An efficient beam-training scheme for the optimally designed subarray structure in mmWave LoS MIMO systems
- Chunlin Xue^{1},
- Shiwen He^{1, 2}Email authorView ORCID ID profile,
- Yongming Huang^{1},
- Yongpeng Wu^{3} and
- Luxi Yang^{1}
DOI: 10.1186/s13638-017-0820-8
© The Author(s) 2017
Received: 6 July 2016
Accepted: 1 February 2017
Published: 14 February 2017
Abstract
This paper studies the fundamental relations between key design parameters of millimeter wave (mmWave) multiple input multiple output (MIMO) communication systems and subarray structures deployed at both the transmitter and the receiver with each radio frequency (RF) chain connected to only a specific subset of the antennas. The concept of effective degrees of freedom (EDoF) is introduced to measure the maximum spatial multiplexing gain available for the MIMO system. An analytical expression for the EDoF with respect to the parameters of antenna configuration and transmission distance is obtained for the line of sight (LoS) scenario. In addition, the upper and lower bounds of the EDoF are further obtained for some special cases. A fast beam training algorithm based on the codebook is developed to reduce the number of training for the designed mmWave system. Extensive simulation results indicate that the proposed scheme reduces the computational load of the exhaustive approach with only minimal loss in performance. Moreover, the proposed design is robust to the geometrical change and misplacement.
Keywords
Millimeter wave (mmWave) Antenna deployment Beam training Line of sight (LoS) Multiple input multiple output (MIMO)1 Introduction
Motivated by the ever increasing growth in multimedia applications and the number of users, millimeter wave (mmWave) has been regarded as an essential technique for the next generation wireless communication network [1]. The multiple gigabit per second (Gbps) date rate requirements of future broadband systems can be satisfied by large swathes of unlicensed spectrums around the mmWave band [2]. Besides, the remarkable advancements in the mmWave hardware make it feasible to adopt mmWave band in many applications [3]. Spectral efficiencies can be further improved by employing multiple antennas, where multiple independent data streams are transmitted and received in parallel through spatial multiplexing without extra bandwidth or transmit power [4].
The advantages of multiple input multiple output (MIMO) techniques rely heavily on the unique propagation characteristics of wireless channels. It is well known that mmWave channels are usually characterized with sparse scattering structures, which are unfavorable for MIMO systems [5, 6]. Most previous research on MIMO techniques are based on the dense scattering environment to enable spatial multiplexing [4]. While mmWave communications usually take place in the strong line of sight (LoS) circumstances where the channel responses are rank deficient, the spatial multiplexing gain can still be obtained by employing carefully designed antenna arrays thanks to the short wavelength [7]. Some efforts have been made on this topic, both for indoor scenarios [8, 9] and outdoor scenarios [10].
On the other hand, the propagation of mmWave suffers from large path loss due to the small wavelength, which causes the sharp attenuation of signal power and results in disabilities for long distance communications [11, 12]. For outdoor applications, the beamforming technique is regarded as a good solution to compensate the path loss by narrow beams with high array gains [13, 14]. Digital beamforming is traditionally designed based on channel state information (CSI) to improve communication quality at the advantages of digital processing techniques, such as interference cancellation and formation of multiple simultaneous beams. Analog beamforming is put forward to overcome the radio frequency (RF) hardware limitations where a network of analog phase shifters is employed to control the phase of the signal at each antenna. Due to the high power consumption of the digital scheme [15, 16] and the possible performance loss of the analog scheme, the hybrid beamforming scheme has been presented in [17, 18] to divide the beamforming operations between the analog and digital domains, where the required number of RF chains is reduced. In [19], beam training algorithms are proposed to design the partially-connected subarray antenna structure, which uses a separate RF chain and analog-to-digital converter (ADC) for each phase shifters network. More recently, [20] and [21] have investigated the beamforming codebook design for millimeter systems and demonstrated its superiority over the exhaustive searching protocol. However, the existing investigations have mainly considered beamforming techniques and the optimal design for antennas placement separately. As a result, the balance between spatial multiplexing gains and array gains may be ignored in mmWave communications.
In this paper, we propose an efficient beam-training scheme for mmWave LoS MIMO communication systems with subarray structures at both the transmitter and the receiver. The subarray structure with directional antenna elements and phase shifters is designed to maximize effective degrees of freedom (EDoF) so that more multiplexing gain is available. An estimation for the separation between subarrays is also performed when the required EDoF are smaller than the number of RF chains. The codebook design method from [20] and [21] is employed in the proposed beam training scheme to apply for the situation where explicit CSI is unavailable. Furthermore, the proposed scheme focuses on the selection strategy of codewords and is aimed at shortening training time as much as possible. In a word, the proposed beam training scheme includes several iterative training steps, where codewords from the codebook are selected and trained at each step. Numerical results show that the proposed scheme has some advantages over the existing counterparts.
The rest of this paper is organized as follows. Section 2 introduces the system architecture and channel models, Section 3 provides the optimal design criterion of the normalized subarray separation product for maximizing EDoF in the subarray structure. Section 4 provides the proposed low complexity beam training scheme. Simulation results of EDoF and capacity performance are presented in Section 5, and conclusions are given in Section 6.
Notation: A is a matrix, a is a vector, a is a scalar, (·)^{ T } and (·)^{ H } denote transpose and Hermitian conjugate transpose, respectively. I _{ N } is the N×N identity matrix, 1 _{ N } is the N×N all-ones matrix, |·| denotes the determinant operation, ∥·∥_{ F } denotes the Frobenius norm operation, ⌊a⌋ is the largest integer that is smaller than or equal to a, ⌈a⌉ is the smallest integer that is larger than or equal to a, diag(a) is a matrix whose diagonal elements are formed by a, \(\mathcal {CN}\left (\mathbf {a},\mathbf {A}\right)\) is a complex Gaussian vector with mean a and covariance matrix A.
2 System model
where ρ denotes the average received signal to noise ratio (SNR) at the input of the receiver.
where H _{LoS} and H _{NLoS} denote the LoS component and the non line of sight (NLoS) component, respectively. K _{ f } denotes the ratio between the power of these two components. In general, mmWave channel H is mainly determined by the LoS component due to the limited number of scatters in the mmWave propagation environment [23, 24]. Therefore, in this paper, we focus on the case where K _{ f }→+∞.
The approximation is obtained via \(\left ({1 + \Delta } \right)^{{1 \left / 2\right.}} \approx 1 + \frac {\Delta }{2}\) on the condition that Δ≪1.
where ω _{ i } is the ith eigenvalue of \(\widetilde {\mathbf {Q}} = {\mathbf {\widetilde {H}}^{H}}\mathbf {\widetilde {H}}\).
3 Subarray structure design for LoS MIMO
3.1 Maximum EDoF criterion
(13) shows that the EDoF is a simple function of the average SNR, the number of transmit RF chains, and the eigenvalues of the \(\widetilde {\mathbf {Q}}\) matrix. When the average SNR and the eigenvalues are large (ρ ω _{ i }≫N), a 3 dB increase in SNR gives approximately a capacity increase of 1 bit/s/Hz for each subchannel.
It is easy to see that the key design parameter of the LoS MIMO communication system with subarray structures is the ratio between the product S _{ t } S _{ r } and λ R. For description convenience, we define the normalized subarray separation product as \({N_{ssp}} = \frac {{{S_{t}}{S_{r}}}}{{\lambda R}}\). When (18) is satisfied, the EDoF of the RF equivalent channel approach the smaller number of RF chains between the transmitter and the receiver and there are min(N,M) data streams that can be transmitted in parallel effectively. The maximum EDoF criterion is expressed as the relationship among the subarray separation, transmission distance, wavelength and the number of RF chains. It is worth noting that the optimal normalized subarray separation product N _{ ssp } is independent of the angle ϕ and the position shift z _{0} in Fig. 2. Thus, the optimal separations can be easily determined only if the information about the transmission distance, the carrier frequency and the number of RF chains are known to the transmitter and the receiver. It is worth pointing out that the criterion written as a function of the product of the subarray separations allows a tradeoff between the antenna array sizes of the transmitter and the receiver. If one end of the link is restricted to a certain area, it can be compensated by deploying a larger antenna array at the other end to avoid performance loss of the system. This is a common situation in distributed MIMO applications.
3.2 An estimation of subarray separations for required EDoF
In practice, each data stream can be assigned for more than one RF chains at the advantage of diversity techniques. So the required EDoF is usually smaller than the number of RF chains and Eq. (18) is a stricter condition for the actual system. A looser condition can be derived by determining the dynamic range of the EDoF for a specific antennas deployment.
When the vector f _{ n } is linearly dependent on every row of the channel submatrix H _{ m,n } and the vector w _{ m } is linearly dependent on every column of the channel submatrix H _{ m,n }, the equality in (19) is achieved.
where EDoF _{ r } denotes the required EDoF.
4 Proposed beam-training scheme
In this section, an efficient beam training scheme is designed for a specific antenna subarray structure where the transmitter and the receiver have the same number of subarrays, i.e., M=N.
Let l _{ j }∈{1,2,⋯,L _{ t }} be the index of the best codeword so that \(\left \| {{\mathbf {H}_{n,n}}\mathbf {c}_{{l_{j}}}^{t}} \right \|_{F}^{2} = \zeta _{*}^{t}\) and \({\mathbf {c}_{{l_{j}}}^{t}}\) be the referenced codeword for the (j+1)th iteration. As \(\mathbf {e}_{B_{1}^{t}}^{t} = \mathbf {c}_{{l_{0}}}^{t}\), the original referenced codeword \(\mathbf {c}_{{l_{0}}}^{t}\) is covered in the first sub-codebook and it is trained in the first iteration. Finally, the iteration training procedure is terminated when \(\Delta _{j}^{t} = 1\) and the beamforming vector for the nth transmit subarray is set as \({\widehat {\mathbf {f}}_{n}} = \mathbf {c}_{{l_{{J_{t}}}}}^{t}\). The transmit beam training algorithm is summarized in Algorithm 1.
Let l _{ j }∈{1,2,⋯,L _{ r }} be the index of the best codeword so that \(\left \| {{{\left ({\mathbf {c}_{{l_{j}}}^{r}} \right)}^{H}}{\mathbf {H}_{m,m}}{\mathbf {f}_{m}}} \right \|_{F}^{2} = \zeta _{*}^{r}\) and \({\mathbf {c}_{{l_{j}}}^{r}}\) be the referenced codeword for the (j+1)th iteration. As \(\mathbf {e}_{B_{1}^{r}}^{r} = \mathbf {c}_{{l_{0}}}^{r}\), the original referenced codeword \(\mathbf {c}_{{l_{0}}}^{r}\) is covered in the first sub-codebook and it is trained in the first iteration. Finally, the receive training procedure is terminated when \(\Delta _{j}^{r} = 1\) and the beamcombining vector for the mth receive subarray is set as \({\widehat {\mathbf {w}}_{m}} = \mathbf {c}_{{l_{{J_{r}}}}}^{r}\).
Assume that L _{ t }=L _{ r }=L, J _{ t }=J _{ r }=J, and \(B_{1}^{t} = B_{1}^{r} = B_{2}^{t} = B_{2}^{r} = \cdots B_{{J_{t}}}^{t} = B_{{J_{r}}}^{r} = {B_{j}}\), the proposed algorithm needs (N+M)JB _{ j } times of training. For the algorithms in [25], the number of training is (K _{ t }+K _{ r }+K ^{2})N for Algorithms 1 and K _{ t }+K _{ r }+K ^{2} N for Algorithms 2, where K _{ t } (K _{ r }) denotes the number of codewords trained in the initial coarse beamforming training phase for the transmitter (receiver), and K denotes the number of codewords trained in the beamforming refinement phase. Considering an exhaustive search using (23), it requires NL ^{2} times of training, which has a sharp increase when the codebook size becomes large.
Complexity comparison with different algorithms
Algorithms | Number of Training | Parameters | N=2 | N=4 |
---|---|---|---|---|
Equation (23) | NL ^{2} | L=32 | 2048 | 4096 |
Algorithm 1 in [25] | (K _{ t }+K _{ r }+K ^{2})N | \(\begin {array}{c} {K_{t}}=8\\ {K_{r}}=8 \\ K = 4 \end {array}\) | 64 | 128 |
Algorithm 2 in [25] | K _{ t }+K _{ r }+K ^{2} N | 48 | 80 | |
Proposed Algorithm | (N+M)JB _{ j } | \(\begin {array}{c} {B_{j}} = 2\\ J = 5\end {array}\) | 40 | 80 |
5 Simulation results
In this section, numerical results are presented to evaluate the effectiveness of the proposed design criterion with the optimal channel quality and the beam training scheme with low complexity. We consider a 45 GHz MIMO system with the subarray structure in Fig. 2 and N=M=4, N _{ t }=N _{ r }=32, thus each subarray has P=Q=8 antennas. The system is optimized at R=100 m, z _{0}=0 and θ=ϕ=0°, which satisfies the optimal normalized subarray separation product \({N_{ssp}} = \frac {{{S_{t}}{S_r}}}{{\lambda R}} = \frac {1}{4}\) in (18). We set S _{ t }=10S _{ r } and d _{ t }=d _{ r }=λ/2 as a practical placement for the antennas. SNR is set to be −5 dB in our simulations.
Parameters for three antenna array structures
Structures | d _{ t }(cm) | D _{ t }(cm) | S _{ t }(cm) | d _{ r }(cm) | D _{ r }(cm) | S _{ r }(cm) |
---|---|---|---|---|---|---|
(1) | 0.33 | 126.77 | 129.10 | 0.33 | 10.58 | 12.91 |
(2) | 14.43 | 14.43 | 115.47 | 14.43 | 14.43 | 115.47 |
(3) | 0.33 | 0.33 | 2.67 | 0.33 | 0.33 | 2.67 |
6 Conclusions
The subarray structure was designed to realize high EDoF MIMO transmission for wireless channels with a strong LoS component. The criterion showed that the optimal subarray separation product is proportional to the transmission distance multiplied by the wavelength and inversely proportional to the number of RF chains multiplied by the cosine of the spherical angle θ at the receiver. The iterative training scheme had the complexity logarithmic with the codebook size and outperformed some existing algorithms. Although LoS channels are in general rank deficient, the communication system with the proposed scheme can afford both array gains and spatial multiplexing gains through carefully designed antenna subarray spacing and appropriate beam training at both the transmitter and the receiver. In addition, the system designed based on the optimal criterion is robust to the geometrical change and the misplacement in practice. And the beam training scheme has the ability to tolerate some abrupt errors occurred in earlier training steps. Therefore, it is feasible to create high rank MIMO systems over LoS channels by special geometrical configurations and effective beamforming schemes. Finally, more research on flexible styles of the antenna deployment such as uniform plan arrays (UPAs) are expected in the future work.
7 Endnote
^{1} A codebook based beam training scheme to determine f _{ n } and w _{ m } will be explained in more details in Section 4.
Declarations
Acknowledgments
This work was supported by National Natural Science Foundation of China under Grants 61471120 and 61422105, Key Laboratory of Cognitive Radio and Information Processing Ministry of Education (Guilin University of Electronic Technology) under Grants CRKL160203.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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