Correlation Analysis of Instantaneous Mutual Information in 2 2 MIMO Systems

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1 Correlaton Analyss of Instantaneous Mutual Informaton n MIMO Systems Shuangquan Wang, Al Abd Center for Wreless Communcatons Sgnal Processng Research Department of Electrcal Computer Engneerng New Jersey Insttute of Technology, Newark, New Jersey 7 Emal: sw7@njtedu; alabd@njtedu Abstract In ths paper, the second-order statstcs such as the autocorrelaton functon correlaton coeffcent of the nstantaneous mutual nformaton IMI are studed, n multple-nput multple-output MIMO tme-varyng Raylegh flat fadng channels, assumng general non-sotropc scatterng envronments Exact closed-form expressons are derved, as well as asymptotc approxmatons n low- hgh-snr regmes Monte Carlo smulatons are provded to verfy the accuracy of those derved analytcal results I INTRODUCTION The ncreasng dem for wreless communcaton over tme-varyng channels has motvated further nvestgaton of the channel dynamcs ther statstcal behavor There are numerous studes on the temporal correlatons of a varety of terrestral [] [4] satellte channels [5] For such mportant quantty as the nstantaneous mutual nformaton IMI, however, only the mean value, whch s the ergodc capacty, has receved much attenton as well as the outage probablty [6][7] Clearly, ergodc capacty outage probablty do not show the dynamc temporal behavor, such as correlatons, of IMI n tme-varyng fadng channels The IMI feedback s used for the rate schedulng n multuser communcaton envronments to ncrease the system throughput [8], where only the perfect feedback s consdered However, t s hard to obtan the perfect feedback n practce, generally the feedbacked IMI s outdated In ths case, the correlaton of IMI can be used to analyze the schedulng performance wth outdated IMI feedbacks Moreover, the correlaton can also be used to mprove the rate schedulng algorthm by explorng the correlaton of IMI To the best of our knowledge, correlaton analyss of IMI n sngle-nput sngle-output SISO systems were reported n [9] For MIMO systems, only some smulaton results regardng correlaton analyss are gven n [], wthout analytcal results, n [], lower upper bounds on the correlaton coeffcent n the hgh sgnal-to-nose rato SNR regme, as well as some approxmatons, are derved wthout exact results In ths paper, the second-order statstcs such as the autocorrelaton functon correlaton coeffcent of IMI are studed n MIMO tme-varyng Raylegh flat fadng channels, where all the subchannels are ndependent dentcally dstrbuted d wth the same temporal correlaton coeffcent, consderng general non-sotropc scatterng propagaton envronments Closed-form expressons smple approxmatons are derved for the autocorrelaton functon ACF correlaton coeffcents of MIMO IMI Monte Carlo smulatons are provded to verfy the accuracy of our closedform expressons approxmate results Notaton: s reserved for matrx Hermtan, for complex conjugate, j for, E[ ] for mathematcal expectaton, I m for the m m dentty matrx, ln for the natural logarthm, F for the Frobenus norm, R[ ] I[ ] for the real magnary parts of a complex number, respectvely, f x for [fx The rest of ths paper s organzed as follows Sec II ntroduces the channel model the IMI rom process n multple-nput multple-output MIMO systems, as well as angle-of-arrval AoA models Sec III s devoted to the dervaton of ACF correlaton coeffcent of IMI n MIMO channels, as well as ther low- hgh-snr asymptotc approxmatons Numercal results are presented n Sec IV, concludng remarks are gven n Sec V II CHANNEL MODEL AND MIMO IMI For smplcty, only MIMO tme-varyng Raylegh flat fadng channels are consdered n ths paper, we assume all 4 suchannels, {h mn t}, m=,n= are d, wth the same temporal correlaton coeffcent, e, E[h mn th pqt τ = δ m,p δ n,q τ, where the Kronecker symbol δ m,p s or accordng as m = p or m p, τ s derved as follows gven by 7 In flat Raylegh fadng channels, h mn t, a zero-mean complex Gaussan rom process, can be represented as [4] h mn t = h I mnt + jh Q mnt = α mn texp[ jφ mn t, where the zero-mean real Gaussan rom processes h I mnt h Q mnt are the real magnary parts of h mn t, respectvely α mn t s the envelope of h mn t Φ mn t s In ths paper, each subchannel represents the rado lnk between each transmt/receve par of antennas

2 [ { px,y= e x+y x+yx + exp x+y ] xy I y x+ x y+ }, x,y 3 4 { = [ k λ 3 k! λ G 3, λ [ }, η + λ,,k+ G 3, λ, η G 3, λ,,,k+ η,,k+ k= { + G 3, η, G 3,,, η, [ G 3,,,3 η, } [,, λ G 3, η, G 3,,, η, 4,, E[I l ]=G 3, η, G 3,,,3 η, +G 3,,, η, 5,, [ E[Il ]=4e η G4, η 3,3,3 G 4, 3,,, η,, + 3,,, G4, η,, +G 4, 3,,, η,, G 4,,,, η,,,,, + G 4, η,, { + G 3,,,, η, G 3,,, η, [ G 3,,,3 η, } 6,, the phase of h mn t At any tme t, α mn t has a Raylegh dstrbuton Φ mn t s dstrbuted unformly over [ π,π Wthout loss of generalty, we assume each subchannel has unt power, e, E[α mnt = Usng the emprcally-verfed [4] multple von Mses probablty densty functon PDF [9, 4 for the angle of arrval AoA at the recever n non-sotropc scatterng envronments shown as Fg [9], the channel correlaton coeffcent τ of h mn t, m,n, s gven by [9, 7 τ = E[h mn th mnt τ, N I κ = P n 4π fd τ + j4πκ n f D τ cos θ n n, I κ n n= where I k z = π π ez cos θ coskθdθ s the k th order modfed Bessel functon of the frst knd θ n s the mean AoA of the n th cluster of scatterers, κ n controls the wdth of the n th cluster of scatterers, P n represents the contrbuton of the n th cluster of scatterers, such that N n= P n =, < P n, N s the number of clusters of scatterers, f D s the maxmum Doppler frequency When κ n =, n, whch corresponds to the sotropc scatterng, 7 reduces to τ = I jπf D τ = J πf D τ [9] Smlar to the SISO systems consdered n [6], we consder a pecewse constant approxmaton for the contnuous-tme fadng channel coeffcent Ht, represented by {HlT s } L l=, where T s s the symbol duraton L s the number of samples In the presence of addtve whte Gaussan nose, f perfect channel state nformaton {HlT s } L l=, s avalable at the recever only, the ergodc channel capacty s gven by [7][] 7 [ C = E ln det I + η, 8 nats/s/hz, where η s the average SNR at the recever sde, H l denotes HlT s In the above equaton, at any gven tme ndex l, ln det I + η s a rom varable as t depends on the fadng parameter H l Therefore I l = lndet I + η, l =,,, 9 s a dscrete-tme rom process wth the ergodc capacty as ts mean We study the second-order statstcs of {I l } l=, such as autocorrelaton correlaton coeffcent, n the followng secton III ACF AND CORRELATION COEFFICIENT OF MIMO IMI In ths secton, frst we concentrate on the ACF of MIMO IMI n 9, whch s defned by =E[I l I l ], [ =E ln det I + η ln det I + η H l H l We assume x x are two unordered egenvalues of H l H l, y y are two unordered egenvalues of H l H l Moreover, x s romly selected from x x, y s romly selected from y y Therefore, reduces to = E = 4E [ m= [ ln ln + η x m + η x ln n= + η y ln + η n ] y, To further smplfy, we need the jont PDF of x y, whch s gven by 3 [3], where = T s <,, τ s presented n 7 The margnal PDF of x s gven by x px = x + e x,x, 3 by ntegratng 3 over the varable y, whch s also gven n [] Combnaton 3, wth the followng Taylor seres of I t [4, pp 97, 8447] I t = k= t k k!, 4 k

3 smplfes to the exact nfnte-summaton closed-form representaton n 4, where G s Mejer s G functon [4, pp 96, 93] λ = 5 Wth I l = m= ln + η x m, the frst moment of Il s shown to be 5, the second moment of I l s E [ Il ] [ =E ln + η x +ln + η x ln + η x, 6 whch reduces to 6 by usng the margnal PDF n 3 the jont PDF of x x [5, px,x = x x e x+x, x,x 7 Therefore the normalzed ACF correlaton coeffcent can be calculated accordng to = E[Il 8 ], = {E[I l ]} E[I l ] {E[I l]}, 9 by pluggng 4 6 nto 8, 4-6 nto 9, respectvely In general, t seems dffcult to further smplfy 4 However, we note that the ntegral Ξk,ω,λ = x k e λx ln + ωxdx, k, can be approxmated by { ωx k+ e λx dx, ω, Ξk,ω,λ x k e λx lnωxdx, ω, usng ln + ωx ωx, ω, ln + ωx ln ωx, ω, respectvely In the followng two subsectons, we obtan asymptotc closed-form expressons for the normalzed ACF,, correlaton coeffcent,, n low- hgh-snr regmes A Low-SNR Regme If η, based on 4, after some basc algebrac manpulatons, we proved that the normalzed ACF the correlaton coeffcent are, respectvely, expressed as 4 +, 5 3 Interestngly, the above results are the same as those n the MIMO wth orthogonal space-tme block code OSTBC transmsson n low-snr regmes, as expected In fact, n low- SNR regmes, I l n 9 can be approxmated as I l η H l F, The utlty accuracy of s confrmed by Monte Carlo smulatons n Sec IV whch s the same as the low-snr approxmaton of I l n MIMO-OSTBC systems [6] Wth sotropc scatterng, we have T s = J πf D T s, the Clarke s correlaton [], ths smplfes 3 to J πf D T s B Hgh-SNR Regme If η, based on 4, after lengthy algebrac calculatons, we have shown that the normalzed ACF the correlaton coeffcent are, respectvely, gven by [ L + π 3 [ln + ln η γ η γ + ] + ], 4 L π 3, 5 where γ = 787 s the Euler-Mascheron constant [4, pp xxx], L x s the dlogarthm functon, defned as x k L x =, x 6 k k= Wth sotropc scatterng, 5 reduces to L [ J πf D T s ] Jπf D T s π 3 7 IV NUMERICAL RESULTS AND DISCUSSION In ths paper, the generalzed power spectrum [9, 8 s used to smulate Raylegh flat fadng channels wth non-sotropc scatterng, accordng to the spectral method [7] To verfy the accuracy of the derved formulas, we consder three types of scatterng envronments: sotropc scatterng, non-sotropc scatterng wth three clusters of scatterers, hghly nonsotropc scatterng wth one cluster of scatterers κ = 4, θ = π 9 For the non-sotropc scatterng, parameters of three [ clusters, [P n ],κ n,θ n ],n =,, are gven by [P,κ,θ ] = 45,, π 8, [P,κ,θ ] = [ ] [ ],, π 8, [P3,κ 3,θ 3 ] = 35,3, 53π 36, respectvely In addton, n all the smulatons, the maxmum Doppler frequency f D s set to Hz, T s = f D seconds In the followng subsectons, smulatons are performed to verfy the ACF correlaton coeffcent of IMI n the above three types of envronments n both low- hgh-snr regmes For evaluatng the approxmaton accuracy of ACF the correlaton coeffcent, we set η = 5 db for low SNR, η = 3 db for hgh SNR A Isotropc Scatterng Ths s the Clarke s model, wth unform AoA The smulaton results are shown n Fg B Non-sotropc Scatterng Ths s a general case, wth an arbtrary AoA dstrbuton Smulatons are carred out, wth the results presented n Fg

4 5 Channel Correlaton Coeffcents R[ T s Smu Theo AoA Dstrbuton Channel Correlaton Coeffcents Smu Theo R[ρ T h s I[ T s AoA Dstrbuton I[ T s Normalzed Autocorrelaton Functon of IMI 8 6 Correlaton Coeffcent of IMI Low SNRSmu Hgh SNRSmu Hgh SNRTheo 8 6 Normalzed Autocorrelaton Functon of IMI 8 6 Correlaton Coeffcent of IMI 4 Low SNRSmu Hgh SNRSmu Hgh SNRTheo Low SNRSmu Hgh SNRSmu Hgh SNRTheo Low SNRSmu Hgh SNRSmu Hgh SNRTheo 5 5 Fg The channel correlaton coeffcent, AoA dstrbuton, as well as the normalzed ACF correlaton coeffcent of IMI, n a MIMO system wth sotropc scatterng Fg 3 The channel correlaton coeffcent, AoA dstrbuton, as well as the normalzed ACF correlaton coeffcent of IMI, n a MIMO system wth hghly non-sotropc scatterng C Hghly Non-sotropc Scatterng Ths s a case where MS receve the sgnal wthn a very narrow beam Smulaton results are presented n Fg 3 In Fgs -3, The upper left rght fgures show the channel correlaton coeffcent AoA dstrbutons; the lower left rght fgures show the normalzed approxmate ACF the approxmate correlaton coeffcent of the IMI I l, respectvely, n both low- hgh-snr regmes In all fgures, Smu means smulaton, Theo means the channel coeffcent τ, the normalzed ACF correlaton coeffcent n low- hgh-snr regmes are calculated accordng to 7,, 3, 4, 5, respectvely From Fgs -3, the followng observatons can be made In both low- hgh-snr regmes, the asymptotc 5 5 Channel Correlaton Coeffcents Smu Theo I[ T s R[ T s Normalzed Autocorrelaton Functon of IMI 4 Low SNRSmu Hgh SNRSmu Hgh SNRTheo AoA Dstrbuton Correlaton Coeffcent of IMI Low SNRSmu Hgh SNRSmu Hgh SNRTheo 5 5 Fg The channel correlaton coeffcent, AoA dstrbuton, as well as the normalzed ACF correlaton coeffcent of IMI, n a MIMO system wth non-sotropc scatterng results of the normalzed ACF correlaton coeffcent derved n Sec III perfectly match the smulaton results n all the three scenaros we consdered The narrower of the angle spread at the recever sde, the larger the correlaton coeffcent It mples that the nstantaneous mutual nformaton has less fluctuaton when the recever takes the sgnal from a narrow angle, whch s the same as the results we obtaned n the SISO systems [9] In all scenaros we consdered, there are obvous gaps, whch can not be gnored, between low- hgh-snr asymptotc approxmatons Therefore, for not so small or large SNRs, we need to resort to the exact formulas n 4-6 for the accurate values of correlatons For example, η = 5dB, the smulaton theoretcal curves, as well as low- hgh-snr approxmatons are shown n Fg 4 for the correlaton coeffcent V CONCLUSION In ths paper, closed-form expressons for the autocorrelaton functon correlaton coeffcent of the nstantaneous mutual nformaton IMI n tme-varyng Raylegh flat fadng channels are derved, n MIMO systems The analytcal expressons, supported by Monte Carlo smulatons, provde useful qualtatve quanttatve nformaton regardng the fluctuatons of IMI In ths paper, we only consdered MIMO systems It s our ongong work to consder the spatally uncorrelated general MIMO system wth M transmtters N recevers, where all the subchannels are ndependent dentcally dstrbuted, wth the same temporal correlaton functon REFERENCES [] W C Jakes, Ed, Mcrowave Moble Communcatons New York: IEEE Press, 994

5 9 8 Correlaton Coeffcent of IMI Low SNR Approx Hgh SNR Approx η=5dbsmu η=5dbtheo Fg 4 The correlaton coeffcent of IMI at SNR η = 5dB, n a MIMO system wth non-sotropc scatterng [] A Abd, K Wlls, H A Barger, M S Aloun, M Kaveh, Comparson of the level crossng rate average fade duraton of Raylegh, Rce, Nakagam fadng models wth moble channel data, n Proc IEEE Veh Technol Conf, Boston, MA,, pp [3] N Youssef, T Munakata, M Takeda, Fade statstcs n Nakagam fadng envronments, n Proc IEEE Int Symp Spread Spec Tech App, Manz, Germany, 996, pp [4] A Abd, J A Barger, M Kaveh, A parametrc model for the dstrbuton of the angle of arrval the assocated correlaton functon power spectrum at the moble staton, IEEE Trans Veh Technol, vol 5, pp , May [5] A Abd, W C Lau, M S Aloun, M Kaveh, A new smple model for l moble satellte channels: Frst- second-order statstcs, IEEE Trans Wreless Commun, vol, pp 59 58, May 3 [6] L H Ozarow, S Shama, A D Wyner, Informaton theoretc consderatons for cellular moble rado, IEEE Trans Veh Technol, vol 43, pp , May 994 [7] D Tse P Vswanath, Fundamentals of Wreless Communcaton Cambrdge, UK: Cambrdge Unversty Press, 5 [8] B M Hochwald, T L Marzetta, V Tarokh, Multple-antenna channel hardenng ts mplcatons for rate feedback schedulng, IEEE Trans Inform Theory, vol 5, pp , 4 [9] S Wang A Abd, On the second-order statstcs of the nstantaneous mutual nformaton of tme-varyng fadng channels, n Proc IEEE Int Workshop Sgnal Processng Advances n Wreless Communcatons, New York, 5, pp [] A Gorgett, M Chan, M Shaf, P J Smth, Level crossng rates MIMO capacty fades: mpacts of spatal/temporal channel correlaton, n Proc IEEE Int Conf Commun, Anchorage, AK, 3, pp [] N Zhang B Vojcc, Evaluatng the temporal correlaton of MIMO channel capactes, n Proc IEEE Global Telecommun Conf, St Lous, MO, 5, pp 87 8 [] İ E Telatar, Capacty of mult-antenna Gaussan channels, European Trans Telecommun, vol, pp , 999 [3] S Wang A Abd, Jont sngular value dstrbuton of two correlated rectangular complex Gaussan matrces ts applcaton, submtted, 6 [4] I S Gradshteyn, I M Ryzhk, A Jeffrey, Eds, Table of Integrals, Seres, Products, 5th ed San Dego, CA: Academc, 994 [5] A M Tulno S Verdú, Rom matrces wreless communcatons, Foundatons Trends n Communcatons Informaton Theory, vol, June 4 [6] S Wang A Abd, On the second-order statstcs of the nstantaneous mutual nformaton n Raylegh fadng channels, submtted, 5 [7] K Acolatse A Abd, Effcent smulaton of space-tme correlated MIMO moble fadng channels, n Proc IEEE Veh Technol Conf, Orlo, FL, 3, pp