1. AoD-Adaptive Subspace Codebook for Channel Feedback in FDD Massive MIMO Systems
Wenqian Shen1, Linglong Dai1, Guan Gui2, Zhaocheng Wang1, Robert W. Heath Jr.3, and Fumiyuki Adachi4
1Department of Electronic Engineering, Tsinghua University, Beijing, China
2College of Communication and Information Engineering, NUPT, Nanjing, China
3Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, USA
4Research Organization of Electrical Communication, Tohoku University, Sendai, Japan
Good morning everyone! I am very happy to share my work about massive MIMO channel feedback. I’m the first author Wenqian Shen from Tsinghua University, China. This work is coauthored with prof. Linglong Dai, prof. Zhaocheng Wang, prof. Guan Gui, prof. Robert Heath, and prof. Fumiyuki Adachi.
In this work, we investigated the challenging problem of massive MIMO channel feedback due to the overwhelming overhead. An angle of departure (AoD)-adaptive subspace codebook was proposed to reduce the required feedback overhead and codebook size.

2. What is Massive MIMO? Conventional MIMO Massive MIMO
Let’s look at some background information. Massive MIMO is an key technique for future 5G communication system due to its high spectrum- and power-efficiency. Compared with conventional MIMO, the number of base station antenna increased an order of magnitude in massive MIMO systems.
Conventional MIMO
M: 2~8, K:1~4 (LTE-A)
Massive MIMO
M: 100~1000, U: 16~64
[1] T. L. Marzetta, “Non-cooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas,” IEEE TWC, 2010.

3. Channel feedback in FDD Massive MIMO
As we know, since the carrier frequency of uplink and downlink channel in FDD systems is different, channel reciprocity does not hold for FDD. Therefore, the base station has to transmit downlink pilots for channel training. Then, users feed the channel state information back to the BS to enable channel adaptive techniques such as precoding.
Channel reciprocity does not hold for FDD

4. CSI quantization based on codebook𝐵 𝐵
Feedback
Channel
𝐡 𝑢
𝐡 𝑢
Q(·)
R(·)
Downlink CSI
Quantized CSI
In general, quantization loss
∝ 2 − 𝐵 𝑀−1 [1]
# of BS antenna (𝑀)
M = 4
M = 128
Feedback bits (𝐵) 8
256
Codebook size ( 2 𝐵 )
2 8
2 256
Before channel feedback, users usually quantize the channel to several digital bits (here we assume B bits) based on a codebook. and feed these B digital bits back to the base station. And base station can recover the quantized CSI.
In general, compared with the ideal case of perfect CSI, the achievable rate loss due to quantization can be limited within a small value when the number of quantization bits B scale linearly with the number of base station antenna M.
With hundreds of base station antennas in massive MIMO systems, the codebook size and the feedback overhead will be overwhelming. Massive MIMO channel feedback is challenging.
Massive MIMO channel feedback is challenging
[1] N. Jindal, ``MIMO broadcast channels with finite-rate feedback,'‘ IEEE TIT, 2006.

5. How can we quantize complete CSI with less overhead ?
Prior work
Sparsity based channel feedback [2-3]
Correlation based channel feedback [4-5]
Statistics based channel feedback [6]
Reciprocity based channel feedback [7]
CSI measurements
Effective CSI
Partial CSI
Several prior works on massive MIMO channel feedback has been proposed.
Sparsity-based channel feedback schemes have been proposed where the sparse channel vector is projected as low-dimensional channel measurements which can be fed back to the BS with less overhead.
In Correlation-based channel feedback schemes and Statistics based channel feedback schemes, low-dimensional effective CSI was fed back to the BS with less overhead.
By utilizing partial reciprocity between uplink and downlink, only partial CSI need to be fed back with less overhead.
Different from the prior works, our objective in this paper is to quantize the complete channel vector with a novel designed codebook which can achieve accurate CSI with reduced number of quantization bits.
How can we quantize complete CSI with less overhead ?
[2] P. H. Kuo, at al., ``Compressive sensing based channel feedback protocols for spatially-correlated massive antenna arrays,” WCNC, 2012.
[3] X. Rao, at al., ``Distributed compressive CSIT estimation and feedback for FDD multi-user massive MIMO systems,” IEEE TSP, 2014.
[4] B. Lee, at al., ``Antenna grouping based feedback compression for FDD-based massive MIMO systems,” IEEE TCOM, 2015.
[5] W. Shen, at al., ``Joint CSIT acquisition based on low-rank matrix completion for FDD massive MIMO systems,” IEEE CL, 2015.
[6] A. Adhikary, at al., ``Joint spatial division and multiplexing-The large-scale array regime, ”IEEE TIT, 2013.
[7] H. Xie, at al., ``A unified transmission strategy for TDD/FDD massive MIMO systems with spatial basis expansion model,” IEEE TVT, 2017.

6. Contributions
Our contributions are outlined in this slide. We first show that Angle coherence time is much longer than channel coherence time. Then, we propose subspace codebook. Finally, we provide the achievable rate analysis.

7. Massive MIMO Downlink Channel
We start from the massive MIMO downlink channel model. We adopt the classical ray-based channel model. The BS have M antennas and user has a single antenna. Between the BS and user u, there are P_U resolvable propagation paths. Each path is characterized by the angle of departure \theat_u, P_u and complex path gain.

8. Massive MIMO Downlink Channel
𝑃 𝑢 is the number of resolvable paths
𝑔 𝑢,𝑖 is the complex gain
𝜃 𝑢,𝑖 is the angle of departure (AoD)
Array response for ULA:
𝐡 𝑢 = 𝑖 𝑃 𝑢 𝑔 u,𝑖 𝐚( 𝜃 u,𝑖 )= 𝐀 𝑢 𝐠 𝑢
The downlink channel vector of user u can be expressed as the sum of P_U path gains g_u multiplexing with the array response. The array response a_theta for uniform linear array is expressed as this equation.
Channel vector h_u can be rewrite in matrix form by stacking a(theta_u,i) into the columns of matrix A_u. The column vector g_u is composed of path gain g_u1 to g_u,Pu.
𝐚( 𝜃 u,𝑖 )= 1 √𝑀 1, 𝑒 −𝑗2𝜋 𝑑 𝜆 sin( 𝜃 u,𝑖 ) ,⋯, 𝑒 −𝑗2𝜋 𝑑 𝜆 (𝑀−1)sin( 𝜃 u,𝑖 ) H

9. Angle coherence time VS. Scatter’s position changes slowly
Rapidly-varying path gain +
Slowly-varying AoD
Slowly-varying AoD
Now we will present the concept of angle of coherence time. Different from the traditional channel coherence time, which is usually less than 1 millisecond, due to the rapidly-varying paths gain, the angle coherence time only depends on the AoD which is slowly-varying due to the constant scatter’s position around the BS.
Therefore, angle coherence time is longer than channel coherence time. This also means that we can estimate the AoD with small average overhead.
Channel coherence time
Angle coherence time
VS.
Angle coherence time is longer than channel coherence time [8]
[8] V. Va, at al., ``The impact of beamwidth on temporal channel variation in vehicular channels and its implications'', IEEE TVT, 2017.

10. Estimate AoDs S(𝜃)= 1 𝐚 𝜃 H 𝐔 2 𝐔 2 H 𝐚(𝜃
Step 1: Channel correlation matrix:
Step 2: Multiple signal classification (MUSIC) algorithm
Singular value decomposition (SVD)
Spatial spectrum of noise space
𝐑 𝑢 =E[ 𝐡 𝑢 𝐡 𝑢 H
𝐑 𝑢 =𝑼𝝨 𝑽 H = 𝐔 1 𝐔 𝜎 1 ⋯ ⋱ ⋮ 𝜎 𝑃 𝑈 ⋮ ⋱ ⋯ 𝑽 H
To estimate the AoD in the angle coherence time, the first step is calculating the channel correlation matrix.
Then, we utilize the MUSIC algorithm in the second step. Through SVD, we can obtain noise space which is expanded by the eigenvectors U_2 corresponding to small eigenvalues.
Finally, the spatial spectrum of noise space (S_theta) is shown as this equation and AoDs can be obtained at the peak point S_theta.
S(𝜃)= 1 𝐚 𝜃 H 𝐔 2 𝐔 2 H 𝐚(𝜃

11. AoD-Adaptive Subspace Codebook
Channel subspace: the column space of 𝑨 𝑢
Quantization vector:
Now we are ready to propose the AoD-adaptive subspace codebook. From the channel model we presented before, the channel vector is only distributed in the column space of A_u, which is composed of array response vectors. This column space is called as channel subspace.
We propose the generate the i-th quantization vector for user u in the channel subspace as c_u,i equals to A_u multiplexed with w_i， w_i is isotopically distributed with unit norm. in this way,…
𝒘 𝒊 is isotopically
distributed
𝐜 𝑢,𝑖 = 1 𝑀 𝑨 𝑢 𝐰 𝑖
Quantization vector is distributed exactly in the channel subspace

12. The rate loss with proposed codebook
𝐡 𝑢 = 𝒉 𝑢 || 𝒉 𝑢 ||
Rate with perfect CSIT
Rate with quantized CSIT
Transmit power
𝛥𝑅(𝛾)= 𝑅 Ideal −𝑅≤​ log 2 ​ 1​+​ U​−​1)𝛾 𝑈 ​𝐸 ‖ 𝐡 𝑢 ‖ 2 ]𝐸[ sin 2 (∡( 𝐡 𝑢 , 𝐡 𝑢
Constant value indicating the rate gap
𝐸 sin 2 ∡ 𝐡 , 𝐡
<𝛽(1− 2 − 𝐵 𝑃−1 )+ 2 − 𝐵 𝑃−1
Let and assume 𝛽=0,
𝛥𝑅(𝛾)≤ log 2 (𝑏
𝐵≥ 𝑃−1 3 SNR+(𝑃−1) log 2 𝐾−1 𝑏−1
Now we analyze the performance of our proposed subspace codebook.
The metric is the per-user rate gap between the ideal case with perfect CSIT and the practical case with quantized CSIT.
We proved that to limit the rate gap less than a constant value, the number of quantization bits B scales linearly with the resolvable path number P.
Depends on AoDs accuracy and 𝛽≪1
Feedback bits B scales linearly with number of resolvable paths P

13. The proposed subspace codebook outperforms the traditional codebook
Simulation Results
Parameter
Value
(M, U, P)
(128,8,3)
AoDs
𝒰[−𝜋/2,𝜋/2]
Feedback bits
𝐵≥ 𝑃−1 3 SNR+(𝑃−1) log 2 𝐾−1 𝑏−1
We show some simulation results. The number of BS antennas is 128, the number of users is 8, and the number of resolvable paths for each user is 3. The AoDs are chosen from uniform distribution. And the number of feedback bits is set as our previous analytical result.
The x-axis is the SNR at users and the y–axis is the per-user rate. The dashed line shows the per-user rate with perfect CSIT. The red line shows our proposed subspace codebook, and the blue line is a traditional channel statistics-based codebook.
We can observe that the proposed subspace codebook outperforms the traditional codebook.
Per-user rate vs. SNR
The proposed subspace codebook outperforms the traditional codebook

14. B scales linearly with P when rate gap is limited
Simulation Results
P is much smaller
than M
# of feedback bits B
vs.
# of paths P
In this figure, we show the required number of feedback bits when the rate gap is limited within a constant value. The x-axis is the number of paths P and the y-axis is the number of feedback bits B. The dashed line is the theoretical results of feedback bits B against the number of path P shown in slides 10. The red diamond is the simulated results. Both the theoretical result and simulated result show that B scales linearly with P. Note that P is the number of resolvable paths between the BS and each user. P is much smaller than the number of the BS antennas. Our proposed subspace codebook can achieve significant reduction of feedback overhead as well as codebook size.
B scales linearly with P when rate gap is limited

15. Conclusions Generate the quantization vector in the channel subspace
Angle coherence time is much longer than channel coherence time
Now we conclude our work. We first show that Angle coherence time is much longer than channel coherence time. Therefore, we propose to Generate the quantization vector in the channel subspace. We have also shown that the required number of Feedback bits scales linearly with resolvable path numbers.
Feedback bits scales linearly with resolvable path number
𝐵≥ 𝑃−1 3 SNR+(𝑃−1) log 2 𝐾−1 𝑏−1

16. Thank you for your attention !
Contact information
Thank you for your attention !
That’s all. Thanks for your attention. You may refer to my homepage for simulation code of this work.
I am very willing to answer your questions.
Wenqian Shen