Efficient Power Allocation for LDPC-Coded MIMO Systems

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1 Effcent Power Aocaton for LDPC-Coded MIMO Systems Laya A Sd. shan Rao & M. Sushanth Ba Vaagdev Coege of Engneerng Waranga Jawahara ehru Technoogca Unversty yderaad Inda E-ma: ayaq0786@gma.com prof_r@redffma.com sushanth_6@yahoo.com Astract - In ths papers an effcent orderng scheme for an ordered successve nterference canceaton detector s determned under the t error rate mnmzaton crteron for mutpe-nput mutpe-output(mimo) communcaton systems usng transmsson power contro. From the convexty of the Q-functon we evauate the choce of sutae quantzaton characterstcs for oth the decoder messages and the receved sampes n Low Densty Party Chec (LDPC)-coded systems usng M-QAM schemes. We derve the orderng strategy that maes the channe gans converge to ther geometrc mean. Based on ths approach the fxed orderng agorthm s frst desgned for whch the geometrc mean s used for a constant threshod usng correaton among orderng resuts. ey words - Mutpe-Input Mutpe-Output (MIMO) QR-decomposton Detecton Orderng Low Densty Party Chec (LDPC) Bt Error Rate(BER). I. ITRODUCTIO The mutpe-nput mutpe-output (MIMO) system has een an actve area of research as we as practca transcever mpementatons for ther great potenta of enhancng the system s performance [][]. The V- BLAST archtecture proposed n [3] and [4] aso referred to as the BLAST-ordered successve nterference canceaton (B-OSIC) detector s regarded as an attractve souton that expots ths potenta. In a B-OSIC recever the data stream wth the strongest sgna-to-nterference-nose rato (SIR) s seected frst and sutracted from the receved sgna and the procedure s successvey performed for the remanng mutpe data streams. For equa power aocaton (PA)[5][6] across the transmt antenna array t s optma n terms of t error rate (BER). Low Densty Party Chec (LDPC) codes are stateof-art error correctng codes [7] and [8] ncuded n severa standards for roadcast transmssons. Iteratve soft-decson decodng agorthms for LDPC codes reach exceent error correcton capaty. Great attenton has een pad n recent terature to the topc of quantzaton for LDPC decoders ut mosty focusng on nary moduatons and anayzng fnte precson of the recever. The current scenaro of error correctng codes s domnated y schemes usng Soft-Input Soft-Output (SISO) decodng. Among them an mportant roe s payed y Low- Densty Party-Chec (LDPC) codes whch permt to approach the theoretca Shannon mt whe ensurng reduced compexty. For the name reason these codes have een ncuded n some recent teecommuncaton standards. We provde the BER mnmzaton condton derved from the convexty of the Q -functon n the PA scheme. It s demonstrated that the orderng strategy whch maes the channe gans converge to ther geometrc average acheves the mproved error performance. By usng ths oservaton we deveop the two orderng agorthms whch are dentca except for the threshod adaptaton. The asc agorthm determnes the detecton-order usng the geometrc mean as a constant threshod whereas the modfed orderng scheme for roust convergence adaptvey updates the threshod y tang nto account the prevous orderng resuts. The comparson of the cumuatve dstruton s conducted to confrm the superorty of the adaptve desgn. There s no cosed form souton for the BER or FER of coded systems except for some trva cases that ncude un-coded systems and orthogona codng [90]. Upper ounds are the conventona method for performance anayss of coded systems. The upper ounds of coded systems are otaned usng the unon ound technque. The remander of the paper s organzed as foows: In secton II we propose system mode of MIMO. In secton III proposed orderng scheme for power aocaton and decoders are mpemented. In secton IV SIC Recever ased on QR decomposton s presented. In secton V numerca anayss and Smuaton resuts are refy dscussed. Fnay we concudes n Secton VI. II. SYSTEM MODEL Fg. depcts a MIMO system wth transmt antennas and receve antennas. The fat-fadng MIMO Internatona Journa of Power System Operaton and Energy Management ISS (PRIT): Voume- Issue- 0 0

Effcent Power Aocaton for LDPC-Coded MIMO Systems channe s expressed y the matrx wth the eement h j representng the channe gan from th transmt antenna to j th receve antenna.

n t ] T s the r dmensona nose vector wth eements foowng compex zero mean Gaussan dstruton wth varance of σ n.e s s the tota transmtted sgna energy on t transmt antenna.

r ˆ = 0 t T () () Es ρ = p R - =.. t. (3) σ n The anayss we have deveoped s qute genera and can e apped to any vaue of M.

For any equa to an even power of a Gray aeng can e adopted to match every sequence of encoded ts to each symo. The LDPC encoder maps each M t word producedd y the source nto an M-t LDPC codeword.

At the recever sde the demapper oc s a maxmum a posteror (MAP) symo-to-t metrc cacuator that s ae to produce an nta ehood vaue for each receved t (such vaues are denoted as ntrnsc or channe

Gray aeng for 6-QAM are used as nput for the susequent teraton whe the atter represent the decoder output and serve to otan an estmated codeword that s suject to the hard decson and the party-chec

The upper tranguar matrx R s dfferenty defned y the detecton-order; The QR-decomposton ased OSIC detecton for BER-mnmzed PA transmsson can e performed usng proposed archtecture.

The ndependenty encoded symos are processed through a dagona PA matrx and then transmtted from t data streams.

2 Effcent Power Aocaton for LDPC-Coded MIMO Systems channe s expressed y the matrx wth the eement h j representng the channe gan from th transmt antenna to j th receve antenna. The receved sgna vector r = [r.r r r] T s wrtten ass E r= s px+n t Where x=[x x..x t ] T denotes the x transmtted sgna vector and n=[n n.n t ] T s the r dmensona nose vector wth eements foowng compex zero mean Gaussan dstruton wth varance of σ n.e s s the tota transmtted sgna energy on t transmt antenna. To express the sgna mode for the MMSE-QR detector an ( r ++ t ) t augmented channe matrx Ĥ an ( r + t ) extended receve vector ˆr and an t zero matrx O t can c e wrtten as orderng r = σ n I QR and r ˆ = 0 t T () () Es ρ = p R - =.. t. (3) σ n The anayss we have deveoped s qute genera and can e apped to any vaue of M. owever for etter evdence n the foowng we w many refer to the specfc case of a 6-QAM consteaton. For any equa to an even power of a Gray aeng can e adopted to match every sequence of encoded ts to each symo. The LDPC encoder maps each M t word producedd y the source nto an M-t LDPC codeword. Each codeword s then passed to the mapper and moduator oc for M-QAM consteaton. The moduated sgna s then transmtted over an Addtve Whte Gaussan ose (AWG) channe. At the recever sde the demapper oc s a maxmum a posteror (MAP) symo-to-t metrc cacuator that s ae to produce an nta ehood vaue for each receved t (such vaues are denoted as ntrnsc or channe messages). These messagess serve as nput for the Sum-Product Agorthm (SPA) that starts teratng and at each teraton produces updated versons of the extrnsc and the a posteror messages. Gray aeng for 6-QAM are used as nput for the susequent teraton whe the atter represent the decoder output and serve to otan an estmated codeword that s suject to the hard decson and the party-chec test. III. PROPOSED ORDERIG SCEME FOR POWER ALLOCATIO AD DECODERS A. Power aocaton schemes n OFDM Fg. : MIMO transmssonn mode wth QR-OSIC detector. The upper tranguar matrx R s dfferenty defned y the detecton-order; The QR-decomposton ased OSIC detecton for BER-mnmzed PA transmsson can e performed usng proposed archtecture. Based on the feedac nformaton of the dagona eements transmsson power P s assgned to each data stream. The ndependenty encoded symos are processed through a dagona PA matrx and then transmtted from t data streams. The QR-OSIC sequentay n accordance wth the recever detects the transmt symos desgnated detecton-order. A. Q-functon n PA Power aocated to the nus of the frequency responsee s ey to e wasted. The performancee of a coded OFDM system can e represented y upper ounds. For smpe anayss we assume that the channe code s near and the moduaton s BPS or QPS. ere we w present an effcent power aocaton agorthm for a codedd OFDM system to mnmze the target BER under a constant tota power constrant. Usng the unon ound technque the upper ounds of BER gve y pber Z. Where s the tota numer of nformaton ts n a frame and s the ammng dstance of the c o c. The par wse error proaty Z s a Q functon (4) Internatona Journa of Power System Operaton and Energy Management ISS (PRIT): Voume- Issue- 0

3 Effcent Power Aocaton for LDPC-Coded MIMO Systems Where Z =Q ρ j j Q(x)= Where ρj ss the channe gan of the j th symo P j s the aocated power at the j th symo d j s the Eucdean dstance etween codeword c0 and c0 c at the j th symo ocaton wth unt symo power and 0 s the nose spectra densty of the AWG. The ojectve of power aocaton s to mnmze the target BER or FER under power constrants gven y p 0 for = = 8 Where P ss the transmsson power of the -th sucarrer P T s the tota transmsson power and s s the numer off moduaton symos n a frame. The sum of exponenta functons and that of Q-functons are convex functons. A detaedd proof of ths statement w e gven nn the next secton. Therefore power optmzaton to mnmze the FER or BER ound under a constant power constrant s a convex optmzaton proem. P Q BER exp.d.p j j /0 p p T -(t /) x e dt Π p. d. p / j j j o j p. d j j. pj/ o j Sums of exponenta functons and Q-functons are og-convex functons. It s dffcut to compute the summatons of the BER ounds for a posse code words especay when there are many code words n the codeoo. As the numer of codewords ncreases the compexty of the optmzaton ncreases. In ths case t s posse to approxmate the BER ounds y usng the t terms whch are nearest to the transmtted data sequencee c 0.The aove BER ounds are otaned assumng nearty of the codng and moduaton. The genera expresson for non-near codng needs the average for the transmtted code words whch ncreases the compexty of the BER computaton. A.. LDPC Consder a transmtterr wth t antennas and a recever wth r antennas n Fgure. The data ts from L ayers are ndvduay encoded y LDPC encoders. The coded ts are t-wse ntereaved pror (5) (6) (7) to Gray mappng to M -QAM symos wth unty average power at the th ayer. The resutng symos are grouped y the symos and fed nto an OSTBC encoder producng an M T code matrx. An OSTBC matrx wth the symos x x...x s constructed as R Where x and xi are the rea part and magnary part of x respectvey. U and V are dsperson matrces of sze M T. whch satsfy the foowng condtons: V V = R I x U + jx V = U U = I V n n M nn V = I M n U U = U U n m m n n m = V V nm mn n m U V = V U nm m n nm Fg. : Group ayered mut-antenna archtecture usng LDPCcoded transmsson and detecton ased on SIC. Where I M denotes the dentty matrx of sze M M. The OSTBC matrces from the L ayers are transmtted n parae over t antennas. That s the transmsson matrx s gven y t t t t = [... L ] (9) and t = ML where A t denotes the transpose of a matrx A. The r T receved matrx Y s gven y Y = + (0) Where s an r t channe matrx whose entres are ndependent and dentcay dstruted (..d.) compex (8) Internatona Journa of Power System Operaton and Energy Management ISS (PRIT): Voume- Issue- 0

4 Effcent Power Aocaton for LDPC-Coded MIMO Systems Gaussan random varaes wth zero mean and unt varance and s the nose matrx whose entres are statstcay ndependent zero-mean compex Gaussan random varaes wth varance σ. We assume that s constant over a oc of T symos and perfecty nown to the recever. IV. SIC RECEIVER BASED O QR DECOMPOSITIO The recever for our system performs the QR decomposton of the channe matrx.e. = QR where Q s a untary matrx wth sze r t and R s an upper tranguar matrx wth sze t t. By mutpyng Q to the receved matrx Y we get Y% = Q Y = R + % () Where Q s the ermtan of Q and % = Q has..d. zero-mean compex Gaussan entres wth varance σ. The matrx R can e wrtten as R R... R L 0 R... R L R = RLL Where R s an M M su matrx of R gven y r ( ) ( )... r M + M + M( ) + M.. R =.. r M M ( ) +... r M M () and r uv s the (u v) entry of R and s zero whenever u > v. The statstca propertes of r j < j t may e drecty deduced y appyng the foowng theorem to the case that m = r and n = t. Theorem : Let e an m n compex Gaussan matrx wth m n whose entres are..d. compex Gaussan random varaes wth zero-mean and unt varance. Denote ts QR decomposton y = QR. The matrx R s upper tranguar wth rea vaued dagona. The entres of R are ndependent of each other. Moreover the square of the th dagona eement of R r.j s of CI- SQUARE dstruton wth the degree of freedom (m +). The off-dagona eements r j for < j n are zero-mean compex Gaussan wth unt varance. The th M T su oc n Ỹ for decodng of the th ayer may e wrtten as L Y = R + R + =+ (3) Where Ñ s the th M T su oc of Ñ. The decodng process s successvey done from the th ayer to the frst ayer y usng the nformaton from the prevousy decoded ayers. That s the decson matrx Z for decodng at the th ayer s gven y L Z ˆ = Y% R (4) =+ Where ˆ s the OSTBC matrx recovered y remappng the codeword decoded n the th ayer to M- QAM symos and re-encodng the symos to the OSTBCs. We assume that there are no errors n decodng of the prevous ayers so that error propagaton does not occur. Then Z n can e wrtten as Z = R + % (5) By denotng the m th rows of Z R and Ñ y Z m r m and n m respectvey and appyng we get the foowng vector equaton z% = % R% + n% (6) where R I R I R I R I z% = [( z z...( z z )] x% M M = [( x x... ( x x )] R I R I n% = [( n n...( n M n M )] and the superscrpts R and I denote the rea part and magnary part respectvey. The equvaent rea channe matrx R. r% U... r% MU r% V... r% M V R % = ru %... r% MU r% V... r% M V (7) R I U U I R V where V U V = I R = R I and U U V V r rr ri m = m m From the propertes of the dsperson matrces n t may e easy checed that Ř has Internatona Journa of Power System Operaton and Energy Management ISS (PRIT): Voume- Issue- 0 3

5 Effcent Power Aocaton for LDPC-Coded MIMO Systems mutuay orthogona rows.e. = R R R I where. denotes the Froenus norm of a matrx. Thus we get the foowng decson statstc: zr zˆ ˆ = = x n + (8) R Where the components of ň are..d. zero-mean rea Gaussan random varaes wth varance σ / R. ote that Ž n (0) s equvaent to the receved vector for QAM symos transmtted over parae Gaussan channes havng R as a channe gan. In addton t s easy shown y Theorem that R s of ch-square dstruton wth the degree of freedom M( r M( )). The th eement ž n ž has the foowng condtona proaty densty functon (pdf): p ρ ( zˆ ˆ x p) exp{ ρ( z x ) } = Π (9) Fg. 3 : Graph ofφ (Ř ) A The BER Performance As n the dervaton of the post-detecton SIR the error rate s aso affected y the channe gans and the transmsson power. A Power Aocaton scheme for the average BER mnmzaton under the assumpton of the QR-decomposton of the channe matrx and no error propagaton n successve canceaton of the data streams has een proposed. The PA scheme for M- QAM moduaton can e expressed as the average BER of the PA can e approxmated wth a consteatonspecfc constant the average BER as we as the postdetecton SIR s determned y the aocated power and the channe gan. Because of the convexty property of the Q-functon the resutng BER s mnmzed y the detecton orderng of the QR-OSIC recever such that a dagona eements of the matrx are equa to ther geometrca average and aternatvey the PA scheme at the transmtter whch maes the product of two varaes and dentca for a data streams. As the rea MIMO channe s characterzed y severa spatatempora propertes the condton () Is not practca n spte of ts optmaty. () Dfferent detecton-order eads to dfferent transmsson power and hence power gan shoud e aso dfferenty assgned. Ths ndcates that an approprate detecton orderng strategy ncorporates wth the PA scheme can acheve the mproved BER performance. Fg. 4 : Comparson of cumuatve destruton Ř -µ V. UMERICAL RESULTS For each of the MIMO systems and for a specfc vaue of SR a quas-statc channe s assumed for the performance evauaton for whch the channe gan s constant over a frame and changed ndependenty from frame to frame. To concentrate our pont on comparng orderng agorthms we postuate the perfect channe estmaton at the recever and error-free PA nformaton at the transmtter. Fg. 4 shows the average BER performance comparson for MIMO systems wth three transmt/receve antennas and the smuaton resuts of four transmt/receve antennas are depcted n Fg. 5. ere the dashed ne ndcates a system wth the BERmnmzed PA scheme whereas the sod ne represents a system wthout the PA. Impementaton of QR recever wth PA and wthout orderng n denoted as QR-SIC. The QR recever wth the PA ut no orderng denoted as QR-SIC w/ PA has smar performance to the open-oop OSIC systems wthout the PA. Internatona Journa of Power System Operaton and Energy Management ISS (PRIT): Voume- Issue- 0 4

Effcent Power Aocaton for LDPC-Coded MIMO Systems Ths demonstrates the mportance of the detecton-order for successve detecton. As expected wthout the PA the B-OSIC outperforms the QR-OSIC recever.

6 Effcent Power Aocaton for LDPC-Coded MIMO Systems Ths demonstrates the mportance of the detecton-order for successve detecton. As expected wthout the PA the B-OSIC outperforms the QR-OSIC recever. Despte the reduced compexty however power controed MIMO systems empoyng the proposed orderng strategy acheve the mproved error performance compared to those wth the B-OSIC agorthm. It s suffcent to confrm the superorty of the proposed desgn ecause the orderng agorthms of prevous studes compy wth the strategy of the B-OSIC. A further performance mprovement n the hgh SR regon can e expaned n terms of the error propagaton snce the PA scheme as we as the proposed QR-OSIC recever s desgned under the assumpton of the error-free decson n prevous detecton stages. SIMULATIO RESULTS Fg. 7: Comparson of OSIC and LDPC codes VI. COCLUSIOS In ths study we nvestgated the QR-OSIC recever desgn for the transmtter-sde power aocated MIMO system. Based on the propertes of the Q-functon and orderng resuts we deveop the effcent power aocaton for LDPC n dgta system. We evauate the choce of sutae quantzaton characterstcs for oth the decoder messages and the receved sampes n Low Densty Party Chec (LDPC)-coded systems usng M- QAM schemes. Because of the post-detecton SIR ncrement the coded systems wth the derved approach can aso e expected to acheve the performance mprovement. Fg. 5: BER performance of MIMO system wth three transmt/receve antennas Fg. 6 : BER performance of MIMO system wth four transmt/receve antennas REFERECES [] G. J. Foschn and M. J. Gans On mts of wreess communcatons n a fadng envronment when usng mutpe antennas Wreess Pers. Commun. vo. 6 no. 3 pp [] A. Pauraj R. aar and D. Gore An Introducton to Space-Tme Wreess Communcatons. Camrdge U..: Camrdge Unversty Press 003. [3] P. W.Wonansy G. J. Foschn G. D. Goden and R. A. Vaenzuea V-BLAST: An archtecture for reazng very hgh data rates over the rchscatterng wreess channe n Proc. ISSSE 98 Psa Itay Oct. 998 pp [4] G. J. Foschn G. D. Goden R. A. Vaenzuea and P. W.Wonansy Smpfed processng for hgh spectra effcency wreess communcaton empoyng mut-eement arrays IEEE J. Se. Areas Commun. vo. 7 no. pp ov Internatona Journa of Power System Operaton and Energy Management ISS (PRIT): Voume- Issue- 0 5

7 Effcent Power Aocaton for LDPC-Coded MIMO Systems [5] Z. Yan. M. Wong and Z. Q. Luo Optma dagona precoder for mut antenna communcaton systems IEEE Trans. Sgna Process. vo. 53 no. 6 pp Jun [6]. Wang and S. D. Bosten Approxmate mnmum BER power aocaton for MIMO spata mutpexng systems IEEE Trans. Commun. vo. 55 no. pp Jan [7] R.G.Gaager Low-densty party chec codes IRE Transacton Info. Theory vo. IT-8 pp. 8 Jan. 96. [8] D.Oh and. parh performance of quantzed mn-sum decodng agorthms for rreguar LDPC codes n proc IEEE ISCAS 007ew Oreeans LA May 007pp [9] D. Garrett L. Davs S. ten Brn B. ochwad and G. nagge. Scon compexty for maxmum ehood MIMO detecton usng spherca decodng. IEEE Journa of Sod-State Crcuts 39: Sept [0] B. M. ochwad and S. ten Brn. Achevng nearcapacty on a mutpe-antenna channe. IEEE Transactons on Communcatons 5: March 003. Internatona Journa of Power System Operaton and Energy Management ISS (PRIT): Voume- Issue- 0 6

Performance Analysis of MIMO SFBC CI-COFDM System against the Nonlinear Distortion and Narrowband Interference

Performance Analysis of MIMO SFBC CI-COFDM System against the Nonlinear Distortion and Narrowband Interference Performance Anayss of MIMO SFBC CI-COFDM System aganst the onnear Dstorton and arrowband Interference YSuravardhana eddy Department of ECE JTUACEAnantapur AP E-ma: suravardhana@gmacom K ama adu Department