Hybrid Beamforming Based Millimeter Wave Massive MIMO Channel Estimation

Jingze Huang; Xuwen Liang; Zhuochen Xie

doi:10.3788/LOP202259.0506002

Journals >Laser & Optoelectronics Progress >Volume 59 >Issue 5 >Page 0506002 > Article

Laser & Optoelectronics Progress
Vol. 59, Issue 5, 0506002 (2022)

Hybrid Beamforming Based Millimeter Wave Massive MIMO Channel Estimation

Jingze Huang^1、2, Xuwen Liang¹, and Zhuochen Xie^1、*

Author Affiliations

¹Innovation Academy for Microsatelite, Chinese Academy of Sciences, Shanghai , 201203, China

²University of Chinese Academy of Sciences, Beijing 100049, China

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DOI: 10.3788/LOP202259.0506002 Cite this Article Set citation alerts

Jingze Huang, Xuwen Liang, Zhuochen Xie. Hybrid Beamforming Based Millimeter Wave Massive MIMO Channel Estimation[J]. Laser & Optoelectronics Progress, 2022, 59(5): 0506002 Copy Citation Text

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Abstract

The millimeter wave massive multiple input multiple output(MIMO) system relies on accurate channel state information. However, the high frequency of the millimeter wave shortens the time slot for channel estimation, so the previous high complexity estimation algorithms that use either the sparsity in the beam domain or the low rank property in the antenna domain alone were no longer feasible. Therefore, this paper proposes a new estimation algorithm, which combines the sparsity and low rank property. The channel estimation is regarded as a matrix complete problem. The inexact Newton method based on Augmented Lagrange alternating direction is used to solve the problem. The simulation result shows that the proposed algorithm has faster convergence speed and higher accuracy than other algorithms.

Keywords

channel estimation convex optimization massive MIMO matrix completion millimeter wave

H_{} ≜ \sum_{k = 1}^{N_{p}} α_{k} a_{R} (ϕ_{R}^{(k)}, θ_{R}^{(k)}) a_{T}^{H} (ϕ_{T}^{(k)}, θ_{T}^{(k)}),

（1）

a_{T} (ϕ_{T}) = \{1, e x p (j π s i n ϕ_{T}), \dots, e x p [j (N_{T} - 1) π s i n ϕ_{T}]\} / \sqrt[]{N_{T}} \in ℂ^{N_{T} \times 1}

，（2）

a_{R} (ϕ_{R}) = \{1, e x p (j π s i n ϕ_{R}), \dots, e x p [j (N_{R} - 1) π s i n ϕ_{R}]\} / \sqrt[]{N_{R}} \in ℂ^{N_{R} \times 1}

。（3）

H = D_{R} Z D_{T}^{H},

（4）

\begin{array}{l} \underset{H, Z}{m i n} τ_{H} {‖H‖}_{*} + τ_{Z} {‖Z‖}_{1} \\ s . t . Ω \circ H = H_{Ω} a n d H = D_{R} Z D_{T}^{H}, \end{array}

（5）

\begin{array}{l} \underset{H, Z, C, Y}{m i n} τ_{H} ‖ H ‖_{*} + τ_{Z} ‖ Z ‖_{1} + \frac{1}{2} ‖ C ‖_{F}^{2} + \\ \frac{1}{2} {‖Ω \circ Y - H_{Ω}‖}_{F}^{2} s . t . Y = H a n d C = Y - D_{R} Z D_{T}^{H}, \end{array}

（6）

\underset{H, Z, C, Y}{m i n} τ_{H} ‖ H ‖_{*} + τ_{Z} ‖ Z ‖_{1} + \frac{1}{2} ‖ C ‖_{F}^{2} + \frac{1}{2} {‖Ω \circ Y - H_{Ω}‖}_{F}^{2} + t r [λ_{1}^{H} (H - Y)] + \frac{ρ}{2} ‖ H - Y ‖_{F}^{2} + t r [λ_{2}^{H} (C - Y + D_{R} Z D_{T}^{H})] + \frac{ρ}{2} {‖C - Y + D_{R} Z D_{T}^{H}‖}_{F}^{2},

（7）

\begin{array}{l} H^{(i + 1)} = a r g \underset{H}{m i n} τ_{H} ‖ H ‖_{*} + \\ \frac{ρ}{2} ‖ H - (Y^{(i)} - \frac{1}{2} λ_{1}^{(i)}) ‖_{F}^{2}, \end{array}

（8）

\begin{array}{l} H^{(i + 1)} = U_{L}^{(i)} d i a g ({\{s i g n (ζ_{j}^{(i)}) \times m a x (ζ_{j}^{(i)}, 0)\}}_{1 \leq j \leq r}) \\ {(U_{R}^{(i)})}^{H}, \end{array}

（9）

\begin{array}{l} \frac{\partial}{\partial Y} \{\frac{1}{2} {‖Ω \circ Y - H_{Ω}‖}_{F}^{2} + t r [(λ_{1}^{(i)})^{H} (H^{(i + 1)} - Y)] + \frac{ρ}{2} ‖ H^{(i)} - Y ‖_{F}^{2} + t r [(λ_{2}^{(i)})^{H} (C^{(i)} - Y + D_{R} Z^{(i)} D_{T}^{H})] + \\ \frac{ρ}{2} {‖C^{(i)} - Y + D_{R} Z^{(i)} D_{T}^{H}‖}_{F}^{2}\} = Ω \circ Y - H_{Ω} - λ_{1}^{(i)} - ρ (H^{(i)} - Y) - λ_{2}^{(i)} - ρ (C^{(i)} - Y + D_{R} Z^{(i)} D_{T}^{H}), \end{array}

（10）

\begin{array}{l} y^{(i + 1)} = {(K_{1} + 2 ρ I)}^{- 1} \\ (λ_{1}^{(i)} + ρ h^{(i + 1)} + H_{Ω} + λ_{2}^{(i)} + ρ c^{(i)} + ρ K_{2} z^{(i)}), \end{array}

（11）

Z^{(i + 1)} = a r g \underset{Z}{m i n} τ_{Z} ‖ Z ‖_{1} + \frac{ρ}{2} {‖\frac{1}{ρ} λ_{2}^{(i)} + C^{(i)} - Y^{(i)} + D_{R} Z D_{T}^{H}‖}_{F}^{2},

（12）

\underset{z}{m i n} τ_{Z} ‖ z ‖_{1} + {‖K_{2} z - k^{(i)}‖}_{2}^{2},

（13）

\underset{x}{m i n} τ_{Z} {‖x‖}_{1} + {‖G_{1} x - R e (k_{}^{(i)})‖}_{2}^{2} + {‖G_{2} x - I m (k_{}^{(i)})‖}_{2}^{2}

，（13）

\underset{x}{m i n} f_{1} (x_{1}) + f_{2} (x_{2}) s . t . x_{1} - x_{2} = 0 |η

，（14）

\begin{matrix} \frac{\partial}{\partial C} (\frac{1}{2} ‖ C ‖_{F}^{2} + t r ((λ_{2}^{(i)})^{H} (C^{(i)} - Y^{(i + 1)} + D_{R} Z^{(i + 1)} D_{T}^{H})) + \frac{ρ}{2} {‖C - Y^{(i + 1)} + D_{R} Z^{(i + 1)} D_{T}^{H}‖}_{F}^{2}) = \\ (1 + ρ) C - ρ (Y^{(i + 1)} - D_{R} Z^{(i + 1)} D_{T}^{H} - \frac{1}{ρ} λ_{2}^{(i)}), \end{matrix}

（15）

C^{(i + 1)} = \frac{ρ}{1 + ρ} (Y^{(i + 1)} - D_{R} Z^{(i + 1)} D_{T}^{H} - \frac{1}{ρ} λ_{2}^{(i)}),

（16）

{λ_{1}}^{(i + 1)} = {λ_{1}}^{(i)} + ρ (Y^{(i + 1)} - H^{(i + 1)}),

（17）

{λ_{2}}^{(i + 1)} = {λ_{2}}^{(i)} + ρ (C^{(i + 1)} - Y^{(i + 1)} + D_{R} Z^{(i + 1)} D_{T}^{H}),

（18）

V_{N M S E} ≜ \frac{‖H - \tilde{H}‖}{‖H‖}

，（19）

V_{A S E} ≜ E \{l o g_{2} d e t \{I_{N_{R}} + [N_{T} N_{R} (σ_{n}^{2} + V_{N M S E} {)]}^{- 1} H H^{H}\}\}

。（20）

Jingze Huang, Xuwen Liang, Zhuochen Xie. Hybrid Beamforming Based Millimeter Wave Massive MIMO Channel Estimation[J]. Laser & Optoelectronics Progress, 2022, 59(5): 0506002

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