CN112803977A - Hybrid precoding method of millimeter wave communication system under beam offset effect - Google Patents

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

Hybrid precoding method of millimeter wave communication system under beam offset effect

Technical Field

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.

Background

Millimeter wave/sub-terahertz (mmWave/sub-THz) communication is an important potential technology of a next generation wireless communication system, and by utilizing rich spectrum resources of a millimeter wave/sub-terahertz frequency band, a communication rate of a few gigabits per second can be realized. On the one hand, in order to achieve a balance between performance and complexity/cost, the prior art proposes hybrid architectures that employ a very small number of Radio Frequency (RF) links; on the other hand, due to the adoption of a large-scale array antenna and the fact that the array antenna works in the mmWave/sub-THz frequency band, the beam direction under each working frequency can change along with the change of the frequency, and the phenomenon is called as a beam offset effect. In order to mitigate the beam offset effect, some existing works consider codebook design and hybrid precoding design to achieve this goal, and these conventional schemes all adopt a set of array response vectors to approach the optimal precoder. These schemes suffer significant performance loss when the number of radio links is limited.

Disclosure of Invention

The invention aims to provide a more appropriate hybrid precoder to overcome the beam offset effect in millimeter wave/sub-terahertz communication. An optimal all-digital encoder is better approached by designing a new set of Radio Frequency (RF) precoding vectors. Each designed base vector has a wider radiation pattern, and can effectively cover the offset beam direction under each frequency. Meanwhile, the basis vector design can construct an infinite norm problem, and is effectively solved by adopting an Alternating Direction Multiplier Method (ADMM). After the new basis vectors are designed, the hybrid precoding design can be obtained by approaching a full-digital precoder through a hybrid analog/digital precoding matrix, and the concept of the scheme for receiving the design of the merging matrix is the same as that of the hybrid precoding design.

The technical scheme of the invention is as follows:

in order to solve the design problem of a hybrid precoding matrix and a receiving combining matrix under millimeter wave/sub-terahertz communication with a beam offset effect, a multi-input multi-output orthogonal frequency division multiplexing system is considered, wherein the total number of subcarriers is P, and the number of base station configuration antennas is N_tAnd a Radio Frequency (RF) link number of M_tThe number of the mobile user side configuration antennas is N_rAnd the number of radio frequency links is M_rAnd satisfy M_t＜＜N_tAnd M_r＜＜N_r. The same RF analog precoder is adopted under each subcarrier

And frequency dependent baseband digital precoder

Here N_sIndicating the number of data streams. Similarly, the same RF analog receiving matrix is adopted under each subcarrier

And frequency dependent baseband digital receiving matrix

The technical scheme comprises the following steps:

and S1, constructing the channel. System center carrier frequency of f_cThe total number of sub-carriers is P, the system bandwidth is B,

representing the channel complex gain, the frequency of the p-th subcarrier can be expressed as

And is provided with

Assuming that the number of scattering paths is L, the corresponding exit angle and incident angle are expressed as { theta [ theta ]) respectively_lAnd psi_lAnd then, the channel matrix under the p-th subcarrier can be represented as:

here, the number of the first and second electrodes,

wherein, tau_lRepresenting 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.

Where lambda is_cIs the central carrier frequency f_cThe corresponding wavelength, c, represents the speed of light. Then the channel transmission arrives at the base station at the first antenna and the mth antenna with a time difference of

And S2, obtaining an optimal all-digital precoding matrix and an optimal receiving matrix. The received signal under the p-th subcarrier is:

wherein the content of the first and second substances,

representing the corresponding channel matrix at the p-th sub-carrier,

represents the corresponding symbol vector under the p sub-carrier and satisfies

Is N_sLine N_sThe identity matrix of the column(s),

means mean 0 and variance σ²Additive complex gaussian noise. Here, the number of the first and second electrodes,

denotes x_pThe complex conjugate transpose of (2).

The achievable spectral efficiency can be expressed as:

wherein

And

the goal of this step is to solve the optimal full digital pre-coding matrix G_pAnd a receiving matrix J_pThen, the hybrid precoding/receiving matrix is obtained by approaching the optimal all-digital precoding/receiving matrix. This is because the above objective function is non-convex and the constraint that the variable modulo is one. Therefore, neglecting the limitation of modulo one of the elements in the analog precoding matrix, considering an all-digital structure, the above problem can be simplified to

When fixing G_pThe most suitable receiving and combining matrix J_pIs composed of

At this time, about G_pHas an objective function of

Then optimum G_pCan be obtained as

G_p＝V_p(:,1:N_s)Δ_p

Here V_p(:,1:N_s) Is a matrix V_pFront N of_sColumn, and V_pIs through the pair channel H_PBy performing singular value decomposition, i.e.

Simultaneous diagonal matrix sigma_pIs shown as

Is a diagonal matrix of power distribution by water injection method and has

The goal of this step is to solve the optimal full digital pre-coding matrix G_pAnd receiving a combining matrix J_p。

S3, designing more appropriate base vector b_n}. Unlike the conventional scheme, the present invention is directed to finding a more optimal basis vector b_nIs used to approximate the optimal full digital precoding matrix, and each base vector b is required_nHave a broad beam radiation pattern to cover the offset beam directions at each frequency. Specifically, the beamforming gain of the basis vector in the required beam direction interval is as large as possible, and the beamforming gain in the other beam directions is as small as possible. How such a set of base vectors b is generated will be described below_n}。

Definition of

Wherein the distribution interval of the actual spatial emergence angle xi is [ -1, +1], then the continuous emergence angle interval can be discretized into a series of lattice points, based on which, an over-complete dictionary can be constructed

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

All n ═ 1.., T

Will D_nThe columns in (a) are removed from the overcomplete dictionary D, and the matrix of the remaining columns is represented as

Then, when designing the nth basis vector b_nThen, the following optimization problem can be constructed

s.t.|b _n,m1, all m

Wherein, b_n,mIs b_nThe mth element of (1), the function | · | | non-woven phosphor_∞Represents an infinite norm, and

and there are vectors with 1's being all 1's.

By solving the optimization problem as described above, a more appropriate set of basis vectors { b }can be obtained_n}。

And S4, completing the mixed precoding and the receiving matrix design. When a set of basis vectors b is obtained in the above manner_nAfter, a hybrid precoding matrix can be obtained by approximating the optimal all-digital precoder. Order to

And

then

s.t.F_RF∈{b_n}

The optimization can be reconstructed as a sparse problem, i.e.

Wherein

Is a matrix of measurements of the position of the object,

to represent

Presence of M_tA non-zero row, since the number of RF links is M_tAnd (4) respectively. Performing sparse recovery on multiple observation vectors based on orthogonal matching algorithm, and estimating

Then, F is taken

M in (1)_tA non-zero row, F_RFTaking corresponding M in B_tA column vector. The hybrid receive matrix design may be derived by the same principles as the hybrid precoding design, see steps S5 and S6.

S5 design basis vectorb _n}：

Definition of

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:

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:

all n ═ 1.., T

Will E_nIs removed from the overcomplete dictionary E, and the matrix of the remaining columns is represented as

When designing the nth basis vectorb _nThen, the following optimization problem is constructed:

s.t.| b _n,m1, all m

Wherein the content of the first and second substances,b _n,mis thatb _nThe mth element of (1), the function | · | | non-woven phosphor_∞Represents an infinite norm, and

and there are vectors with 1 being all 1;

for the optimization problem, the solution is carried out by adopting an alternative direction multiplier method until a base vector b is obtained_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

And

then

s.t.W_RF(:,i)∈{b _n}

Wherein, W_RF(:,i)∈{b _nDenotes W_RFEach column of (a) belongs to a limited setb _nRe-constructing the optimization into a sparse problem, i.e.

Wherein

Is a matrix of measurements of the position of the object,

to represent

Presence of M_rA non-zero row, since the number of RF links is M_rA plurality of; performing sparse recovery on multiple observation vectors based on orthogonal matching algorithm, and estimating

Then, baseband digital receiving matrix

Get

M in (1)_rA non-zero row, analog receiving matrix W_RFTaking corresponding M in B_rA column vector. Thus, the design of the hybrid precoding and the reception matrix is completed.

The hybrid precoding method has the beneficial effects that the hybrid precoding method provided by the invention can effectively relieve the beam offset effect and is superior to the existing scheme.

Drawings

FIG. 1 shows the relationship between the spectral efficiency and the signal-to-noise ratio of each method under the experimental condition M_r＝M_t＝N_s；

FIG. 2 is the relationship between the spectral efficiency and the signal-to-noise ratio of each method, and the experimental condition is M_r＝M_t＝2N_s；

FIG. 3 is a graph showing the relationship between the spectral efficiency of each method and the number of RF links under the experimental condition M_r＝M_tAnd SNR is 10 dB;

FIG. 4 is a diagram showing the relationship between the spectral efficiency of each method and the system bandwidth under the experimental condition M_r＝M_t＝N_sAnd SNR 10 dB.

Detailed Description

The invention is described in detail below with reference to the drawings and simulation examples to prove the applicability of the invention.

In the invention, the design problems of mixed precoding and receiving matrix under millimeter wave/sub-terahertz communication with beam offset effect are considered, and for a multi-input multi-output orthogonal frequency division multiplexing system, the number of antennas configured by a base station in the system is N_tAnd a Radio Frequency (RF) link number of M_tThe number of the mobile user side configuration antennas is N_rAnd the number of radio frequency links is M_rAnd satisfy M_t＜＜N_tAnd M_r＜＜N_r. The same RF analog precoding matrix is adopted under each subcarrier

And frequency dependent baseband digital precoding matrix

Here N_sIndicating the number of data streams. Similarly, the same RF analog receiving matrix is adopted under each subcarrier

And frequency dependent baseband digital receiving matrix

The received signal under the p sub-carrier is

Wherein the content of the first and second substances,

representing the corresponding channel matrix at the p-th sub-carrier,

represents the corresponding symbol vector under the p sub-carrier and satisfies

Is N_sLine N_sThe identity matrix of the column(s),

means mean 0 and variance σ²Additive complex gaussian noise. Here, the number of the first and second electrodes,

denotes x_pThe complex conjugate transpose of (2).

Let f_cRepresenting the system center carrier frequency, the frequency of the p-th sub-carrier can be represented as

At this time

The total number of subcarriers in the system is P and the bandwidth is B. Simultaneously, the time delay of the transmitting end and the receiving end is tau_lThe base station antenna spacing d being equal to half the wavelength of the central carrier frequency, i.e.

Where lambda is_cIs the wavelength corresponding to the center carrier frequency, and c represents the speed of light. Then the channel transmission arrives at the base station at the first antenna and the mth antenna with a time difference of

For a single-antenna mobile ue, the channel with the mth antenna of the base station can be expressed as

Considering the mobile ue configuring the array antenna, the corresponding channel matrix at the p-th sub-carrier can be expressed as

Wherein the content of the first and second substances,

further, the achievable spectral efficiency can be expressed as:

wherein

And

with the goal of maximizing spectral efficiency, hybrid precoding and receive matrix design can be constructed as an optimization problem as follows

|F_RF(i,j)|＝|W_RF(k, j) | 1, all i, j, k

G_p＝F_RFF_p,J_p＝W_RFW_p

The above optimization problem is difficult to solve, in that the objective function is non-convex, and the constraint that each element of the precoding matrix is modeled as one is simulated. To simplify the problem, consider an all-digital architecture, which can be reduced to

When fixing G_pThe most suitable receiving matrix J_pIs composed of

At this time, about G_pHas an objective function of

Then optimum G_pCan be obtained as

G_p＝V_p(:,1:N_s)Δ_p

Here V_p(:,1:N_s) Is a matrix V_pFront N of_sColumn, and V_pIs through the pair channel H_PBy performing singular value decomposition, i.e.

Simultaneous diagonal matrix sigma_pIs shown as

Is a diagonal matrix of power distribution by water injection method and has

Order to

And

then a hybrid precoding matrix is found to approximate the optimal all-digital precoding matrix, i.e.

s.t.F_RF∈γ^RF

Wherein upsilon^RFRepresenting a feasible set of RF pre-coding. For conventional schemes, the basis vector is taken as the array response vector, i.e. the vector is the vector of the response

s.t.F_RF∈{a_BS(φ_n)}

Here, the

And { phi_nIs a set of exit angles, which are finite lattice points discretizing a continuous angular interval.

Unlike the conventional scheme, the present invention is directed to finding a more optimal basis vector b_nIs used to approximate the optimal full digital precoding matrix, and each base vector b is required_nHave a broad beam radiation pattern to cover the offset beam directions at each frequency. In particular, it is the basis vectors that are being claimedThe beam forming gain of the beam direction interval is as large as possible, and the beam forming gain in other beam directions is as small as possible. How such a set of base vectors b is generated will be described below_n}。

Definition of

Wherein the distribution interval of the actual spatial emergence angle xi is [ -1, +1], then the continuous emergence angle interval can be discretized into a series of lattice points, based on which, an over-complete dictionary can be constructed

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

All n ═ 1.., T

Will D_nThe columns in (a) are removed from the overcomplete dictionary D, and the matrix of the remaining columns is represented as

Then, when designing the nth basis vector b_nThen, the following optimization problem can be constructed

s.t.|b _n,m1, all m

Wherein, b_n,mIs b_nThe mth element of (1), the function | · | | non-woven phosphor_∞Representing an infinite norm. The optimization problem can be translated into

b_n＝z_n

|z _n,m1, all m

|C_n(k, k) | 1, all k

Here, the

Is a diagonal matrix with diagonal elements modulo one, C_n(k, k) represents C_nThe k-th diagonal element of (1). The above optimization problem can further be equivalently written as

s.t.|z _n,m1, all m

|C_n(k, k) | 1, all k

The dual variables can be updated simultaneously as

And variable { b_n,q_n,z_n,C_nThe update can be obtained alternately as follows:

1) update b_nB and b_nThe objective function of the correlation is

At this time b_nIs solved as

2) Updating q_nQ and q_nThe objective function of the correlation is

The infinite norm minimization problem can be paired with

Cutting to obtain q_n。

3) Updating z_nAnd z_nThe objective function of the correlation is

s.t.|z _n,m1, all m

Can be obtained as

Wherein · represents the angle value of the complex variable.

4) Update C_nC and C_nThe objective function of the correlation is

s.t.|C_n(k, k) | 1, all k

Then the diagonal matrix C_nThe k-th diagonal element of (a) can be found as

All k

Wherein (. sub. (x))_kRepresenting the angle value of the kth element of the vector x.

When a set of basis vectors b is obtained in the above manner_nAfter, a hybrid precoding matrix can be obtained by approximating the optimal all-digital precoder, i.e.

s.t.F_RF∈{b_n}

The optimization can be reconstructed as a sparse problem, i.e.

Wherein

Is a matrix of measurements of the position of the object,

to represent

Presence of M_tA non-zero row, since the number of RF links is M_tAnd (4) respectively. Performing sparse recovery on multiple observation vectors based on orthogonal matching algorithm, and estimating

Then, F is taken

M in (1)_tA non-zero row, F_RFTaking corresponding M in B_tA column vector. The hybrid receive matrix design may be derived by the same principles as the hybrid precoding design, see steps S5 and S6. .

In simulation, a point-to-point downlink broadband millimeter wave MIMO-OFDM system is considered, and the number of base station configuration antennas is N_t256, the number of antennas configured at the mobile ue is N_r128. System center carrier frequency of f_c28GHz and bandwidth B4 GHz, the total number of subcarriers is set to P512. Simultaneous exit angle [ theta ]_lAnd angle of incidence { psi_lAre randomly distributed in

Then there is sin (theta)_l)∈[-1,+1]And sin (psi)_l)∈[-1,+1]. Delay per channel τ_lUniformly distributed in 0 to 100 nanoseconds, and the channel composite gain is

And is

And

meanwhile, the communication distance D is 60 meters, the channel fading index alpha is 2, and the light speed is c. Number of data streams N_sThe number of base vectors is 3, T is 64, N is 512 and d is 8.

The signal-to-noise ratio is defined as

In the performance analysis, the invention (I-SSP) is compared with the traditional scheme (C-SSP), and the base vector of the traditional scheme C-SSP is the array response vector, and meanwhile, the performance curve of a full digital optimal encoder (called full-digital) is also added in the simulation. The adopted index is spectral efficiency (spectral efficiency); .

FIG. 1 depicts the spectral efficiency of each method versus SNR with experimental conditions set to M_r＝M_t＝N_s. It can be observed from fig. 1 that the proposed I-SSP scheme has significant performance advantages over the conventional C-SSP scheme. This verifies that the scheme of the present invention can effectively alleviate the beam offset effect. The experimental conditions set for FIG. 2 are M_r＝M_t＝2N_sThe advantages of the proposed scheme I-SSP are also demonstrated.

FIG. 3 depicts the frequency spectrum efficiency of each method versus the number of Radio Frequency (RF) links, with experimental conditions set to M_r＝M_tAnd SNR 10 dB. As can be observed from fig. 3, the higher the number of Radio Frequency (RF) links, the better performance can be obtained. The number of RF links is limited in millimeter wave communication in consideration of the cost and power consumption of the RF links. Also, it can be noted that the proposed I-SSP has a greater performance advantage than the conventional C-SSP when the number of RF links is limited.

Next, FIG. 4 depicts the relationship between spectral efficiency and system bandwidth with experimental conditions set to M_r＝M_t＝N_sAnd SNR 10 dB. It can be seen from fig. 4 that as the bandwidth becomes larger, the performance distance between the all-digital scheme and the hybrid precoding scheme becomes larger, because the beam offset effect becomes more and more significant as the bandwidth increases. Meanwhile, experimental results show that the scheme I-SSP provided by the patent always keeps the advantages of the traditional scheme C-SSP for different system bandwidths.

In conclusion, the invention researches the design of hybrid precoding and receiving matrixes in the millimeter wave MIMO-OFDM system under the beam offset effect. In order to alleviate the beam offset effect, a new objective function is constructed to design a more appropriate set of RF basis vectors. After the basis vectors are designed, the hybrid precoding/receiving matrix can be designed by approximating the optimal all-digital coding matrix. Simulation results show that compared with the traditional scheme, the scheme provided by the patent can effectively relieve the beam offset effect.

Claims (1)

1. Hybrid precoding method of millimeter wave communication system under beam offset effect, in millimeter wave communication system, base station configuration antenna number is N_tAnd a Radio Frequency (RF) link number of M_tThe number of the mobile user side configuration antennas is N_rAnd the number of radio frequency links is M_rAnd satisfy M_t＜＜N_tAnd M_r＜＜N_r(ii) a The total number of subcarriers in the system is P, and the same RF analog precoding matrix is adopted under each subcarrier

And frequency dependent baseband digital precoding matrix

Here N_sRepresenting the number of data streams; similarly, the same RF analog receiving matrix is adopted under each subcarrier

And frequency dependent baseband digital receiving matrix

Characterized in that the hybrid precoding method comprises the following steps:

s1, constructing a channel: system center carrier frequency of f_cTotal number of subcarriers is P, system bandwidth is B, order

Representing the channel complex gain, the frequency of the p-th subcarrier is represented as:

assuming that the number of scattering paths is L, the corresponding exit angle and incident angle are expressed as { theta [ theta ]) respectively_lAnd psi_lAnd if so, the channel matrix under the p-th subcarrier is represented as:

wherein, tau_lRepresenting 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.

λ_cIs the central carrier frequency f_cThe 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

S2, obtaining an optimal all-digital precoding matrix and a receiving combination matrix:

the received signal under the p-th subcarrier is:

wherein the content of the first and second substances,

representing the corresponding channel matrix at the p-th sub-carrier,

represents the corresponding symbol vector under the p sub-carrier and satisfies

Is N_sLine N_sThe identity matrix of the column(s),

means mean 0 and variance σ²The additive complex gaussian noise of (a) is,

denotes x_pThe complex number conjugate transpose;

the achievable spectral efficiency is expressed as:

wherein the optimal full digital pre-coding matrix

Receiving a combining matrix

Considering an all-digital architecture, the above problem is simplified to:

when fixing G_pThe most suitable receiving and combining matrix J_pIs composed of

At this time, about G_pHas an objective function of

Finding the optimum G_pComprises the following steps:

G_p＝V_p(:,1:N_s)Δ_p

here V_p(:,1:N_s) Is a matrix V_pFront N of_sColumn, and V_pIs through the pair channel H_PBy performing singular value decomposition, i.e.

Simultaneous diagonal matrix sigma_pIs shown as

Is a water injection method power distribution diagonal matrix, and has:

obtaining an optimal all-digital precoding matrix G_pAnd receiving a combining matrix J_p；

S3 design basis vector b_n}：

Definition of

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:

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:

all n ═ 1.., T

Will D_nThe columns in (a) are removed from the overcomplete dictionary D, and the matrix of the remaining columns is represented as

When designing the nth basis vector b_nThen, the following optimization problem is constructed:

s.t.|b_n,m1, all m

Wherein, b_n,mIs b_nThe m-th element of (a) is,function | · | non-conducting phosphor_∞Represents an infinite norm, and

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 b_n}；

S4, obtaining a set of base vector sets b_nAfter the precoding matrix is obtained, a hybrid precoding matrix is obtained by approaching to the optimal full-digital precoder; order to

And

then

s.t. F_RF(:,i)∈{b_n}

Wherein, F_RF(:,i)∈{b_nDenotes F_RFEach column of (a) belongs to a finite set b_nRe-constructing the optimization into a sparse problem, i.e.

Wherein

Is a matrix of measurements of the position of the object,

to represent

Presence of M_tA non-zero row, since the number of RF links is M_tA plurality of; performing sparse recovery on multiple observation vectors based on orthogonal matching algorithm, and estimating

Late, baseband digital precoding matrix

Get

M in (1)_tA non-zero row, analog precoding matrix F_RFTaking corresponding M in B_tA column vector;

s5 design basis vectorb _n}：

Definition of

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:

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:

all n ═ 1.., T

Will E_nIs removed from the overcomplete dictionary E, and the matrix of the remaining columns is represented as

When designing the nth basis vectorb _nThen, the following optimization problem is constructed:

s.t. |b _n,m1, all m

Wherein the content of the first and second substances,b _n,mis thatb _nThe mth element of (1), the function | · | | non-woven phosphor_∞Represents an infinite norm, and

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

And

then

s.t. W_RF(:,i)∈{b _n}

Wherein, W_RF(:,i)∈{b _nDenotes W_RFEach column of (a) belongs to a limited setb _nRe-constructing the optimization into a sparse problem, i.e.

Wherein

Is a matrix of measurements of the position of the object,

to represent

Presence of M_rA non-zero row, since the number of RF links is M_rA plurality of; performing sparse recovery on multiple observation vectors based on orthogonal matching algorithm, and estimating

Then, baseband digital receiving matrix

Get

M in (1)_rA non-zero row, analog receiving matrix W_RFGetBCorresponding M in_rA column vector.

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