Beamforming matrix quantization with variable feedback rate
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Abstract
We propose a new technique to quantize and feedback the parameters when a beamforming matrix is compressed with the Givens Rotation (GR). We suggest to feedback the parameters with variable feedback rate, and use efficient source coding and codebook to quantize the GR parameters. The variable feedback rate means that the number of bits used to represent the quantized beamforming matrix is based on the value of the matrix itself. And due to the nonuniform distribution of the GR parameters, source coding and code book can be designed to quantize those parameters in a more effective manner. Compared with the fixed feedback rate scheme, the proposed method delivers a better performance without incurring additional feedback bandwidth.
Keywords
Channel State Information Quantization Error MIMO Channel Mobile Client Time Division DuplexIntroduction
Multiple transmit and receive antennas system has been adopted in several communication standards in order to achieve a higher throughput. The openloop multipleinput multipleoutput (MIMO) technique has already been shown to achieve a high performance gain. With the availability of either the full or partial channel state information (CSI) at the transmitter, we can achieve further performance gain or receiver complexity reduction. Such closedloop schemes have been considered in many communication standards for application of beamforming or multiuser precoding.
However, CSI estimation for the downlink channel at the base station is not possible in Frequency Division Duplex systems. It is also not straightforward to implement CSI estimation in Time Division Duplex (TDD) systems due to the mismatch in the radio front end. Hence in general, the CSI will be estimated at the mobile clients and be sent back to the base station. For example, in the 802.11n wireless LAN system, when the system is operated in TDD mode, the channel can either be estimated by the transmitter through calibration or the channel is fed back by the receiver [1]. This unfortunately requires a high and undesirable feedback bandwidth.
Another popular way to reduce the amount of CSI feedback is through differential encoding [2, 3]. However, such technique suffers accumulated error propagation. Therefore, the mobile client often computes the beamforming matrix, which is usually a unitary matrix and “compress” such a matrix before feeding back to the base station. The “compression” can significantly reduce the feedback bandwidth requirement. And 802.11 ac wireless LAN system [4] adopts this methodology to feedback the beamforming matrix for single and multiuser MIMO. Although we only use unitary beamforming matrix as an example in this article, the techniques that are discussed in this article apply to the feedback of channel matrix as well. For example, we may perform singular value decomposition (SVD) on the channel matrix, and feedback the eigenvalues and both the left and right eigenvectors matrices, which are both unitary.
There have been several proposals in the literature to compress the beamforming vector. One is codebook based such as the vector quantization (VQ) scheme proposed in [5, 6, 7, 8, 9, 10], and another is by using the Givens Rotation (GR) [1, 4, 11, 12, 13]. Compared with the GRbased scheme, the VQ approach requires a higher storage, as a set of codebooks is needed for a particular antenna setting. It has a higher complexity than the GR approach, especially when the number of codewords in the codebook increases. Due to these reasons, the GR approach has been adopted in the 802.11n and 802.11 ac standards [1, 4].
In this article, we investigate an effective approach to quantize and feedback the GR parameters that compress the beamforming matrix. The proposed scheme is capable of achieving a better performance, in the absence of extra bandwidth, than existing techniques that quantize and feedback the GR parameters.
Signal model
MIMO model
where $\tilde{n}$ has the same statistics as n (as U is a unitary matrix). Since $\overline{D}$ is a diagonal matrix, eigenbeamforming leads to simple decoding, as the MIMO channel can be treated as a number of parallel subchannels.
In practice, due to the limited bandwidth in the feedback channel, W has to be quantized, and the base station receives the quantized version of W, denoted by $\tilde{W}$. We assume that the channels are estimated accurately, and there is no error or delay in the feedback channel. With these assumptions we consider only the impact of quantization error due to limited feedback bandwidth. Hence, $\tilde{W}$, instead of W, will be used as the beamforming matrix. In this article, we propose an effective method to quantize W and it will be shown that we can achieve a better performance than that of existing methods using the same average number of feedback bits.
GR model
Hence, the 3 × 2 unitary matrices W can fully be described by just six parameters: ϕ_{11}ϕ_{21}ψ_{21}ψ_{31}ϕ_{22}, and ψ_{32}. A 3 × 1 unitnorm vector only needs four parameters, namely ϕ_{11}ϕ_{21}ψ_{21}, and ψ_{31}. Whereas for 2 × 1 and 2 × 2 cases, two parameters, ϕ_{11} and ψ_{21}, will be sufficient. The full details can be founded in [1, 4].
There are four combinations of bits assigned to the GR parameters in the IEEE 802.11n draft. They can be summarized as follows in the format of (b_{ ψ }b_{ ϕ }), namely (1,3), (2,4), (3,5), and (4,6), where b_{ ψ } represents the number of bits assigned to ψ, and b_{ ϕ } represents the number of bits assigned to ϕ. Parameter ψ has a range from 0 to π/2 whereas ϕ spans over a range from 0 to 2π [1, 4].
Methods
 A.
Dynamic bit assignment:The bits assigned to the GR parameter ϕ can be made dependent on the value of ψ. When the resolution is “sparse” (which can be predetermined based on the value of ψ), we use more bits for the quantization of ϕ; when the resolution is “crowded”, we use fewer bits for the quantization of ϕ. In other words, the bit assignment to ϕ is adaptively adjusted based on the value of ψ.
 B.
Efficient source coding:Due to the nonuniform distribution of the GR parameters ψ, efficient source coding such as the Huffman code [14] can be used to efficiently encode the GR parameter ψ and hence reducing the number of feedback bits required.
 C.
Codebook design:Due to the same reason of nonuniform distribution, instead of quantizing the GR parameter ψ in a uniform manner, codebook can be designed so as to quantize the parameter in a more effective manner.
Depending on the receiver structure or the design criteria, we can apply each of these ideas separately or jointly. We will illustrate each of the above in more details.
Dynamic bit assignment

When r_{2} is large (?_{21} is large), r_{1} will be small; ?_{11} can have a lower resolution.

When r_{2} is small (?_{21} is small), r_{1} will be large; ?_{11} will need a higher resolution.
To achieve a lower quantization error, we can assign different number of bits to ϕ_{11} according to the value of r_{1}. For example, as shown in Figure 2, we assign two bits to ϕ_{11} when r_{1} is small (i.e., ψ_{21} is large, the upper circle), and we assign 4 bits to ϕ_{11} when r_{1} is large (i.e., ψ_{21} is small, the lower circle). It can be seen that the distance between the points are more evenly distributed in this case. Depending on the assignment of ψ_{21}, we can make the probability of having two cases (upper or lower circle) equal. Hence, in this scenario, the total number of bits representing w is 1 + 2 = 3 or 1 + 4 = 5, which is 4 bits on average (if the probability ψ_{21} to be large and small, hence the probability of upper and lower circle are the same).
Efficient source coding
The distributions of the quantized version of ψ when using four levels of granularity are shown in Figure 4g–i. So, we should use less bits to source code those values with higher occurring probability, and more bits to source code those values with lower occurring probability. One possibility is the use of Huffman source coding [14].
Codebook design
Due to the nonuniform and asymmetric distribution of some of the parameters, instead of quantizing the GR parameters uniformly, we can design a codebook so as to reduce the quantization error. For example, instead of quantizing ψ_{31} uniformly with 2 bits using the value [11.25, 33.75, 56.25, 78.75], we can use a codebook [8, 25, 41, 62] that is also 2 bits. As shown in Figure 4, ψ_{31} has a distribution that concentrate to the lefthand side (i.e., higher chances for smaller value), hence our codebook of [8, 25, 41, 62] also tends to have a lower value than the uniform codebook of [11.25, 33.75, 56.25, 78.75].
The above three techniques can be combined and optimized by certain design criteria, which can be a function of receiver. We will perform two case studies in the following sections, one based on the techniques A and C, and another based on the techniques A and B.
Case studies
Depending on the training symbol placement and the receiver design, we consider two cases. In the first case, a simple receiver is not retrained with the beamforming matrix, hence the receiver does not take the mismatch of the quantized beamforming matrix into account, and it simply uses a parallel decoder. On the other hand, in the second case, the receiver is retrained with the updated beamforming matrix, hence the mismatch between the quantized beamforming matrix and the channel is taken into account. It uses a more complicated receiver, such as an MMSE receiver.
Receiver with simple parallel decoder
Due to quantization, $V\tilde{W}$ is no longer an identity matrix, therefore such a simple receiver should be highly sensitive to the quantization error.
Number of bits allocation for three transmit antennas beamforming vector
Bit allocation for ϕ _{ 21 } and ϕ _{ 11 } when the average number of feedback bit is 8
Bit representative of  Bits allocated for ϕ_{21}and ϕ_{11}  Total number of feedback bits  

ψ _{31} ^{*}  ψ _{21} ^{*}  b_{ϕ}_{21}  b_{ϕ}_{11}  
0  0  3  4  9 
0  1  4  3  9 
1  0  2  3  7 
1  1  3  2  7 
Bit allocation for ϕ _{ 21 } and ϕ _{ 11 } when the average number of feedback bit is 12
Bit representative of  Bits allocated for ϕ_{21}and ϕ_{11}  Total number of feedback bits  

ψ _{31} ^{*}  ψ _{21} ^{*}  b_{ϕ}21  b_{ϕ}_{11}  
00  00  3  6  13 
00  01  4  5  13 
00  10  5  4  13 
00  11  6  3  13 
01  00  3  5  12 
01  01  4  4  12 
01  10  4  4  12 
01  11  5  3  12 
10  00  3  5  12 
10  01  4  4  12 
10  10  4  4  12 
10  11  5  3  12 
11  00  3  4  11 
11  01  3  4  11 
11  10  4  3  11 
11  11  4  3  11 
where · in (20) denotes the dot product operation.
MSE readings for quantization of 3 × 1 beamforming vector based on the traditional fixed rate feedback approach versus that of the newly proposed scheme based on variable rate feedback are as follows: MSE of 0.11 versus 0.091 (for 8 bits feedback) and MSE of 0.03 versus 0.028 (for 12 bits feedback). MAD readings are as follows: MAD of 0.31 versus 0.282 (for 8 bits feedback) and MAD of 0.162 versus 0.156 (for 12 bits feedback). Hence, the proposed scheme always achieves a lower MSE and MAD than the traditional scheme for both cases of average 8 or 12 bits feedback.
Receiver with MMSE detector
In this case, we consider a three transmit antennas and two streams of data. Due to the nonuniform distribution of the parameters ψ as shown in Figure 4, we can make use of Huffman coding [14] to encode the quantized value of ψ. Hence, we make use of the techniques “dynamic bit assignment” and “efficient source coding” discussed earlier in this case study.
Huffman code for GR parameters
GR parameters ψ_{21}and ψ_{32}  

Quantized value of ψ _{ 21 } or ψ _{ 32 }  Probability (i)  Huffman code  Bits for ψ _{ 21 } or ψ _{ 32 } (ii)  Ave bits for ψ _{ 21 } or ψ _{ 32 } (i) × (ii) 
11.25  0.14714  110  3  0.44142 
33.75  0.35496  0  1  0.35496 
56.25  0.35146  10  2  0.70292 
78.75  0.14644  111  3  0.43932 
1.93862  
GR parameters ψ _{ 31 }  
Quantized value of ψ _{ 31 }  Probability (i)  Huffman code  Bits for ψ _{ 31 } (ii)  Ave bits for ψ _{ 31 } (i) × (ii) 
11.25  0.2722  10  2  0.5444 
33.75  0.47748  0  1  0.47748 
56.25  0.2299  110  3  0.6897 
78.75  0.02042  111  3  0.06126 
1.77284 
Bits assignment for GR parameters ϕ _{ 11 } and ϕ _{ 21 }
Bits for ϕ_{11}(i)  Bits for ϕ_{21}(ii)  Condition  Probability (iii)  Ave bits for ϕ_{11}and ϕ_{21}((i) + (ii)) * (iii) 

3  3  ψ_{21} = 33.75, 56.25  (0.35496 + 0.35146) * (0.2722 + 0.47748) = 0.5296  3.1776 
ψ_{31} = 11.25, 33.75  
2  2  Otherwise  1 – 0.5296 = 0.4704  1.8816 
5.0592 
The newly proposed scheme can be considered as a hybrid of the traditional GR approach and VQbased approach, i.e., we have a code book for the GR parameters ϕ and ψ. However, the new scheme has a lower storage requirement than those based on VQ codebooks.
The assignment of the number of bits and the codebook design in these case studies are just for illustration, there could be other assignment methods that lead to better performance, and different receiver or different system design may lead to different design criteria.
Conclusions
In this article, a simple quantization scheme has been presented for the unitnorm beamforming vector or unitary beamforming matrix based on variablerate feedback. The basic idea is to provide for higher resolution in the dense area and lower resolution in the sparse area. The idea can directly be applied to the existing GR approach allocating variable bits to the ϕ parameter according to the value of ψ. Due to the nonuniform distribution of the GR parameter ψ, the performance can be further improved if we incorporate into the system efficient source coding and codebook design for GR parameters. Results show that the proposed scheme can achieve a lower MSE and lower MAD. The BER performance of the closeloop MIMO system based on the proposed quantization scheme also outperforms that of existing schemes.
The proposed idea is not restricted to the use of eigenbeamformer or GR which have been used as the baseline for comparison. Our proposed method gives a better accuracy when compressing a unitnorm vector or unitary matrix, and such accuracy plays an important role in many communications system including precoding for multiuser MIMO.
Notes
Acknowledgment
This study was partly supported by the Singapore University Technology and Design (grant no. SUTDZJU/RES/02/2011). Zhaoyang Zhang’s was supported in part by the National Key Basic Research Program of China (No. 2012CB316104) and Zhejiang Provincial Natural Science Foundation of China (No. LR12F01002).
Supplementary material
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