Subspace Estimation and Decomposition for Hybrid Analog-Digital Millimetre-Wave MIMO systems

Transcription

1 Subspace Estimation and Decomposition for Hybrid Anaog-Digita Miimetre-Wave MIMO systems Hadi Ghauch, Mats Bengtsson, Taejoon Kim, Mikae Skogund Schoo of Eectrica Engineering and the ACCESS Linnaeus Center, Roya Institute of Technoogy (KTH Department of Eectronic Engineering, City University of Hong Kong Abstract In this work, we address the probem of channe estimation and precoding / combining for the so-caed hybrid miimeter wave (mmwave MIMO architecture. Our proposed channe estimation scheme expoits channe reciprocity in TDD MIMO systems, by using echoing, thereby aowing us to impement Kryov subspace methods in a fuy distributed way. The atter resuts in estimating the right (resp. eft singuar subspace of the channe at the transmitter (resp. receiver. Moreover, we aso tacke the probem of subspace decomposition whereby the estimated right (resp. eft singuar subspaces are approximated by a cascade of anaog and digita precoder (resp. combiner, using an iterative method. Finay we compare our scheme with an equivaent fuy digita case and concude that a reativey simiar performance can be achieved, however, with a drasticay reduced number of RF chains times ess (i.e., massive savings in cost and power consumption. Keywords Miimeter wave MIMO systems, sparse channe estimation, hybrid architecture, anaog-digita precoding, subspace decomposition, Arnodi iteration, subspace estimation. I. INTRODUCTION Communication in the miimeter wave (mmwave band is one of the strongest candidates to answer the fundamenta chaenge of the exponentiay increasing demand for data rates, in ceuar networks. It has the distinct advantage of expoiting the arge unused spectrum bands, thereby offering up to 200 times more spectrum than conventiona ceuar systems. Furthermore, the resuting antenna size/spacing at such frequencies is extremey sma, thus impying that a arge number of such antennas can be assumed at both the base station and the user (e.g. tens to hundreds. The socaed hybrid architecture, first reported in [1], [2], and ater studied in [3], [4], has been receiving increasing interest. In the atter, the number of RF chains at the transmitter and receiver is drasticay smaer than the number of antennas. Moreover, both the precoding and combining are done in two stages, digita and anaog. However, many fundamenta questions surrounding the atter architecture have to be answered, namey, how to estimate the arge mmwave channe, and design the digita / anaog precoders and combiners. Though an agorithm was proposed in [4] for that purpose, the atter requires knowedge of the number of propagation paths (i.e. propagation environment, it exhibits reativey eevated compexity, and buids an estimate of the entire channe that is then used to design the precoding / combining. Rather than estimating the entire channe, our proposed Kryov subspace method expoits the reciprocity of the channe in TDD MIMO systems, and directy estimates the right (resp. eft singuar subspace at the transmitter (resp. receiver - required for optima transmission. Moreover, we propose an agorithm for subspace decomposition, whereby each of the estimated subspaces is approximated by a cascade of the digita and anaog precoder, whie satisfying the constraints of the hybrid architecture. We underine the fact that this proposed approach is perfecty appicabe to conventiona MIMO systems, i.e. fuy digita, as we. We note that some parts of this works are based on [5], therefore some discussions / derivations / proofs / agorithms are omitted here. In the foowing, we use bod upper-case etters to denote matrices, and bod ower-case denote vectors. Furthermore, for a given matrix A, [A] i:j denotes the matrix formed by taking coumns i to j, of A, A 2 F its Frobenius norm, A its determinant, A its conjugate transpose. [A] i,j = a i,j denotes eement (i, j in a matrix A, and [a] i eement i in a vectors a. Whie I n denotes the n n identity matrix, 1 n denotes the n 1 vector of ones. Finay, we et {n} {1,..., n}, and S p,q = { X C p q X ij = 1/ p, (i, k {p} {q} }. II. SYSTEM MODEL Assume a singe user MIMO system with M and N transmit antennas at the BS and MS, respectivey, where each is equipped with r RF chains, and sends d independent data streams (d r min(m, N. The downink (DL received signa, after fitering, is given by, x (r = U W HF Gx (t + U W n (r (1 where H C N M is the compex channe - assumed to be sowy bock-fading, F C M r is the anaog precoder, G C r d the digita precoder, x (t is the d-dimensiona transmit signa with covariance matrix E[x (t x (t ] = (P s /di d and n (r is the AWGN noise at the receiver, with E[n (r n (r ] = σri 2 N. Simiary, W C N r and U C r d are the anaog and digita combiner, respectivey. In addition to requiring both the anaog precoder and combiner to have constant moduus eements, i.e., F S M,r and W S N,r (since the atter represent phase shifters, a tota power constraint must sti satisfied, i.e., F G ρ 2 d (where we assume that ρ = 1 w..o.g.. 1 We aso assume a TDD system where channe reciprocity hods, and denote the SVD of H as, [ ] [ ] Σ1 0 Γ 1 H = [Φ 1, Φ 2 ] = Φ 0 Σ 1 Σ 1 Γ 1 + Φ 2 Σ 2 Γ 2 (2 2 Γ 2 where Γ 1 C M d and Φ 1 C N d are unitary, and Σ 1 C d d is diagona with the d-argest singuar vaues of H (reca that Γ 1 Γ 2 = 0 and Φ 1 Φ 2 = 0. In view of carifying the aim of our work, we present the foowing intuitive resut. Proposition 1. Given the signa mode in (1, the optima anaog and digita precoder / combiner that maximize the achievabe user rate are such that F G = Γ 1 and W U = Φ 1 (assuming waterfiing power aocation is performed over the resuting effective channe. 1 Simiary, expoiting channe reciprocity, the upink received signa is given by x (t = G F H W Ux (r +n (t where y (t is the M-dimensiona signa at the transmitter and n (t is the AWGN noise at the transmitter, such that E[n (t n (t ] = σ 2 t I N

2 Though the atter resut is expected, it is reminiscent of the we-known optima transmission strategy for cassica MIMO, where the transmitter uses right singuar vectors, Γ 1, for precoding, and receiver uses eft singuar vectors, Φ 1, for combining: the above proposition suggests that this structure sti maximizes the achievabe rate in the hybrid architecture, provided one is abe to approximate F G by Γ 1, and W U by Φ 1 (and assuming that waterfiing is empoyed. Since no a priori CSI is assumed to be avaiabe at neither the transmitter nor the receiver, our aim is firsty to propose an agorithm to estimate Γ 1 at the transmitter, i.e. Γ 1, and Φ 1 at the receiver, i.e. Φ 1. This done, we shed ight on the probem of subspace decomposition, and present an agorithm for approximating the estimated subspaces, Γ 1 by F G and Φ 1 by W U. We describe our scheme in the context of conventiona MIMO systems, i.e. fuy digita, and ater extend it to the hybrid architecture. III. EIGENVALUE ALGORITHMS AND SUBSPACE ESTIMATION With this mind, the aim of subspace estimation agorithms is to obtain Γ 1 at the transmitter (keeping in mind that Γ 1 is nothing but the dominant eigenvectors of H H, and Φ 1 at the receiver. We note that eigenvaue agorithms such as the Power Method or Subspace Iteration, we known from numerica anaysis, were used in [6] for that same purpose. In this work we resort to Kryov subspace methods, to achieve the atter goa. One such method is the we-known Arnodi Iteration (the variant we use here is detaied in [7] whereby one starts with a random vector q 1, and recursivey buids Q m [q 1,..., q m ] C M m (m M such that Q m(h HQ m = T m, Q mq m = I m where T m C m m is an upper Hessenberg matrix, and the resuting Q m is an orthonorma basis for the Kryov subspace in question. Consequenty, the eigenpairs of T m are eigenpairs of H H, and the desired subspace Γ 1 can be computed by finding the eigenpairs of T m - which can be found efficienty. A more carefu examination quicky reveas that impementing the atter method in a distributed way requires the transmitter to have the sequence {H Hq 1,, H Hq m }. Without any prior channe knowedge, this can be accompished using the echoing mechanism that was empoyed in [6], whereby the transmitter sends q in the DL, and it is echoed back by the receiver using Ampify-and-Forward (A-F, as foows, //DL : s = Hq + w (r //UL : p = H s + w (t = H Hq + H w (r + w (t (3 After the echoing phase, the transmitter has a noisy estimate, p, of H Hq, as seen from (3. We note that incorporating noise, i.e., w (r and w (t in the agorithm formuation, aows us to extend the origina formuation of the Arnodi Iteration, to account for externa distortion, and provide bounds on the estimation error (further detais are provided in [5], where we derive bounds on the estimation error of the subspaces in question. Steps 2.a - 3.a foow the conventiona Arnodi iteration. Finay, computing the estimate of Γ 1 (steps 4.a - 4.c foows immediatey from the fact that the eigenvectors of T m, at the output of the Arnodi iteration, approximate the Ritz eigenvectors of H H [7]. The above steps are summarized in the Subspace Estimation using Arnodi Iteration (SE-ARN procedure beow. Subspace Estimation using Arnodi Iteration (SE-ARN procedure Γ 1 = SE-ARN (H, d Set m (m M; Random unit-norm q; Q = [q 1 ] for = 1, 2,..., m do // transmitter-initiated echoing: estimate H Hq 1.a s = Hq + w (r 1.b p = H s + w (t // Gram-Schmidt orthogonaization 2.a t m, = q mp, m = 1,..., 2.b r = p m=1 q m t m, 2.c t +1, = r 2 // Update Q 3.a Q = [Q, q +1 = r /t +1, ] end for // Compute Γ 1 4.a 4.b 1 T m = Θ Λ Θ Γ 1 = Q m Θ 1:d 4.c Γ 1 = qr( Γ 1 end procedure IV. HYBRID PRECODING FOR MMWAVE MIMO SYSTEMS In this section we extend the previous framework to fit the hybrid architecture, and highight the major chaenges. We first start by presenting some preiminaries that wi ater be used throughout this section. A. Preiminaries: Subspace Decomposition We assume that d of the r avaiabe RF chains are used, i.e., F C M d and G C d d (more on that, ater in this section. In conventiona MIMO systems, once the estimates, Γ 1 and Φ 1, are obtained they can immediatey be used as transmit and receive fiters, respectivey. However, in the case of the hybrid architecture, as Proposition 1 suggests, Γ 1 needs to be expressed as F G (moreover Φ 1 needs to be expressed as W U, but we restrict the discussion to the transmitter, for brevity, whie satisfying both the maximum power and hardware constraints. Using the Frobenius norm a distance metric - a rather simpe engineering heuristic, we formuate the subspace decomposition (SD probem as foows, min h 0(F, G = Γ 1 F G 2 F F, G s. t. h 1 (F, G = F G 2 F d (4 F S M,d 1 Bock Coordinate Descent for Subspace Decomposition: Due to the couped nature of (4, Bock Coordinate Decent (BCD stands out as an attractive approach, whereby F and G are iterativey updated, such that the sequence {h 0 (F k, G k } k is non-increasing. We wi subsequenty show that the updates resuting from the BCD method impicity enforce a power constraint (consequenty, the atter can be dropped from (4. Reaxing the hardware constraint on F, we first fix G and optimize F, and vice versa. Note that that resuting sub-probems probems are instances of a non-homogeneous convex QCQP that can be soved using standard Lagrangian techniques, to yied the foowing soutions, F k+1 = Γ 1 G k (G k G k 1 (5 G k+1 = (F k+1 F k+1 1 F k+1 Γ 1 (6 Note that our earier assumption that ony d RF chains are used, ensures that (G k G k in (5 is invertibe. Moreover, using

3 architecture this impies that both transmitter and receiver need to be abe to approximate any digita beamforming vector q, by F G, where f is a vector and g is a scaar. When d = 1, it can be shown that (4 reduces to the probem beow. Lemma 1. Consider singe dimension SD probem, { min f, g h o(f, g = f 2 2 g 2 2gR(f γ 1 s. t. [f] i = 1/ M e jφi, i where g R + and [ γ 1 ] i = r i e jθi. Then the probem admits a gobay optimum soution given by, [f ] i = 1/ M e jθi, i and g = γ 1 1 / M (7 Fig. 1: Average subspace distance Γ 1 F G 2 F simpe manipuations (and assuming w..o.g. that Γ 1 2 F it can be shown that = 1 F k+1 G k+1 2 F d, k impying that the power constraint is indeed enforced. Reca that F k+1 in (5 does not necessariy satisfy the hardware constraint. It can be shown that its (unique Eucidean projection on the set S M,d, i.e., F k+1 Π S [F k+1 ] = argmin U F k+1 2 F U S M,d is given by [ F k+1 ] m,n = (1/ Me jφm,n, (m, n, where φ m,n arg([f k+1 ] m,n. The corresponding agorithm, Bock Coordinate Descent for Subspace Decomposition (BCD-SD is shown beow. Note that the atter projection makes convergence caims extremey difficut to make. We note that the Bock Coordiate Descent for Subspace Decomposition (BCD-SD procedure [F, G] = BCD-SD ( Γ 1, ρ Start with arbitrary G 0 for k = 0, 1, 2,... do F k+1 Π S [ Γ 1 G k (G k G k 1 ] G k+1 (F k+1 F k+1 1 F k+1 Γ 1 end for end procedure authors in [3] formuated the same probem as (4 after a series of approximations to the mutua information, and proposed a variation on the we-known Orthogona Matching Pursuit (OMP, whereby the coumns of F are iterativey recovered in a greedy manner. We thus compare its average performance with our proposed method, for a case where Γ 1 C M d is such that M = 64, r = 10 (for severa vaues of d. The reason for the massive performance gap in Fig. 1 is that our proposed method attempts to find a ocay optima soution to (4 (though this cannot be shown due to the projection step. Moreover, OMP is hated after r iterations, since it recovers the coumns of F one at at time, whereas our proposed method runs unti reaching a stabe point. 2 Beamforming case: The case where d = 1 in (4 is of particuar importance. Reca that echoing received vectors is the mechanism at the heart of our approach. For the hybrid Proof: Refer to [5] for proof Moreover, the approximation error e γ 1 fg is such that, [e] i = r i γ 1 1 /M e jθi, i {M}. (8 B. Echoing in Hybrid Architecture 1 Motivation: For the sake of simpicity, we negect noise from our formuations, and focus on other sources of distortion. Reca that the proposed scheme requires {H Hq } m =1 at the transmitter. Though this can be easiy done in conventiona MIMO systems (using the transmitter-initiated echoing mechanism in (3, the A-F step required by the receiver is not possibe in the hybrid architecture. 2 With this in mind, one can naivey attempt to emuate the A-F step in transmitterinitiated echoing, described in (3, as foows: decompose q at the transmitter, into f g, i.e. q = f g +e, and send f g over the DL; processes the received signa in the downink, with the anaog combiner, i.e., s = W (H f g ; appy same fiter to process the transmit signa in the UL, i.e., W s. Finay, the received signa the the transmit antennas is processed with the anaog precoder F. The resuting signa at the transmitter is, p = F H W W H(q e (9 It is cear from (9 that p is no onger a good estimate of H Hq. Firsty, the fact that the signas at the receiver (resp. transmitter need to processed with the anaog combiner W (resp. precoder F distorts the desired estimate of H Hq. Moreover, the appication of F C M r in (9 impies that the dimension of the estimate is reduced from M to r. We dub such distortions Anaog-Processing Impairments (API. In addition, the estimate of H Hq is further distorted by the decomposition error, e, emanating from decomposing q at the transmitter (which we refer to a Decomposition-Induced Distortion (DID The above impairments are a by-product of the constraints imposed by the hybrid architecture, and wi individuay be investigated and addressed. 2 Canceation of Anaog-Processing impairments: Our proposed method for mitigating anaog-processing impairments (API reies on the simpe idea of taking mutipe measurements at both transmitter and receiver, using carefuy chosen anaog precoders / combiners, such that W W and F F approximate an identity matrix. In the DL, q is approximated by f g, and f g is sent over the DL channe 3, K r times (where K r = N/r, each ineary 2 Reca that digitay processing the baseband signa is ony possibe after the appication of the anaog precoder / combiner (and possiby the digita precoder / combiner [3]. 3 Instead of using ony one RF chain to send f g over the DL, we use a the avaiabe d RF chains, thereby resuting in an array gain factor of d. We aso make use of this observation in the UL sounding.

4 processed with an anaog combiner {W,k C N r } Kr k=1, to obtain the digita sampes {s,k } Kr k=1. Moreover, the anaog combiners are taken from the coumns of a Discrete Fourier Transform (DFT matrix, i.e, [W,1,..., W,Kr ] = D r, (10 where D r C N N is a normaized N N DFT matrix. The same anaog combiners, {W,k } k, are used to ineary combine {s,k } k, to form s. The above steps are summarized in the Repetition-Aided (RAID Echoing procedure beow. Combining the above equations, we rewrite s as, ( Kr s = W,k W,k H(d f g = dh f g (11 k=1 where equaity foows from the fact that {W,k } k are coumns of a DFT matrices. Note that the effect of processing the received signa with the anaog combiner has been competey suppressed. The exact same process is used in the UL: s Repetition-Aided (RAID echoing // DL phase q = f g + e (t s,k = W,k H(d f g, k {K r N/r} s = K r k=1 W,k s,k // UL phase s = w ũ + e (r z,m = F,m H (d w ũ, m {K t M/r} p = K t m=1 F,m z,m is first decomposed into w ũ, i.e. s = w ũ + e (r, d RF chains are used to send it over the UL, K t times (where K t = M/r, and each observation is ineary processed with an anaog precoder {F,m C M r } Kt m=1, where the atter is taken from the coumns of a DFT matrix. The process for the UL is summarized in the RAID echoing procedure. We combine the atter steps to rewrite p as, ( Kt p = F,m F,m H (d w ũ = dh w ũ (12 m=1 Thus, the output of the RAID procedure is as foows, p = dh w ũ = dh ( s e (t = dh (dh f g e (t = d 2 H Hq d 2 H He (t dh e (r (13 where e (t (resp. e (r is the transmitter-side DID (resp. receiver-side DID resuting from decomposing the digita transmitted signa at the transmitter (resp. receiver. It is quite insightfu to compare p in the atter equation with (9. We can ceary see that impairments originating from processing the received signas with both W and F, have competey been suppressed: in (13, p indeed is the desired estimate, i.e., H Hq, corrupted by distortions. Note that empoying this process reduces the hybrid architecture into a conventiona MIMO channe: any transmitted vector in the DL, ( f g, can be received in a MIMO-ike fashion, as seen from (11, at a cost of K r channe uses (the same hods for the UL, as seen from (12. 3 Decomposition-Induced Distortion (DID: We investigate the effect of transmitter-side DID, e (t, and receiver-side DID, e (r, that distort p, at the output of the RAID procedure in (13. It can be easiy verified that e (t ony distorts the magnitude of H Hq, not its phase, and consequenty its effect is minima and can be negected. Since this caim cannot be made for the receiver-side DID, e (r, we provide a mechanism for mitigating the atter, however, at the cost of additiona communication overhead. The detais of the atter technique are further eaborated in [5], but omitted here due to space imitations. C. Proposed Agorithm We now formuate our agorithm for Subspace Estimation and Decomposition (SED in the mmwave architecture (shown in Agorithm 1: estimates of the right / eft singuar subspaces, Γ 1 and Φ 1, can be obtained by using the SE-ARN procedure (Sect. III, keeping in mind that the echoing phase (Steps 1.a and 1.b is now repaced by the RAID echoing procedure (Sect. IV-B2. Then, the muti-dimensiona subspace decomposition procedure, BCD-SD in Sect. IV-A2, is then used to approximate each of the estimated singuar spaces, by a cascade of anaog and digita precoder / combiner. Note that the tota communication overhead required by the agorithm is Ω = 2m(M + N/r channe uses. Agorithm 1 Subspace Estimation and Decomposition (SED for Hybrid Architecture // Estimate Γ 1 and Φ 1 Γ 1 = SE-ARN (H, d Φ 1 = SE-ARN (H, d // Decompose Γ 1 and Φ 1 [F, G ] = BCD-SD ( Γ 1, ρ [W, U ] = BCD-SD ( Φ 1, ρ V. NUMERICAL RESULTS Though our approach is not restricted to any particuar channe mode, for our numerica evauations, we adopt the prevaent channe mode in the mmwave iterature, where ony L scatterers are assumed to contribute to the received signa (an inherent property of their poor scattering nature, H = MN L L i=1 β i a r (χ (r i a t(χ (t i (14 where χ (r i and χ (t i are anges of arriva at the MS, and anges of departure at the BS (AoA / AoD of the i th path, respectivey (both assumed to be uniform over [ π/2, π/2], β i is the compex gain of the i th path such that β i CN (0, 1, i. Finay, a r (χ (r i and a t (χ (t i are the array response vectors at both the MS and BS, respectivey (assumed to be uniform inear arrays. We assume that the number of RF chains scaes with the number of antennas, e.g., M/r = 8 and N/r = 4. Though it remains to be seen whether it is achievabe, we use the foowing user rate as a our metric [8], R = og 2 I d + Ps U W HF GG F H W U(U W W U 1 dσr 2 where H is the channe estimate resuting from our proposed method. Note that an agorithm for mmwave MIMO channe estimation was proposed in [4]. However, since many of its underying detais are not present in the paper, we opt to use a simpe Independent Sounding sounding scheme: the anaog precoder and combiner are first seected by exhaustivey sounding DFT codebooks at both transmitter and receiver, then the digita precoder and combiner are chosen as right and eft singuar vectors of the effective channe estimate. We adjust the number of iterations for our scheme, m, such that the resuting

5 mance of the independent sounding scheme is highy unstabe, and very much dependent on the size of the codebooks. We next investigate the scaabiity of our proposed scheme, by scaing up M and N (assuming N = M/2 for simpicity, whie keeping everything ese fixed, i.e. d = 2, m = 6, and consequenty Ω = 144. Fig 3 ceary shows that the agorithm are abe to harness the significant array gain inherent to arge antenna systems, whie keeping the overhead the same. Though the performance might not be good enough to offset the overhead, for the case, it surey does for the The key to this impressive resut is to have M/r and N/r fixed, as M, N increase. Consequenty, our resuts indeed suggest that the performance achieved by conventiona MIMO systems can sti be maintained in the hybrid architecture, with a drasticay reduced number of RF chains ( 4 to 8 times ess, thereby resuting in massive savings in terms of cost and power consumption. Fig. 2: Average user rate of proposed schemes over SCM channes (M = 64, N = 32, m = 2d Fig. 3: Average user rate for different M, N (N = M/2, d = 2, L = 4, m = 6 communication overhead is simiar to that of the benchmark scheme. In addition, we use a perfect CSI, fuy digita case (i.e. the capacity of equivaent MIMO channe with perfect CSIT / CSIR as an upper bound. A curves are averaged over 500 channe reaizations. In view of having a more reaistic performance evauation, we adopt the Spatia Channe Mode (SCM detaied in [9], and modify its parameters to emuate mmwave channes described above (where a sma vaue of Ω is desired. Fig. 2 shows the user rate of such a system, with M = 64, N = 32, m = 2d, for severa vaues of d (each resuting in different vaues for Ω. We can ceary see that our scheme yieds a significanty high throughput in this reaistic simuation setting (especiay for d = 3, whie sti keeping the overhead at a reativey ow eve. Interestingy, we can see that the benchmark scheme offers a surprisingy poor performance, except for the case when d = 3 (since in this case, the receiver codebook consists of the entire DFT matrix. This does suggest that the perfor- VI. CONCLUSION We proposed an agorithm for estimating the right and eft subspaces for arge MIMO systems, expoiting echoing and the inherent reciprocity in TDD MIMO channes. We first detaied the agorithm within the context of conventiona MIMO systems, and then extended it to fit the many operationa constraints of the hybrid architecture. Moreover, we highighted the importance of the subspace decomposition probem, and provided an iterative agorithm for that purpose. Finay, our simuations showed the performance of our proposed approach it very simiar to their fuy digita counterpart. REFERENCES [1] A. Sayeed and N. Behdad, Continuous aperture phased MIMO: Basic theory and appications, in Communication, Contro, and Computing (Aerton, th Annua Aerton Conference on, pp , Sept [2] V. Venkateswaran and A.-J. van der Veen, Anaog beamforming in MIMO communications with phase shift networks and onine channe estimation, Signa Processing, IEEE Transactions on, vo. 58, pp , Aug [3] O. E Ayach, S. Rajagopa, S. Abu-Surra, Z. Pi, and R. Heath, Spatiay sparse precoding in miimeter wave MIMO systems, Wireess Communications, IEEE Transactions on, vo. 13, pp , March [4] A. Akhateeb, O. E Ayach, G. Leus, and R. Heath, Channe estimation and hybrid precoding for miimeter wave ceuar systems, Seected Topics in Signa Processing, IEEE Journa of, vo. 8, pp , Oct [5] H. Ghauch, T. Kim, M. Bengtsson, and M. Skogund, Bind subspace estimation and decomposition for arge miimeter wave MIMO systems, Manuscript in preparation, preprint avaiabe at [6] T. Dah, N. Christophersen, and D. Gesbert, Bind MIMO eigenmode transmission based on the agebraic power method, Signa Processing, IEEE Transactions on, vo. 52, pp , Sept [7] Y. Saad, Numerica Methods for Large Eigenvaue Probems, Manchester University Press, no. Second Edition, pp , [8] D. Baum and H. Bocskei, Information-theoretic anaysis of MIMO channe sounding, Information Theory, IEEE Transactions on, vo. 57, pp , Nov [9] Spatia channe mode for mutipe input mutipe output (MIMO simuations, 3GPP TR V10.0, Mar 2011.