CN109714091B - Iterative hybrid precoding method based on hierarchical design in millimeter wave MIMO system - Google Patents

Iterative hybrid precoding method based on hierarchical design in millimeter wave MIMO system Download PDF

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CN109714091B
CN109714091B CN201910073932.1A CN201910073932A CN109714091B CN 109714091 B CN109714091 B CN 109714091B CN 201910073932 A CN201910073932 A CN 201910073932A CN 109714091 B CN109714091 B CN 109714091B
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邹卫霞
张然
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Beijing University of Posts and Telecommunications
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Abstract

The invention discloses an iterative hybrid precoding method based on hierarchical design in a millimeter wave MIMO system. The millimeter wave hybrid precoding method comprises the following steps: step a, establishing a mixed precoding problem model of a millimeter wave system; step b, introducing the idea of layered design to design the digital-analog mixed precoding; c, assuming the known optimal analog pre-coding, designing digital pre-coding to further improve the spectral efficiency of the system; step d, providing a design strategy of iterative solution, and deducing an optimal solution type solution of the simulation pre-coding; and e, outputting the optimal digital pre-coding matrix and the optimal analog pre-coding matrix. In a millimeter wave MIMO system, the invention provides an iterative hybrid precoding method based on a layered design, so as to realize the system performance which can approach the unconstrained digital precoding.

Description

Iterative hybrid precoding method based on hierarchical design in millimeter wave MIMO system
Technical Field
The invention relates to the field of wireless communication, in particular to an iterative hybrid precoding method based on hierarchical design in millimeter wave MIMO.
Background
With the continuous progress of mobile intelligent terminals and the rapid development of mobile internet, the demand of people for wireless network capacity is increasing day by day. In order to meet the requirements of people on the mobile internet and the business environment thereof, the next generation mobile communication technology (5G) aims to construct a ubiquitous and everything-interconnected omnibearing information ecosystem, promote the transformation of social and economic bodies in an intelligent manner and improve the productivity and sustainable development of the country. Compared with the previous generation mobile communication system (4G), 5G is required to be applicable to more scenes and provide a more elegant experience for users. 11/10.2017, the frequency use plan of the 5G system in the middle frequency range of 3000-plus 5000MHz is released internationally in China, and meanwhile, the 5G system needs to work in a high frequency range (a millimeter wave frequency range above 24 GHz) and a low frequency range (a frequency range below 3000 MHz) so as to meet the pursuit of users for high performance. Under such circumstances, the millimeter wave band with abundant spectrum resources is becoming the focus of attention in the industry.
However, the serious path loss becomes an urgent problem to be solved in the millimeter wave communication research. Preferably, when the millimeter wave wavelength is shorter, the corresponding antenna units can be densely arranged in a smaller space, so that a large-scale antenna array is formed, the serious path loss of the millimeter wave frequency band can be resisted by using the beam gain formed by the large-scale antenna, and the spectrum efficiency of the system can be improved by using the precoding technology based on the large-scale antenna. In a conventional MIMO system, all-digital precoding is performed by a digital encoder in a baseband part, and a coded data stream is converted into a radio frequency analog signal through a radio frequency chain and transmitted by an antenna. Under the structure, each antenna needs to be provided with a radio frequency chain, and the system can work normally. This structure can effectively use digital precoding to obtain beamforming gain when the number of antennas is small. However, in the millimeter wave communication system, the number of antennas is greatly increased, and if the digital precoding processing method is still adopted, the number of required radio frequency chains is also obviously increased, thereby causing the energy consumption and hardware cost of the system to be too high. Therefore, a hybrid precoding structure is proposed in the millimeter wave system to solve the above problem. In this scheme, hybrid precoding may utilize fewer radio frequency chains to convert the signal processed by digital precoding into an analog signal and transmit the analog signal to an analog encoder for further processing to form a transmittable baseband transmit signal, thereby completing the hybrid precoding process.
The research results of the existing hybrid precoding find that many related documents indicate that the hybrid precoding is one of the key technologies of millimeter wave communication. In fact, the key to the hybrid precoding problem is to solve the optimization problem of analog precoding matrix non-convex. Therefore, the important research of hybrid precoding is how to design the analog precoding matrix under the constant modulus constraint. The problem of hybrid precoding is firstly equivalent to the Euclidean distance minimum optimization problem by using a spatial precoding in millimeter wave MIMO system published by IEEE Transaction on Wireless Communications in 2014 by Omar El Ayach and the like, and an orthogonal matching tracking algorithm based on an array response vector is provided, so that a hybrid precoding design scheme with constant modulus constraint is simplified. Although the algorithm can obtain better system performance, the implementation complexity is higher due to the inversion operation in the algorithm. Wei-Lun Hung, hussei selem, j.chen et al further studied codebook-based low complexity hybrid precoding algorithms in which the design of the analog part is done according to a preset codebook, such as an array response codebook, a DFT codebook, etc. Besides codebook-based algorithms, heuristic hybrid precoding algorithms are also one of the important directions of research. Xianghao Yu et al defined the corresponding Riemann manifold according to the constant modulus constraint in the analog precoding problem in the IEEE Journal of Selected pics in Signal Processing published in 2016, and proposed a hybrid precoding algorithm based on the conjugate gradient method, however, the search process still leads to higher complexity. The second partial connection structure, i.e., each rf chain is connected to only a portion of the antenna. Xinyu Gao et al, in 2016, IEEE Journal on Selected Areas in Communications published an Energy-Efficient Hybrid analysis and Digital Precoding for MmWave MIMO Systems With Large Antenna Arrays, proposed a Hybrid Precoding algorithm based on successive interference cancellation based on the idea of multi-user signal detection techniques. The algorithm assumes that the digital precoding matrix is a diagonal matrix and only provides power allocation, thus causing a certain system performance loss.
Disclosure of Invention
In view of the foregoing defects in the prior art, an object of the present invention is to provide an iterative hybrid precoding method based on a hierarchical design, which aims to solve the problem of improving the system spectrum efficiency by using a hybrid precoding technology in a millimeter wave MIMO system.
In order to solve the problems, the invention provides the following technical scheme:
an iterative hybrid precoding method based on layered design in a millimeter wave MIMO system comprises the following steps:
establishing a mixed precoding problem model of a millimeter wave system;
step two, introducing the idea of layered design and designing the digital-analog mixed precoding;
and step three, assuming that the optimal analog precoding is known, designing digital precoding to further improve the frequency spectrum of the system
Efficiency;
a design strategy of iterative solution is proposed, and an approximate optimal solution type solution of the simulation pre-coding is deduced;
and fifthly, outputting the optimal digital precoding and analog precoding matrix.
The technical scheme adopted by the embodiment of the invention also comprises the following steps: the hybrid precoding model in the step one is as follows:
Figure GDA0002824302420000031
Figure GDA0002824302420000032
Figure GDA0002824302420000033
in the above formula, FRFRepresenting analog precoding, FBBRepresenting digital pre-coding. Since analog precoding is achieved by phase shifters, the constant modulus constraint, i.e., if, must be satisfiedRF(i, j) | 1. In addition to this, the present invention is,
Figure GDA0002824302420000034
representing the power constraint, N, of the transmitting endsIs the number of data streams.
The technical scheme adopted by the embodiment of the invention also comprises the following steps: and in the second step, the idea of layered design is introduced to design millimeter wave hybrid precoding. Specifically, digital precoding is designed first assuming that analog precoding is known. Secondly, based on the obtained digital precoding, an analog precoding matrix is designed.
The technical scheme adopted by the embodiment of the invention also comprises the following steps: in the third step, digital precoding is designed to improve the spectral efficiency of the system, assuming that analog precoding is known. In order to further simplify the design of hybrid precoding,and decoupling digital and analog precoding under power constraints, we assume
Figure GDA0002824302420000035
Wherein
Figure GDA0002824302420000036
Virtual variables, then the design of digital precoding can be expressed as:
Figure GDA0002824302420000037
Figure GDA0002824302420000038
in the formula (I), the compound is shown in the specification,
Figure GDA0002824302420000039
is defined as an equivalent channel matrix. Because the above formula has only one variable, we can use water filling algorithm to obtain virtual variable
Figure GDA0002824302420000041
Figure GDA0002824302420000042
Wherein, VeFirst N of right singular matrix representing equivalent channel matrixsAnd (4) columns. PeA diagonal matrix is represented whose diagonal elements are the power allocated for each data stream. Thus, the digital precoding matrix can be expressed as:
Figure GDA0002824302420000043
the technical scheme adopted by the embodiment of the invention also comprises the following steps: in the fourth step, a design strategy of iterative solution is provided, and a simulation pre-programming is deducedA near-optimal solution to the code. According to the conclusions in the existing documents, when the number of antennas tends to infinity, the analog precoder satisfies
Figure GDA0002824302420000044
Therefore, in the case that the number of radio frequency chains is equal to the number of data streams, the optimal digital precoding matrix satisfies
Figure GDA0002824302420000045
The corresponding analog precoding design can be expressed as:
Figure GDA0002824302420000046
Figure GDA0002824302420000047
in the above formula due to FRFIs not changed in the column arrangement
Figure GDA0002824302420000048
As a result of (1), thus can
Figure GDA0002824302420000049
Is rewritten as
Figure GDA00028243024200000410
Wherein (F)RF)-jIs FRFRemoving j-th column fjThe latter sub-matrix. Therefore, the design of analog precoding can be further simplified as:
Figure GDA00028243024200000411
in the formula (II)
Figure GDA00028243024200000412
Is an auxiliary matrix. If a suitable initial matrix F is selectedRFAnd assume (F)RF)-jIs fixed, the simulated precoding optimization problem can be decomposed into a series of sub-problems, and the jth sub-problem can be expressed as:
Figure GDA0002824302420000051
Figure GDA0002824302420000052
the above formula can be further simplified as follows:
Figure GDA0002824302420000053
Figure GDA0002824302420000054
defining an intermediate matrix
Figure GDA0002824302420000055
The main function of the above formula can be simplified to
Figure GDA0002824302420000056
In the formula fj(-i) denotes removal of fjThe ith element fj(i) The remaining subvectors of the latter. Mj(i, i) represents a matrix MjRow i and column i of (1), Mj(-, i) represents the matrix MjM, a sub-vector of the ith column vector excluding the element of the ith rowj(-i, -i) denotes the removal matrix MjThe ith row and ith column of the submatrix,
Figure GDA0002824302420000057
the representation takes the real part. According to the above formula for fj(i) The design of (c) can be expressed as:
Figure GDA0002824302420000058
Figure GDA0002824302420000059
thus, with respect to fj(i) The optimal closed-form solution of (c) can be expressed as:
Figure GDA00028243024200000510
in the formula, for an arbitrary complex variable x, there are
Figure GDA00028243024200000511
Thus, based on fj(i) The optimal solution of the method is that an iterative method is utilized to obtain the optimal simulation precoding matrix.
The technical scheme adopted by the embodiment of the invention also comprises the following steps: in said fourth step, the digital precoding F is outputBBAnd analog precoding FRF。
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Fig. 1 is a block diagram of a hybrid precoding structure in a millimeter wave MIMO system according to an embodiment of the present invention;
FIG. 2 is a flow chart of an embodiment of the present invention;
FIG. 3 is a comparison diagram of performance simulation results based on a narrow-band channel according to an embodiment of the present invention;
FIG. 4 is a comparison diagram of performance simulation results based on broadband channels according to an embodiment of the present invention;
FIG. 5 is a diagram illustrating simulation results based on iteration count according to an embodiment of the present invention;
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the invention.
The embodiment of the invention adopts a millimeter wave MIMO system with a mixed structure, and the structure of the antenna is a shared antenna structure. The transmitting end has
Figure GDA0002824302420000061
Number of radio frequency chains, NtA root antenna; the receiving end has
Figure GDA0002824302420000062
Number of radio frequency chains, NrRoot antenna, transmitting end and receiving end pass through NsThe data streams are communicated, and the relationship among the three is as follows:
Figure GDA0002824302420000063
Figure GDA0002824302420000064
as shown in fig. 1.
The first embodiment is as follows: first, an application principle of millimeter wave hybrid precoding design is described with respect to a narrowband channel.
In order to solve the problem of hybrid precoding in the millimeter wave MIMO system, an iterative hybrid precoding method based on a hierarchical design is designed, and a flowchart of the implementation case is shown in fig. 2. The method comprises the following steps:
establishing a mixed precoding problem model of a millimeter wave system;
step two, introducing the idea of layered design and designing the digital-analog mixed precoding;
and step three, assuming that the optimal analog precoding is known, designing digital precoding to further improve the frequency spectrum of the system
Efficiency;
a design strategy of iterative solution is proposed, and an approximate optimal solution type solution of the simulation pre-coding is deduced;
and fifthly, outputting the optimal digital precoding and analog precoding matrix.
Step one, establishing a millimeter wave hybrid precoding problem model based on a narrow-band channel model, wherein an objective function is as follows:
Figure GDA0002824302420000071
Figure GDA0002824302420000072
Figure GDA0002824302420000073
in the above formula, FRFRepresenting analog precoding, FBBRepresenting digital pre-coding. Since analog precoding is achieved by phase shifters, the constant modulus constraint, i.e., if, must be satisfiedRF(i, j) | 1. In addition to this, the present invention is,
Figure GDA0002824302420000074
representing the power constraint, N, of the transmitting endsIs the number of data streams.
And step two, introducing a layered design idea and designing millimeter wave hybrid precoding. Specifically, digital precoding is designed first assuming that analog precoding is known. Secondly, based on the obtained digital precoding, an analog precoding matrix is designed.
And step three, assuming that analog precoding is known, designing digital precoding to improve the spectral efficiency of the system. To further simplify the design of hybrid precoding and decouple digital and analog precoding under power constraints, we assume that
Figure GDA0002824302420000075
Wherein
Figure GDA0002824302420000076
Virtual variables, then the design of digital precoding can be expressed as:
Figure GDA0002824302420000077
Figure GDA0002824302420000078
in the formula (I), the compound is shown in the specification,
Figure GDA0002824302420000079
is defined as an equivalent channel matrix. Because the above formula has only one variable, we can use water filling algorithm to obtain virtual variable
Figure GDA00028243024200000710
Figure GDA00028243024200000711
Wherein, VeFirst N of right singular matrix representing equivalent channel matrixsAnd (4) columns. PeA diagonal matrix is represented whose diagonal elements are the power allocated for each data stream. Thus, the digital precoding matrix can be expressed as:
Figure GDA00028243024200000712
and step four, utilizing the strategy design of iterative solution and deducing the approximate optimal solution type solution of the simulation pre-coding. According to the conclusions in the existing documents, when the number of antennas tends to infinity, the analog precoder satisfies
Figure GDA00028243024200000713
Thus, in the case where the number of radio frequency chains equals the number of data streamsThe optimal digital precoding matrix satisfies
Figure GDA0002824302420000081
The corresponding analog precoding design can be expressed as:
Figure GDA0002824302420000082
Figure GDA0002824302420000083
in the above formula due to FRFIs not changed in the column arrangement
Figure GDA0002824302420000084
As a result of (1), thus can
Figure GDA0002824302420000085
Is rewritten as
Figure GDA0002824302420000086
Wherein (F)RF)-jIs FRFRemoving j-th column fjThe latter sub-matrix. Therefore, the design of analog precoding can be further simplified as:
Figure GDA0002824302420000087
in the formula, we define
Figure GDA0002824302420000088
Is an auxiliary matrix. If a suitable initial matrix F is selectedRFAnd assume (F)RF)-jBeing fixed, we can decompose the simulated precoding optimization problem into a series of sub-problems, and the jth sub-problem can be expressed as:
Figure GDA0002824302420000089
Figure GDA00028243024200000810
the above formula can be further simplified as follows:
Figure GDA00028243024200000811
Figure GDA00028243024200000812
defining an intermediate matrix
Figure GDA00028243024200000813
The main function of the above equation can be simplified as:
Figure GDA00028243024200000814
in the formula fj(-i) denotes removal of fjThe ith element fj(i) The remaining subvectors of the latter. Mj(i, i) represents a matrix MjRow i and column i of (1), Mj(-, i) represents the matrix MjM, a sub-vector of the ith column vector excluding the element of the ith rowj(-i, -i) denotes the removal matrix MjThe ith row and ith column of the submatrix,
Figure GDA0002824302420000091
the representation takes the real part. According to the above formula for fj(i) The design of (c) can be expressed as:
Figure GDA0002824302420000092
Figure GDA0002824302420000093
thus, with respect to fj(i) The optimal closed-form solution of (c) can be expressed as:
Figure GDA0002824302420000094
in the formula, for an arbitrary complex variable x, there are
Figure GDA0002824302420000095
Thus, based on fj(i) The optimal solution of the method is that an iterative method is utilized to obtain the optimal simulation precoding matrix.
Step five, in the millimeter wave MIMO system, outputting digital precoding FBBAnd analog precoding FRF。
Example two: in the case of a wideband channel, the application principle of the above millimeter wave hybrid precoding design scheme is further described. The specific steps are the same as in the first embodiment.
Step one, establishing a millimeter wave hybrid precoding problem model based on a broadband channel model, wherein an objective function is as follows:
Figure GDA0002824302420000096
Figure GDA0002824302420000097
Figure GDA0002824302420000098
in the formula (I), the compound is shown in the specification,
Figure GDA0002824302420000099
indicating the transmission rate on the k-th carrier.
And step two, introducing the idea of layered design, and designing the broadband channel based hybrid precoding. Specifically, digital precoding is designed first assuming that analog precoding is known. Secondly, based on the obtained digital precoding, an analog precoding matrix is designed.
Step three, assuming that the analog precoding is known, and designing the digital precoding to improve the spectral efficiency of the system, the design of the digital precoding can be expressed as:
Figure GDA0002824302420000101
Figure GDA0002824302420000102
the same method as the first embodiment can be used to obtain the optimal solution of digital precoding:
Figure GDA0002824302420000103
and step four, providing a design strategy of iterative solution, and deducing an approximate optimal solution type solution of the simulation pre-coding. The corresponding analog precoding design problem is:
Figure GDA0002824302420000104
Figure GDA0002824302420000105
the main function of the above formula can be simplified as:
Figure GDA0002824302420000106
wherein (b) is derived from the Jensen inequality.
The same method as the first embodiment can be used to obtain the optimal solution of the analog word pre-coding:
Figure GDA0002824302420000107
step five, in the millimeter wave MIMO system, outputting digital precoding FBBAnd analog precoding FRF。
The application effect of the present invention will be described in detail below with reference to the simulation result.
1) Simulation conditions
The invention adopts a large-scale MIMO system, the transmitting end and the receiving end both adopt area arrays, and the number of the transmitting end antennas is Nt144, the number of receiving end antennas is Nr36. The number of the radio frequency chains of the transmitting end and the number of the radio frequency chains of the receiving end are both
Figure GDA0002824302420000111
The cluster channel model is adopted in the text, and the channel model is assumed to be N cl10 clusters, each cluster having N thereinray5 paths and the angle of departure and angle of arrival obey 0,2 pi]The distribution is uniform, and the values are continuous. For fairness, the same total power constraint is enforced for all precoding solutions. The signal-to-noise ratio is defined as
Figure GDA0002824302420000112
2) Emulated content and conclusions
The hybrid precoding method of the embodiment of the invention adopts the idea of layered design, simplifies the design of hybrid precoding and improves the frequency spectrum efficiency of the system. In addition, the method provided by the invention can be applied to narrowband and wideband channel models. To reduce the hardware cost of the mm-wave MIMO system, we assume that the number of radio frequency chains is equal to the number of data streams. As shown in fig. 3, in the case of a narrowband channel model, an optimal digital precoding method and an OMP algorithm are respectively simulated, and the performances of the two algorithms are used as upper and lower baselines of the system performance. To show more clearly that our algorithm can achieve closer to optimal performance, we also simulated the existing PE-AltMin algorithm for comparison. Furthermore, as shown in fig. 4, in the case of wideband channels, we simulate the hybrid precoding method of the present invention. For comparison, the PE-AltMin algorithm and the OMP algorithm are simulated at the same time. Simulation results show that under the condition of a broadband channel, the method can still achieve better performance than a PE-AltMin algorithm and an OMP algorithm. In fig. 5, we measure the convergence of the method of the present invention based on the number of iterations, and the simulation result shows that the convergence speed of the method of the present invention is very fast.
Specific embodiments of the present invention have been described above in detail. The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features herein disclosed, as defined by the following claims.

Claims (3)

1. An iterative hybrid precoding method based on layered design in a millimeter wave MIMO system is characterized by comprising the following steps:
establishing a mixed precoding problem model of a millimeter wave MIMO system;
step two, introducing the idea of layered design and designing the digital-analog mixed precoding;
assuming a known optimal analog pre-coding matrix, designing a digital pre-coding matrix to further improve the spectral efficiency of the system;
a design strategy of iterative solution is proposed, and an approximate optimal solution type solution of the simulation pre-coding matrix is deduced;
outputting an optimal digital pre-coding matrix and an optimal analog pre-coding matrix;
wherein, the hybrid precoding problem model in the step one is as follows:
Figure FDA0002824302410000011
Figure FDA0002824302410000012
Figure FDA0002824302410000013
in the above formula, FRFRepresenting an analog precoding matrix, FBBRepresenting a digital precoding matrix, I being an identity matrix, p being an average received power,
Figure FDA0002824302410000014
is the variance of the noise, H represents the channel between the base station side and the user side, since the analog precoding is achieved by phase shifters, the constant modulus constraint, i.e., if, must be satisfiedRF(i, j) | 1, and further,
Figure FDA0002824302410000015
representing the power constraint, N, of the transmitting endsIs the number of data streams;
in the second step, the idea of layered design is introduced to design millimeter wave hybrid precoding, specifically, firstly, a digital precoding matrix is designed on the assumption that an analog precoding matrix is known; secondly, designing an analog pre-coding matrix based on the obtained digital pre-coding matrix;
in the third step, in order to further simplify the design of the hybrid precoding matrix and decouple the digital precoding matrix and the analog precoding matrix under the power constraint, a virtual variable is introduced
Figure FDA0002824302410000016
Suppose that
Figure FDA0002824302410000017
The design of digital precoding can be expressed as:
Figure FDA0002824302410000018
Figure FDA0002824302410000019
in the formula (I), the compound is shown in the specification,
Figure FDA0002824302410000021
is defined as an equivalent channel matrix, and because the above formula has only one variable, the water filling algorithm can be used to obtain the virtual variable
Figure FDA0002824302410000022
Figure FDA0002824302410000023
Wherein, VeFirst N of right singular matrix representing equivalent channel matrixsColumn, PeRepresents a diagonal matrix whose diagonal elements are the power allocated for each data stream, and thus, the digital precoding matrix can be represented as:
Figure FDA0002824302410000024
2. the iterative hybrid precoding method based on hierarchical design as claimed in claim 1, wherein according to step four, a design strategy for iterative solution is proposed and a simulated precoding moment is derivedThe optimal solution of the array is obtained according to the conclusion in the existing literature that the analog precoding matrix meets the requirement when the number of antennas tends to infinity
Figure FDA0002824302410000025
Therefore, in the case that the number of radio frequency chains is equal to the number of data streams, the optimal digital precoding matrix satisfies
Figure FDA0002824302410000026
Wherein, mu2＝1/NtDenotes the normalization factor, NtFor the number of antennas at the transmitting end, the corresponding analog precoding matrix design can be expressed as:
Figure FDA0002824302410000027
Figure FDA0002824302410000028
in the above formula due to FRFIs not changed in the column arrangement
Figure FDA0002824302410000029
As a result of (1), thus can
Figure FDA00028243024100000210
Is rewritten as
Figure FDA00028243024100000211
Wherein (F)RF)-jIs FRFRemoving j-th column fjThe following sub-matrices, therefore, the design of the analog precoding matrix can be further simplified as:
Figure FDA00028243024100000212
in the formula (II)
Figure FDA0002824302410000031
For the auxiliary matrix, if an appropriate initial matrix F is selectedRFAnd assume (F)RF)-jIs fixed, the analog precoding matrix optimization problem can be decomposed into a series of sub-problems, and the jth sub-problem can be expressed as:
Figure FDA0002824302410000032
Figure FDA0002824302410000033
the above formula can be further simplified as follows:
Figure FDA0002824302410000034
Figure FDA0002824302410000035
defining an intermediate matrix
Figure FDA0002824302410000036
The main function of the above equation can be simplified as:
Figure FDA0002824302410000037
in the formula (f)j(-i) denotes removal of fjThe ith element fj(i) The remaining sub-vector ofj(i, i) tableDisplay matrix MjRow i and column i of (1), Mj(-, i) represents the matrix MjM, a sub-vector of the ith column vector excluding the element of the ith rowj(-i, -i) denotes the removal matrix MjThe ith row and ith column of the submatrix,
Figure FDA0002824302410000038
representing a real part; according to the above formula for fj(i) The design of (c) can be expressed as:
Figure FDA0002824302410000039
Figure FDA00028243024100000310
thus, with respect to fj(i) The optimal closed-form solution of (c) can be expressed as:
Figure FDA00028243024100000311
in the formula, for an arbitrary complex variable x, there are
Figure FDA00028243024100000312
Thus, based on fj(i) The optimal solution of (2) can be used for obtaining the optimal simulation precoding matrix by using an iterative method.
3. The iterative hybrid precoding method based on the hierarchical design as claimed in claim 2, wherein according to step five, in the mmwave MIMO system, a digital precoding matrix F is outputBBAnd an analog precoding matrix FRF。
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