1. March 2006
doc.: IEEE yy/xxxxr0
5/11/2019
[Beamforming through Subspace Tracking for MIMO-OFDM Using Limited Feedback ]
Date:
Authors:
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Jae Son, Nokia Inc.
Jae Son, Nokia Inc.

2. March 2006
doc.: IEEE yy/xxxxr0
5/11/2019
Abstract
We propose a subspace tracking precoding scheme as a novel alternative explicit beamforming method to simplify and reduce feedback information bits and computational complexity. Precoding matrix codebook construction is based on IEEE (P802.16e-2005, ), and a simple but effective subspace tracking table construction is proposed. Computer simulation results are provided to show the competitive performance of the proposed precoding scheme.
Jae Son, Nokia Inc.
Jae Son, Nokia Inc.

3. Conventional MIMO-OFDM Precoding System
March 2006
doc.: IEEE yy/xxxxr0
5/11/2019
Conventional MIMO-OFDM Precoding System J
Precoding M
Ant. 1
Ant. 2
Ant. M
Ant. N N
Equalization
Channel
Estimation
Precoding Matrix
Selection
Codebook
Look-up
Codebook Index, I(k)
J: # of Spatial Streams
M: # of TX Antennas
N: # of RX Antennas
MxJ Precoding matrix for kth subcarrier of nth symbol
Jx1 Input symbol vector for kth subcarrier of nth symbol
Mx1 Transmitted symbol vector for kth subcarrier of nth symbol
Jae Son, Nokia Inc.
Jae Son, Nokia Inc.

4. Conventional MIMO-OFDM Precoding System
5/11/2019
Conventional MIMO-OFDM Precoding System
A precoding method is finding a beamforming matrix among a group of unitary matrices called a codebook.
Thus, the feedback information provided to the transmitter is a codebook index not CSI nor a beamforming matrix itself. So, a significantly less amount of feedback bits is required compared to other information format.
For a MIMO OFDM system, channel coherence bandwidth enables a subcarrier clustering (by grouping several adjacent subcarriers) implementation further reducing feedback information and overall computation.
The total number of feedback bits per cluster would be where L represents the codebook size.
The precoding matrix selection process depends on implementation, but often cited metrics are minimum mean square error, minimum singular value, maximum capacity, etc.
Jae Son, Nokia Inc.

5. Precoding Matrix Codebook Construction
5/11/2019
Precoding Matrix Codebook Construction
There are many ways of constructing a precoding matrix codebook
Grassmannian beamforming codebook construction [1].
Hochward’s unitary systematic codebook construction [2].
IEEE Std e codebook [3].
The proposed codebook is based on codebook construction adopted by IEEE Std (P802.16e-2005, ).
To simplify the codebook construction procedure and storage requirement, the following fixed codebook sizes are proposed.
# of TX antennas: 2 => L = 8.
# of TX antennas: 3 => L = 32.
# of TX antennas: 4 => L = 64.
Jae Son, Nokia Inc.

6. Precoding Matrix Codebook Construction
5/11/2019
Precoding Matrix Codebook Construction
The entire codebook construction is based on three unitary vectors, v(2,1,8), v(3,1,32), and v(4,1,64).
v(Nr, Nss, L)
Nr is # of TX antennas.
Nss is # of spatial streams.
L is the codebook size.
v(2,1,8)
Nr\Index
000
001
010
011
100
101
110
111 1
0.7940
0.7941
0.3289
0.5112 2
j0.1818
j0.6051
j0.5298
j0.0689
j0.6740
j0.7160
j0.3481
Jae Son, Nokia Inc.

7. Precoding Matrix Codebook Construction
5/11/2019
Precoding Matrix Codebook Construction
v(3,1,32) and v(4,1,64) will be constructed as follows
where
is the phase of the first entry of Nr L U S
v(3,1,32) 3 32
[ ]
[ j0.6409, j0.4974, j0.2360]
v(4,1,64) 4 64
[ ]
[ j0.0738, j0.4326, j0.5445, j0.1599]
Jae Son, Nokia Inc.

8. Precoding Matrix Codebook Construction
5/11/2019
Precoding Matrix Codebook Construction
The following table provides the codebook construction methods for all MIMO configurations.
Nc is the number of spatial streams.
represents the null space vectors of where they can be obtained from eigenvectors of corresponding to the zero eigenvalues.
represents taking last two column vectors of
Nc\Nr
2 (L=8)
3 (L=32)
4 (L=64) 1 2 3 4
Jae Son, Nokia Inc.

9. 5/11/2019
Subspace Precoding
We propose a subspace precoding scheme that further reduces feedback information and precoding selection computation.
Due to the channel coherence BW, it is likely that the desired precoding matrix, Q(k+1), of the adjacent subcarrier/cluster (k+1) may be similar to the precoding matrix, Q(k), of the subcarrier (k).
This indicates that Q(k+1) can be found from a smaller subspace (whose subspace size is W<L) that is closely related to Q(k).
Define the precoding matrix subspace of based on some matrix correlation metric.
Ex.
Thus, the precoding matrix search of the adjacent subcarrier (or cluster) will be conducted within this subspace, and its feedback information will represent the matrix index in the subspace. Hence, this will reduce precoding matrix search operations and the number of feedback bits.
Jae Son, Nokia Inc.

10. Subspace Precoding Q_1 Q_3 Q_7 … Q_9 Q_2 Q_L Q_6 … Q_11 L Q_L Q_2 Q_4
5/11/2019
Subspace Precoding
The column index of a selected precoding matrix within the subspace will constitute feedback information. 1 2 W
Subspace
Q_1
Q_3
Q_7 …
Q_9
Q_2
Q_L
Q_6 …
Q_11 L
Q_L
Q_2
Q_4 …
Q_5
Original Codebook W
Subspace Tracking Table
Jae Son, Nokia Inc.

11. Subspace Tracking Table Construction
5/11/2019
Subspace Tracking Table Construction
The row index of the subspace tracking table represents the precoding matrix index in the precoding matrix codebook.
The column index of the subspace tracking table represents the subspace tracking feedback index where its table contents represent the indices of precoding matrices in the subspace.
Each row of the subspace tracking table will be obtained as follows
Compute L metrics for th row,
Sort them by an ascending order.
Collect the first W sorted indices. They constitute the contents of the th row in the subspace tracking table.
Repeat the procedure for all rows of the table.
Jae Son, Nokia Inc.

12. Subspace Tracking Table Construction
5/11/2019
Subspace Tracking Table Construction
An Example of a Subspace Tracking Table for v(2,1,8) when W=4. 00 01 10 11
1 (000) 1 4 5 2
2 (001) 6
3 (010) 3
4 (011) 7
5 (100) 8
6 (101)
7 (110)
8 (111)
Jae Son, Nokia Inc.

13. Feedback Information Frame Format
5/11/2019
Feedback Information Frame Format
The feedback Information frame can be structured by the following vector format.
Nc specifies the number of columns in the precoding matrix
Nr specifies the number of rows in the precoding matrix
Ng specifies a subcarrier cluster size.
Nssi specifies a subspace size.
J specifies the feedback index for the first cluster (which will be the row index of the selected precoding matrix in the codebook).
I(1) specifies the feedback index for the second cluster (which is the column index of the selected precoding matrix in the subspace tracking table).
I(K) specifies the subspace feedback index for the K+1 th cluster.
where Nr Ng
Nssi J
I(1) Nc
I(K) .
Jae Son, Nokia Inc.

14. An Example 5/11/2019 Subspace Tracking Table L = 16 W = 4 I(K)
Freq
J: 4bits (1111)
I(1): 2bits (00)
I(25): 2bits
P_16 => P_3 => P4 => …
Precoding matrix selection
I(2): 2bits (11) 15 14 10 3
P_16 … 4 1 5 16
P_3 12 9 7
P_2 13 8
P_1 11 01 00
0000
0001
0010
1111
I(K)
Date Field Feedback Frame
L = 16
W = 4
Subspace Tracking Table
Jae Son, Nokia Inc.

15. 5/11/2019
An Example
The codebook size is L=16, and the subspace size is W=4.
The first cluster precoding matrix would be found from the original precoding matrix codebook. Based on some precoding matrix selection metric (such as MSE, capacity, etc), let us assume that a precoding matrix, Q_16, is chosen as a precoding matrix for the first subcarrier cluster. So, the first feedback index, J, will be 1111 (=16).
Instead of finding an adjacent precoding matrix from the codebook, it will be found from the subspace tracking table as shown in the example. Since the index of the previous matrix selection is 16, the next precoding matrix would be found among matrices in the 16th row of the subspace tracking table. Let us assume that a precoding matrix, Q_3, is selected by the selection algorithm for the second subcarrier cluster. Then, the feedback index will be 00 which corresponds its column index in the table.
This in turns indicates that the precoding matrix of the third subcarrier cluster can be found from the third row of the subspace tracking table. Again, if Q_4 was selected based on the selection algorithm, then the feedback index for the third subcarrier cluster will be 11.
This procedure will continue for the rest of the data subcarrier clusters.
Jae Son, Nokia Inc.

16. Implementation Overhead Comparison
5/11/2019
Implementation Overhead Comparison
Feedback Bits
Eigen Beamforming:
where is a cluster size and is the number of quantization bits.
Proposed Precoding Method:
where W is the subspace size.
Computational Complexity
Eigen Beamforming
complex number multiplications per cluster
Proposed Precoding Method
complex number multiplications per cluster when the selection algorithm is based on the matrix computation of
However, this number can be lowered depending on selection algorithm implementation.
Jae Son, Nokia Inc.

17. Simulation Simulation Settings Mixed Mode Channel B & D MIMO 2x2
5/11/2019
Simulation
Simulation Settings
Mixed Mode
Channel B & D
MIMO 2x2
Spatial Streams: 1 and 2
Coding & Modulation: CC ½, QPSK
Subspace size: 4
Cluster size: 1, 2, 4
Packet size: 1K Byte/packet
Precoding Selection Algorithm:
Jae Son, Nokia Inc.

18. 5/11/2019
Simulation
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19. 5/11/2019
Simulation
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20. Simulation 5/11/2019 Feedback Bits Comparison
Cluster size = 1
# of Feedback Bits for Subspace Tracking: 3+51*2 = 105 Bits
# of Feedback Bits for SVD: 2x2x8x2x52 = 3328 Bits (Assume 8 bits quantization)
Cluster size = 2
# of Feedback Bits for Subspace Tracking: 3+25*2 = 53 Bits
# of Feedback Bits for SVD: 2x2x8x2x26 = 1664 Bits (Assume 8 bits quantization)
Cluster size = 4
# of Feedback Bits for Subspace Tracking: 3+12*2 = 27 Bits
# of Feedback Bits for SVD: 2x2x8x2x13 = 832 Bits (Assume 8 bits quantization)
Complex Number Multiplications per Cluster
Subspace Tracking: 2*(8+4*51)/52*8 = 66
SVD: 12*8 = 96
Subspace Tracking: 2*(8+4*25)/26*8 = 66
Subspace Tracking: 2*(8+4*12)/13*8 = 69
Jae Son, Nokia Inc.

21. Summary Significant feedback information bits reduction.
5/11/2019
Summary
Significant feedback information bits reduction.
Simple & computationally efficient look-up table approach.
802.16e standard (Section ) based codebook construction .
Competitive performance against SVD beamforming (less than 2 dB performance loss).
No auxiliary feedback procedure required such as RF calibration.
Jae Son, Nokia Inc.

22. 5/11/2019
References
[1] David J. Love, Robert W. Heath Jr., and Thomas Strohmer, 'Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems', IEEE Trans. Information Theory, Vol. 49, No. 10, October 2003, pp
[2] Bertrand M. Hochwald, Thomas L. Marzetta, Thomas J. Richardson, Wim Sweldens, and Rudiger Urbanke, 'Systematic Design of Unitary Space-Time Constellations', IEEE Trans. Information Theory, Vol. 46, No. 6, pp , September 2000.
[3] E-2005 IEEE Standard for Local and metropolitan area networks Part 16:Air Interface for Fixed and Mobile Broadband Wireless Access Systems Amendment for Physical and Medium Access Control Layers for Combined Fixed and Mobile Operation in Licensed Bands
Jae Son, Nokia Inc.