BER Analysis and Optimization of Generalized Spatial Modulation in Correlated Fading Channels

Transcription

1 BER Aalysis ad Optimizatio of Geeralized Spatial odulatio i Correlated Fadig Chaels Thomas Hadte, Adreas üller ad Joachim Speidel Istitute of Telecommuicatios, Uiversity of Stuttgart, Germay {hadte, mueller, speidel}@iue.ui-stuttgart.de Abstract We propose ad aalyze a geeralized spatial modulatio scheme, where depedig o the bits to be trasmitted a certai data symbol as well as a certai beamformig vector is chose. The receiver the has to estimate both the used beamformig vector ad the trasmitted data symbol for beig able to recostruct the origially trasmitted bit sequece. We aalyze our scheme theoretically by derivig a very tight upper boud o the average bit error rate BER) for arbitrary symbol costellatios ad beamformig codebooks i correlated Rayleigh fadig chaels. Based o this boud, we determie a desig criterio for optimizig the used beamformig codebook i presece of spatial correlatio at the trasmitter-side. Simulatio results are show to be i excellet agreemet with our aalytical calculated BERs ad they illustrate the sigificat performace gais that ca be obtaied with our geeralized scheme compared to covetioal spatial modulatio. I. ITRODUCTIO The ever icreasig demad for higher data rates i wireless commuicatio systems requires iovative trasmissio schemes achievig a high spectral efficiecy. I this regard, multiple-iput multiple-output IO) systems have become a key techology for achievig this target ad a great variety of correspodig trasmissio schemes has bee proposed ad thoroughly aalyzed i literature withi the past few years. With the well-kow vertical Bell Labs layered spacetime V-BLAST) architecture, for example, complex data symbols are simultaeously trasmitted usig differet atea elemets, thus directly icreasig the achievable data rates by a factor of i the ideal case []. However, due to the iheret iter-chael iterferece ICI) of such systems, rather complex receiver structures like maximum likelihood L) or ordered successive iterferece cacelatio OSIC) receivers are required for achievig a good performace. I order to mitigate the computatioal requiremets at the receiver-side while still achievig a good performace, recetly so-called spatial modulatio has bee proposed [], [3]. Spatial modulatio is a fudametally ovel trasmissio techique for IO systems, where depedig o the iput bits both a sigle trasmit atea elemet as well as a complex data symbol is chose. Hece, the actual iformatio to be trasmitted is cotaied i the trasmitted data symbol as well as i the selected trasmit atea elemet. Sice oly oe trasmit atea is active per poit i time, ICI is completely avoided, thus allowig sigificatly less complex receiver structures, which ow have to estimate both the active trasmit atea ad the actual data symbol [], [3]. The performace of spatial modulatio has bee thoroughly ivestigated for certai low-complexity receiver structures i [3] ad it has bee show that it is possible to achieve a comparable performace to covetioal V-BLAST systems with a remarkably lower complexity. Besides, the optimal detectio algorithm of a spatial modulatio system has recetly bee preseted i [4] ad aalytical expressios for the average BER have bee obtaied based o the correspodig pairwise error probabilities PEP). However, the derived results are oly valid for real-valued sigal poit costellatios ad scearios without ay spatial correlatio at the trasmitter- or receiver-side. Recetly, i [5] the authors proposed so-called geeralized space-shift keyig as a special case of spatial modulatio, where the data to be trasmitted is exclusively ecoded i the selectio of a certai set of trasmit ateas, without performig ay symbol modulatio. I this paper, we first geeralize the origial spatial modulatio scheme itroduced i [] by ot selectig a sigle trasmit atea elemet based o the bits to be trasmitted, but rather a certai beamformig vector from a give codebook. I a ext step, we aalyze the correspodig performace by derivig a very tight upper boud o the average BER, which i cotrast to the results i [4] is valid for arbitrary complex-valued data symbols as well as arbitrarily correlated chaels. Based o this result, we determie a simple criterio for optimizig the BER performace by choosig a appropriate log-term beamformig codebook depedig o the spatial correlatio properties at the trasmitter-side. Simulatio results show that it is possible to reduce the average BERs tremedously compared to a covetioal spatial modulatio system. The paper is orgaized as follows: Sectio II itroduces the system model ad the detectio algorithm. I Sectio III, we derive the tight BER boud, whereas the optimizatio criterio is preseted i Sectio IV. Simulatio results are show i Sectio V ad Sectio VI fially cocludes the paper. Throughout this paper, we use the followig otatio. Bold lower case letters represet vectors, while bold upper case letters deote matrices. The Hermitia trasposed, the trace ad the determiat of a matrix A are idicated by A H, tr A) ad det A), respectively. We use for the Kroecker product ad vec A) stacks all colums of A ito a vector. Furthermore, E [.] represets the expected value of a radom variable RV) ad a complex Gaussia distributio with mea m ad variace σ is deoted by C m, σ ).The-th partial derivative with respect to variable s is idicated by s /9/$5. 9 IEEE

2 II. SYSTE ODEL Figure shows the cosidered IO system with trasmit ad receive ateas. The covetioal) spatial mapper maps a certai umber of iput bits b oto a vector symbol qe i, where e i = [......] T deotes the i-th uit vector havig oe o-zero etry at the i-th positio.the modulatio symbol q ca be draw from ay real- or complexvalued symbol costellatio. For illustratig the basic priciple i detail, a exemplary mappig process is show i Fig. for a IO system with four trasmit ateas ad a QPSK costellatio. Before every trasmissio iterval, four bits are take from the iput stream, where the first two bits are used for selectig a uit vector e i ad hece oe out of four available ateas whereas the other two bits are mapped to a complex QPSK symbol q. I the case of covetioal spatial modulatio as itroduced i [], the sigal qe i would the be directly trasmitted, thus havig always oly oe active trasmit atea. The umber of simultaeously trasmitted bits per chael use or equivaletly the spectral efficiecy is clearly give by η = η symb + log ), ) where η symb deotes the umber of bits per data symbol q ad depeds o the applied sigal poit costellatio. As a geeralizatio, we propose to select a geeral beamformig vector p i from a give codebook based o the first iput bits rather tha a uit vector e i. Hece, the actual trasmit sigal is geerally give by qp i ad the symbol q is ot ecessarily trasmitted over oly oe sigle atea elemet. This approach ca be easily implemeted by multiplyig the output of the covetioal spatial mapper with a precodig matrix P =[p,..., p ] as depicted i Fig., which i fact represets the codebook of available beamformig vectors. Clearly, if P is chose as the idetity matrix of size, our geeralized scheme reduces to covetioal spatial modulatio as a special case. Besides, it is quite obvious that the liear trasformatio doe by P eeds to be ivertible i.e. det P) ) ad that the total mea trasmit power must ot be affected by P i.e. tr P H P ) = ). I the followig, we always assume that the vector s = qp i is trasmitted through a correlated frequecy-flat Rayleigh fadig chael H = A H H W B, where the matrices A ad B are the square roots of the correlatio matrices R rx = A H A ad R tx = B H B, respectively. Furthermore, the etries of H W b Spatial apper qe i Geeralized Spatial apper ˆb P qp i Spatial Demapper ˆq ê i H + L Detector r Fig.. b...] [...,, e e e 3 e 4 e j π 4 e j 3π 4 e j 7π 4 e j 5π 4 e j 3π 4 qe i Exemplary mappig process iside the Spatial apper =4, QPSK). are modeled as idepedet, complex Gaussia RVs accordig to C, ). It is well-kow that autocorrelatio matrix R HH of H is the give by R HH = R tx R rx. The elemets of the AWG vector are assumed to be idepedet ad idetically distributed accordig to C,σ). The received sigal vector r is the give by r = Hs + ad a maximum likelihood L) detector [4] estimates the trasmitted symbol vectors accordig to ˆq, ê i ) = arg mi q,ei) r HPqe i. Provided that the mea symbol power is ormalized i.e., E [ q ] =) ad that P fulfills the metioed prerequisites, the average SR per receive atea is geerally give by σ. Due to the isertio of the prefilter matrix P, geerally more tha oe TX atea is active per poit i time, resultig i a higher trasmitter ad receiver complexity. However, this approach has the big advatage that the available trasmit power ca be equally distributed amog all atea elemets while covetioal spatial modulatio assigs the power to oly oe atea elemet. This has for example the potetial to sigificatly reduce the requiremets o the used amplifiers. I fact, if P is chose as a discrete Fourier trasform matrix [P] p,q =exp j π ) p ) q ) p, q =,...,, ) every atea elemet is always active ad trasmits at the same costat power level. Clearly, without spatial correlatio at the trasmitter-side, ay uitary matrix P achieves the same performace as covetioal spatial modulatio sice the statistical properties of H W ad H W P are idetical. However, i presece of spatial correlatio, this is geerally o loger the case ad P might be chose appropriately i order to achieve a more reliable trasmissio. We follow this idea i Sectio IV, where we derive a desig criterio for the matrix P for such a sceario. III. PERFORACE AALYSIS I the followig, we derive a tight upper boud o the average BER of our geeralized spatial modulatio scheme based o the PEP P x ˆx), which deotes the probability that the receiver decides more likely i favor of a vector symbol ˆx tha i favor of the actually trasmitted vector symbol x. Havig a L detector at the receiver-side, this PEP ca be formulated as P x ˆx) =Prob [ r x Hx > r x Hˆx ], 3) Fig.. Cosidered system model. Please ote that the receiver complexity is higher tha i [4], sice r eeds to be compared to qhp i istead of qh i.

3 where r x = Hx + deotes the oisy received sigal. Please ote that the defiitio of the PEP i 3) takes oly two vector symbols x ad ˆx ito accout. As the Euclidea distace from r x to all other vector symbols x ˆx of the sigal space is ot cosidered, there might actually be aother vector symbol which is more likely detected tha ˆx. Followig the aalysis i [6], the PEP accordig to 3) coditioed o H ca be rewritte as P x ˆx H) =Q H x ˆx), 4) σ where Q ) deotes the Gaussia Q-fuctio. Averagig 4) over the distributio of H may be accomplished usig a alterative expressio for the Q-fuctio as i [7], where it is show that the ucoditioal PEP ca be obtaied by P x ˆx) = π ) Φ π 4σ si dθ, 5) θ with Φs) deotig the momet-geeratig fuctio GF) of the RV Hδ, where δ = x ˆx. Clearly, Hδ ca be expressed as a complex quadratic form like i [7] as Hδ = vec H H ) H H W R s I δδ H) R s vec H H ) W, 6) with the short-had otatio R s = R rx R tx.usigthe results derived i [8], the GF of 6) ca the be expressed as Φs) = ). 7) det I sr H s I δδ H ) R s athematical maipulatios as i [7, eqs. 7)-9)] lead to Φs) = sψ i= j= i ξ j, 8) where the parameters ψ i ad ξ j deote the eigevalues of the matrices δδ H R tx ad R rx, respectively. Clearly, δδ H R tx has for ay δ ad R tx always rak oe due to the factor δδ H. For that reaso, there is geerally oly oe sigle o-zero eigevalue ψ, which ca be determied by ψ = ψ i = tr δδ H ) R tx = δ H R tx δ, 9) i so that Φs) ca be rewritte as P Φs) =, ) j j= sψξ j ) where ξ,...,ξ P deote the P distict eigevalues of R rx, which occur with multiplicities,..., P. Combiig ) with 5), it is evetually possible to derive exact aalytical closed-form expressios for the correspodig PEPs. Expadig Φs) accordig to ) ito partial fractios, we obtai Φs) = P j j= m= A j,m sψξ j ) m ) where the parameters A j,m deote the correspodig expasio coefficiets, which ca geerally be determied as A j,m = j m)! ψξ j ) j m j m P ) s j m. sψξ ν ) ν ν= ν j s= ψξ j Combiig ) with 5) ad usig the itegratio result preseted i [9, App. B], we fially obtai the exact aalytical closed-form expressio for the PEP P x ˆx) = = P j j= m= P j ψξj A j,m μ j= m= m ν= 4σ A j,m π ) m π m +ν ν si θ si θ + ψξj 4σ )[ μ )] ν ψξj, 4σ ) m dθ 3) 4) where the short-had otatio μ x) is defied as μ x) = ) +, x. 5) x If we have spatial correlatio at the trasmitter-side oly, the PEP expressio of 4) ca be simplified as R rx correspods to the idetity matrix ad thus P = ad the eigevalue ξ is equal to oe ad has a multiplicity of =. It ca be show that i this case the -th expasio coefficiet A, equals oe ad all other coefficiets A,m m are equal to zero. Hece, we obtai for the PEP ) ψ P x ˆx) =μ +k ) [ )] k ψ 4σ k μ 4σ k= 6) with μ x) accordig to 5). A. Havig aalytical closed-form expressios for the average PEP, we ca derive a accurate approximatio for the average BER []. Cosiderig oly a sigle vector symbol x, itca easily be see that its average BER is geerally bouded by P e,bit x) d x, ˆx) P x ˆx), 7) η where d x, ˆx) deotes the Hammig distace ad idicates the umber of differeces i the bits associated with the vector symbols x ad ˆx, respectively. Averagig P e,bit x) over all η differet symbols x leads to the total bit error probability. P tot e,bit E [ η ] d x, ˆx) P x ˆx) = η η x d x, ˆx) P x ˆx). 8)

4 B. Theoretical BER Limits of Spatial odulatio The asymptotic performace of ay space-time code is usually quatified i terms of the correspodig diversity order ad codig gai. I this regard, the diversity order specifies how fast the average BER decays with icreasig SR i the high SR regime whereas the codig gai is reflected by a certai SR shift. I fact, spatial modulatio basically represets just a special space-time code ad hece it ca be aalyzed i exactly the same way. I geeral, the maximum achievable diversity order is determied by the umber of ozero eigevalues ψ i ad ξ j. Assumig a receive correlatio matrix R rx with full rak, the RX diversity is equal to. However, as ca be see from 9), there is always oe ozero eigevalue ψ. Therefore, spatial modulatio ca ever achieve a higher diversity order tha. Cosiderig a system without chael codig, spatial modulatio therefore experieces geerally a performace pealty compared to most other space-time codes, which are capable of exploitig trasmit diversity as well. This also holds for the well-kow Alamouti code [], for example, which trasmits the iformatio i two cosecutive time itervals over two trasmit ateas, thus achievig a diversity order of resultig i a lower BER i the high SR regime. However, at low ad medium SR, the BER of spatial modulatio ca evertheless be lower due to a higher codig gai. IV. OPTIIZATIO OF THE PREFILTER ATRIX As already metioed before, spatial modulatio caot exploit ay trasmit diversity ad therefore the oly way to improve its performace by choosig a appropriate beamformig codebook P is to optimize the codig gai. Keepig i mid that the eigevalues ξ j of R rx are idepedet of the prefilter parameters, oly ψ may be iflueced. I the followig, δ is cosidered to be the differece betwee two sigal vectors from the Spatial apper. I presece of a prefilter P, ψ is give by ψ = δ H P H R tx Pδ. 9) Clearly, miimizig the average BER is equivalet to a joit miimizatio of all PEPs. As it ca be see from 4) ad 5), the PEPs are miimized if their correspodig ψ is maximized. Assumig that all PEPs cotribute equally to the average BER, the followig optimizatio criterio is adequate { } P = arg max P mi {ψ δ)} δ. ) As the prefilter has to fulfill the power costrait, the optimizatio i ) is subjected to tr PP H) =. Please ote that the miimum operator i ) is take over every sigal vector differece, which yields i geeral η η ) differet combiatios. The expoetially growig amout of combiatios i ) requires a computer aided evaluatio. However, a prefilter sample may be rapidly checked for its performace by calculatig its ψ mi = mi {ψ δ)}. The higher ψ mi is, the lower the BER will be. δ Furthermore, it should be oted that this optimizatio criterio is ivariat with respect to the applied mappig sice the impact of d x, ˆx) has bee eglected. V. SIULATIO RESULTS I the followig, we preset selected performace results to illustrate that our aalytically calculated average BERs are i good agreemet with simulated values ad how the performace might be sigificatly improved by choosig a appropriate codebook matrix P. I this regard, we assume that the receiver performs a L detectio ad that for the selectio of a appropriate modulatio symbol q a stadard Gray mappig is used. Besides, spatial correlatio at the trasmitter ad receiver-side is modeled usig a expoetial correlatio model with two idepedet parameters ρ tx ad ρ rx, respectively. Hece, the correspodig correlatio matrices R tx ad R rx are geerally give by { ρ p q tx q p [R tx ] p,q = [ ρ q p] p, q =,..., ) tx q>p { ρ p q rx q p [R rx ] p,q = [ρrx q p ] p, q =,...,. ) q>p Figure 3 shows the average BER of covetioal spatial modulatio P = I) with =4TX ad =4RX ateas as well as QPSK, resultig i a total spectral efficiecy of bit 4 s Hz. Both simulatio ad aalytical results are plotted for the o-correlated, TX or RX correlated case. First of all, it ca be see that the deviatio betwee the simulated ad the aalytical results is almost egligible, especially i the high SR regio. Clearly, this holds for all cosidered correlatio coefficiets ρ tx ad ρ rx. oreoever, it is obvious that i presece of spatial correlatio either at the trasmitteror the receiver-side, the performace degrades cosiderably compared to the totally ucorrelated case, resultig i a effective SR loss of more tha 6.5 db i the high SR regio, for example. Figure 4 shows the BER performace of a IO scheme with =TX ad =4RX ateas. Covetioal spatial modulatio with QPSK is compared to a Alamouti scheme with 8-PSK, so that the spectral efficiecy is equal to 3 bit s Hz i both cases. Cosiderig the totally ucorrelated case, spatial modulatio clearly exhibits a smaller BER tha the Alamouti scheme i the low SR regio. However, with icreasig SR the Alamouti scheme beefits from its higher diversity order, therefore geerally leadig to a better performace. The itersectio of both BER curves will move to higher SR, whe η icreases because the Alamouti scheme suffers from the higher modulatio order. I case of TX correlatio, the Alamouti scheme has a lower BER tha covetioal spatial modulatio, yieldig a gai of 4 db at a BER of 3. Figure 5 shows the BER of a spatial modulatio scheme with =TX, =4RX ateas ad QPSK with ad without spatial correlatio at the trasmitter-side. I order to improve the performace, i this case, the codebook matrix P has bee optimized by meas of a exhaustive search ad is

5 ρ tx=.ρ rx=. Aalytical ρ tx=.ρ rx=. Simulatio ρ tx=.ρ rx=.9 Aalytical ρ tx=.ρ rx=.9 Simulatio ρ tx=.9ρ rx=. Aalytical ρ tx=.9ρ rx=. Simulatio Average SR [db] Fig. 3. BER as a fuctio of SR for covetioal spatial modulatio with 4 TX, 4 RX ateas ad QPSK. Simulatio ad aalytical results are compared Sp. od. ρ tx=. Aalytical Sp. od. ρ tx=. Simulatio Sp. od. ρ tx=.9 Aalytical Sp. od. ρ tx=.9 Simulatio Alamouti ρ tx=. Simulatio Alamouti ρ tx=.9 Simulatio Average SR [db] Fig. 4. Compariso betwee spatial modulatio ad a Alamouticode, usig a IO scheme with TX ad 4 RX ateas. To achieve η =3 bit s Hz spatial modulatio uses QPSK ad the Alamouti scheme uses 8-PSK. give by ).5+j.5.5+j.5 P =. 3).5 j.5.5 +j.5 Obviously, without prefilter ad spatial correlatio, there is a sigificat loss of 6 db at BER 3 compared to the ucorrelated case. With the prefilter of 3), i cotrast, this loss is oly up to about db, what illustrates the sigificat gais that ca be gaied by meas of our geeralized spatial modulatio scheme. VI. COCLUSIO We have proposed a geeralized spatial modulatio scheme, where the bits to be trasmitted are ecoded through the choice of a certai beamformig vector as well as a particular data symbol. Furthermore, we have performed a theoretical aalysis of our approach by derivig a very tight upper boud o the average BER for arbitrary sigal poit costellatios ad beamformig codebooks i spatially correlated Rayleigh fadig chaels. As codebooks provide ew degrees of freedom, we have determied a simple optimizatio criterio for desigig a appropriate beamformig codebook, reducig the BER i presece of spatial correlatio at the trasmitter-side. Fially, simulatio results were show to be i good agreemet with our aalytically calculated BERs ad they illustrated that our geeralized approach ca yield SR gais i the order of several dbs compared to covetioal spatial modulatio. REFERECES [] P. W. Woliasky, G. J. Foschii, G. D. Golde, ad R. A. Valezuela, V-BLAST: A architecture for realizig very high data rates over the rich-scatterig wireless chael, URSI It. Symp. o Sigals, Systems, ad Electroics, pp. 95 3, Sep-Oct 998. [] R. esleh, H. Haas, C. W. Ah, ad S. Yu, Spatial modulatio - A ew low complexity spectral efficiecy ehacig techique, It. Cof. o Commu. ad etw. i Chia, 6., Oct. 6. [3] R. esleh, H. Haas, S. Siaovic, C. W. Ah, ad S. Yu, Spatial modulatio, IEEE Tras. o Veh. Techol., vol. 57, o. 4, pp. 8 4, July ρ tx=. w/o Prefilter Aalytical) ρ tx=. w/o Prefilter Simulatio) ρ tx=.9 w/o Prefilter Aalytical) ρ tx=.9 w/o Prefilter Simulatio) ρ tx=.9 w/ Prefilter Aalytical) ρ tx=.9 w/ Prefilter Simulatio) Average SR [db] Fig. 5. Performace of spatial modulatio with ad without prefilter uder differet TX correlatio properties. The IO scheme features TX, 4 RX ateas ad QPSK symbols. ρ rx = [4] J. Jegaatha, A. Ghrayeb, ad L. Szczeciski, Spatial modulatio: Optimal detectio ad performace aalysis, IEEE Commu. Lett., vol., o. 8, pp , Aug. 8. [5], Geeralized space shift keyig modulatio for mimo chaels, IEEE It. Symp. o Persoal, Idoor ad obile Radio Commu., 8., Sept. 8. [6]. Simo ad.-s. Alouii, Digital Commuicatio over Fadig Chaels, d ed. Wiley Itersciece, 5. [7] A. Hedayat, H. Shah, ad A. osratiia, Aalysis of space-time codig i correlated fadig chaels, IEEE Tras. o Wireless Commu., vol. 4, o. 6, pp , ov. 5. [8] G. L. Turi, The characteristic fuctio of hermitia quadratic forms i complex ormal variables, Biometrika, vol. 47, o. -, pp. 99, 96. [9].-S. Alouii ad A. Goldsmith, A uified approach for calculatig error rates of liearly modulated sigals over geeralized fadig chaels, IEEE Tras. o Commu., vol. 47, o. 9, pp , Sep 999. [] J. G. Proakis, Digital Commuicatios, 4th ed. cgraw-hill, 4. [] S. Alamouti, A simple trasmit diversity techique for wireless commuicatios, IEEE Joural o Selected Areas i Commu., vol. 6, o. 8, pp , Oct 998.