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approximative matrix inverse computations for very large mimo and applications to linear pre coding systems prabhu hemanth rodrigues joachim edfors ove rusek fredrik published in 2013 link to publication citation ...

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          Approximative Matrix Inverse Computations for Very-large MIMO and Applications to
          Linear Pre-coding Systems
          Prabhu, Hemanth; Rodrigues, Joachim; Edfors, Ove; Rusek, Fredrik
          Published in:
          [Host publication title missing]
          2013
          Link to publication
          Citation for published version (APA):
          Prabhu, H., Rodrigues, J., Edfors, O., & Rusek, F. (2013). Approximative Matrix Inverse Computations for Very-
          large MIMO and Applications to Linear Pre-coding Systems. In [Host publication title missing] (pp. 2710-2715).
          IEEE - Institute of Electrical and Electronics Engineers Inc..
          Total number of authors:
          4
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                  Approximative Matrix Inverse Computations for
                     Very-large MIMO and Applications to Linear
                                                      Pre-coding Systems
                                      Hemanth Prabhu, Joachim Rodrigues, Ove Edfors, and Fredrik Rusek
                                     Department of Electrical and Information Technology, Lund University, Sweden
                                      {Hemanth.Prabhu, Joachim.Rodrigues, Ove.Edfors, Fredrik.Rusek}@eit.lth.se
                Abstract—In    very-large   multiple-input    multiple-output   slower rate than inner products of the propagation vectors
             (MIMO) systems, the base station (BS) is equipped with very        with themselves when the number of antennas grow, i.e.,
             large number of antennas as compared to previously considered      the user channels are asymptotically orthogonal. In [3], mea-
             systems. There are various advantages of increasing the number     surements in a realistic propagation environment for large
             of antennas, and some schemes require handling large matrices
             for joint processing (pre-coding) at the BS. The dirty paper       array of antennas at a BS (up to 128 antennas at the BS
             coding (DPC) is an optimal pre-coding scheme and has a             and 26 different single antennas users) are performed. It
             very high complexity. However, with increasing number of           was shown that by using reasonably large antenna arrays it
             BS antennas, linear pre-coding performance tends to that of        is possible to decorrelate single user channels. Furthermore,
             the optimal DPC. Although linear pre-coding is less complex        in [4], residential area measurements for very-large MIMO
             than DPC, there is a need to compute pseudo inverses of
             large matrices. In this paper we present a low complexity          system were performed, showing linear pre-coding sum rates
             approximation of down-link Zero Forcing (ZF) linear pre-coding     of up to 98% of those achieved by dirty paper coding (DPC),
             for  very-large  multi-user   MIMO systems. Approximation          for BS to Mobile Station (MS) antenna ratios as low as 10.
             using a Neumann series expansion is opted for inversion of
             matrices over traditional exact computations, by making use
             of special properties of the matrices, thereby reducing the cost      Although there is a clear benefit of scaling up the number
             of hardware. With this approximation of linear pre-coding,         of BS antennas, including an almost (near) optimal linear pre-
             we can significantly reduce the computational complexity for        coding, the hardware cost and signal processing complexity
             large enough systems, i.e., where we have enough BS antenna
             elements. For the investigated case of 8 users, we obtain 90%      can be very high. When using linear pre-coding, such as
             of the full ZF sum rate, with lower computational complexity,      Zero-Forcing (ZF), we can operate at the above mentioned
             when the number of BS antennas per user is about 20 or more.       BS/MS antenna ratios around 10 and the main source of
                                   I. INTRODUCTION                              complexity for ZF pre-coding becomes the inverse of a K×K
                Multiple-Input Multiple-Output (MIMO) techniques for            matrix, where K is the number of users. The assumption of
             wireless communication offer high data rates and reliabil-         a significantly higher number of antenna elements does not
             ity through the utilization of multiple transmit and receive       affect the dimensions of this matrix, but it does offer the
             antennas. These techniques are becoming more mature and            opportunity to carry out the matrix inverse by much simpler
             have been incorporated in advanced standards like LTE (Long        means than outright inversion.
             Term Evolution) Release 10 [1] to meet the International
             Mobile Telecommunications-Advanced (IMT-A) requirements               There are various hardware implementations for matrix
             of gigabits-per-sec data rates. Basically, the more antennas the   inversion using different algorithms, QR-Gram Schmidt [5],
             transceivers are equipped with, the better performance can         QR-Givens Rotation [6], and Gauss-Jordan [7]. While these
             be obtained in terms of data rate, diversity (reliability) and     methods are generic and work well for any type of matrix,
             spectral efficiency.                                                we can exploit the special structure of matrices appearing in
                In [2], a Multi-User (MU) MIMO system with (an assump-          very-large MIMO to reduce the complexity of the linear pre-
             tion of) unlimited number of base station (BS) antennas in         coding and make it more hardware efficient. To meet these
             a multi-cell environment is investigated. It is shown that all     objectives, approximations (using Neumann series) of matrix
             the effects of uncorrelated noise and fast fading disappear, as    inversion is opted rather than computing the exact inverse.
             does the intra-cell interference. The assumption of an unlim-      We describe the general setting in which our pre-coding is
             ited number of BS antennas greatly simplifies the theoretical       assumed to operate, discuss the linear class of pre-coders, and
             analysis. However, it is obvious that in a practical system the    finally focus on complexity of the ZF pre-coder, when using
             numberofantennas cannot be arbitrarily large due to physical,      QR-Gram Schmidt and Neumann series expansion to perform
             cost, and power constraints.                                       matrix inversion. We derive and compare complexity both in
                The theoretical analysis in [2], assumes that inner products    terms of the number of operations and the energy required to
             between propagation vectors of different users grow at a           perform the computations.
                                                                                  where P is a K × K diagonal matrix for power allocation.
                                                                                  Thesumrate is maximized by optimizing the power allocation
                                                                                  under the constraint that Tr(P) = 1, where Tr(·) is the trace
                           Pre-Coding                                             operator.
                                                                                    Although optimal sum rate is achieved by DPC, this
                            (Base Station)                                        approach is too resource expensive to be implemented in
                           Central Processing                                     hardware and is used as a benchmark for ZF, Minimum Mean
                                     Unit
                                                                                  Square Error (MMSE) and low complexity approximations of
                                                                                  ZF. The pre-coding matrix F can be decomposed as
                                                                                                                1     √
                                                                                                         F = √ W P;                             (4)
                                                                                                                 γ
             Fig. 1: System model of a MU-MIMO system with M-antenna
             base station and K single-antenna mobiles stations.                  where W represents a particular linear pre-coding algorithm,
                                                                                  and γ = ||Wp(P)||2 , is a power normalization factor, where
                                                                                                       F
                                                                                  || • ||F is Frobenius Norm.
                               II. SYSTEM DESCRIPTION                             A. ZF pre-coding scheme
                The system model and the pre-coding in the following                ZF linear pre-coding transmits user signals towards the
             section is in line with the corresponding description in [8].        intended user with nulls steered in the direction of other users.
             A MU-MIMO system model consists of a BS equipped with                The ZF pre-coder is given as
             Mantennas, which is simultaneously serving K single users
             antenna, as shown in Fig.1. The received vector y of size                                      W =H†;                              (5)
                                                                                                              ZF
             K×1is described as                                                             †       H       H −1
                                          √                                       where H     = H (HH )            is the pseudo-inverse of the
                                     y =    ρHz+n;                         (1)    channel matrix H. A perfect CSI at the transmitter and nulling
                                                                                  makes this scheme interference free, and the sum rate is given
             where H is the K ×M propagationmatrix of complex-valued              as
                                                                                                                K                 
             channel coefficients, z is an M × 1 transmit vector, and n is                                      X              ρP
                                                                                              C = max              log    1+ i :                (6)
             an additive noise vector with Independent and Identically Dis-                     ZF                    2         γ
                                                                                                      Tr(P)=1 i=1
             tributed (IID) zero-mean and unit-variance complex Gaussian
             random variables. The scalar ρ is a measure of the Signal-to-        As the number of BS antennas M increases, H tends to
             Noise Ratio (SNR) of the link, which is proportional to the          have nearly orthogonal columns as the terminals are not
             transmitted power divided by the noise-variance. Furthermore,        correlated due to their physical separation. This assures that
             it also absorbs various normalizing constants. The total trans-      the performance of ZF pre-coding will be close to that of
             mit power is normalized and independent of the number of             optimal DPC pre-coding. However, ZF pre-coding requires
             antennas M, the transmit vector z satisfies E{||z||2} = 1.            computation of the pseudo-inverse (in (5)), which requires the
                The pre-coding process at the transmit side is specified as        computationally expensive inversion of a K ×K matrix.
                                         z = Fx;                           (2)    B. MMSE pre-coding scheme
                                                                                    MMSE pre-coding can trade interference reduction for
             where F is a M ×K pre-coding matrix, x a K × 1 vector                signal power inefficiency. The MMSE pre-coder is given as
             containing user symbols, as described in [4].
                                                                                                                              
                Although the very-large MU-MIMO model is similar to a                           W        =HH HHH+αI −1 ;                        (7)
                                                                                                  MMSE
             standard MIMO model, the increased number of BS antennas             where α = K=ρ. At low SNRs (large α) the MMSE ap-
             has several consequences. Things that were random before,                                                                          H
                                                                                  proaches a Matched Filter (MF) pre-coder, i.e., W        =H ,
             now start to look deterministic. For example, the distribution                                                           MF
             of the singular values of the channel matrix approaches a            and at high SNRs (low α) it approaches the ZF pre-coder.
             deterministic function [9]. Another observed property is that        C. Low Complexity Pre-Coding
             very wide (or tall) matrices tend to be very well conditioned.
                                                                                    A problem with both ZF and MMSE pre-coding is the
                          III. LINEAR PRE-CODING SCHEMES                          inverse operation of the K ×K matrix. Since the complexity
                The optimal sum rate in the downlink of a MU-MIMO                 for both linear pre-coders is similar (when α is not large in
             system with prefect channel state information (CSI) can be           (7)), in this paper we analyse impact of low complexity (ap-
             achieved by the interference pre-subtraction coding technique        proximations) only on ZF pre-coder. A standard and expensive
             called DPC [10]. The optimal sum rate is given as                    approach would be to compute the exact inverse of the matrix
                                                                                  Z(,HHH)in
                       C       = max log det(I +ρHHPH);
                         DPC                 2                             (3)                        †      H      H −1        H     −1
                                 Tr(P)=1                                                   W =H =H (HH ) =H (Z) :                               (8)
                                                                                             ZF
             However, as the number of BS and MS antennas (M and                     The modification is described as follows. The scalar multi-
             K) increases, the eigenvalues of the matrix Z converges to           plication by δ=(M + K) in (12) is represented as a diagonal
             a fixed deterministic distribution known as the Marchenko-            matrix
             Pastur distribution. Now following the analysis in [8], the                                D = δ I :
                                                                                                          MP     M+K K
             largest and the smallest eigenvalues of Z converge to
                                                                              Using this notation, (12) is rewritten as
                                      1    2                        1    2
                 λ    (Z) −→    1+√          ; λ    (Z) −→    1−√          ;                              L
                  max                           min
                                       β                             β                            −1     X                   n
                                                                                                Z     ≈      (I   −D Z) D ;                    (13)
                                                                                                               K      MP         MP
             where (β = M=K), as M and K grows to infinity. By scaling                                    n=0
             the Z matrix with a factor  β , the eigenvalues are found          the accuracy of the approximation,for a given number of terms
                                            1+β
             in the region                                                        (L), depend on the size of the eigenvalues of (I − DMPZ).
                                                       √                      Thesmaller their magnitude, the faster the convergence. Given
                          λ         β Z −→ 1+2 β                  ;               this, we want to pre-condition our matrix Z so that it will lead
                           max    1+β                   1+β
                                                                                  to a fast convergence for a finite M and K system.
                                                       √                         Now, assume that we want to pre-condition it with a diag-
                          λ         β Z −→ 1−2 β                  :         (9)
                            min   1+β                   1+β                       onal matrix D, with non-zero diagonal entries. In principle,
             Hence,theeigenvaluesofI −β=(1+β)Z = I −Z=(M+K)                       we would like to calculate the eigenvalues of (I − DZ) and
                                   √      K       √           K                   optimize D so that the magnitudes of the eigenvalues are as
             lie in the range [−2    β=(1+β);2 β=(1+β)], where I             is   small as possible. This, however, is a complex and non-trivial
                                                                          K
             an K×K identity matrix. By asymptotically increasing β, the          task. We will therefore use Gershgorins circle theorem [12] to
             eigenvalues of I    −Z=(M+K)lie in the range                         derive an upper bound of the magnitude of the eigenvalues.
                               K
                          "       √   √ #                                     By keeping this bound small, by selecting D, we can also
                    lim       −2     β    ;  2    β       −→ [−0;0]:       (10)   guarantee that the magnitude of the eigenvalues are small.
                  β→+∞           1+β           1+β                                   In this derivation of the ”optimal” D we will assume that the
                                                                                  Hermitian matrix Z = HHH is diagonallydominant,meaning
             Therefore, as β grows, the faster is the convergence of              that the magnitude of the diagonal elements z      are larger than
                                                                                                                                  ii
                                                    
                                              1        n                          the sum of the magnitude of the off-diagonal elements in the
                             lim    I  −           Z     ≃0 :              (11)                                                    P
                            n→∞ K M+K                        K                    same row, zij;i 6= j, namely that |zii| >              |zij|. The
                                                                                                                                     i6=j
             It is known that if a matrix satisfies (11), its inverse can be       largest magnitude of any eigenvalue of (I − DZ) is upper
                                                                                  bounded by
             expressed by Neumann series [11] as
                                          L                                                                                X !
                                                                  n
                                         X                                              max|λ | ≤ max |1−d z |+d                 |z |   ;      (14)
                         −1        δ                     δ                                      i                  i ii     i      ij
                       Z    ≈                  I   −          Z     ;      (12)           i            i
                                M+K             K     M+K                                                                    i6=j
                                         n=0
                                                                                  and under the condition of a diagonally dominant Z, the
             with equality when L grown to infinity, and δ < 1 is a                smallest upper bound is obtained if d         = 1=z . For this
             attenuation factor introduced, since for finite M and K the                                                      i         ii
                                                                                  selection of D we also have that max|λ | < 1, which
                                                                                                                                  i
             eigenvalues of channel realizations may lie outside the range                                                   i
             specified in (9). For a feasible implementation of a matrix           guarantees convergence of the Neumann series. Hence, our
             inversion using Neumann series the number of iterations (L)          final approximation of the inverse of a diagonally dominant
             needs to be finite (or small).                                        Z is a matrix D = diag(1=z11;1=z22;::::;1=zkk), the inverse
                Theinverseof Z is approximatedby a summation of powers            can be expressed using Neumann series as
             of a matrix (or matrix multiplications) (12), which has a                                       L
                                                3                                                     −1    X                n
                                                                                                    Z    ≈      (I   −DZ) D:                   (15)
             complexity order O((L − 1) · K ). Although the complexity                                            K
             order can be equal or higher (depending on L) than computing                                   n=0
             the exact inverse (direct inversion, QR based etc), matrix              Afast (or accelerated) way to compute the series (15) and
             multiplications are preferable in hardware compared to exact         (12), up to L = 2P −1 terms, where P is an integer, is to use
             inversion.                                                           the identity
                The convergence of (11) is based on the fact that the
                                                                                            L                       P−1                      
             eigenvalues lie in the range given by (9) as M and K grows                    X                n         Y                     n
                                                                                     −1                                                    2
                                                                                   Z    ≈      (I   −DZ) D=               (I +(I −DZ) ) D;
             asymptotically. However, for practical systems with finite M                          K
                                                                                           n=0                        n=0
             and K the eigenvalues may lie outside this range. In addition                                                                     (16)
             to what is described in [8], we introduce one modification of         which leads to a numerical complexity proportional to the
             the Neumann series inversion. It is based on the fact that the       logarithm of the number or terms in the truncated series. In
             closer the eigenvalues of our matrix are to 1, the faster the        terms of number of matrix multiplications, the brute force
             convergence of the series in (12).                                   computation of the inverse using (15) (or (12)) would require
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...Approximative matrix inverse computations for very large mimo and applications to linear pre coding systems prabhu hemanth rodrigues joachim edfors ove rusek fredrik published in link publication citation version apa h j o f pp ieee institute of electrical electronics engineers inc total number authors general rights unless other specific re use are stated the following apply copyright moral publications made accessible public portal retained by or owners it is a condition accessing that users recognise abide legal requirements associated with these may download print one copy any from purpose private study research you not further distribute material profit making activity commercial gain freely url identifying read more about creative commons licenses https creativecommons org take down policy if believe this document breaches please contact us providing details we will remove access work immediately investigate your claim department information technology lund university sweden eit ...

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