On Ergodic Rates and Optimal Array Geometry in Line-of-Sight Massive MIMO
MM
Prabhu Chandhar, Danyo Danev, and Erik G. Larsson
YS
Div. of Communication Systems, Dept. of Electrical Engineering (ISY), Link¨oping University (LiU), SE-581 83 Link¨oping, Sweden
Introduction
Results
Maximal-Ratio Combining (MRC) Receiver • Output of the MRC detector:
• LoS Massive MIMO:
Uniform Linear Array
ˆ y=G ˆ G(√pu ⊙ q) + G ˆ Hn r =G H
– High probability of LoS propagation in future cellular communications [1]
H
Mx = 50, My = 1 50
• Received signal of the k-th terminal:
( ) √ √ √ ˆH ˆH ˆH rk = E{ puk g puk g k g k }qk + k g k − E{ puk g k g k } qk | {z }
• Contributions: – Derivation of a lower bound on achievable ergodic rate of high mobility LoS Massive MIMO
+
– Design of optimal array geometry
−50
a
∑K
√
j=1,j̸=k
|
0
{z
ˆH puj g k g j qj
}
ˆH +g k n | {z } c
b
• System Setup:
Ω (dB)
– Drone communications [2]
– a, b, and c are independent of the first term – Uplink of a Massive MIMO system – Rectangular array at the BS – Spherically uniformly distributed terminal locations
−100 −150 −200
• Ergodic rate achieved by the k-th terminal: ( ) √ 2 ˆH |E{ puk g k g k }| lb Rk ≥ Rk , log2 1 + var(a) + var(b) + var(c)
−250 −300 0
0.5
Geometric Model
1
1.5 δ /λ
2
2.5
3
x
z
Power Control
Moving direction
Uniform Rectangular Array
θDk
• Channel inversion power control:
φDk z
puk
y x
Ω in normal scale when Mx = 5, My = 5
ρu = βk
– ρu : SNR
dk θAk
• Spherically uniformly distributed within a spherical shell
φAk
δx
M x δx
Distribution of terminal locations
• Distribution of the distances: y
fdj (r) =
δy
x
M y δy
R3
3r2 , Rmin ≤ r ≤ R, j = 1, 2, ..., K 3 − Rmin
– Rmin : Inner radius – R: Outer radius
• δx , δy : element spacing on x and y-axis • Mx , My : number of elements on x and y-axis
• Distribution of the elevation angle:
• p ∈ {1, 2, ..., Mx }, q ∈ {1, 2, ..., My }, l = (q − 1)Mx + p • ϕk ∈ [0, 2π]: azimuth angle, θk ∈ [0, π]: elevation angle • When the terminal is far away from the array, ( ) dkl ≈ dk − sin θk (p − 1)δx cos ϕk + (q − 1)δy sin ϕk
fθj (θ) =
sin θ , 0≤θ≤π 2
Uniform Rectangular Array
• Distribution of the azimuth angle:
Ω in dB scale when Mx = 5, My = 5
1 fϕj (ϕ) = , 0 ≤ ϕ ≤ 2π 2π
– dk : Radial distance from the element at the origin Ergodic Rate Ergodic rate lower bound:
Channel Model
• M × 1 channel vector from the k-th terminal to the BS:
M ) 2 2 2 2 2 1 +M+(K −1)(1+ Ω ) 4(R +Rmin+RRmin) −M+ (Kρu+1) 4(R +Rmin+RRmin) 2 2 ρu M ηρ ρ u p 3(R+Rmin ) 9(R+Rmin)
g k = [gk1 gk2 .... gkM ]T , k = 1, 2, ..., K – gkl =
i2πdkl √ βkl e− λ , l = 1, 2, ..., M
– βkl =
η : d2kl
Case I: Terminals are on the surface of the sphere (
Rklb
– λ: carrier wavelength free space pathloss component
• M × K channel matrix: G = [g 1 g 2 ... g K ]
lb Rk =log2 1+ (
)
→ log2 1 +
M (K − 1)(1 +
Ω M
)+
(Kρu +1)R2 ηρu ρp
+
With very high pilot and data SNRs, ) ( M , ρp → ∞, ρu → ∞. Rklb → log2 1 + Ω (K − 1)(1 + M ) Case II: Rmin is close to the BS
Channel Estimation
ˆ = G + √1 W G ρp – ρp : normalized pilot SNR
Rklb → log2 1+
3M M +4(K − 1)(1 +
( Rklb → log2 1 +
Ω M
u +1)R )+ ρ4u + 4(Kρ 3ηρu ρp
3M M + 4(K − 1)(1 +
Conclusions
)
(
• Maximum likelihood estimate:
, Rmin → R.
1 ρu
2
, Rmin → 0.
) Ω M)
, ρp → ∞, ρu → ∞.
– W ∈ CM ×K : estimation error – [W ]ij : i.i.d CN (0, 1)
Uplink Data Transmission
M x −1 ∑
My −1
∑
u̸=0,u=1−Mx v̸=0,v=1−My
√ y = G( pu ⊙ q) + n T
– q = [q1 , q2 , ..., qK ] : vector of symbols simultaneously transmitted by the K terminals T
– pu = [pu1 , pu2 , ..., puK ] : vector of transmit power of the K terminals – n ∼ CN (0, I M )
{ (Mx − |u|)(My − |v|) )} ( √ 2 u2 δx2 + v 2 δy2 . × sinc λ 2
• M × 1 received signal vector at the BS: • Ω is zero when:
λ δx∗ =n , n ≥ My and 2 λ ∗ δy =m , m ≥ Mx 2 • In case of ULA: δx∗ = n λ2
• In high mobility LoS Massive MIMO, for the spherically uniformly distributed terminal locations inside a spherical volume, – the ergodic rate is maximized for antenna spacing equal to integer multiples of one-half wavelength
Optimal Antenna Spacing Ω=
• Considering a 3-dimensional geometric model, we derived a lower bound on uplink ergodic rate of LoS Massive MIMO system with an estimated CSI
References [1] K. Haneda et al., “5G 3GPP-like Channel Models for Outdoor Urban Micro-cellular and Macro-cellular Environments,” To Appear in Proceedings of the IEEE Vehicular Technology Conference (VTC 2016-Spring), May 2016. [2] P. Chandhar, D. Danyo, and E. G. Larsson, “Massive MIMO as Enabler for Communications with Drone Swarms,” in Proceedings of The International Conference on Unmanned Aircraft Systems (ICUAS), pp. 347–354, Jun. 2016.
