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CN112803977B - Hybrid precoding method for millimeter wave communication system under beam shift effect - Google Patents

Hybrid precoding method for millimeter wave communication system under beam shift effect Download PDF

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CN112803977B
CN112803977B CN202110016299.XA CN202110016299A CN112803977B CN 112803977 B CN112803977 B CN 112803977B CN 202110016299 A CN202110016299 A CN 202110016299A CN 112803977 B CN112803977 B CN 112803977B
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方俊
万千
陈智
李鸿彬
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University of Electronic Science and Technology of China
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Abstract

The invention belongs to the technical field of wireless communication, and particularly relates to a hybrid precoding method of a millimeter wave communication system under a beam offset effect. Conventional Radio Frequency (RF) basis vectors employ array response vectors, and the beam direction varies with frequency when beam offset effects are present, so the present invention is directed to designing basis vectors that are more appropriate than conventional solutions to mitigate the beam offset effects. The base vector designed by the invention has a wider radiation directional diagram, can cover the offset beam direction caused by different frequencies, simultaneously has the gain as large as possible in the required beam interval, and has the beam forming gain as small as possible in other intervals. The basis vector design can be finally constructed as an infinite norm minimization problem and can be solved by an alternating direction multiplier method. Based on the designed basis vectors, hybrid precoding design is performed. Experiments show that the hybrid precoding method provided by the invention can effectively relieve the beam offset effect and is superior to the existing scheme.

Description

波束偏移效应下毫米波通信系统的混合预编码方法Hybrid precoding method for millimeter wave communication system under beam shift effect

技术领域technical field

本发明属于无线通信技术领域，具体涉及一种波束偏移效应下毫米波通信系统的混合预编码方法。The invention belongs to the technical field of wireless communication, and in particular relates to a hybrid precoding method of a millimeter wave communication system under the effect of beam shift.

背景技术Background technique

毫米波/亚太赫兹(mmWave/sub-THz)通信是下一代无线通信系统的重要潜在技术，通过利用毫米波/亚太赫兹频段丰富的频谱资源，可以实现每秒几个G比特的通信速率。一方面，为了在性能和复杂度/成本上达到一个平衡，现有技术提出采用极少射频(RF)链路数的混合构架；另一方面，由于采用了大规模阵列天线以及工作在mmWave/sub-THz频段，使得各工作频率下的波束方向会随着频率的变化而变化，这种现象称之为波束偏移效应。为了缓解波束偏移效应，部分现有工作考虑了码本设计以及混合预编码设计来达到此目的，这些传统方案均采取一组阵列响应向量来逼近最优的预编码器。在射频链路数有限时，这些方案会性能损失明显。Millimeter-wave/sub-THz (mmWave/sub-THz) communication is an important potential technology for next-generation wireless communication systems. By utilizing the abundant spectrum resources of the millimeter-wave/sub-THz frequency band, a communication rate of several gigabits per second can be achieved. On the one hand, in order to achieve a balance between performance and complexity/cost, the prior art proposes a hybrid architecture with very few radio frequency (RF) links; In the sub-THz frequency band, the beam direction at each operating frequency changes with the frequency, which is called the beam shift effect. In order to alleviate the beam shift effect, some existing works consider codebook design and hybrid precoding design to achieve this goal. These traditional schemes all adopt a set of array response vectors to approximate the optimal precoder. When the number of RF links is limited, these schemes suffer significant performance losses.

发明内容SUMMARY OF THE INVENTION

本发明的目的在于提出更合适的混合预编码器，克服毫米波/亚太赫兹通信中的波束偏移效应。通过设计一组新的射频(RF)预编码向量来更好逼近最优的全数字编码器。所设计的每个基向量都具有较宽的辐射方向图，可以有效覆盖各频率下的偏移波束方向。同时，基向量设计可以构建成一个无穷范数问题，通过采用交替方向乘子法(ADMM)来有效解决。当新的基向量设计好后，混合预编码设计可通过混合模拟/数字预编码矩阵逼近全数字预编码器而获得，接收合并矩阵的设计同混合预编码设计的方案思路相同。The purpose of the present invention is to propose a more suitable hybrid precoder to overcome the beam shift effect in millimeter wave/terahertz communication. The optimal all-digital encoder is better approximated by designing a new set of radio frequency (RF) precoding vectors. Each designed basis vector has a wide radiation pattern, which can effectively cover the offset beam direction at each frequency. At the same time, the basis vector design can be constructed as an infinite norm problem, which is effectively solved by adopting the alternating direction multiplier method (ADMM). After the new basis vector is designed, the hybrid precoding design can be obtained by approximating the all-digital precoder by the hybrid analog/digital precoding matrix. The design of the receiving combining matrix is the same as that of the hybrid precoding design.

本发明的技术方案为：The technical scheme of the present invention is:

面向具有波束偏移效应的毫米波/亚太赫兹通信下的混合预编码矩阵和接收合并矩阵设计问题，考虑多输入多输出正交频分复用系统，其中子载波总数为P，基站配置天线数为Nt和射频(RF)链路数为Mt，移动用户端配置天线数为Nr和射频链路数为Mr，且满足Mt＜＜Nt和Mr＜＜Nr。各子载波下都采用相同的RF模拟预编码器

Figure BDA0002886816560000011
和与频率有关的基带数字预编码器
Figure BDA0002886816560000012
此处Ns表示数据流数目。同样，各子载波下都采用相同的RF模拟接收矩阵
Figure BDA0002886816560000021
和与频率有关的基带数字接收矩阵
Figure BDA0002886816560000022
技术方案包括以下步骤：For the design of hybrid precoding matrix and receive combining matrix in mmWave/terahertz communication with beam shift effect, consider a MIMO OFDM system, where the total number of subcarriers is P, and the number of antennas configured at the base station is considered. is N t and the number of radio frequency (RF) links is M t , the mobile user terminal configures the number of antennas as N r and the number of radio frequency links as Mr , and satisfies M t <<N t and Mr <<N r . The same RF analog precoder is used for each subcarrier
Figure BDA0002886816560000011
and frequency dependent baseband digital precoders
Figure BDA0002886816560000012
Here N s represents the number of data streams. Similarly, the same RF analog receive matrix is used for each sub-carrier
Figure BDA0002886816560000021
and the frequency dependent baseband digital receive matrix
Figure BDA0002886816560000022
The technical solution includes the following steps:

S1、构建信道。系统中心载频为fc，子载波总数为P，系统带宽为B，

Figure BDA0002886816560000023
表示信道复合增益,第p个子载波的频率可以表示为
Figure BDA0002886816560000024
且有
Figure BDA0002886816560000025
假设散射路径数为L，对应的出射角与入射角分别表示为{θl}和{ψl}，则第p个子载波下的信道矩阵可以表示为：S1. Build a channel. The system center carrier frequency is f c , the total number of sub-carriers is P, and the system bandwidth is B,
Figure BDA0002886816560000023
Represents the channel composite gain, and the frequency of the p-th subcarrier can be expressed as
Figure BDA0002886816560000024
and have
Figure BDA0002886816560000025
Assuming that the number of scattering paths is L, and the corresponding outgoing angle and incident angle are expressed as {θ l } and {ψ l }, respectively, the channel matrix under the p-th subcarrier can be expressed as:

Figure BDA0002886816560000026
Figure BDA0002886816560000026

此处，here,

Figure BDA0002886816560000027
Figure BDA0002886816560000027

其中，τl表示发射端与接收端的延时，基站天线间距d等于中心载频的半波长，即

Figure BDA0002886816560000028
此处λc是中心载频fc对应的波长，c表示光速。那么信道传输到达基站第一根天线与第m根天线的时间差为
Figure BDA0002886816560000029
Among them, τ l represents the delay between the transmitting end and the receiving end, and the base station antenna distance d is equal to the half wavelength of the center carrier frequency, that is,
Figure BDA0002886816560000028
Here λ c is the wavelength corresponding to the center carrier frequency f c , and c represents the speed of light. Then the time difference between the channel transmission reaching the first antenna of the base station and the mth antenna is
Figure BDA0002886816560000029

S2、获得最优全数字预编码矩阵和接收矩阵。第p个子载波下的接收信号为：S2. Obtain an optimal all-digital precoding matrix and a receiving matrix. The received signal under the p-th subcarrier is:

Figure BDA00028868165600000210
Figure BDA00028868165600000210

其中，

Figure BDA00028868165600000211
表示第p个子载波下的对应信道矩阵，
Figure BDA00028868165600000212
表示第p个子载波下的对应信符号向量且满足
Figure BDA00028868165600000213
Figure BDA00028868165600000214
为Ns行Ns列的单位矩阵，
Figure BDA00028868165600000215
表示均值为0和方差为σ2的加性复高斯噪声。此处，
Figure BDA00028868165600000216
表示xp的复数共轭转置。in,
Figure BDA00028868165600000211
represents the corresponding channel matrix under the p-th subcarrier,
Figure BDA00028868165600000212
represents the corresponding signal symbol vector under the p-th subcarrier and satisfies
Figure BDA00028868165600000213
Figure BDA00028868165600000214
is an identity matrix with N s rows and N s columns,
Figure BDA00028868165600000215
Represents additive complex Gaussian noise with mean 0 and variance σ 2 . here,
Figure BDA00028868165600000216
Represents the complex conjugate transpose of x p .

可达频谱效率可以表示为：The attainable spectral efficiency can be expressed as:

Figure BDA0002886816560000031
Figure BDA0002886816560000031

其中

Figure BDA0002886816560000032
和
Figure BDA0002886816560000033
本步骤目标就是求解出最优全数字预编码矩阵Gp和接收矩阵Jp，之后步骤就是通过逼近最优全数字预编码/接收矩阵来求得混合预编码/接收矩阵。这是因为以上目标函数是非凸的以及变量模为一的约束限制。故忽略模拟预编码矩阵中元素的模为一的限制，考虑一个全数字结构，以上问题可以简化为in
Figure BDA0002886816560000032
and
Figure BDA0002886816560000033
The goal of this step is to solve the optimal all-digital precoding matrix G p and the receiving matrix J p , and the next step is to obtain the hybrid precoding/receiving matrix by approximating the optimal all-digital precoding/receiving matrix. This is due to the above objective function being non-convex and the constraint that the variables are modulo one. Therefore, ignoring the limitation that the modulus of the elements in the analog precoding matrix is one, considering an all-digital structure, the above problem can be simplified as

Figure BDA0002886816560000034
Figure BDA0002886816560000034

Figure BDA0002886816560000035
Figure BDA0002886816560000035

当固定Gp，最合适的接收合并矩阵Jp为When Gp is fixed, the most suitable receive combining matrix Jp is

Figure BDA0002886816560000036
Figure BDA0002886816560000036

此时，有关Gp的目标函数为At this time, the objective function of Gp is

Figure BDA0002886816560000037
Figure BDA0002886816560000037

Figure BDA0002886816560000038
Figure BDA0002886816560000038

那么最优Gp可以求得为Then the optimal Gp can be obtained as

Gp＝Vp(:,1:Ns)Δp G p =V p (:,1:N s ) Δp

此处Vp(:,1:Ns)是矩阵Vp的前Ns列，且Vp是通过对信道HP进行奇异值分解得来，即

Figure BDA0002886816560000039
同时对角矩阵Σp表示为
Figure BDA00028868165600000310
Figure BDA00028868165600000311
是注水法功率分配对角矩阵，且有Here V p (:,1:N s ) is the first N s columns of the matrix V p , and V p is obtained by performing singular value decomposition on the channel HP , that is
Figure BDA0002886816560000039
At the same time, the diagonal matrix Σ p is expressed as
Figure BDA00028868165600000310
Figure BDA00028868165600000311
is the power distribution diagonal matrix of the water injection method, and has

Figure BDA0002886816560000041
Figure BDA0002886816560000041

本步骤目标就是求解出最优全数字预编码矩阵Gp和接收合并矩阵Jp。The goal of this step is to solve the optimal all-digital precoding matrix G p and the receiving combining matrix J p .

S3、设计更合适的基向量{bn}。与传统方案不同的是，本发明致力于寻找更优的基向量{bn}来逼近最优的全数字预编码矩阵，同时要求每个基向量bn都具有宽波束辐射方向图，以此能够覆盖各频率下的偏移波束方向。具体来说，就是基向量在所要求的波束方向区间的波束赋形增益尽可能大，在其它波束方向的波束赋形增益尽可能小。以下将介绍如何产生这样的基向量集合{bn}。S3. Design a more suitable basis vector {b n }. Different from the traditional scheme, the present invention is devoted to finding a better base vector {b n } to approximate the optimal all-digital precoding matrix, and at the same time requires each base vector b n to have a wide beam radiation pattern. Can cover the offset beam direction at each frequency. Specifically, the beamforming gain of the base vector in the required beam direction interval is as large as possible, and the beamforming gain in other beam directions is as small as possible. The following describes how to generate such a set of basis vectors {b n }.

定义definition

Figure BDA0002886816560000042
Figure BDA0002886816560000042

其中实际空间出射角ξ的分布区间是[-1,+1]，则连续的出射角区间可以离散化为一系列的格点，基于此，可以构建一个过完备字典The distribution interval of the actual space exit angle ξ is [-1,+1], then the continuous exit angle interval can be discretized into a series of lattice points. Based on this, an overcomplete dictionary can be constructed.

Figure BDA0002886816560000043
Figure BDA0002886816560000043

其中总格点数为N，令N＝T×d，此处T和d都是整数，且T和d意味着整个出射角区间分成T个区间且每个区间的格点数为d，那么第n个区间对应的矩阵为The total number of grid points is N, let N=T×d, where T and d are both integers, and T and d mean that the entire exit angle interval is divided into T intervals and the number of grid points in each interval is d, then the nth The matrix corresponding to each interval is

Figure BDA0002886816560000044
所有n＝1,...,T
Figure BDA0002886816560000044
all n=1,...,T

将Dn中的列从过完备字典D中移除，剩余列构成的矩阵表示为

Figure BDA0002886816560000045
Remove the columns in D n from the overcomplete dictionary D, and the matrix formed by the remaining columns is expressed as
Figure BDA0002886816560000045

那么，当设计第n个基向量bn时，可以构建如下优化问题Then, when designing the nth basis vector b n , the following optimization problem can be constructed

Figure BDA0002886816560000046
Figure BDA0002886816560000046

s.t.|bn,m|＝1，所有mst|b n,m |=1, all m

其中，bn,m是bn的第m个元素，函数||·||∞表示无穷范数，与

Figure BDA0002886816560000051
且有1为全1的向量。Among them, b n,m is the mth element of b n , the function ||·|| ∞ represents the infinity norm, and
Figure BDA0002886816560000051
And there is a vector whose 1s are all 1s.

通过求解如上优化问题，可以得到一组更加合适的基向量{bn}。By solving the above optimization problem, a set of more suitable basis vectors {b n } can be obtained.

S4、完成混合预编码和接收矩阵设计。当通过以上方式获得一组基向量集合{bn}后，可以通过逼近最优全数字预编码器来获得混合预编码矩阵。令

Figure BDA0002886816560000052
和
Figure BDA0002886816560000053
则S4, complete the hybrid precoding and receiving matrix design. After obtaining a set of basis vectors {b n } in the above manner, a hybrid precoding matrix can be obtained by approximating the optimal all-digital precoder. make
Figure BDA0002886816560000052
and
Figure BDA0002886816560000053
but

Figure BDA0002886816560000054
Figure BDA0002886816560000054

s.t.FRF∈{bn}stF RF ∈{b n }

Figure BDA0002886816560000055
Figure BDA0002886816560000055

该优化可以重新构建成一个稀疏问题，即This optimization can be restructured as a sparse problem, i.e.

Figure BDA0002886816560000056
Figure BDA0002886816560000056

Figure BDA0002886816560000057
Figure BDA0002886816560000057

Figure BDA0002886816560000058
Figure BDA0002886816560000058

其中

Figure BDA0002886816560000059
是测量矩阵，
Figure BDA00028868165600000510
表示
Figure BDA00028868165600000511
存在Mt个非零行，这是由于射频链路数为Mt个。基于正交匹配算法对多观测向量进行稀疏恢复，当估计出
Figure BDA00028868165600000512
后，F取
Figure BDA00028868165600000513
中的Mt个非零行，FRF取B中相对应的Mt个列向量。混合接收矩阵设计可以通过与混合预编码设计的相同原理得到，参见步骤S5和S6。in
Figure BDA0002886816560000059
is the measurement matrix,
Figure BDA00028868165600000510
express
Figure BDA00028868165600000511
There are Mt non-zero rows due to the Mt number of radio frequency links. The multi-observation vector is sparsely restored based on the orthogonal matching algorithm. When the estimated
Figure BDA00028868165600000512
After that, F takes
Figure BDA00028868165600000513
M t non-zero rows in F RF take the corresponding M t column vectors in B. The hybrid receive matrix design can be obtained by the same principle as the hybrid precoding design, see steps S5 and S6.

S5、设计基向量{b n}：S5. Design basis vector { b n }:

定义definition

Figure BDA00028868165600000514
Figure BDA00028868165600000514

其中实际空间出射角ξ的分布区间是[-1,+1]，则连续的出射角区间可以离散化为一系列的格点，据此构建一个过完备字典：The distribution interval of the actual space exit angle ξ is [-1,+1], then the continuous exit angle interval can be discretized into a series of lattice points, and an overcomplete dictionary is constructed accordingly:

Figure BDA00028868165600000515
Figure BDA00028868165600000515

其中总格点数为N，令N＝T×d，此处T和d都是整数，且T和d意味着整个出射角区间分成T个区间且每个区间的格点数为d，那么第n个区间对应的矩阵为：The total number of grid points is N, let N=T×d, where T and d are both integers, and T and d mean that the entire exit angle interval is divided into T intervals and the number of grid points in each interval is d, then the nth The matrix corresponding to each interval is:

Figure BDA0002886816560000061
所有n＝1,...,T
Figure BDA0002886816560000061
all n=1,...,T

将En中的列从过完备字典E中移除，剩余列构成的矩阵表示为

Figure BDA0002886816560000062
Remove the columns in En from the overcomplete dictionary E, and the matrix formed by the remaining columns is expressed as
Figure BDA0002886816560000062

当设计第n个基向量b n时，构建如下优化问题：When designing the nth basis vector b n , the following optimization problem is constructed:

Figure BDA0002886816560000063
Figure BDA0002886816560000063

s.t.|b n,m|＝1，所有mst| b n,m |=1, all m

其中，b n,m是b n的第m个元素，函数||·||∞表示无穷范数，与

Figure BDA0002886816560000064
且有1为全1的向量；Among them, b n,m is the mth element of b n , the function ||·|| ∞ represents the infinity norm, and
Figure BDA0002886816560000064
And there is a vector whose 1 is all 1;

对于如上优化问题，采用交替方向乘子法求解，到基向量{bn}；For the above optimization problem, the alternating direction multiplier method is used to solve the problem to the basis vector {b n };

S6、获得一组基向量集合{b n}后，通过逼近最优全数字接收合并矩阵来获得混合接收矩阵；令

Figure BDA0002886816560000065
和
Figure BDA0002886816560000066
则S6. After obtaining a set of basis vector sets { b n }, a hybrid receiving matrix is obtained by approximating the optimal all-digital receiving combining matrix; let
Figure BDA0002886816560000065
and
Figure BDA0002886816560000066
but

Figure BDA0002886816560000067
Figure BDA0002886816560000067

s.t.WRF(:,i)∈{b n}stW RF (:,i)∈{ b n }

其中，WRF(:,i)∈{b n}表示WRF的每一列都属于有限集合{b n}，将该优化重新构建成一个稀疏问题，即where W RF (:,i)∈{ b n } indicates that each column of W RF belongs to the finite set { b n }, and the optimization is reconstructed as a sparse problem, namely

Figure BDA0002886816560000068
Figure BDA0002886816560000068

Figure BDA0002886816560000069
Figure BDA0002886816560000069

其中

Figure BDA00028868165600000610
是测量矩阵，
Figure BDA00028868165600000611
表示
Figure BDA00028868165600000612
存在Mr个非零行，这是由于射频链路数为Mr个；基于正交匹配算法对多观测向量进行稀疏恢复，当估计出
Figure BDA00028868165600000613
后，基带数字接收矩阵
Figure BDA00028868165600000614
取
Figure BDA00028868165600000615
中的Mr个非零行，模拟接收矩阵WRF取B中相对应的Mr个列向量。至此，完成了混合预编码以及接收矩阵的设计。in
Figure BDA00028868165600000610
is the measurement matrix,
Figure BDA00028868165600000611
express
Figure BDA00028868165600000612
There are M r non-zero rows, because the number of radio frequency links is M r ; the multi-observation vector is sparsely restored based on the orthogonal matching algorithm, when the estimated
Figure BDA00028868165600000613
After the baseband digital receive matrix
Figure BDA00028868165600000614
Pick
Figure BDA00028868165600000615
The M r non-zero rows in the analog receive matrix W RF take the corresponding M r column vectors in B. So far, the hybrid precoding and the design of the receiving matrix are completed.

本发明的有益效果为，本发明所提出的混合预编码方法能够有效缓解波束偏移效应，并且优于现有方案。The beneficial effect of the present invention is that the hybrid precoding method proposed by the present invention can effectively alleviate the beam shift effect, and is superior to the existing solution.

附图说明Description of drawings

图1为各方法频谱效率与信噪比的关系,实验条件为Mr＝Mt＝Ns；Figure 1 shows the relationship between the spectral efficiency and the signal-to-noise ratio of each method, and the experimental conditions are Mr = M t = N s ;

图2为各方法频谱效率与信噪比的关系，实验条件为Mr＝Mt＝2Ns；Fig. 2 shows the relationship between the spectral efficiency and the signal-to-noise ratio of each method, and the experimental condition is Mr =M t = 2N s ;

图3为各方法频谱效率与射频链路数的关系，实验条件为Mr＝Mt和SNR＝10dB；Figure 3 shows the relationship between the spectral efficiency of each method and the number of radio frequency links , and the experimental conditions are Mr = M t and SNR = 10dB;

图4为各方法频谱效率与系统带宽的关系，实验条件为Mr＝Mt＝Ns和SNR＝10dB。FIG. 4 shows the relationship between the spectral efficiency of each method and the system bandwidth, and the experimental conditions are Mr= Mt = Ns and SNR= 10dB .

具体实施方式Detailed ways

下面结合附图和仿真示例对本发明进行详细的描述，以证明本发明的实用性。The present invention is described in detail below in conjunction with the accompanying drawings and simulation examples to prove the practicability of the present invention.

本发明考虑具有波束偏移效应的毫米波/亚太赫兹通信下的混合预编码和接收矩阵设计问题，对于多输入多输出正交频分复用系统，系统中基站配置天线数为Nt和射频(RF)链路数为Mt，移动用户端配置天线数为Nr和射频链路数为Mr，且满足Mt＜＜Nt和Mr＜＜Nr。各子载波下都采用相同的RF模拟预编码矩阵

Figure BDA0002886816560000071
和与频率有关的基带数字预编码矩阵
Figure BDA0002886816560000072
此处Ns表示数据流数目。同样，各子载波下采用相同的RF模拟接收矩阵
Figure BDA0002886816560000073
和与频率有关的基带数字接收矩阵
Figure BDA0002886816560000074
第p个子载波下的接收信号为The present invention considers the hybrid precoding and receiving matrix design problems under the millimeter wave/terahertz communication with beam shift effect. For the multi-input multi-output orthogonal frequency division multiplexing system, the base station in the system configures the number of antennas as N t and radio frequency The number of (RF) links is M t , the number of antennas configured by the mobile user terminal is N r and the number of radio frequency links is Mr , and M t <<N t and Mr <<N r are satisfied . The same RF analog precoding matrix is used for each subcarrier
Figure BDA0002886816560000071
and the frequency-dependent baseband digital precoding matrix
Figure BDA0002886816560000072
Here N s represents the number of data streams. Similarly, the same RF analog receive matrix is used for each sub-carrier
Figure BDA0002886816560000073
and the frequency dependent baseband digital receive matrix
Figure BDA0002886816560000074
The received signal under the p-th subcarrier is

Figure BDA0002886816560000075
Figure BDA0002886816560000075

其中，

Figure BDA0002886816560000076
表示第p个子载波下的对应信道矩阵，
Figure BDA0002886816560000077
表示第p个子载波下的对应信符号向量且满足
Figure BDA0002886816560000078
Figure BDA0002886816560000079
为Ns行Ns列的单位矩阵，
Figure BDA00028868165600000710
表示均值为0和方差为σ2的加性复高斯噪声。此处，
Figure BDA00028868165600000711
表示xp的复数共轭转置。in,
Figure BDA0002886816560000076
represents the corresponding channel matrix under the p-th subcarrier,
Figure BDA0002886816560000077
represents the corresponding signal symbol vector under the p-th subcarrier and satisfies
Figure BDA0002886816560000078
Figure BDA0002886816560000079
is an identity matrix with N s rows and N s columns,
Figure BDA00028868165600000710
Represents additive complex Gaussian noise with mean 0 and variance σ 2 . here,
Figure BDA00028868165600000711
Represents the complex conjugate transpose of x p .

令fc表示系统中心载频，则第p个子载波的频率可以表示为Let f c denote the system center carrier frequency, then the frequency of the p-th subcarrier can be expressed as

Figure BDA0002886816560000081
Figure BDA0002886816560000081

此时

Figure BDA0002886816560000082
系统总子载波数为P且带宽为B。同时令发射端与接收端的延时为τl，基站天线间距d等于中心载频的半波长，即
Figure BDA0002886816560000083
此处λc是中心载频对应的波长，c表示光速。那么信道传输到达基站第一根天线与第m根天线的时间差为
Figure BDA0002886816560000084
对于单天线移动用户端而言，与基站第m根天线之间的信道可以表示为at this time
Figure BDA0002886816560000082
The total number of subcarriers in the system is P and the bandwidth is B. At the same time, let the delay between the transmitter and the receiver be τ l , and the base station antenna distance d is equal to the half wavelength of the center carrier frequency, that is,
Figure BDA0002886816560000083
Here λ c is the wavelength corresponding to the center carrier frequency, and c represents the speed of light. Then the time difference between the channel transmission reaching the first antenna of the base station and the mth antenna is
Figure BDA0002886816560000084
For a single-antenna mobile user terminal, the channel with the mth antenna of the base station can be expressed as

Figure BDA0002886816560000085
Figure BDA0002886816560000085

考虑移动用户端配置阵列天线，那么第p个子载波下的对应信道矩阵可以表示为Considering that the mobile user terminal is configured with an array antenna, the corresponding channel matrix under the p-th subcarrier can be expressed as

Figure BDA0002886816560000086
Figure BDA0002886816560000086

其中，in,

Figure BDA0002886816560000087
Figure BDA0002886816560000087

进而，可达频谱效率可以表示为：Furthermore, the attainable spectral efficiency can be expressed as:

Figure BDA0002886816560000088
Figure BDA0002886816560000088

其中

Figure BDA0002886816560000089
和
Figure BDA00028868165600000810
目标是为了最大化频谱效率，则混合预编码和接收矩阵设计可以构建成如下优化问题in
Figure BDA0002886816560000089
and
Figure BDA00028868165600000810
The goal is to maximize spectral efficiency, then the hybrid precoding and receive matrix design can be formulated as the following optimization problem

Figure BDA0002886816560000091
Figure BDA0002886816560000091

Figure BDA0002886816560000092
Figure BDA0002886816560000092

|FRF(i,j)|＝|WRF(k,j)|＝1,所有i,j,k|F RF (i,j)|=|W RF (k,j)|=1, all i,j,k

Gp＝FRFFp,Jp＝WRFWp G p =F RF F p , J p =W RF W p

以上优化问题很难求解，在于目标函数是非凸的，以及模拟预编码矩阵各元素模为一的限制。为了简化问题，考虑一个全数字结构，以上问题可以简化为The above optimization problem is difficult to solve because the objective function is non-convex and the restriction that the elements of the simulated precoding matrix are modulo one. To simplify the problem, consider an all-digital structure, the above problem can be reduced to

Figure BDA0002886816560000093
Figure BDA0002886816560000093

Figure BDA0002886816560000094
Figure BDA0002886816560000094

当固定Gp，最合适的接收矩阵Jp为When Gp is fixed, the most suitable receiving matrix Jp is

Figure BDA0002886816560000095
Figure BDA0002886816560000095

此时，有关Gp的目标函数为At this time, the objective function of Gp is

Figure BDA0002886816560000096
Figure BDA0002886816560000096

Figure BDA0002886816560000097
Figure BDA0002886816560000097

那么最优Gp可以求得为Then the optimal Gp can be obtained as

Gp＝Vp(:,1:Ns)Δp G p =V p (:,1:N s ) Δp

此处Vp(:,1:Ns)是矩阵Vp的前Ns列，且Vp是通过对信道HP进行奇异值分解得来，即

Figure BDA0002886816560000098
同时对角矩阵Σp表示为
Figure BDA0002886816560000099
Figure BDA00028868165600000910
是注水法功率分配对角矩阵，且有Here V p (:,1:N s ) is the first N s columns of the matrix V p , and V p is obtained by performing singular value decomposition on the channel HP , that is
Figure BDA0002886816560000098
At the same time, the diagonal matrix Σ p is expressed as
Figure BDA0002886816560000099
Figure BDA00028868165600000910
is the power distribution diagonal matrix of the water injection method, and has

Figure BDA0002886816560000101
Figure BDA0002886816560000101

令

Figure BDA0002886816560000102
和
Figure BDA0002886816560000103
则寻找混合预编码矩阵来逼近最优全数字预编码矩阵，即make
Figure BDA0002886816560000102
and
Figure BDA0002886816560000103
Then find the hybrid precoding matrix to approximate the optimal all-digital precoding matrix, namely

Figure BDA0002886816560000104
Figure BDA0002886816560000104

s.t.FRF∈γRF stF RF ∈ γ RF

Figure BDA0002886816560000105
Figure BDA0002886816560000105

其中ΥRF表示可行的RF预编码集合。对于传统方案而言，基向量取阵列响应向量，即where YRF denotes the set of feasible RF precodings. For the traditional scheme, the basis vector takes the array response vector, i.e.

Figure BDA0002886816560000106
Figure BDA0002886816560000106

s.t.FRF∈{aBS(φn)}stF RF ∈{a BS (φ n )}

Figure BDA0002886816560000107
Figure BDA0002886816560000107

此处

Figure BDA0002886816560000108
且{φn}是一组出射角，该组出射角是将连续角度区间离散化后的有限格点。here
Figure BDA0002886816560000108
And {φ n } is a set of outgoing angles, which are finite lattice points after discretizing the continuous angle interval.

与传统方案不同的是，本发明致力于寻找更优的基向量{bn}来逼近最优的全数字预编码矩阵，同时要求每个基向量bn都具有宽波束辐射方向图，以此能够覆盖各频率下的偏移波束方向。具体来说，就是基向量在所要求的波束方向区间的波束赋形增益尽可能大，在其它波束方向的波束赋形增益尽可能小。以下将介绍如何产生这样的基向量集合{bn}。Different from the traditional scheme, the present invention is devoted to finding a better base vector {b n } to approximate the optimal all-digital precoding matrix, and at the same time requires each base vector b n to have a wide beam radiation pattern. Can cover the offset beam direction at each frequency. Specifically, the beamforming gain of the base vector in the required beam direction interval is as large as possible, and the beamforming gain in other beam directions is as small as possible. The following describes how to generate such a set of basis vectors {b n }.

定义definition

Figure BDA0002886816560000109
Figure BDA0002886816560000109

其中实际空间出射角ξ的分布区间是[-1,+1]，则连续的出射角区间可以离散化为一系列的格点，基于此，可以构建一个过完备字典The distribution interval of the actual space exit angle ξ is [-1,+1], then the continuous exit angle interval can be discretized into a series of lattice points. Based on this, an overcomplete dictionary can be constructed.

Figure BDA00028868165600001010
Figure BDA00028868165600001010

其中总格点数为N，令N＝T×d，此处T和d都是整数，且T和d意味着整个出射角区间分成T个区间且每个区间的格点数为d，那么第n个区间对应的矩阵为The total number of grid points is N, let N=T×d, where T and d are both integers, and T and d mean that the entire exit angle interval is divided into T intervals and the number of grid points in each interval is d, then the nth The matrix corresponding to each interval is

Figure BDA0002886816560000111
所有n＝1,...,T
Figure BDA0002886816560000111
all n=1,...,T

将Dn中的列从过完备字典D中移除，剩余列构成的矩阵表示为

Figure BDA0002886816560000112
Remove the columns in D n from the overcomplete dictionary D, and the matrix formed by the remaining columns is expressed as
Figure BDA0002886816560000112

那么，当设计第n个基向量bn时，可以构建如下优化问题Then, when designing the nth basis vector b n , the following optimization problem can be constructed

Figure BDA0002886816560000113
Figure BDA0002886816560000113

s.t.|bn,m|＝1，所有mst|b n,m |=1, all m

其中，bn,m是bn的第m个元素，函数||·||∞表示无穷范数。该优化问题可以转化为Among them, b n,m is the mth element of b n , and the function ||·|| ∞ represents the infinity norm. This optimization problem can be transformed into

Figure BDA0002886816560000114
Figure BDA0002886816560000114

Figure BDA0002886816560000115
Figure BDA0002886816560000115

bn＝zn b n =z n

|zn,m|＝1，所有m|z n,m |=1, all m

|Cn(k,k)|＝1，所有k|C n (k,k)|=1, all k

此处

Figure BDA0002886816560000116
是对角元模为一的对角矩阵，Cn(k,k)表示Cn的第k个对角元。上述优化问题进一步可以等价写为here
Figure BDA0002886816560000116
is a diagonal matrix whose diagonal elements are modulo one, and C n (k,k) represents the kth diagonal element of C n . The above optimization problem can be further equivalently written as

Figure BDA0002886816560000117
Figure BDA0002886816560000117

s.t.|zn,m|＝1，所有mst|z n,m |=1, all m

|Cn(k,k)|＝1，所有k|C n (k,k)|=1, all k

同时对偶变量可更新为At the same time the dual variable can be updated as

Figure BDA0002886816560000118
Figure BDA0002886816560000118

Figure BDA0002886816560000119
Figure BDA0002886816560000119

和变量{bn,qn,zn,Cn}可以按如下方式交替更新获得：and variables { bn , q n , z n , C n } can be obtained by alternately updating as follows:

1)更新bn:与bn相关的目标函数为1) Update b n : the objective function related to b n is

Figure BDA0002886816560000121
Figure BDA0002886816560000121

此时bn的解为At this time, the solution of b n is

Figure BDA0002886816560000122
Figure BDA0002886816560000122

2)更新qn:与qn相关的目标函数为2) Update q n : the objective function related to q n is

Figure BDA0002886816560000123
Figure BDA0002886816560000123

该无穷范数最小化问题可以对

Figure BDA0002886816560000124
裁剪得到qn。This infinite norm minimization problem can be solved for
Figure BDA0002886816560000124
Crop to get q n .

3)更新zn:与zn相关的目标函数为3) Update z n : the objective function related to z n is

Figure BDA0002886816560000125
Figure BDA0002886816560000125

s.t.|zn,m|＝1，所有mst|z n,m |=1, all m

可以获得解为can be solved as

Figure BDA0002886816560000126
Figure BDA0002886816560000126

其中∠(·)表示复变量的角度值。where ∠(·) represents the angle value of the complex variable.

4)更新Cn:与Cn相关的目标函数为4) Update C n : the objective function related to C n is

Figure BDA0002886816560000127
Figure BDA0002886816560000127

s.t.|Cn(k,k)|＝1，所有kst|C n (k,k)|=1, all k

则对角矩阵Cn的第k个对角元可以求得为Then the k-th diagonal element of the diagonal matrix C n can be obtained as

Figure BDA0002886816560000128
所有k
Figure BDA0002886816560000128
all k

其中(∠(x))k表示向量x的第k个元素的角度值。where (∠(x)) k represents the angle value of the kth element of the vector x.

当通过以上方式获得一组基向量集合{bn}后，可以通过逼近最优全数字预编码器来获得混合预编码矩阵，即After obtaining a set of basis vectors {b n } in the above manner, the hybrid precoding matrix can be obtained by approximating the optimal all-digital precoder, namely

Figure BDA0002886816560000131
Figure BDA0002886816560000131

s.t.FRF∈{bn}stF RF ∈{b n }

Figure BDA0002886816560000132
Figure BDA0002886816560000132

该优化可以重新构建成一个稀疏问题，即This optimization can be restructured as a sparse problem, i.e.

Figure BDA0002886816560000133
Figure BDA0002886816560000133

Figure BDA0002886816560000134
Figure BDA0002886816560000134

Figure BDA0002886816560000135
Figure BDA0002886816560000135

其中

Figure BDA0002886816560000136
是测量矩阵，
Figure BDA0002886816560000137
表示
Figure BDA0002886816560000138
存在Mt个非零行，这是由于射频链路数为Mt个。基于正交匹配算法对多观测向量进行稀疏恢复，当估计出
Figure BDA0002886816560000139
后，F取
Figure BDA00028868165600001310
中的Mt个非零行，FRF取B中相对应的Mt个列向量。混合接收矩阵设计可以通过与混合预编码设计的相同原理得到，参见步骤S5和S6。。in
Figure BDA0002886816560000136
is the measurement matrix,
Figure BDA0002886816560000137
express
Figure BDA0002886816560000138
There are Mt non-zero rows due to the Mt number of radio frequency links. The multi-observation vector is sparsely restored based on the orthogonal matching algorithm. When the estimated
Figure BDA0002886816560000139
After that, F takes
Figure BDA00028868165600001310
M t non-zero rows in F RF take the corresponding M t column vectors in B. The hybrid receive matrix design can be obtained by the same principle as the hybrid precoding design, see steps S5 and S6. .

仿真中，考虑点对点下行链路宽带毫米波MIMO-OFDM系统，基站配置天线数为Nt＝256，移动用户端配置天线数为Nr＝128。系统中心载频为fc＝28GHz和带宽为B＝4GHz，子载波总数设置为P＝512。同时出射角{θl}和入射角{ψl}随机分布在

Figure BDA00028868165600001311
则有sin(θl)∈[-1,+1]和sin(ψl)∈[-1,+1]。每条信道的延迟τl均匀分布在0到100纳秒，信道复合增益为
Figure BDA00028868165600001312
且
Figure BDA00028868165600001313
和
Figure BDA00028868165600001314
同时通信距离D为60米，信道衰落指数α取2，光速为c。数据流数为Ns＝3，基向量个数为T＝64，且有N＝512和d＝8。In the simulation, considering the point-to-point downlink wideband millimeter-wave MIMO-OFDM system, the number of antennas configured by the base station is N t =256, and the number of antennas configured by the mobile user terminal is N r =128. The center carrier frequency of the system is f c =28GHz and the bandwidth is B=4GHz, and the total number of sub-carriers is set to P=512. Simultaneously the exit angle {θ l } and the incident angle {ψ l } are randomly distributed in
Figure BDA00028868165600001311
Then there are sin(θ l )∈[-1,+1] and sin(ψ l )∈[-1,+1]. The delay τ l of each channel is uniformly distributed from 0 to 100 nanoseconds, and the channel composite gain is
Figure BDA00028868165600001312
and
Figure BDA00028868165600001313
and
Figure BDA00028868165600001314
At the same time, the communication distance D is 60 meters, the channel fading index α is taken as 2, and the speed of light is c. The number of data streams is N s =3, the number of basis vectors is T=64, and there are N=512 and d=8.

信噪比定义为The signal-to-noise ratio is defined as

Figure BDA00028868165600001315
Figure BDA00028868165600001315

在性能分析中，本发明(improved spatially-sparse-precoding，简称I-SSP)将与传统方案(conventional spatially-sparse-precoding，简称C-SSP)比较，且传统方案C-SSP的基向量是阵列响应向量，同时，在仿真中，也加入了全数字最优编码器(称之为full-digital)的性能曲线。所采用的指标是频谱效率(spectral efficiency)；。In the performance analysis, the present invention (improved spatially-sparse-precoding, referred to as I-SSP) will be compared with the traditional scheme (conventional spatially-sparse-precoding, referred to as C-SSP), and the basis vector of the traditional scheme C-SSP is an array The response vector, at the same time, in the simulation, also joined the performance curve of the fully digital optimal encoder (called full-digital). The index used is spectral efficiency;

图1描述了各方法频谱效率与SNR的关系，实验条件设置为Mr＝Mt＝Ns。从图1可以观察出，所提出I-SSP方案相对于传统C-SSP方案有着明显性能优势。这验证了本发明方案可以有效缓解波束偏移效应。图2所设置的实验条件为Mr＝Mt＝2Ns，同样展示出了所提出方案I-SSP的优势。Fig. 1 describes the relationship between spectral efficiency and SNR of each method, and the experimental conditions are set as Mr =M t =N s . It can be observed from Figure 1 that the proposed I-SSP scheme has obvious performance advantages over the traditional C-SSP scheme. This verifies that the solution of the present invention can effectively alleviate the beam shift effect. The experimental condition set in FIG. 2 is Mr =M t = 2N s , which also shows the advantages of the proposed scheme I-SSP.

图3描述了各方法频谱效率与射频(RF)链路数的关系，实验条件设置为Mr＝Mt和SNR＝10dB。从图3可以观测出，当射频(RF)链路数越多，就可以获得越好的性能。考虑到RF链路的成本和功耗，所以RF链路数在毫米波通信中是有限的。同时，可以注意到，当RF链路数有限时，所提出的I-SSP相较于传统C-SSP有着较大性能优势。Figure 3 depicts the relationship between the spectral efficiency of each method and the number of radio frequency (RF) links , and the experimental conditions are set as Mr = M t and SNR = 10dB. It can be observed from Figure 3 that when the number of radio frequency (RF) chains is increased, the better performance can be obtained. Considering the cost and power consumption of RF links, the number of RF links is limited in mmWave communications. At the same time, it can be noticed that the proposed I-SSP has a large performance advantage over the traditional C-SSP when the number of RF links is limited.

接下来，图4描述了频谱效率与系统带宽的关系，实验条件设置为Mr＝Mt＝Ns和SNR＝10dB。从图4中可以发现，当带宽变大时，全数字方案于混合预编码方案之间的性能间距越来越大，这是因为随着宽带增加，波束偏移效应越来越明显。同时实验结果表明，对于不同系统带宽，本专利所提出方案I-SSP一直保持着对传统方案C-SSP的优势。Next, Fig. 4 depicts the relationship between spectral efficiency and system bandwidth, and the experimental conditions are set as Mr= Mt = Ns and SNR= 10dB . It can be found from Figure 4 that when the bandwidth becomes larger, the performance gap between the all-digital scheme and the hybrid precoding scheme becomes larger and larger, because as the bandwidth increases, the beam shift effect becomes more and more obvious. At the same time, the experimental results show that, for different system bandwidths, the scheme I-SSP proposed in this patent has always maintained the advantages over the traditional scheme C-SSP.

综上所诉，本发明研究了在波束偏移效应下的毫米波MIMO-OFDM系统中的混合预编码和接收矩阵设计。为了缓解波束偏移效应，构建了一个新的目标函数来设计更合适的RF基向量集合。当设计好基向量之后，可以通过逼近最优全数字编码矩阵来设计混合预编码/接收矩阵。仿真结果表明，本专利所提出的方案，相对于传统方案，可以有效缓解波束偏移效应。To sum up, the present invention studies the hybrid precoding and receiving matrix design in the millimeter-wave MIMO-OFDM system under the beam shift effect. To mitigate the beam shift effect, a new objective function is constructed to design a more suitable set of RF basis vectors. After designing the base vector, the hybrid precoding/receiving matrix can be designed by approximating the optimal all-digital coding matrix. The simulation results show that, compared with the traditional scheme, the scheme proposed in this patent can effectively alleviate the beam shift effect.

Claims (1)

1. Hybrid precoding method for millimeter wave communication system under beam offset effect, millimeter wave communication system, and millimeter wave communication systemIn a wave communication system, a base station configures an antenna number NtAnd a Radio Frequency (RF) link number of MtThe number of the mobile user side configuration antennas is NrAnd the number of radio frequency links is MrAnd satisfy Mt＜＜NtAnd Mr＜＜Nr(ii) a The total number of subcarriers in the system is P, and the same RF analog precoding matrix is adopted under each subcarrier
Figure FDA0002886816550000011
And frequency dependent baseband digital precoding matrix
Figure FDA0002886816550000012
Here NsRepresenting the number of data streams; similarly, the same RF analog receiving matrix is adopted under each subcarrier
Figure FDA0002886816550000013
And frequency dependent baseband digital receiving matrix
Figure FDA0002886816550000014
Characterized in that the hybrid precoding method comprises the following steps:
s1, constructing a channel: system center carrier frequency of fcTotal number of subcarriers is P, system bandwidth is B, order
Figure FDA0002886816550000015
Representing the channel complex gain, the frequency of the p-th subcarrier is represented as:
Figure FDA0002886816550000016
Figure FDA0002886816550000017
assuming that the number of scattering paths is L, the corresponding exit angle and incident angle are respectively shownShown as { thetalAnd psilAnd if so, the channel matrix under the p-th subcarrier is represented as:
Figure FDA0002886816550000018
Figure FDA0002886816550000019
wherein, taulRepresenting the delay between the transmitting and receiving ends, the base station antenna spacing d being equal to half the wavelength of the central carrier frequency, i.e.
Figure FDA00028868165500000110
λcIs the central carrier frequency fcThe corresponding wavelength, c represents the speed of light; the time difference between the arrival of the channel transmission at the first antenna and the m-th antenna of the base station is
Figure FDA00028868165500000111
S2, obtaining an optimal all-digital precoding matrix and a receiving combination matrix:
the received signal under the p-th subcarrier is:
Figure FDA0002886816550000021
wherein,
Figure FDA0002886816550000022
representing the corresponding channel matrix at the p-th sub-carrier,
Figure FDA0002886816550000023
represents the corresponding symbol vector under the p sub-carrier and satisfies
Figure FDA0002886816550000024
Figure FDA0002886816550000025
Is NsLine NsThe identity matrix of the column(s),
Figure FDA0002886816550000026
means mean 0 and variance σ2The additive complex gaussian noise of (a) is,
Figure FDA0002886816550000027
denotes xpThe complex number conjugate transpose;
the achievable spectral efficiency is expressed as:
Figure FDA0002886816550000028
wherein the optimal full digital pre-coding matrix
Figure FDA0002886816550000029
Receiving a combining matrix
Figure FDA00028868165500000210
Considering an all-digital architecture, the above problem is simplified to:
Figure FDA00028868165500000211
Figure FDA00028868165500000212
when fixing GpThe most suitable receiving and combining matrix JpIs composed of
Figure FDA00028868165500000213
At this time, about GpHas an objective function of
Figure FDA00028868165500000214
Figure FDA00028868165500000215
Finding the optimum GpComprises the following steps:
Gp＝Vp(:,1:Ns)Δp
here Vp(:,1:Ns) Is a matrix VpFront N ofsColumn, and VpIs through the pair channel HPBy performing singular value decomposition, i.e.
Figure FDA0002886816550000031
Simultaneous diagonal matrix sigmapIs shown as
Figure FDA0002886816550000032
Figure FDA0002886816550000033
Is a water injection method power distribution diagonal matrix, and has:
Figure FDA0002886816550000034
obtaining an optimal all-digital precoding matrix GpAnd receiving a combining matrix Jp；
S3 design basis vector bn}：
Definition of
Figure FDA0002886816550000035
The distribution interval of the actual spatial emergence angle xi is [ -1, +1], and then the continuous emergence angle interval can be discretized into a series of lattice points, so that an over-complete dictionary is constructed:
Figure FDA0002886816550000036
wherein the total lattice number is N, let N be T × d, where T and d are both integers, and T and d mean that the whole emergence angle interval is divided into T intervals and the lattice number of each interval is d, then the matrix corresponding to the nth interval is:
Figure FDA0002886816550000037
all n ═ 1.., T
Will DnThe columns in (a) are removed from the overcomplete dictionary D, and the matrix of the remaining columns is represented as
Figure FDA0002886816550000038
When designing the nth basis vector bnThen, the following optimization problem is constructed:
Figure FDA0002886816550000039
s.t.|bn,m1, all m
Wherein, bn,mIs bnThe mth element of (1), the function | · | | non-woven phosphor∞Represents an infinite norm, and
Figure FDA00028868165500000310
and there are vectors with 1 being all 1; for the optimization problem, the alternative direction multiplier method is adopted to solve to obtain a base vector bn}；
S4, obtaining a set of base vector sets bnAfter the multiplication, the hybrid pre-coding is obtained by approaching to the optimal full-digital pre-coderA code matrix; order to
Figure FDA0002886816550000041
And
Figure FDA0002886816550000042
then
Figure FDA0002886816550000043
s.t. FRF(:,i)∈{bn}
Figure FDA0002886816550000044
Wherein, FRF(:,i)∈{bnDenotes FRFEach column of (a) belongs to a finite set bnRe-constructing the optimization into a sparse problem, i.e.
Figure FDA0002886816550000045
Figure FDA00028868165500000415
Figure FDA0002886816550000046
Wherein
Figure FDA0002886816550000047
Is a matrix of measurements of the position of the object,
Figure FDA0002886816550000048
to represent
Figure FDA0002886816550000049
Presence of MtA non-zero row, since the number of RF links is MtA plurality of; performing sparse recovery on multiple observation vectors based on orthogonal matching algorithm, and estimating
Figure FDA00028868165500000410
Late, baseband digital precoding matrix
Figure FDA00028868165500000411
Get
Figure FDA00028868165500000412
M in (1)tA non-zero row, analog precoding matrix FRFTaking corresponding M in BtA column vector;
s5 design basis vectorb n}：
Definition of
Figure FDA00028868165500000413
The distribution interval of the actual spatial emergence angle xi is [ -1, +1], and then the continuous emergence angle interval can be discretized into a series of lattice points, so that an over-complete dictionary is constructed:
Figure FDA00028868165500000414
wherein the total lattice number is N, let N be T × d, where T and d are both integers, and T and d mean that the whole emergence angle interval is divided into T intervals, the lattice number of each interval is d, and the matrix corresponding to the nth interval is:
Figure FDA0002886816550000051
all n ═ 1.., T
Will EnFrom overcompleteRemoved from dictionary E, and the matrix of the remaining columns is represented as
Figure FDA0002886816550000052
When designing the nth basis vectorb nThen, the following optimization problem is constructed:
Figure FDA0002886816550000053
s.t. |b n,m1, all m
Wherein,b n,mis thatb nThe mth element of (1), the function | · | | non-woven phosphor∞Represents an infinite norm, and
Figure FDA0002886816550000054
and there are vectors with 1 being all 1; for the optimization problem, the method of alternative direction multiplier is used to solve the problem until the basic vectorb n}；
S6, obtaining a set of basis vector setsb nAfter the sum of the received signals is multiplied, a hybrid receiving matrix is obtained by approaching to the optimal full digital receiving merging matrix; order to
Figure FDA0002886816550000055
And
Figure FDA0002886816550000056
then
Figure FDA0002886816550000057
s.t. WRF(:,i)∈{b n}
Wherein, WRF(:,i)∈{b nDenotes WRFEach column of (a) belongs to a limited setb nRe-constructing the optimization into a sparse problem, i.e.
Figure FDA0002886816550000058
Figure FDA0002886816550000059
Wherein
Figure FDA00028868165500000510
Is a matrix of measurements of the position of the object,
Figure FDA00028868165500000511
to represent
Figure FDA00028868165500000512
Presence of MrA non-zero row, since the number of RF links is MrA plurality of; performing sparse recovery on multiple observation vectors based on orthogonal matching algorithm, and estimating
Figure FDA00028868165500000513
Then, baseband digital receiving matrix
Figure FDA00028868165500000514
Get
Figure FDA00028868165500000515
M in (1)rA non-zero row, analog receiving matrix WRFGetBCorresponding M inrA column vector.
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