Channel inversion in MIMO systems over Rician fading

Transcription

1 This full tet aer was eer reviewed at the direction of IEEE Communications Society subject matter eerts for ublication in the IEEE Globecom roceedings Channel inversion in MIMO systems over Rician fading Damith Senaratne and Chintha Tellambura Deartment of Electrical and Comuter Engineering, University of Alberta, Edmonton, AB, Canada HimalASuraweera Deartment of Electrical and Comuter Engineering, National University of Singaore, Singaore Abstract We eamine the distribution of the er virtualchannel received signal-to-noise ratio over a multile-inut multile-outut channel modeled by a rank- non-central Wishart matri The eact robability density function is derived for all ossible non-centrality matrices; and verified through simulation for selected systems Identities for eact analytic results on the outage robability, the average symbol error rate, and the moment generating function are also derived Inde Terms MIMO, channel inversion, non-central Wishart distribution, outage robability, symbol error rate I INTRODUCTION A multile-inut multile-outut MIMO system [] with N t transmit antennas and N r receive antennas is modeled by an N r N t channel matri With erfect transmit channelstate-information, the transmit recoding and receiver reconstruction matrices can be chosen such that the channel reduces to a set of eigenmodes that can be indeendently coded, modulated, and ower allocated for Channel Inversion [] ower allocation scheme seeks to maintain the er-channel instantaneous received signal-tonoise ratio SNR identical, and the total instantaneous transmit ower a constant all the time In other words, it causes ower i,i n to be allocated for the channels having resective instantaneous gains λ i so that i i P is constant; and λ i i identical across the channels i n Channel-inversion is caacity-wise inferior to the classical water-filling ower allocation scheme Yet, it is unique among the ower allocation schemes in assuring the fairness across the channels with resect to achievable caacity Channel-inversion has further advantages in the MIMO contet [4] [6] It is imlementable with non-iterative transmitter rocessing and unitary receiver rocessing [4], [6] Hence, no noise enhancement occurs at the receiver When the channel rank is limited by the number of receiver antennas, receiver rocessing is not required at all; and the resulting virtual channels corresond one-to-one with the receiver antennas This, makes channel-inversion suitable for multi-user MIMO downlink channels [7] References [8] [] eamine variants of channel inversion that seek to imrove the caacity Zero forcing at the transmitter [3] too inverts the channel It maintains the er virtual-channel received ower and the average transmit ower steady On the other hand channel inversion, being a ower allocation scheme, kees the total instantaneous transmit ower steady, and the instantaneous er virtualchannel received ower identical Analyzing MIMO channels in fading requires results from random matri theory [] The eigenmodes of a MIMO channel comrising indeendent and identically distributed Rayleigh faded transmit-receive links, for instance, have the gains governed by the eigenvalue distribution associated with the Central Wishart distribution [] Similarly, non-central Wishart Distribution corresonds to Rician fading, which arises owing to line-of-sight conditions [3] This work generalizes [4] on Rayleigh fading, including it in the case the non-centrality matri has both eigenvalues zero The eact robability density function df of a factor roortional to er virtual-channel received SNR is derived, for non-central Wishart matrices of order n having rank- All ossible non-centrality matrices are considered The df result is then used to derive imortant erformance metrics including the outage robability, and the average symbol error rate The moment generating function too is derived All required identities are roven here even though some eressions are omitted for brevity To further the insights, the first order aroimation of the df characterizing the high SNR behavior is also rovided The aer is organized as follows: Sections II and III resent the system model and the mathematical formulation Section IV highlights how certain erformance metrics can be obtained In Section V, the results are verified for selected MIMO configurations through Monte-Carlo simulation Section VI concludes the aer Proof of Theorem is anneed Notation: P [ ] is the robability assignment; and E Λ }is the eectation oerator with resect to a random variable Λ The df, the cumulative distribution function cdf and the moment generating function mgf of Λ are given by f Λ, F Λ and M Λ resectively C n,m reresents the sace sanned by n m comle Gaussian matrices H and F designate the conjugate transose and the Frobenius norm of a matri eig yields the eigenvalues of a square matri Modified Bessel Functions of the First Kind and the Second Kind [5, 96] having order ν are denoted by I ν and K ν resectively Q reresents the Gaussian Q-function [5, Eq 63], while Γ denotes the Gamma function [5, 6] G m n q a,,a n,a n+,,a reresents the b,,b m,b m+,,b q Meijer G function [6, 49] L }is the Lalace Transform //$6 IEEE

2 This full tet aer was eer reviewed at the direction of IEEE Communications Society subject matter eerts for ublication in the IEEE Globecom roceedings n Transmitter N t n MIMO channel H Receiver N r Fig System Model: A MIMO channel between a transmitter having n antennas, and a receiver having antennas The line-of-sight aths are indicated; while isotroic scattering is assumed The results also hold in the reverse direction for N t,n r n II SYSTEM MODEL Consider a MIMO system having N t transmit antennas and N r receive antennas, where minn t,n r see Fig Let n man t,n r This scenario can occur in any MIMO channel with antennas at one end; and at least antennas in the other It occurs even in certain multiuser MIMO configurations, such as in the downlink channel from a multi-antenna base station to two single-antenna mobile stations it simultaneously communicates with Without further loss of generality let N t n and N r Suose the n channel matri H is of form H ah s + bh sc, where H s is a deterministic secular line-of-sight comonent, H sc C,n a random scatter comonent, and a +b The secular comonent is governed by the directional gains of the antennas, resence of dominant multi-aths, etc K a H s F gives the Rician factor [7] Ω a b H sc b H F s H H s is the non-centrality matri Let, λ,λ } eig HH H and ω,ω ω > ω } eig Ω The joint distribution of unordered eigenvalues is given by [7, Eq 5] f λ,λ λ,λ e ω+ω λ λ λ λ n ω ω ω ω n e λ+λ I n ω λ I n ω λ I n ω λ I n ω λ The secular comonent too being continuously distributed, the case ω ω is not likely in ractice However, it can occur with non-zero robability in the model since H s is considered to be deterministic Corresonding df is given by the limiting oeration ω ω on Assume erfect transmit CSI, and the channel inversion ower allocation scheme Given total transmit SNR γ, er channel received SNR is given by b γ Λ, where Λ λ λ 3 λ + λ In this aer, we will resent new eact outage robability and SER results for the channel The major mathematical challenge in establishing them is deriving the statistical distribution of Λ III MATHEMATICAL FORMULATION Theorem : the df of Λ Let λ,λ be the eigenvalues of a rank- non-central comle Wishart matri having n degrees of freedom, and non-centrality matri Ω whose eigenvalues are ω,ω ω ω > } The df of Λ in 3 is given by: i g j,i j ω,ω f Λ j + n! i j + n! j! i j! i j i+n i +n e i+n K n+j K n+j, 4 where g i,j α, β e α+β α β α i β j α j β i, α β i jα i+j e α, α β 5 Proof: See Aendi When the non-centrality matri has one non-zero eigenvalue ie ω,ω, g j,i j ω,ω in 4 is non-zero only when j,i j or j,i j Thus the df simlifies as follows Corollary : Let λ,λ be the eigenvalues of a rank- non-central comle Wishart matri having n degrees of freedom, and non-centrality matri Ω whose eigenvalues are ω,ω ω >,ω } The df of Λ in 3 is given by: f Λ i i+n ω i e ω n! i + n! i! i +n e i+n K n+i K n K n+i +K n 6 Proof: Omitted Setting ω in 6 yields 7, which is identical to the df of corresonding central Wishart distribution ie Rayligh fading Its equivalence to [4, Thm ] can be easily established using [8, Eq 3475] n n f Λ e n n! n! K n + K n +K n //$6 IEEE

3 This full tet aer was eer reviewed at the direction of IEEE Communications Society subject matter eerts for ublication in the IEEE Globecom roceedings IV APPLICATIONS The df f Λ eressions given by 4, 6 or 7 allow erformance metrics such as the outage robability to be derived Required identities are roven here A Outage Probability An outage occurs when the instantaneous received SNR falls below an accetable SNR threshold γ th Hence, its robability, ie the outage robability, can be given in terms of the cdf F Λ f Λtdt, which reduces to be a weighted sum of integrals of the form tμ e t K ν tdt It may be simlified as follows using the identities [9, 9344], and [8, Eq 734] t μ e t K ν tdt π t μ G 4t 5 dt ν, ν π 4 4 μ+ t μ G t 5 dt ν, ν π 4 μ+ G 3 4, μ+5 8 μ + ν +,μ ν +, The Meijer G function is available in oular comutational software such as MATLAB, Male and Mathematica B Average Symbol Error Rate The SER is the robability a transmitted symbol is incorrectly decoded at the receiver It is deendent on the instantaneous channel-state; the average SER obtained through averaging over all ossible channel-states characterizes the overall quality of the channel Under many modulation schemes this } can be reresented or aroimated as E Λ αq βλ This, given by α β e βt π t F Λ tdt, is a weighted sum of integrals of the form t 5 e βt G 3 4t, ρ dt μ, ν, Further simlification is ossible using [8, Eq 7343] t 5 e βt G 3 4t, ρ dt μ, ν, } L t 5 G 3 4t, ρ s μ, ν, β 4 β 5 G3 3 5,, ρ 9 β μ, ν, C Moment Generating Function Similarly, the mgf of Λ, defined as E } Λ e sλ s e st F Λ tdt, can be obtained [8, Eq 7343] as a weighted sum of terms of form e st s G 3 4t, ρ dt 4 μ, ν, s G 3 3,, ρ s μ, ν, The mgf itself is not a erformance metric Its imortance lies mainly in the fact that the mgf is a starting oint for comuting many numerical erformance results Incidentally, the mgf result gives the eact symbol error rate for differential hase shift keying modulation scheme D Diversity Order The df result is a weighted sum of e μ K ν terms Using [5, Eq 969], and with some maniulations, the first order aroimation of 4 can be obtained Corollary : Let λ,λ be the eigenvalues of a rank- non-central comle Wishart matri having n degrees of freedom, and non-centrality matri Ω whose eigenvalues are ω,ω ω ω > } The first order aroimation of the df of Λ in 3 given by f Λ a n, where e ω n+ω n! a, ω ω > n +ω e ω n+ω e ω ω ω n!, ω >ω > Proof: Omitted The above is sufficient to derive the diversity order and the coding gain [] Diversity order is n, as would have been with zero-forcing [] Similar results can be comuted for 6 and 7 V NUMERICAL RESULTS Figures through 5 verify the analytical results, for selected MIMO configurations, via semi-analytic Monte-Carlo simulation conducted with 6 simulation oints Fig corresonds to rank- non-centrality matrices having unequal eigenvalues ω 4,ω Close agreement of analytic and simulation results suorts the roof to establish the validity of 4 Although not as evident with the cdf or average SER, near-zero behavior of a df curve ie lim f Λ, reflects the diversity order Reaffirming the results derived in Section IV-D, the df curves for larger n sloe downward more raidly, at the vicinity of A non-centrality matri having all zero eigenvalues need not be a zero matri But a zero matri always has all eigenvalues zero Therefore 7 is identical to what already derived and verified for Rayleigh fading in [4, Thm ] The cdf result for non-centrality matrices having unequal eigenvalues ω 4,ω is shown in Fig 3 Shown in dotted lines are the curves corresonding to case ω,ω, which encomasses the Rayleigh fading scenario It highlights the fact that the resence of a secular comonent imroves the outage robability The increase of the diversity order with n is aarent in the sloe of the cdf as, which is the sloe of the outage curve as normalized equivalent SNR tends to infinity Fig 4 deicts the average SER for binary hase shift keying BPSK modulation, for a 3 MIMO system, and a noncentrality matri given by ω 6 K/,ω, thereby indirectly establishing the validity of 6 K is the Rician factor in db The rank- non-centrality matri of this case arises from directional gain of the transmit and receive antenna arrays [7] Corresonding curve for the Rayleigh fading scenario is also given The fact that increasing the Rician factor K imroves the error erformance can clearly be seen Fig 5 shows the average SER for BPSK modulation, for the case ω ω The asymtotic SER curves comuted by //$6 IEEE

4 This full tet aer was eer reviewed at the direction of IEEE Communications Society subject matter eerts for ublication in the IEEE Globecom roceedings n MIMO, with ω [4,] 3 MIMO, with ω [6 K, ] K 5 db K db K 5 db f Λ Average SER 3 4 Rayleigh fading Fig The df, analytic vs simulated, of Λ in 3 over a n MIMO system, in Rician fading modeled by a rank- non-centrality matri having eigenvalues [4, ] Average transmit SNR γ db Fig 4 The average SER for BPSK ie α,β b γ, analytic vs simulated, for a 3 MIMO system, in Rician fading modeled by a rank- non-centrality matri arameterized [7] by θ t o,θ r o,d} F Λ 3 4 n MIMO, with ω [4,] Average SER n MIMO, with ω [,] Average transmit SNR γ db Fig 3 The cdf, analytic solid lines vs simulated, of Λ in 3 over a n MIMO system, in Rician fading modeled by a rank- non-centrality matri having eigenvalues [4, ] The cdf for Rayleigh fading is shown in dotted lines ɛ i for n for n TABLE I VALUE OF i AT WHICH THE TERMS OF THE OUTER SUM OF 4 FALL BELOW ɛ e ω +ω IN MAGNITUDE FOR ω [, 4] Fig 5 The average SER for BPSK ie α,β b γ, analytic solid lines vs simulated, for a n MIMO system, in Rician fading modeled by a rank- non-centrality matri having equal eigenvalues of, forb Asymtotic SER curves are shown in dotted lines using are shown in dotted lines Diversity order, deicted by the asymtotic sloe is seen increasing with n As each eression 4, 6 has an infinite summation, it is interesting to find out their rate of convergence, which determines their usefulness Table I resents, for ω 4,ω and si combinations of n,, thevalueofi beyond which the absolute value of the i th term of 4 falls below a threshold ɛ e ω+ω From Table I one can deduce that the convergence rate imroves with n, and degrades with The observations deviating from this attern eg for ɛ 3, 5,n are not uneected; they arise when the first term of the sum itself is smaller than ɛ However, the observations raise a concern Convergence acceleration would be required for larger and large Rician factors at higher recision //$6 IEEE

5 This full tet aer was eer reviewed at the direction of IEEE Communications Society subject matter eerts for ublication in the IEEE Globecom roceedings VI CONCLUSION The eact robability density result in [4] has been etended for the non-central Wishart scenario, for all forms of the non-centrality matri Identities that rovide eact analytic results for the outage robability, the average SER, and others have been derived The results were verified through Monte- Carlo simulation for selected configurations Numerical results rovide further insights on the effect of the Rician factor Being associated with the distribution of the harmonic mean of the eigenvalues of a non-central Wishart matri, the results could be useful for other alications as well VII APPENDIX Proof Theorem : Letλ,λ be the eigenvalues of a rank- non-central comle Wishart matri having n degrees of freedom, and non-centrality matri Ω having eigenvalues ω,ω ω >ω } The joint df of λ,λ is given by Let Λ λλ λ +λ λ F Λ F λ λ f Λ Substituting from, where t + e ωω t f λ,λ λ λ t +, f λ λ dλ dt t + t f Λ ω ω ω ω n J ω,ω, n, n J ω,ω, n, n J ω,ω, n, n + J ω,ω, n, n, Ja, b, k, l l e t + k+l t l+ e t+ t I n at + I n bt + dt Using [5, 96], and then [9, 347], we get Ja, b, k, l e i + k + l + n where c i,j i i j i+k+l+n c j,i j i+k+l+n K n/+j+k, 3 a i+n/ b j+n/ i + n! j + n! i! j! Noting a, b ω,ω }; k, l n/,n/ }; and k + l n, Eqns and 3 can be simlified further, to obtain 4 where g i,j ω,ω is given by the case ω ω of 5 Case: ω ω : Eqn does not hold as is when ω ω ; but the limit ω ω does Similarly, 4 holds true in the limiting form lim ω ω g i,j ω,ω e ω ω lim ω i j ωj ωi ω ω ω ω We get the case ω ω of 5 using the L Hosital s rule ACKNOWLEDGMENT This work is suorted in art by the Alberta Ingenuity Fund through the icore ICT Graduate Student Award; and the Research Grant Numbers R & R of the National University of Singaore, Singaore REFERENCES [] G J Foschini, Layered sace-time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Labs technical journal, vol 5, no, 4 59, 996 [] A J Goldsmith and P P Varaiya, Caacity of fading channels with channel side information, IEEE Trans Inf Theory, vol 43, no 6, , Nov 997 [3] RUNabar,OOyman,HBölcskei, and A J Paulraj, Caacity scaling laws in MIMO wireless networks, in Proc Allerton Conference on Communication, Control, and Comuting, Monticello, IL, Oct 3, [4] T Haustein, C von Helmolt, E Jorswieck, V Jungnickel, and V Pohl, Performance of MIMO systems with channel inversion, in Proc IEEE Vehicular Technology Conference, VTC Sring, Birmingham, AL, May, [5] E Jorswieck, G Wunder, V Jungnickel, and T Haustein, Inverse eigenvalue statistics for Rayleigh and Rician MIMO channels, in MIMO: Communications Systems from Concet to Imlementations, IEE Seminar on, Dec, 3 [6] V Jungnickel, T Haustein, E Jorswieck, and C von Helmolt, A MIMO WLAN based on linear channel inversion, in MIMO: Communications Systems from Concet to Imlementations, IEE Seminar on, Dec, [7] H Lee, K Lee, B M Hochwald, and I Lee, Regularized channel inversion for multile-antenna users in multiuser MIMO downlink, in Proc Communications, International Conference on, Beijing, China, May 8, [8] V Jungnickel, T Haustein, V Pohl, and C von Helmolt, Link adatation in a multi-antenna system, in Proc Vehicular Technology Conference, VTC Sring, vol, Jeju, Korea, Ar 3, [9] C B Peel, B M Hochwald, and A L Swindlehurst, A vector-erturbation technique for near-caacity multiantenna multiuser communication-art I: channel inversion and regularization, IEEE Trans Commun, vol 53, no, 95, Jan 5 [] C Masouros and E Alsusa, Dynamic linear recoding for the eloitation of known interference in MIMO broadcast systems, IEEE Trans Wireless Commun, vol 8, no 3, , Mar 9 [] A M Tulino and S Verdu, Random Matri Theory and Wireless Communications, st ed, ser Foundations and Trends in Communications and Information Theory Hanover, MA: now ublishers, 4, vol [] I E Telatar, Caacity of multi-antenna Gaussian channels, Bell Laboratories, Lucent Technologies, Tech Re, Oct 995 [3] J Kermoal, L Schumacher, K Pedersen, P Mogensen, and F Frederiksen, A stochastic MIMO radio channel model with eerimental validation, IEEE J Sel Areas Commun, vol, no 6, 6, Aug [4] D Senaratne and C Tellambura, Performance analysis of channel inversion over MIMO channels, in Proc IEEE Global Communication Conference, Honolulu, HI, Dec 9 [5] M Abramowitz and I Stegun, Handbook of Mathematical Functions Dover Publications, Inc, New York, 97 [6] L C Andrews, Secial Functions of Mathematics for Engineers, nd ed New York, NY: McGraw-Hill, Inc, 99 [7] P J Smith and L M Garth, Eact caacity distribution for dual MIMO systems in Ricean fading, IEEE Commun Lett, vol 8, no, 8, Jan 4 [8] The Wolfram functions site Wolfram Research, Inc [Online] Available: htt://functionswolframcom [9] I Gradshteyn and I Ryzhik, Table of Integrals, Series, and Products, 7th ed Academic Press, [] Z Wang and G Giannakis, A simle and general arameterization quantifying erformance in fading channels, IEEE Trans Commun, vol 5, no 8, , Aug 3 [] J Winters, J Salz, and R Gitlin, The imact of antenna diversity on the caacity of wireless communication systems, IEEE Trans Commun, vol 4, no 34, 74 75, Feb //$6 IEEE