006 IEEE ith Iteratioal Symposium o Spread Spetrum Tehiques ad Appliatios Solvig the ZF Reeiver Equatio for MIMO Systems Uder Variable hael oditios Usig the Blok Fourier Algorithm João arlos Silva, Rui Diis, Atóio Rodrigues Istituto Superior Téio/Istituto de Teleomuiações/ APS-IST Torre orte 11-11, Av. Roviso Pais 1, 1049-001 Lisboa, Portugal, joao.arlos.silva@lx.it.pt, rdiis@ist.utl.pt, atoio.rodrigues@lx.it.pt Abstrat The ZF (Zero Forig) algorithm is oe of the best liear reeivers for DS-DMA (Diret Sequee-ode Divisio Multiple Aess). owever, for the ase of MIMO/BLAST (Multiple Iput, Multiple Output / Bell Laboratories Layered Spae Time) with high loadig, the pereived omplexity of the ZF reeiver is take as too big, ad thus other types of reeivers are employed, yieldig worse results. I this paper, we ivestigate the omplexity of the solutio to the MIMO ZF reeiver s equatios usig Blok-Fourier algorithms, for both steady ad usteady hael situatios. Keywords- Zero Forig, MIMO,Blok Fourier, usteady haels. I. ITRODUTIO The ZF algorithm (explaied i detail i [1]), or algorithms based o them, are equalizers that are essetial for ompesatig the various error soures that are preset i the wireless ommuiatio lik, suh as ISI (Iter-Symboli Iterferee) ad MAI (Multiple Aess Iterferee), whih beome more sigifiat as the loadig of the system is ireased, with speial iidee o MIMO systems. The Blok-Fourier algorithms, preseted i [],[3] for the ZF algorithm, are oly suitable for ostat hael oditios. I this work ew versios of these algorithms are derived, apable of dealig with detetio i usteady haels with speeds up to 100km/h. These ew algorithms are based i the partitioed blok-fourier algorithms of earlier works [],[3], but extra steps were added to take i osideratio the hael hage from partitio to partitio. Iside eah partitio the hael is osidered ostat (thus providig approximate, yet reasoable results). The ew algorithms, although more omputatioally expesive tha the oes preseted i earlier works, are ot as expesive as the Gauss or holesky algorithms (that provide exat results). The paper is orgaized as follows. Setio II desribes the Blok Fourier (BF) algorithm. Setio III itrodues the Zero Forig (ZF) algorithm, alogside the otio of matrix partitioig for solvig the system equatio. Simulatio results Fraiso eras, uo Souto, Sérgio Jesus ISTE / Istituto de Teleomuiações/ADETTI Torre orte 11-08, Av. Roviso Pais 1, 1049-001 Lisboa, Portugal fraiso.eras@lx.it.pt, uo.souto@lx.it.pt, sj3@lix.pt are disussed i setio IV, ad the olusios are give o Setio V. II. BLOK FOURIER ALGORITM The BF algorithm a be easily applied for irulat matries. Sie i our ase, the matries we deal with are blokirulat, we will exploit this struture. A. Diagoalizig irulat Matries A irulat matrix is a square Toeplitz matrix with eah olum beig a rotated versio of the olum to the left of it: 1 1 3 = 3 1 4 3 1 The iterestig property of irulat matries is that its eigevetors matrix is equal to the orthoormal DFT matrix F of orrespodig dimesio. F a be writte as: F = * F () where F is the o-orthoormal DFT matrix: 1 1 1 1 (3) 1 ω ω ω 4 ( ) F = 1 ω ω ω ( ) ( )( ) 1 ω ω ω j π / with ω = e ; j =. Usig this property a irulat matrix, a be deomposed i: = F = (4) where Λ is a diagoal matrix that otais the eigevalues of.the Λ matrix a be easily omputed from: = diag( )&(:,1)) (5) where diag( x ) represets the diagoal matrix whose diagoal elemets are take from the x vetor. Substitutig by (4) i the liear system: x = b (6) ad solvig for x results i: (1) 0-7803-9780-0/06/$0.00 006 IEEE 87
x = F )E (7) Equatio (7) a be omputed effiietly with three disrete Fourier trasforms ad the iversio of the diagoal matrix. The omplete solutio would require 3 omplex multipliatio/additio pairs for the three DFT matrix/vetor multipliatios; omplex divisios to ivert ; ad omplex multipliatios to multiply by Fb. Usig the ooley & Tukey Fast Fourier Trasform algorithm the omplex multipliatio/additio operatio pairs eeded to ompute a size DFT [] is: log (8) FFT = This meas that equatio (7) a be omputed spedig oly 3 log + omplex floatig poit operatios (osiderig eah multipliatio/additio pair as oe operatio). The memory requiremets to solve the system usig suh algorithm are also very modest. It s oly eessary to keep i memory two size vetors: the b vetor ad the first olum of. All operatios a be made i-plae as the solutio vetor x replaes b. Further eoomy a be ahieved if is sparse (bad or blok diagoal for example). B. Appliatio to Blok-irulat Matries A blok-irulat matrix a be visualized as a irulat matrix where eah elemet is a matrix istead of a salar value. osider a blok-irulat matrix omposed by P bloks of size P Q. If ^ is blokirulat it must satisfy: ( PQ ) i, j = (9) ( PQ ) i, j. This meas that eah with i = (( i + P ) m od P ) + 1 j = (( j + Q ) m od Q ) + 1 elemet of is repeated P rows below ad Q olums to the right of it. Idies that exeed the P lies or olums wrap aroud to the first lies ad first olums, respetively. From ow o we will osider oly square blok matries with bloks, so the idex will be omitted for simpliity. The blok dimesios of the matrix will be represeted i subsript betwee urve brakets, ad i the ase of square bloks matries, oly a dimesio represeted. To deal with blok irulat matries we eed to itrodue the blok-fourier trasformatio. The blok-fourier matrix is defied as: F = F I (10) ( K) K where F is a o-orthoormal DFT matrix of dimesio as defied i (); I is the K size idetity matrix; ad K deotes the Kroeker produt. Similar to the last sub-setio, a blok-irulat matrix a also be deomposed usig blok-fourier trasforms: = F ( P) ) (11) ( Q) where Λ is a blok-diagoal matrix omputed from: ( PQ ) = diag( PQ )( ( Q ) ( PQ )(:,1 : Q)) (1) where diag ( ) x represets the blok-diagoal matrix whose blok-elemets are the P Q sized bloks of x. Similarly to the irulat systems, a blok-irulat system a also be effiietly solved usig the blok-fourier deompositio. The blok-irulat system: ( P ) b (13) P P P P with ( P) ^ ; x ^ ; b ^. It a be solved with: F ( P) ( P) )( P) E (14) If the bloks are ot square the Moore-Perose pseudoiverse oept a be used. osider a blok-irulat matrix P ^, with P Q sized bloks. ( PQ ) b, (15) P ^ ; x ^ ; P b ^ ; Q P. The system (15) a be solved usig the Moore-Perose pseudo-iverse of the omplex matrix : x = b (16) if is ivertible. This solutio is the least squares solutio (ZF) to the system (15), as previously show. Applyig the blok-fourier deompositio of (11) to (16) results: ( F ) ( F( P ) ( PQ ) ( Q )) ( P ) ( PQ ) ( Q ) ( P ) ( PQ ) ( Q ) x = (PQ) with defied as i (1). Equatio (17) a be simplified, osiderig that ( Q) (PQ) (PQ) (PQ) ( P) ( K ) ( K ) (17) F = F : x = F ) E (18) This solutio a be omputed with oly three blok-fourier trasforms, the iversio of ad the multipliatio (PQ) (PQ) of two blok diagoal matries by a olum vetor. The multipliatio ( P ) (PQ) (PQ) (PQ) ( ) ) E (19) must be performed from right to left to miimize the umber of operatios required, sie a matrix-vetor multipliatio is faster tha a matrix-matrix oe. Regardig that: (PQ ) (PQ ) P P (PQ ), ( P ) (0) the multipliatio (19) requires ( P + ) pairs of omplex multipliatios/additios. From the defiitio of blok-fourier trasform i (10) it a be show that give y = F( K ) x (1) K K K with xy, ^ ; F( K ) ^,wehave: y( i: K:( ) K + i) = F x ( i: K:( ) K+ i) () with 1 i K, where F represets a o-orthoormal DFT matrix of dimesio asdefiedi(3). This simply meas that eah blok-fourier trasform of bloksize K a be deomposed i K Fourier trasforms of size. Furthermore, eah Fourier trasform a be exeuted idepedetly of eah other ad thus advatage of parallel hardware implemetatios a be take. Takig this ito osideratio, reallig equatio (8) ad osiderig that are eeded two blok-fourier trasforms of blok-size Q ad oe of blok-size P to ompute (18), the umber of operatios required for that three Fourier trasforms is: log 3 blok FFT = ( Q + P) (3) 88
Due to fiite preisio roud errors, the R matries may ot k be ermitia positive defiite eve if is ermitia positive defiite. This a be orreted simply by removig the imagiary part of the diagoal elemets ad zeroig all other elemets that have omplex modulus below some threshold value, before applyig the holesky fatorizatio. This ew simplified versios require the same umber of floatig poit operatios as derived before beause ull elemets operatios were ot osidered from the begiig. III. ZF DETETOR The solutio of the Zero-Forig detetor is ˆ d = T T T e (4) This is the shortest legth least squares solutio of: e = T d (5) where T is ot square, i geeral. Figure 1 (left) represets the struture of T. Figure 1 Blok Struture of the T matrix (left), Exteded T matrix, ostat hael (right) T is a blok matrix with V bloks disposed alog its diagoal. All bloks are equal i a ostat hael oditio. Eve if the hael varies slowly it a be a reasoably approximatio to osider all V blok equal as we will ivestigate later. I a ostat hael oditio, it s easy to exted matrix T to beome blok-irulat, simply by addig extra blok-olums to it, as show i Figure 1 (right). The elemets below the last V blok wrap aroud to the top of the a olums. The umber of extra olums eeded to make T blok-irulat is: E = = / P (6) The resultig matrix is blok-irulat with bloks of P Q size. This ew matrix will be represeted as T.ow equatio (4) a be trasformed i: ˆ d = T T T e, (7) T( PQ ) d ^ ; P ^ ; ˆ P e ^ ad solved with the Fourier method, as doe for equatio (15) i last setio. The e vetor a be obtaied from e by paddig at its ed with ( + 1) P zeros, ad ˆd a be extrated from the first Q elemets of ˆd. There are two approximatios i the trasformatio of T i T :all V bloks were made equal ad extra olums/lies were added to the matrix. If all bloks were made equal to the first blok, the approximatio would be ireasigly worse (diretly orrelated to the speed) towards the last blok. A better approximatio would be expeted if a middle blok was used. Let us use the middle blok of T if is eve or the left-middle blok if is odd: V = V ( / ) (8) Usig this method, better approximatios i a wider etral rage a be attaied, as show i Figure. Relative Error 10 1 pp 10 0 10-1 10-10 -3 10-4 10-5 100 Km/h 10 Km/h 1Km/h 10-6 0 00 400 600 800 1000 100 1400 1600 1800 000 Symbol Idex Figure ostat hael approximatio relative error (PedestriaA, miimum load, 4x4 ateas) Middle blok. evertheless, the approximatio is very poor for the firsts ad lasts elemets of the ˆd vetor. The last pitures make obvious that the methods preseted i [],[3], although valid for a ostat hael oditio, are ot very useful if the hael hages, eve if low speeds are osidered. We will first oetrate o ostat hael oditios. First we will determie the error level itrodued by the additio of extra olums/lies to the T matrix that trasform it i the blok-irulat matrix T, as explaied before ad the error of the blok-fourier algorithm. A. ostat hael oditios Table 1 show the error level itrodued i the determiatio of vetor ˆd by eah phase of the detetio proess i 1 ostat hael situatios. Eah value is the maximum omplex modulus of the relative differee betwee the estimated ˆd ad real d obtaied i 100 rus with distit radom d vetors. The Estimatio olum idiates the maximum error of the estimatio if equatio (4) is solved diretly. The irulat olum shows the maximum error itrodued by the extesio of the T matrix to beome T relative to the estimated solutio. The Fourier1 ad Fourier olums show the maximum error itrodued by the use of the Fourier algorithms 1 ad ompared with the irulat system solutio. Sie a SF=16 is osidered, the miimum load situatio orrespods to 0 iterferers while the maximum load situatio orrespod to 15 iterferes (with eah user havig oly 1 physial hael). As a be see the errors are very low for all the tested matries ad are oly slightly above the floatig poit preisio used. Furthermore the T matrix extesio is ot the mai error ause. It a be see that the error level is ostat alog the etire symbol vetor ad o begiig either ed high error levels appear, sie o multipath iterferee from adjaet bloks is beig osidered. Exludig the midamble from the detetio ad splittig the proess i two idepedet detetios, eah oe ivolvig oly data symbols, some errors a be itrodued i the begiig of the seod data huk. This a 89
be orreted by iludig some symbols of the mid-amble i the seod detetio proess. Table 1 Maximum absolute error itrodued by eah phase of the Zero-Forig detetor Table umber of operatios required by the Blok-Fourier algorithm B. Partitioig The algorithms proposed i last sub-setio reveal already may parallel paths that ould be exploited for parallel proessig i adequate hardware. evertheless the algorithm remais globally sequetial sie it oly determies the estimated ˆd vetor at the ed. Figure 3 illustrates a approah to split the exteded Zero- Forig equatio (7) i smaller systems. Figure 3 represets SQ = T T ad the estimated vetor ˆd. The idea is to split the S matrix i smaller oes. This a be a reasoable Q approximatio beause the S has the greater values Q oetrated aroud the diagoal, ad dereasig modulus as we get far away from the diagoal. This meas that eah elemet of vetor ˆd depeds maily of the same idex value of vetor T ( PQ e ad it depeds less ad less of the values of it ) as we get farther from that same idex value. Sie eah partitio just approximates well the middle values, the ˆd values of the begiig ad ed of eah partitio must be disarded. I the simulatios it will be disarded the first l ad last l + elemets of eah ˆd partitio. ote that sie the blok-fourier algorithm will be applied at eah partitio, ad beause eah partitio has to be approximated as a blok-irulat matrix, high error will appear also i the first elemets of the first partitio ad i the last elemets of the last partitio. This would ot happe if eah partitio would be solved with a exat method like Gauss or holesky. This is the reaso why those elemets are also disarded i Figure 3. Figure 3 Partitioig blok-fourier algorithm The overlappig legth must be arefully seleted aordigly the preisio required. The bigger the overlappig the better the approximatio but more expesive will be the omputatio. I the derivatio of the algorithms a prelap l ad postlap l + were defied i a similar way as doe i [3] + but we shall always use l = l, sie there is o advatage of defiig dissimilar overlappig legths. The partitio legth a be seleted aordig the umber of iteded partitios ad the overlappig seleted, ad this is usually determied by the hardware struture.. Usteady hael oditios The blok-fourier algorithm preseted so far works well just uder ostat hael oditios or with very slow hagig speeds. Eve at 1 Km/h sigifiat errors arise. The stadard blok-fourier algorithm aot be adapted for usteadyhael oditios beause it just works for blok-irulat matries, but the partitioed versio a be easily adapted just by usig i eah partitio a differet blok of the origial T matrix. If we use the middle blok i eah partitio of the origial T matrix to ostrut eah exteded approximate T, the blok-fourier algorithm a be used for eah partitio as doe i the last sub-setio, ad ˆd obtaied from the middle elemets of eah partitio omputatio. The overlappig legth must be seleted aordigly the preisio required as for the ostat hael. The bigger the overlappig the better the approximatio but more expesive will be the omputatio. The partitio legth ow must be also seleted aordig the preisio required: bigger partitios will approximate quikly hagig haels worst, but smaller partitios will require too muh omputatioal power. Partitio ad overlappig legth a also be determied by the hardware struture available, if some kid of parallel proessig is available i a already developed hardware platform. Also, very small partitios or very log overlappig a beome iompatible makig the required error level just ot attaiable for high speed haels. IV. SIMULATIO RESULTS The Mote arlo was employed; 100 radom data vetors were reated ad used with eah of the T matries to reate the orrespodet e vetors from e = T d. The all the 90
algorithms were applied i tur to estimate d from eah e vetor. Fially the estimated ad origial d vetors were ompared ad the wrog bits out up. This proedure was repeated for differet veloities (0,1,10 ad 100 km/h) ad ateas ofiguratio (SISO ad MIMO x). Table 3 preset the BER results for 1 ad ateas. The Est algorithm represets the exat solutio of the ZF equatio, i.e. its solutio with a algorithm that does ot ilude extra approximatios further from the umeri floatig poit preisio of the simulator. As expeted the Est algorithm always estimated the orret data vetor, sie o oise was osidered i the simulatio. The Fourier algorithm orrespods to the upartitioed Fourier algorithm desribed earlier. The Fourier_m uses the middle blok of the T matrix while Fourier uses the first blok. As expeted, the use of the middle blok gives the best results, exept for low speeds, were the both are equivalet. FourierP_ refers to the partitioed Fourier algorithm for ostat haels while FourierP_v is the orrespodet usteady hael versio. As expeted FourierP_v gives always better results, exept for very low speeds, were FourierP_ a give equivalet results with less floatig poit operatios. The umbers after FourierP_ or FourierP_v idiate the umber of bloks used i eah partitio ad the pre-lap ad post-lap bloks umber. For example FourierP_v_008_00_00 meas a 8 bloks partitio with bloks pre-lap ad bloks post-lap algorithm. From Table 3, it a be oluded that is advatageous to redue the size of partitios, espeially for high speeds (as expeted, sie i eah partitio the hael is approximated as ostat). It s also easy to see that, for a partiular partitio size, better results are attaied as larger laps are used. For low speeds the size of the partitios does ot have a so high ifluee i the orretess of the estimatio, but greater lap sizes are also advatageous as otied for high speeds. The ojutio of these two fators makes hard to fid the best algorithm for high speeds, sie small partitios a ot have large overlaps. I a similar way, for low speeds, large overlaps imply very large partitio ad a ompromise as to be made i eah situatio. evertheless, it s lear that for high speeds the most importat fator is the size of the partitios, while for low speeds the overlap size is the key fator. V. OLUSIOS The Blok-Fourier algorithms preseted i [],[3] for the zeroforig algorithm uder ostat hael oditios where also tested uder usteady hael situatios, havig revealed useless i oditios of medium or high speeds. ew versios of those algorithms, apable of dealig with detetio i usteady haels with speeds util 100km/h were derived ad tested. These ew algorithms where based i the partitioed blok- Fourier algorithms of [],[3], but extra steps were added to take i osideratio the hael hage from partitio to partitio. Iside eah partitio the hael is osidered ostat. The ew algorithms, although more omputatioally expesive tha the origial blok-fourier oes, are ot so expesive as the Gauss or holesky oes (eve if optimized versios were osidered). The best algorithm must be seleted aordig the hael oditios: for almost ostat haels ostat blok- Fourier algorithms ould be used with good results, while for high speeds the ew blok-fourier algorithms proposed must be used, preferably with small sized partitios. Table 3 ZF simulatios results for 1 ad ateas (BER) VI. REFEREES [1] S. Kay, Fudametals of Statistial Sigal Proessig: Estimatio Theory, Eglewood liffs, J: Pretie-all, 1993, pg391 [] M. Vollmer, M. aardt, J. Gotze, omparative Study of Joit- Detetio Tehiques for TD-DMA Based Mobile Radio Systems, IEEE Sel. Areas omm., vol.19, o.8, August 001. [3] R. Mahauer, M. Iurasu, F. Jodral, FFT Speed Multiuser Detetio for igh Rate Data Mode i UTRA-FDD, IEEE VTS 54th, vol.1, 001. 91