On the Spectral Efficiency of Space-Constrained Massive MIMO with Linear Receivers Jiayi Zhang 1 Linglong Dai 2 Michail Matthaiou 3 Christos Masouros 4 and Shi Jin 5 1 School of Electronics and Information Engineering, Beijing Jiaotong University 2 Department of Electronic Engineering, Tsinghua University 3 School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast 4 Department of Electronic and Electrical Engineering, University College London 5 National Mobile Communications Research Laboratory, Southeast University Kuala Lumpur, Malaysia May 26, 2016 Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 1 / 24
Outline Introduction 1 System Model 2 Performance Analysis 3 MRC Receivers ZF Receivers MMSE Receivers Conclusions 4 Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 2 / 24
Introduction I Favorable propagation: the channel vectors between the different UEs and the BS become asymptotically orthogonal with a large antenna array. ◮ The inter-element spacing is more than half a wavelength ◮ Suffer from less spatial correlation Space constrained: the dense deployment of a massive number of antennas in a limited physical space. ◮ The inter-element spacing is less than half a wavelength ◮ Suffer from increased spatial correlation Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 3 / 24
System Model I We consider the uplink of a single-cell massive MIMO system with M BS antennas and K single-antenna UEs. The received vector y ∈ C M × 1 given by y = √ p u Gx + n , (1) where p u is the average power of each UE, x ∈ C K × 1 denotes the zero-mean Gaussian transmit vector from all K UEs with unit average power, and the elements of n represent the additive white Gaussian noise (AWGN) with zero-mean and unit variance. Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 4 / 24
System Model II The channel matrix between the BS and UEs can be written as G = AHD 1 / 2 , where H ∈ C P × K is the propagation response matrix standing for small-scale fading, and D ∈ C K × K denotes a diagonal matrix whose k th diagonal element ζ k models the large-scale fading (including geometric attenuation and shadow fading) of the k th UE. We assume that large-scale fading ζ k are constant. Moreover, A ∈ C M × P is the transmit steering matrix, with P denoting a large but finite number of incident directions in the propagation channel. For the sake of analytical simplicity, we assume that all UEs are seen from the same set of directions with cardinality P . Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 5 / 24
System Model II Considering the widely used uniform linear antenna array, we can write A as A = [ a ( θ 1 ) , a ( θ 2 ) , . . . , a ( θ P )] , (2) where a ( θ i ), for i = 1 , 2 , . . . , P denotes a length- M normalized steering vector as 1 � T � 1 , e − j 2 π d λ sin θ i , . . . , e − j 2 π d λ ( M − 1) sin θ i a ( θ i ) = √ (3) , P where d is the antenna spacing, λ denotes the carrier wavelength, and θ i represents the direction of arrival (DOA). The normalized total antenna array space d 0 at the BS can be 1 expressed as d 0 = dM λ . In (3), we use the factor P to normalize the √ steering vector a ( θ i ). Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 6 / 24
System Model III The linear receiver matrix T ∈ C M × K is used to separate the received signal into K streams by r = T H y = √ p u T H Gx + T H n . (4) The k th element of the received signal vector is given by K r k = √ p u t H k g k x k + √ p u � t H k g l x l + t H (5) k n . l � = k The achievable uplink SE, R k , of the k th UE is given by � � �� k g k | 2 p u | t H R k = E log 2 1 + . (6) k g l | 2 + � t k � 2 � K l � = k | t H p u The uplink sum SE can be then defined as K � R = in bits/s/Hz . (7) R k k =1 Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 7 / 24
MRC Receivers I Proposition 1 For space-constrained massive MIMO systems with MRC receivers, the approximated sum achievable SE is given by � P � M 2 + β 2 p u � ζ k K i R MRC ≈ i =1 � log 2 1 + (8) , K P β 2 k =1 � � i + M ζ k p u ζ l l � = k i =1 where β i is the ith eigenvalue of the matrix A H A . Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 8 / 24
MRC Receivers II 10 9.5 Sum Spectral E ffi ciency (bits/s/Hz) 9 8.5 d 0 = 4 , 6 , 8 8 7.5 Monte-Carlo simulation Analytical approximation 7 50 100 150 200 250 300 350 400 450 500 Number of BS Antennas M Figure 1: Simulated and analytical approximation of the sum SE of massive MIMO with MRC receivers against the number of BS antennas ( P = 12 and K = 6). Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 9 / 24
MRC Receivers III 10 9.5 Sum Spectral E ffi ciency (bits/s/Hz) 9 8.5 d 0 = 4 , 6 , 8 8 7.5 Monte-Carlo simulation Analytical approximation 7 50 100 150 200 250 300 350 400 450 500 Number of BS Antennas M The sum SE saturates with an increasing number of BS antennas for MRC receivers For the same number of BS antennas, a monotonic increase in the sum SE is achieved as d 0 becomes larger The gap between the curves decreases as d 0 increases, which implies that the effect of constrained space becomes less pronounced. Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 10 / 24
ZF Receivers I Proposition 2 For space-constrained massive MIMO systems with ZF receivers, the achievable sum SE is lower bounded as P � | Y n | � � K K | Y P − K +1 | n = P − K +2 R ZF � � L = log 2 1+ p u ζ k exp ζ n ψ ( K )+ − ψ ( n )+ , i < j ( β j − β i ) i < j ( β j − β i ) � P � P k =1 n � = k (9) where ψ ( · ) is the digamma function, and Y n denotes a P × P matrix whose entries are � β q − 1 q � = n , , p [ Y n ] p , q = (10) β q − 1 ln β p , q = n . p Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 11 / 24
ZF Receivers II Proposition 3 For space-constrained massive MIMO systems with ZF receivers, the achievable sum SE is upper bounded as P � � � | Y n | | ∆ 2 | pu | ∆ 1 | K − 1 n = P − K +2 − K K log2 + � ψ ( n )+ , � K − 1 i < j ( β j − β i ) i < j ( β j − β i ) ln 2 i < j ( β j − β i ) Γ( K − i ) � P � K i =1 Γ( K − i +1) � P � P n =1 i =1 where ∆ 1 = [ Ξ 1 Φ 1 ] is a P × P matrix with entries [ Ξ 1 ] p , q = β q − 1 , q =1 , 2 ,..., P − K , p [ Φ 1 ] p , q = β q p Γ( q − P + K +1) , q = P − K +1 ,..., P , and ∆ 2 = [ Ξ 2 Φ 2 ] is a P × P matrix with entries [ Ξ 2 ] p , q = β q − 1 , q =1 , 2 ,..., P − K +1 , p [ Φ 2 ] p , q = β q p Γ( q − P + K ) , q = P − K +2 ,..., P . Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 12 / 24
ZF Receivers III 18 16 Sum Spectral E ffi ciency (bits/s/Hz) K = 4 14 12 10 K = 2 8 ZF Lower Bound 6 ZF Upper Bound Monte-Carlo Simulation 4 10 20 30 40 50 60 70 80 90 100 Number of BS Antennas M Figure 2: Simulated and analytical approximation of the sum SE of massive MIMO with ZF receivers against the number of BS antennas ( P = 12 and d 0 = 4). Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 13 / 24
ZF Receivers IV 18 16 Sum Spectral E ffi ciency (bits/s/Hz) 14 K = 4 12 10 8 K = 2 ZF Lower Bound 6 ZF Upper Bound Monte-Carlo Simulation 4 10 20 30 40 50 60 70 80 90 100 Number of BS Antennas M All lower bounds can predict the exact sum SE for all the considered cases, which validate their tightness The upper bounds are relatively looser, due to the large variance of the involved random variables Adding more antennas significantly improves the sum SE by suppressing thermal noise Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 14 / 24
ZF Receivers V 18 17 Sum Spectral E ffi ciency (bits/s/Hz) 16 K = 4 15 14 ZF Lower Bound 13 ZF Upper Bound Monte-Carlo Simulation 12 K = 2 11 10 3 4 5 6 7 8 9 10 Total Antenna Array Space d 0 = dM λ Figure 3: Simulated and analytical approximation of the sum SE of massive MIMO with ZF receivers against the total antenna array space d 0 = dM ( M = 100 and P = 12. λ Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 15 / 24
ZF Receivers IV 18 17 Sum Spectral E ffi ciency (bits/s/Hz) 16 K = 4 15 14 ZF Lower Bound 13 ZF Upper Bound Monte-Carlo Simulation 12 K = 2 11 10 3 4 5 6 7 8 9 10 Total Antenna Array Space d 0 = dM λ The sum SE does improve with increased total physical space, particularly for the case of more UEs Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 16 / 24
MMSE Receivers I Proposition 4 For space-constrained massive MIMO systems with MMSE receivers, the exact sum SE is given by P P K log 2 e R MMSE = � � β n − 1 e 1 /β l p u � P l i < j ( β j − β i ) l =1 n = P − K +1 � 1 � × D l , n E n − P + K , (11) β l p u where D l , n is the ( l , n ) th cofactor of a P × P matrix D with the ( p , q ) th entry [ D ] p , q = β q − 1 , and E x ( y ) is the exponential integral function. p Zhang et al. (BJTU) jiayizhang@bjtu.edu.cn IEEE ICC2016 17 / 24
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