A Channel Matching Based Hybrid Analog-Digital Strategy for Massive Multi-User MIMO Downlink Systems

Transcription

1 A Channel Matching Based Hybrid Analog-Digital Strategy for Massive Multi-User MIMO Downlin Systems Jianshu Zhang, Martin Haardt Communications Research Laboratory Ilmenau University of Technology, Germany Ilya Soloveychi, Ami Wiesel School of Computer Science and Engineering The Hebrew University of Jerusalem, Israel Abstract In this paper we study the downlin of a hybrid analogdigital massive multi-user MIMO (MU-MIMO) system An efficient hybrid strategy is proposed, where the analog beamforming matrices are determined using a channel matching criterion while the digital beamforming consists of pre-filters and post-filters The digital post-filters are computed using traditional linear MU-MIMO strategies together with water-filling based power allocation using the effective channels Simulation results show that the proposed hybrid analog-digital solutions achieve a good performance compared to their corresponding unconstrained digital solutions I INTRODUCTION Massive MIMO provides significant MIMO gains while maintaining a high energy efficiency [1] When combined with millimeter wave (mmwave) technology, it will additionally benefit from large chuns of underutilized spectrum in the mmwave band [2] Hence, mmwave massive MIMO communication is a potential technique for future wireless networs [3] Assume that the number of deployed RF chains is significantly smaller than the number of antenna elements To fully exploit the MIMO gain, hybrid analog precoding schemes, realized using phase shifters or switches in the RF domain [4], and digital precoding schemes, implemented in the digital baseband domain as in conventional MIMO, provide a promising performance Analog circuits using only phase shifters impose constant modulus constraints on the analog beamforming matrices and thus create new challenges for the required signal processing [5], [6] Several papers have recently studied such hybrid precoding design problems, including [7], [8], [9], [10] In [7] a compressed sensing based hybrid precoding is proposed to approximate the optimal unconstrained solution of a point-to-point MIMO system and is further refined in [8] In [9] a low complexity codeboo based hybrid precoding scheme is introduced for the mmwave multi-user MIMO (MU- MIMO) downlin channel In [10] a low complexity hybrid transmit strategy is proposed for the same scenario However, only single antenna receivers and the zero forcing (ZF) transmit strategy are considered in [10] and [9] Hence, this motivates us to develop a more general hybrid analog-digital strategy for the massive MU-MIMO downlin channel In this paper we develop hybrid precoders and decoders for such a massive MU-MIMO downlin channel to achieve a higher spatial multiplexing gain The RF precoding is implemented using only phase shifters, ie, only constant modulus entries are allowed Consequently, we get the same phase shifts for all users in the equivalent baseband domain To circumvent these non-convex constraints, we design the analog and digital beamforming matrices sequentially First, analog precoders and decoders are designed based on a per-user channel matching criterion Then digital pre-filters are applied at both the transmitter side and the receiver sides Finally, optimal digital postfilters are derived based on four alternative linear transmit strategies for the MU-MIMO downlin channel followed by an optimal power allocation via the water-filling algorithm The applied linear transmit strategies include the zero-forcing (ZF) method [11], the MMSE method [12], [13], the bloc diagonalization (BD) technique [14] and the regularized BD (RBD) technique [15] Sufficient conditions under which the unconstrained digital solutions can be realized using hybrid precoding and decoding solutions are derived Simulation results show that the proposed hybrid transmit and receive strategy achieves a good performance compared to its pure digital counterpart Notation: The expectation, ran, conjugate, and Hermitian transpose are denoted by E{ }, ran( ), { }, and { } H, respectively The Euclidean norm of a vector and the Frobenius norm are denoted by 2 and F, respectively A diagonal matrix and a bloc diagonal matrix are created by diag{c} and bldiag{a n} N n=1, respectively The operator (A) computes the angle of A element-wise II PROBLEM FORMULATION We begin with a brief review of a MU-MIMO downlin system, where a multi-antenna base station (BS) transmits data to multiple multi-antenna user equipments (UEs) The BS has M T transmit antennas with N T = M T RF chains The -th UE has M receive antennas and N = M RF chains ( {1,,K}) We define N R = K =1 N and M R = K =1 M Moreover, we consider a flat fading channel The channel between the BS and the -th UE is denoted as H M M T Let s N s, denote the data vector to be transmitted to the -th UE and define N s = K =1 N s, Let F M T N s, and W H N s, M represent the corresponding precoder and decoder for the -th UE Then the estimated received data at the -th UE is expressed as ŝ[m] = W H H F s +W H l=1,l H F l s l +W H z, (1) where z M denotes zero mean circularly symmetric complex Gaussian (ZMCSCG) noise with covariance matrix E{z z H } = σni 2 M for all Let F = [ ] F 1 F K M T N s and s = [ s T 1 sk] T T N s A transmit power constraint at the BS has to be fulfilled such that E { Fs 2} 2 = F 2 F, where it is assumed that s has zero mean and E{s s H } = I Ns, Let us find matrices F and W,, such that the sum rate is maximized subject to the transmit power constraint at the BS, ie, max F,W, st R n, = log 2 I Ns, +R s, R 1 =1 l=1,l n, W H H F l F H l H H W +σ 2 nw H W

2 ) H, K R s, = W H H F (W H H F F 2 F (2) =1 Assuming that perfect channel state information (CSI) is available at both the BS and the UE, dirty paper precoding achieves the capacity of the MU-MIMO downlin channel and thus provides optimal solutions to (2) [16] However, since it has a high implementation complexity [17], in practice low complexity linear methods are preferred, eg, ZF [11], MMSE [12], [13], and channel bloc diagonalizing based methods such as BD [14] and RBD [15] All these linear algorithms approximately solve (2) by first suppressing the inter-user interference Thereby, the system will be decoupled into K parallel single-user MIMO sub-systems Then each UE designs its precoders and decoders assuming that the residual interference can be treated as noise Armed with this understanding, we now turn to the more challenging hybrid massive MIMO formulation In contrast to the classical model, this formulation assumes that the precoding and decoding are achieved by hybrid analog and digital schemes The number of RF chains is assumed to be much smaller than the number of antenna elements, ie, M T N T N s and M N N s, [7] A baseband precoding matrix F B, N T N s, is applied on a peruser basis Afterwards, the precoded digital signal is upconverted into an analog signal followed by analog circuits, where an RF precoder F A M T N T is implemented using analog phase shifters Hence, constant modulus constraints should be fulfilled for each element of F A, ie, (F A) a,b = 1 for all a {1,,M T} and b {1,,N T} Similarly, hybrid combiners are used at each UE, where the digital combiner is denoted by W H B, N s, N and the corresponding analog combiner is represented by W H A, N M Again, the entries of W A, have unit magnitudes This formulation leads to two additional constraints on (2) {F A,W A, } F (RF) (3a) W = W A, W B,, F = F AF B,, (3b) where F (RF) represents a general set of matrices or vectors with constant modulus entries Thus, the hybrid precoding and decoding matrices are restricted in two challenging manners First, the RF matrices have only constant modulus entries Second, different UEs are coupled via their joint RF solution at the BS The goal of this paper is to develop efficient linear solutions to problem (2) subject to the non-convex constraints in (3), which is in general NP-hard III PROPOSED EFFICIENT HYBRID STRATEGY In this section we introduce the proposed hybrid analog-digital algorithm First, the RF precoders and decoders are designed based on a channel matching criterion Then optimal digital precoders and decoders are computed according to design criteria traditionally used for MU-MIMO downlin systems, ie, BD, RBD, ZF, and MMSE Let us define a ran-min(n,n T, ) truncated SVD of H as H U s, Σ s, V H s,, where U s, M N, V s, M T N T,, and Σ s, R N N T, has non-zero elements only on its main diagonal Moreover, we have K =1 N T, = N T Using the channel matching criterion, our proposed design of analog precoders and decoders is then calculated by W A, = e j (Us,), F A, = e j (Vs,), (4) where F A = [ F A,1 F A,2 F A,K ] Without loss of generality, the digital precoders and decoders are decomposed into W B, = W P, W D,, F B, = F P F D, diag{ γ }, (5) where W P, N N and F P N T N T are square pre-filtering matrices The vector γ = [ ] T γ,1 γ,ns, represent the power loading onto the eigenmodes of the equivalent channel Let us calculate the economy-sized SVD of W A, as W A, = U A, Σ A, V H A,, where U A, M N, V A, N N, and Σ A, is a diagonal matrix containing the singular values of W A,, and the economysized SVD of F A F A = U A Σ A V H A where U A M T N T, V A N T N T, and Σ A is a diagonal matrix containing the singular values of F A Note that ran(w A, ) = N and ran(f A) = N T are assumed for our definitions Then we propose W P, = V A, Σ 1 A,,, F P = V AΣ 1 A (6) Only local CSI is required for the calculation of W A, and W P, The proposed pre-filters in (6) provide two advantages At the receiver side, they whiten the equivalent noise vectors such that W H P,W H A,E{z z H }W A, W P, = U H A,σ 2 nu A, = σ 2 ni N At the transmitter side, the actual transmit power is not a function of the analog precoder any more, ie, F 2 F = F D diag{ γ} 2 F, where F D = [ ] F D,1 F D,K N T N s and γ = [ ] γ T T 1 γk T Post-digital decoders and precoders W H D, C N s, N and F D, N T N s, will be designed for an equivalent MU-MIMO downlin channel, where for the -th UE the effective channel is given by H = W H P,W H A,H F AF P = U H A,H U A N N T and the equivalent noise vector is expressed as z = W H P,W H A,z = U H A,z In the following we derive the design criteria dependent digital decoders W H D, and precoders F D, BD Based Design: Taing the -th UE as an example, to fulfill the requirement of the BD approach in [14], we decompose F D, into F D, = F D1, F D2,, (7) where F D1, is responsible for inter-user interference avoidance and (F D2, diag{γ }) optimizes the performance of the interference-free sub-system of the -th UE Let us define the combined effective channel matrix H for all UEs except the -th UE as H = [ H T 1 H T 1 H T +1 H T K] T (N R N ) N T Define ran( H ) = L and the SVD of H as H = Ũ Σ Ṽ H = Ũ Σ [Ṽ (1) ] H Ṽ (0) (8)

3 where Ṽ (0) N T (N T L ) contains the last (N T L ) right singular vectors The matrix Ṽ (0) forms an orthogonal basis for the null space of H Therefore, we choose Define the SVD of H F D1, as F D1, = Ṽ (0) (9) H F D1, = Ū Σ V H (10) Then the precoder F D2, and the decoder W D, are given by F D2, = V, W D, = Ū, (11) Note that the BD algorithm is applicable only if H has a non-empty null space for all, ie, N T > max (N R N ) if H is a full ran matrix RBD Based Design: Instead of nulling the inter-user interference, the RBD algorithm [15] allows a residual amount of interference in order to balance it with the noise enhancement especially in the low SNR regime Similarly as in (7), we decompose F D, into F D, = F D1, F D2, Let us define F D1 = [ F D1,1 F D1,K], s = [ s T 1 s T K] T, (12) and z = [ z T 1 T z K] T Assume that FD2, is a unitary matrix and equal power allocation is used, eg, γ = β1 Ns, for all [15], where β is used to fulfill the transmit power constraint RBD designs F D1, by solving the following optimization problem [15] min F D1, F =1, =1 ( ) H F D1, 2 F + σ2 nn R F D1, 2 F (13) Using the definition in (8) and inspired by [18], the optimal solution to (13) without scaling is given by ( ) 1 F D1, = Ṽ Σ T Σ + σ2 nn 2 R I NT (14) The remaining digital precoders and decoders are designed in the same way as in case of BD Therefore, the RBD solutions of F D2, and W D, can also be represented by (11) ZF Based Design: The ZF criterion eliminates the interference while ignoring the noise [11] Define F D, = β F D1, (15) Using the definition in (12) and define W D, = β 1 W D1,, the ZF optimization problem is formulated as min E{ s H ef D1 s 2 2}, st β F D1 2 F (16) β,f D1,W D1,, where H e = [ H T 1 W D 1,1 H T KW D 1,K] T N s N T (17) When W D1, is fixed, using the method of Lagrangian multipliers a closed-form solution for F D1 and β is obtained as F D1 = H H e ( H e H H e ) 1, β = PT F D1 2 F (18) Inserting this optimal solution bac into (17), the resulting problem is not a function of W D1, and therefore an arbitrary W D1, is optimal To avoid coloring the noise, we set WD H 1, = [ I Ns 0 ] Note that the involved channel inversion in (18) requires that N T N s if W H D, H is a full ran matrix MMSE Based Design: Using the definitions in (15) and (17), the MMSE optimization problem is formulated as min β,f D1,W D,, E{ s β H ef D1 s W D z 2 2} st β F D1 2 F, (19) where W D = bldiag{wd,} H K =1 Problem (19) is a non-convex optimization problem [12], [13] For implementation simplicity, we resort to a suboptimal sequential design First, the BS computes F D1 using the MMSE criterion while assuming that WD, H = β 1[ I Ns 0 ] Again, by applying the Lagrangian method a closedform solution is obtained as ( F D1 = H H e H e + σ2 nn s I NT ) 1 H H e, β = PT F D1 2 F (20) Next, for fixed F D1, we compute W D, by solving (19) The optimal solution is computed by [12] W D, = ( H F D, F H D, H H +R nn, ) 1 H F D,, (21) where R nn, = K l=1 l H F D,l F H D,l H H + σ 2 ni Ns, Note that it is difficult for the -th UE to obtain R nn, exactly with only local CSI Optimal Power Allocation: After applying the derived analog and digital beamforming matrices, problem (2) subject to the nonconvex constraints in (3) reduces to a power allocation problem A general formulation of this optimization problem is expressed as max γ,i st N s, =1 i =1 N s, =1 i =1 log 2 (1+σ 2 n σ,i γ,i ) α,i γ,i (22) The optimal solution of problem (22) is obtained using the waterfilling algorithm [19], ie, γ,i = 1 α,i max( 1 ν σ2 n α,i is the water level such that K =1 σ,i,0) and ν Ns, i =1 max(1 ν σ2 n α,i σ,i,0) = When BD is used, σ,i is the (i,i )-th element of the matrix Σ, which is defined in (10), and α,i = 1,,i When ZF or MMSE is used, we set σ,i = 1,,i, and α,i is the (i,i )-th element of the matrix FD,F H D, When RBD is used, we set γ = P T K=1 F D, F1 2 Ns, for all IV UNCONSTRAINED DIGITAL SOLUTIONS AND THEIR EQUIVALENT HYBRID REALIZATIONS FOR SPECIAL CASES In this section we discuss optimal unconstrained digital solutions and their exact hybrid analog-digital realizations for some special cases Let F full, M T M and W full, M M represent optimal solutions to problem (2), which are obtained by using BD, RBD, ZF, or MMSE together with equal power allocation Let F opt, M T N s, and W opt, M N s, contain the first N s, columns of F full, and W full,, respectively Then F opt, = F opt, diag{ γ } andw opt, are our proposed unconstrained digital solutions, where γ denotes the optimal power allocation Note that when ZF and MMSE are used, the proposed unconstrained digital solutions are meaningful only if H has full ran and M T M R

4 Inspired by Theorem 1 of [20], we show that optimal hybrid solutions, which achieve the performance of their corresponding unconstrained digital solutions exactly, can be constructed for some special cases Lemma 1: Let us define F opt = [ ] F opt,1 F opt,k C M T N s and ran(f opt) = r An exact matrix decomposition exists such that F opt = F A,opt F B,opt only if N T 2M T r r, M T +2r where F A,opt M T N T F (RF) A closed-form solution for this decomposition can be constructed if N T 2r Proof: Let us define the economy-sized SVD of F opt as F opt = U sσ sv H s, where U s MT r Then we rewrite U s as U s = [ [ ] ] bldiag{νj} r j=1 0 Φ 1 Φ r Φ res, (23) }{{} 0 0 }{{} F A,opt C M T N T Ω C N T r where Φ j MT 2 F (RF), ν j 2 for j {1,,r}, and the elements of Φ res M T (N T 2r) F (RF) have random phases This formulation requires that the j-th column of U s, denoted by u s,j, should be factorized as u s,j = Φ j ν j A closed-form solution for this decomposition is provided by Theorem 1 in [20] Lastly, F B,opt = ΩΣ svs H If F A,opt and Ω do not have the specific structure as in (23), in general this decomposition holds only if the degrees of freedom of (F A,opt Ω) are not less than the degrees of freedom of U s Noticing that one complex variable can be treated as two real variables, the number of free real-valued variables in (F A,opt Ω) is (M TN T + 2N Tr) The matrix U s has orthonormal columns, ie, u H s,ju s, j = 0, j j, j {1, r} and u s,j 2 2 = 1, j These conditions create r 2 constraints thus consume r 2 degrees of freedoms The remaining free variables of U s are (2M Tr r 2 ) We should have 2M Tr r 2 M TN T+2N Tr, which can be reformulated as N T 2M T r M T +2r r This conclusion can be also used for realizing unconstrained digital decoders if N 2r, where r = ran(w opt, ) V SIMULATION RESULTS The proposed algorithms are evaluated using Monte-Carlo simulations The maximum allowable power is fixed to unity The SNR is thus defined as SNR = 1/σ 2 n All the simulation results are obtained by averaging over 1000 channel realizations The simulated channel is a mmwave channel with a geometric channel model of L = 8 scatterers Each scatterer contributes to a single propagation path between the BS and the UE The-th UE s channel is modeled as H = 1 L L l=1 α l, a R,l, (θ R,l, )a H T,l,(θ T,l, ), where α l, is the random complex gain of the l-th path of the -th UE, with zero-mean and E{ α l, 2 } = 1 [8] The variables {θ R,l,,θ T,l, } [0,2π) denote the angle of arrival and the angle of departure of the l-th path of the -th UE, respectively Finally, a R,l ( ) and a T,l ( ) are the array steering vectors of the BS and the UEs, respectively As in [7], a uniform linear array (ULA) geometry is used at both ends The inter-element spacing of the ULA is equal to half of the wavelength Fig 1 and Fig 2 compare the performance of the proposed hybrid analog-digital strategy using different linear design criteria Fig 1 Achievable sum rate [bits/s/hz] Unconstrained RBD Unconstrained BD Hybrid RBD Hybrid BD Hybrid MMSE Hybrid ZF N s, =5 N s, = SNR [db] Fig 1 Achievable sum rate vs SNR for K = 8, M T = 256, N T = 64, M = 16, N = 8, and N T, = 8, Achievable sum rate [bits/s/hz] Unconstrained RBD Unconstrained BD Hybrid RBD Hybrid BD Hybrid MMSE Hybrid ZF N s, Fig 2 Achievable sum rate vs the number of spatial streams N s, for K = 8, M T = 256, N T = 64, M = 16, and N = N T, = 8, SNR = 0 db illustrates that the hybrid solutions have almost the same performance as the corresponding unconstrained digital solutions especially when RBD or BD is used The performance difference increases as the number of spatial steams increases Among the four linear strategies, the RBD based design provides the best performance while the ZF based design performs the worst Moreover, when N s, = 2 the settings of N T and N fulfill the requirement of Lemma 1 Then the unconstrained solutions can be achieved exactly using hybrid analogdigital designs Fig 2 shows that the gap between the unconstrained digital solutions and the hybrid solutions enlarges when the number of spatial streams increases VI CONCLUSION In this paper we have developed a general approach to realize hybrid analog-digital beamforming in a massive MU-MIMO downlin scenario The proposed approach can be applied regardless of the number of antennas at the BS and at the UE A multiple-stream transmission can be achieved for each UE Furthermore, unconstrained digital solutions have been proposed for the same scenario We have shown that the unconstrained digital solutions have equivalent hybrid analog-digital realizations for some special cases Simulation results show that the proposed hybrid analog-digital solutions have almost the same performance as their corresponding unconstrained digital counterparts The MMSE based solution provides a better performance especially in the low SNR regime while the RBD based design achieves a higher data rate especially in the high SNR regime

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