Wth the above, t s possble to convert ths SNR gap nto a dstance quotent ρ, computed as ρ = γ) /α. Thus, we acheve a trade-off of nvestng an amount of addtonal energy requred for mult-layered CQAM compared to lowlayer QAM) at the gan of a second layer of recepton at recevers wthn a porton ρ of the overall dstance. For path loss exponents α rangng between and 4, ths trade-off between ρ and addtonal energy s vsualzed n Fgure 7. For example, at α = 3, the energy cost for a full 4-bt recepton at % of the dstance ρ =.) s approxmately. db. VII. CONCLUSION In ths paper, we present a novel perspectve on the classfcaton and parametrzaton of practcal herarchcal modulaton schemes. We ntroduce generalzed Cantor sets and CPAM, a pulse ampltude modulaton scheme based on Cantor sets. Ths scheme s parametrzed through an approprate choce of scalng factors f. We show that CPAM classfes arbtrary herarchcal PAM schemes and present a formula for the average SNR gap compared to QPSK transmsson at % [db] Coverage and cost of mult layered CQAMf,) transmsson 8 6 4 Path loss exponent α= α=3 α=4 % % 4% 6% 8% % Porton of dstance covered wth second layer recepton 4 bts) Fg. 7. Trade-off between dstance quotent ρ = D /D and addtonal energy requred for CQAMf, ) at Ps CQAM = 3. symbol energy, a low-complexty demodulaton procedure, as well as a natural descrpton of the boundary ponts of the decson regons. 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