A Genetic Algorithm for Designing Constellations with Low Error Floors

Transcription

1 A Genetic Algorithm for Deigning Contellation with Low Error Floor Matthew C. Valenti and Raghu Doppalapudi Wet Virginia Univerity Morgantown, WV Don Torrieri U.S. Army Reearch Lab Adelphi, MD Abtract The error floor of bit-interleaved coded modulation with iterative decoding BICM-ID) can be minimized for a particular contellation by maximizing the harmonic mean of the quared-euclidian ditance of ignal whoe label differ in jut one bit poition. Thi problem ha been formulated a an intance of the Quadratic Aignment Problem QAP) and olved uing the Reactive Tabu Search RTS). In thi paper, we propoe a genetic algorithm for olving the ymbol labeling problem and how that it uually yield deign that are iomorphic to thoe obtained uing RTS. We then extend the algorithm to optimize not only the labeling of the ignal point, but alo their location in the ignal pace. Uing thi approach, new contellation of cardinality 16,, and 64 are evolved that have an error floor lower than that of QAM or PSK modulation. The new contellation exhibit gain of 1. and 0.88 db over the bet 16-QAM and 64-QAM contellation reported previouly. I. INTRODUCTION Bit-interleaved coded-modulation BICM) i a tandard approach for coding in modern wirele ytem involving a binary encoder followed by a bit interleaver and a nonbinary modulator [1]. Relative to trelli-coded modulation TCM), BICM increae diverity by replacing a ymbol interleaver with a bit interleaver, and hence ha improved performance over fading channel. Due to it imple implementation and uperior performance over fading channel, BICM ha become a tandard feature in all propoed future-generation cellular, atellite, and wirele networing ytem. Performance in a BICM ytem can be improved by allowing the demodulator and decoder to iteratively exchange extrinic information. Such ytem are nown a BICM with iterative decoding and demodulation BICM-ID) []. Plot of the bit error rate for BICM-ID ytem generally exhibit a waterfall region, which i characterized by a rapid decreae in the bit error rate a the ignal-to-noie ratio increae, and an error-floor region, in which the bit error rate decreae much more lowly. For a given ignal et, the performance in both region i determined by the labeling map, which pecifie the log M) bit equence aociated with each of the M ymbol in the ignal et. In thi paper, we are concerned with deigning contellation and labeling map that reduce the error floor. Method for generating good labeling map that provide low error floor for a given ignal et have been previouly decribed in [] [5]. In [4] it i hown that the minimization of the o-called error-free feedbac EFF) bound, which i an approximation of the actual error floor, can be formulated a an intance of the combinatorial optimization problem nown a the quadratic aignment problem QAP) [6]. In particular, the goal i to maximize the harmonic mean of the Euclidian ditance of ignal whoe label differ in jut one bit poition. The optimization i done over the M! poible labeling map. In [4], Reactive Tabu Search RTS) [7] wa ued to perform thi optimization. However, the optimization can be olved in other way, and in thi paper we propoe a genetic algorithm for optimizing the labeling map of an arbitrary ignal et. The error floor depend not only on the labeling map, but alo on the location of the ignal point. Previou wor ha only focued on the deign of the labeling map and aumed a fixed modulation type, typically QAM or PSK. However, the error floor can be reduced further by moving the contellation point. We modify the genetic algorithm to deign not only the labeling map, but alo the placement of the point in the contellation. By evolving the deign acro multiple generation, new modulation format are found with error floor that are lower than previouly nown reult involving QAM or PSK. The remainder of the paper i organized a follow. Section II provide a model of the ytem under conideration, while Section III derive the error-free feedbac bound and identifie the main contributor to the bound. Section IV formulate the problem of minimizing the EFF bound for a given contellation a a combinatorial optimization problem, and Section V propoe a genetic algorithm for olving the problem. Section VI modifie the genetic algorithm to allow the point in the contellation to be moved. Numerical reult howing the EFF bound for optimized QAM, PSK, and evolved contellation with M = {16,, 64} are given in Section VII and compared againt the imulated error rate of actual BICM-ID ytem. Finally, Section VIII conclude the paper. II. SYSTEM MODEL A BICM-ID ytem i hown in Fig. 1. In the tranmitter, a vector u of meage bit i paed through a rate r c binary convolutional encoder to produce a codeword c. The codeword i bitwie interleaved by a permutation matrix Π to produce the bit-interleaved codeword c = c Π. The bitinterleaved codeword i then paed through a modulator to produce the vector of complex ymbol X L where X

2 i a contellation of ymbol with cardinality M and average energy E. The overall rate of the ytem i R = mr c, where m = log M, and the average energy per information bit i E b = E /R. The modulator elect ymbol from X baed on the contellation labeling map and the correponding group of m conecutive bit at the modulator input. Each ignal in X i labeled with a unique m-bit equence b. Define the integer repreentation of a length-m binary vector b a qb) = =0 b m 1. 1) Let x = [x 0, x 1,..., x ] be a vector containing each ignal in X. The ignal in vector x are indexed according to a natural mapping o that the label b of x i atifie qb) = i. Let x be an interleaved verion of the vector x uch that x = x µ) where µ) i a permutation function. Let µ = [µ0), µ1),..., µm 1)] be a vector containing the integer 0 through M 1 permuted according to the function µ). When the input to the modulator i b, then the modulator elect ymbol = x qb) = x µqb)) for tranmiion. Thu, the vector µ pecifie the labeling map for a particular ordered ignal et x. Each coded ymbol pae through a frequency-nonelective channel with complex-fading amplitude c. In thi paper, we aume uncorrelated Rayleigh fading uch that the c are i.i.d. zero-mean complex Gauian with unit power. The output of the channel i y = c + n where n i a ample of a white complex-gauian proce with variance / per dimenion. At the receiver, a demapper within the demodulator procee each received ymbol to produce a vector z of bit lielihood. Thi vector provide extrinic information that i deinterleaved and paed to the decoder. The oft-output decoder produce extriniic information that i interleaved and provided to the demapper a a vector v of a priori information. The output of the demodulator for bit i [8] z = log x X 0) py x ) exp [β j x )v j ] j py x ) exp [β j x )v j ] j where the function β j x ) return the j th bit of the label of x and X b) i the et of all ymbol in X labeled with b = b. III. CRITERIA FOR LOW ERROR FLOORS The union bound on the bit error probability i [1] P b 1 W I d)f p d, E ) c d=d f where d f i the minimum ditance of the convolutional code, W I d) i the total input weight of error event in the convolutional code at Hamming ditance d, and f p d, E / ) i ) ) Fig. 1. u c c Encoder Π Modulator u z z Decoder Π -1 v v Π Demodulator Model of a BICM-ID ytem. Π indicate a bitwie interleaver. the pairwie error probability PEP). The PEP depend on the choice of contellation X and it labeling map µ. Let the function g x) return the ymbol whoe label i identical to the label of ymbol x except for the th poition, which i complemented. The ignal x and g x) form an errorfree feedbac EFF) ignal et. In the following, {x, g x)} i called an EFF pair and g x) i aid to be the EFF companion of x. The EFF bound on BER i found from ) under the aumption that the demodulator i provided with perfect a priori information by the decoder, in which cae the LLR of the unnown bit ha the form z = log py x ) log py g x )) 4) where x i one of the ymbol whoe th poition i labeled with a one, i.e. β x ) = 1. The deciion rule for bit involve jut one EFF pair for that bit, and the identity of the pair depend on the a priori information fed bac from the decoder. Becaue the decoder doe not alway provide perfect a priori information to the demodulator, and becaue ) i an upper bound, the EFF bound i not actually a bound but rather i an approximation. However, the EFF bound i uually able to accurately predict the performance at high SNR. Conider how to determine f p 1, E / ), i.e. the probability of a olitary bit error. Aume without lo of generality that a bit b = 1 i tranmitted. An error will occur if the correponding LLR z i negative f p 1, E ) = P r[z < 0 b = 1] 5) Since there are m bit labeling each ymbol, the PEP i found by averaging the m bit error probabilitie f p 1, E ) = 1 m =0 c P r[z < 0 b = 1] 6) The error probability for each bit may be found by averaging over the ymbol that are labeled with b = 1 P r[z > 0 b = 1] = 1 n y P r[z < 0 x ]. 7)

3 Let P z x z) denote the conditional CDF of z given that ymbol x wa ent. Subtituting 7) into 6) allow the PEP to be expreed a f p 1, E ) = 1 m =0 P z x 0). 8) The PEP can be found uing moment generating function technique [1], [9]. Define φ z x ) = L { p z x z)} to be the Laplace tranform of the pdf of z given that ymbol x wa ent. In Rayleigh fading [10], φ z x ) = ) x g x ). 9) From the integration property of the Laplace tranform and the fact that the CDF i the integral of the pdf { P z x z) = φz L 1 x ) } 10) The PEP can thu be found by ubtituting 10) into 8), reulting in f p 1, E ) { } ψ) = L 1 11) z=0 where ψ) = 1 m =0 φ z x ). 1) When d > 1, the PEP i the probability that the um of d independent bit LLR i le than zero given that the correponding tranmitted bit are all one. Replacing z in the firt line of 6) with thi um and conditioning on the event that all bit are one reult in f p d, E ) [ d 1 ] d 1 = P r z i < 0 b i = 1 1) where i i ued to index bit error event, which will generally be in different ymbol. By uing the convolution property of the Laplace tranform, the PEP may be found uing f p d, E ) { } = L 1 [ψ)] d 14) z=0 Once the PEP i found for each d, the the bit error probability i found by ubtituting into ). Aymptotically, a, the BER can be approximated by [1], [9] [ ] df E P b κ 15) where κ i a contant and d h = 1 m =0 d h x g x ) 1 16) i the harmonic mean of the quared-euclidian ditance between ignal in each EFF ignal et. Taing the bae-10 logarithm of 15), defining X db = 10 log 10 X, and recalling that E = RE b yield log 10 P b d f 10 [ Rd h )db + Eb ) db ] + κ db. 17) A plot of log 10 P b veru E b / ) db will be a traight line with negative lope given by d f, which i the diverity gain. The horizontal offet of the curve i controlled by d h, with larger value of d h tranlating into lower error rate. From 16), it i clear that the choice of ignal et X and ymbol labeling map µ influence the harmonic mean, and from 17) the harmonic mean determine the offet of the BER curve. Thu, to minimize the EFF bound for a given outer code, it i neceary to maximize d h. For a given X, the maximization i performed over the et of all poible µ. Such maximization i dicued in Section IV and V. If one i free to choe not only µ but alo the ignal et X, then the maximization i over both function, a decribed in Section VI. IV. LABEL MAPPING OPTIMIZATION Maximizing the harmonic mean given by 16) over all µ i equivalent to finding the minimum of a cot function: min µ =0 x X 1) x g x). 18) A dicued in [4], the minimization given by 18) can be formulated a an intance of the Quadradic Aignment Problem QAP) [6]. Define the flow matrix F uch that element f i,j = 1 if the label of x i and x j are different in jut one bit poition. Since the vector x i indexed according to a natural labeling, f i,j = 1 whenever the binary expanion of the integer i and j have a Hamming ditance of one. Otherwie, f i,j = 0. Define the ditance matrix D with element { xi x d i,j = j, i j 19) 0, i = j For a given F and D, the cot function in 18) may be found a f i,j d i,j. 0) The previou expreion i the cot function when a natural mapping i applied. If another mapping i applied by permuting x to obtain x, then a new ditance matrix D mut be found. However, ince x = x µ), D i merely a columnand-row permuted verion of D. Thu, the cot function under mapping µ can be expreed a [4] f i,j d µi),µj). 1)

4 The optimization i to minimize the above expreion with repect to all poible mapping function permutation) µ. The QAP i NP-hard and may be olved in a variety of way [6]. Exact olution can be found uing dynamic programming, cutting plane technique, and branch and bound procedure. For M, uch technique are generally not tractable. For larger problem, heuritic earch method have been conidered, including RTS [7], genetic algorithm [11] [1], and imulated annealing [14]. Conventional genetic algorithm [11] are not competitive with RTS in term of accuracy and efficiency. However, with recent modification [1], genetic algorithm have emerged a a very promiing candidate for olving the QAP. In [4], the problem of maximizing d h wa olved uing RTS. In thi paper, our goal i to maximize d h by uing a genetic algorithm. V. A GENETIC ALGORITHM FOR MAPPING OPTIMIZATION The genetic algorithm ued to optimize the mapping i baed on the algorithm propoed in [1]. The algorithm i initialized with a population of N randomly drawn mapping function µ i, 0 i N 1. The mapping function are indexed in increaing order of cot o that mapping µ 0 i the bet. Whenever the population change, it i reindexed. Breeding require the election of two parent µ j and µ. The firt parent i elected at random from the entire population except for the bet 1 j N 1), while the econd i elected at random from thoe mapping that are better than the firt 0 j). Thi preferential parent election rule wa not contemplated in [11] [1] and improve the rate of convergence. The two parent produce two children µ j and µ. The children are initially empty vector of length M. A child direct parent i the parent with the ame index, while it indirect parent i the parent with a different index. The children are generated by randomly electing λ croover point, where λ i a fixed number for all breeding. The croover point indicate the poition of each child that are inherited from it indirect parent. At the croover point, value from µ are copied into µ j and value from µ j are copied into µ. The remaining poition of a child µ l are inherited from it direct parent µ l, where l = {j, }. Whenever poible, value of parent µ l that are not already in it child µ l are copied into µ l at the ame poition. The remaining value that could not be copied from parent to child into the ame poition are copied into the cloet open poition. The cot of each of the two children are evaluated. If a child µ l ha a lower cot than it parent µ l, then the parent i replaced with the child. If the child i not better than the parent, then an optional mutation proce i tarted. A mutant i generated from µ l by exchanging the value tored in randomly choen pair of poition in µ l. The mutation rate ρ i the probability that any given poition in µ l will become exchanged with ome other randomly choen poition. During the mutation proce, a total of L mutant are created and the cot of each i evaluated. If any of the mutant have a lower quared harmonic mean ary -ary 64-ary 0.5 PSK QAM generation Fig.. Harmonic mean of the quared-euclidian ditance between EFF ignal pair after each generation of the genetic algorithm. cot than parent µ l, then the parent i replaced with the lowet cot mutant. During each generation, a certain number of breeding occur. While thi number i arbitrary, we have et the number of breeding per generation to equal N, the overall population ize. A propoed in [1], our algorithm periodically perform a culling operation once every T c generation. During a culling generation, each child µ l i no longer compared againt it direct parent. Intead, the child i compared againt the current wort mapping µ N 1 and if the child i better than the wort mapping, it will replace the wort mapping. It i poible that the population produced by thi genetic algorithm will contain duplicate of the ame mapping. During culling generation, thee duplicate are removed and replaced with new random mapping. The genetic algorithm wa run to optimize the mapping of PSK and QAM with M = {16,, 64}. For each cae, the population wa N = 100, the number of croover point λ = M/5, the mutation rate ρ = 0.0, and the culling period T c = M. The number of mutant L per child wa variable. Initially, L = 0. If after T c generation a new bet mapping i not identified, then L i incremented up to a maximum of 0. On the other hand, if a new mapping i identified within T c generation, then L i decremented. The value of d h after each generation, up to generation , i hown in Fig.. The final value of d h along with the generation that the bet mapping wa identified i lited in Table I. Alo hown are the value of d h obtained uing RTS. A can be een, the genetic algorithm ha found the ame optimal value a wa found uing RTS with the exception of 64-QAM, in which cae the genetic algorithm reult i lightly inferior to that of RTS. The contellation labeling map found by the genetic algorithm for M = 16 are hown in Fig.. VI. CONSTELLATION OPTIMIZATION In the previou ection, the contellation X wa fixed and only the labeling map µ wa optimized. However, d h depend not only on the labeling map but alo on the contellation. We

5 TABLE I VALUE OF d h OBTAINED USING RTS AND THE PROPOSED GENETIC ALGORITHM. ALSO LISTED IS THE NUMBER OF GENERATIONS REQUIRED FOR THE GA TO CONVERGE. M Modulation d h from RTS d h from GA generation 16 QAM PSK ,49 QAM PSK QAM ,41 PSK , Fig.. Contellation labeling map that maximize d h found uing the genetic algorithm. 16-QAM i on the left, and 16-PSK i on the right. ee to increae d h beyond the value poible for QAM and PSK by moving ignal point in the contellation. Ideally, the goal i to perform the following optimization min X,µ = x g x ). ) Becaue the number of poible X i infinite, thi optimization i more challenging than the one given by 18). We propoe a heuritic method for performing the optimization in ). The optimization begin with a tandard contellation. Becaue PSK ha a better potential d h than QAM, it i a more appropriate tarting contellation. Several generation of the genetic algorithm are run on the PSK contellation to obtain an initial mapping. Next, the contellation i modified in an attempt to increae d h. Thi can be done by firt picing an EFF pair whoe ditance i i at the minimum d e, where d e = min x g x ). ) 0 If there are multiple EFF pair that are ditance d e apart, then a pair i elected at random. The two point are then forced to be further apart. Let α > 1 repreent a cale factor, uch that the elected EFF pair i forced to be ditance αd e apart with the ame centroid. When thi adjutment i made, the average energy of the contellation will typically increae, and o it mut be renormalized to E. Alo, after the pair i puhed apart, the previouly choen labeling map might no longer be optimal, and thu it i redeigned by running one generation of the genetic algorithm with an initial population of mapping et to the final population of the previouly optimized contellation. The proce continue iteratively, with Fig ary contellation optimized uing a genetic algorithm. Tilted axe have been uperimpoed to emphaize the ymmetry that ha evolved. TABLE II SIGNAL VALUES FOR THE CONSTELLATION SHOWN IN FIG. 4. Labeling Real Imaginary Labeling Real Imaginary each iteration coniting of moving one pair of point apart, renormalizing the contellation, and genetically reoptimizing the contellation labeling map. Starting with the 16-PSK contellation hown in Fig., the optimization wa run with 1000 iteration and α = The reulting contellation i hown in Fig. 4. The optimization reulted in d h =.6841, which i coniderably larger than the bet d h found with QAM and PSK. The ignal value for the contellation are lited in Table II. Inpecting the contellation reveal an intereting geometry. Tilted axe have been uperimpoed on the figure to emphaize the ymmetry in the evolved contellation. Each quadrant of the tilted contellation contain four ymbol whoe label differ in exactly two bit poition; thu, there are no EFF pair in the ame quadrant. Of the four ymbol in each quadrant, the firt two ymbol are nearly colocated, a third ymbol i ditance 0.1 from the colocated pair, and a fourth ymbol i ditance 0.6 from the colocated pair. The fourth ymbol EFF companion all lie on the oppoite quadrant for intance the EFF companion of 8 are {0, 9, 10, 1}). For all other ymbol, two of their EFF companion are in one quadrant and the other two EFF companion are in another quadrant. Contellation for M = and M = 64 were optimized in a imilar fahion. The reulting contellation had d h =.5477 for M = 16 and d h =.5178 for M = 64 which are again better than the value found for PSK and QAM. Note that for the genetically-deigned contellation, d h actually decreae with increaing M, which i oppoite the trend oberved for QAM and PSK modulation. Thu the relative gain achieved by evolving the contellation decreae with increaing M.

6 10-7 M= QAM PSK GA Simulated EFF bound QAM PSK M= Genetic Deign BER BER 10-6 M= E b /N o in db Fig. 5. The EFF bound on BER when uing a 7, 5) convolutional code. For each M, QAM ha the highet error floor and the contellation deigned with the genetic algorithm GA) ha the lowet error floor. VII. NUMERICAL RESULTS The EFF bound wa calculated for the PSK and QAM contellation decribed in Section V a well a the geneticallydeigned contellation decribed in Section VI. The bound are hown in Fig. 5 for the rate-1/ convolutional code with octal generator 7, 5) and d f = 5. While all curve have the ame lope due to the common value of d f, there are gain in moving from one modulation choice to another. In general, there i a gain in moving to a larger alphabet and there are gain in moving from QAM to PSK and from PSK to the genetically-deigned contellation. For intance, the genetically-deigned contellation provide gain of 1. and 0.88 db, repectively, over the 16-QAM and 64-QAM contellation. Thee trend are the oppoite of the trend oberved with uncoded modulation or BICM without feedbac, indicating that BICM-ID achieve it gain in a much different manner. Indeed, the contellation hown in Fig. 4 would perform quite poorly without coding and iterative decoding. To demontrate the tightne of the EFF bound at high SNR, the three 16-ary BICM-ID ytem were imulated. The imulation ued a codeword length of N = 4, 000 bit over an uncorrelated Rayleigh fading channel. The reult of the imulation are hown in Fig. 6. Alo hown on the plot are the correponding EFF bound. A can be een, the EFF bound accurately predict the performance at high SNR. However, it can alo be een that lowering the EFF bound come at a cot. The waterfall region, i.e. the range of SNR characterized by a rapid drop in BER, occur at a higher SNR for each modulation that ha a lower EFF bound. VIII. CONCLUSIONS Evolutionary computing may be ued to not only optimize the labeling map of a contellation, but alo to optimize the placement of the point in the contellation. By applying an appropriate genetic algorithm, a new contellation may be E b /N o in db Fig. 6. Simulated bit error performance uing 16-ary modulation, a 7, 5) covolutional code, and 4, 000-bit codeword. evolved for BICM-ID with a predicted error floor that i on the order of 1 db better than the bet nown QAM contellation. The lower error floor of the evolved contellation come at the cot of the waterfall region being hifted to a higher SNR. REFERENCES [1] G. Caire, G. Taricco, and E. Biglieri, Bit-interleaved coded modulation, IEEE Tran. Inform. Theory, vol. 44, pp , May [] X. Li and J. A. Ritcey, Bit-interleaved coded modulation with iterative decoding, IEEE Commun. Letter, vol. 1, pp , Nov [] J. Tan and G. L. Stüber, Analyi and deign of ymbol mapper for iteratively decoded BICM, IEEE Tran. Wirele Comm., vol. 4, pp , Mar [4] Y. Huang and J. A. Ritcey, Optimal contellation labeling for iteratively decoded bit-interleaved pace-time coded modulation, IEEE Tran. Inform. Theory, vol. 51, pp , May 005. [5] D. Torrieri and M. C. Valenti, Contellation labeling map for low error floor, in Proc. IEEE Military Commun. Conf. MILCOM), Orlando, FL, Oct [6] E. Cela, The Quadratic Aignment Problem: Theory and Algorithm. Kluwer Academic Pre, [7] R. Battiti and G. Tecchiolli, The reactive tabu earch, ORSA J. Comp., vol. 6, pp , [8] M. C. Valenti and S. Cheng, Iterative demodulation and decoding of turbo coded M-ary noncoherent orthogonal modulation, IEEE J. Select. Area Commun., vol., pp , Sept [9] A. Chindapol and J. A. Ritcey, Deign, analyi, and performance evaluation of BICM-ID with quare QAM contellation in Rayleigh fading channel, IEEE J. Select. Area Commun., vol. 19, pp , May 001. [10] S. Benedetto and E. Biglieri, Principle of Digital Tranmiion with Wirele Application. Kluwer, [11] C. Fleurent and J. Ferland, Genetic hybrid for the quadradic aignment problem, DIMACS Serie in Dicrete Mathematic and Theoretical Computer Science, pp , [1] Z. Drezner, A new genetic algorithm for the quadratic aignment problem, INFORMS J. Computing, vol. 15, pp. 0 0, 00. [1] Y. Wu and P. Ji, Solving the quadratic aignment problem by a genetic algorithm with a new replacement trategy, Int. J. Humanitie and Social Science, vol. 1, no., pp , 007. [14] D. T. Conolly, An improved annealing mechanim for the QAP, European J. of Operational Reearch, vol. 46, pp , 1990.