CN112737649A - Millimeter wave channel estimation method based on angle grid optimization and norm constraint - Google Patents

Millimeter wave channel estimation method based on angle grid optimization and norm constraint Download PDF

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CN112737649A
CN112737649A CN202011573510.XA CN202011573510A CN112737649A CN 112737649 A CN112737649 A CN 112737649A CN 202011573510 A CN202011573510 A CN 202011573510A CN 112737649 A CN112737649 A CN 112737649A
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CN112737649B (en
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胡安中
黄秋芳
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Hangzhou Dianzi University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H04B7/0426—Power distribution
    • H04B7/043—Power distribution using best eigenmode, e.g. beam forming or beam steering
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/06—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station
    • H04B7/0613—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission
    • H04B7/0615—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal
    • H04B7/0617—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal for beam forming
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    • H04—ELECTRIC COMMUNICATION TECHNIQUE
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    • H04B7/00—Radio transmission systems, i.e. using radiation field
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Abstract

The invention discloses a millimeter wave channel estimation method based on angle grid optimization and norm constraint. In the existing channel estimation method based on compressed sensing, the channel estimation problem is converted into a sparse signal reconstruction problem by quantizing angle domain parameters into angle grids, and a classical orthogonal matching pursuit algorithm is adopted for solving. The method cannot restrict the sparsity of the channel, and the algorithm has high calculation complexity. The method of the invention firstly comprisesThe divided angle grid is optimized according to the known receiving signal and the wave beam forming matrix, and l is introduced2And constraining the sparsity of the channel parameters by the norm so as to obtain an optimization function of the channel estimation. At the moment, the channel estimation problem is converted into a convex optimization problem, and a global optimal solution is obtained by directly solving the minimum value, so that the channel matrix is restored. The method can optimize the angle grid and restrain the sparsity of the millimeter wave channel, effectively reduces the algorithm computation complexity and has better realizability.

Description

Millimeter wave channel estimation method based on angle grid optimization and norm constraint
Technical Field
The invention belongs to the technical field of wireless communication, in particular to a single-user millimeter wave large-scale multiple-input multiple-output (MIMO) system adopting hybrid beam forming, and relates to a method for optimizing a diagonal grid and optimizing the diagonal grid through I2The norm adds sparsity constraint to the channel parameters.
Background
In the millimeter wave communication system, the quality of communication depends on the wireless Channel to a great extent, and in order to transmit signals better, acquiring Channel State Information (CSI) is important for fully playing the performance of the millimeter wave massive MIMO system. Therefore, channel estimation becomes an indispensable element in the design of communication systems. Because the number of pilot frequencies required by the traditional channel estimation is in direct proportion to the number of antennas at a transmitting end, and a large-scale MIMO system just needs to be provided with a large number of antennas at the transmitting end and a receiving end, the traditional channel estimation method causes huge pilot frequency overhead; meanwhile, millimeter waves have the defects of large path loss, low signal-to-noise ratio of a receiving end and the like, and the traditional channel estimation method can cause low estimation precision. It is very challenging to acquire CSI in a millimeter wave system with a large number of antennas using conventional channel estimation methods. Due to the limited scattering effect, the millimeter wave channel has only a few spatial paths and exhibits sparsity, so researchers have proposed to solve the channel estimation problem of the millimeter wave massive MIMO system by using a Compressed Sensing (CS) algorithm based on the sparsity of the millimeter wave channel, and such an algorithm also has the problems of high complexity, low precision and the like in practice.
In the conventional channel estimation based on compressed sensing, Angle of departure (Angle of departure, AoD)/Angle of arrival (Angle of arrival, AoA) is quantized into a non-uniformly distributed grid, and a redundant dictionary is formed by array response vectors with a fine quantized Angle grid as a sparse transformation basis of CS, at this time, the channel estimation problem is transformed into a sparse signal reconstruction problem, which can be solved by Orthogonal Matching Pursuit (OMP). However, the channel estimation method based on the OMP algorithm has problems: the calculation complexity of the algorithm is high; the sparsity of the channel cannot be constrained.
Disclosure of Invention
The invention aims to provide a millimeter wave channel estimation method based on angle grid optimization and norm constraint, aiming at the problems that the traditional channel estimation method based on an OMP algorithm is large in calculation amount and high in complexity and cannot constrain the sparsity of channel parameters.
The application scenario of the method of the invention is as follows: the transmitting end and the receiving end both adopt a single-user millimeter wave large-scale MIMO communication system with a hybrid beam forming structure; the antenna Array is a Uniform Linear Array (ULA) containing tens or hundreds of antennas.
Considering a single-user millimeter wave large-scale MIMO communication system adopting a hybrid beam forming structure; the transmitting terminal has NTRoot antenna, receiving end having NRThe root antenna, the transmitting end and the receiving end are all provided with NRF≤min(NT,NR) A radio frequency chain. Transmitting end sending
Figure BDA0002861253990000021
A pilot beam is defined as
Figure BDA0002861253990000022
Received at the receiving end
Figure BDA0002861253990000023
A beam is defined as
Figure BDA0002861253990000024
Defining the p-th transmission beam
Figure BDA0002861253990000025
A received vector is
Figure BDA0002861253990000026
Figure BDA0002861253990000027
Then:
Figure BDA0002861253990000028
xpwhich means that the pilot symbols are transmitted,
Figure BDA0002861253990000029
in order to be a matrix of channels,
Figure BDA00028612539900000210
is a noise vector and
Figure BDA00028612539900000211
then set up
Figure BDA00028612539900000212
Can obtain the product
Figure BDA00028612539900000213
Comprises the following steps:
Figure BDA00028612539900000214
wherein
Figure BDA00028612539900000215
Representing diagonal elements as
Figure BDA00028612539900000216
The block-wise diagonal matrix of (a),
Figure BDA00028612539900000217
set ypCan obtain the receiving signal matrix of the receiving end
Figure BDA00028612539900000218
Comprises the following steps:
Y＝WHHFX + N (equation 3)
Figure BDA00028612539900000219
Is a noise matrix and
Figure BDA00028612539900000220
matrix array
Figure BDA00028612539900000221
Is as { xpIs a matrix of diagonal elements, usually set
Figure BDA00028612539900000222
Where P is the pilot power. The precoding matrixes of the transmitting end and the receiving end are respectively F ═ FRFFBBAnd W ═ WRFWBB，
Figure BDA00028612539900000223
Respectively representing Radio Frequency (RF) beamforming matrices at a transmitting end and a receiving end,
Figure BDA00028612539900000224
Figure BDA00028612539900000225
respectively representing the baseband precoding matrix of the transmitting end and the receiving end. Vectorizing Y to
Figure BDA00028612539900000226
Figure BDA00028612539900000227
Wherein vec (H) is a vector obtained after H vectorization, and N is a vector obtained after N vectorization.
The channel model can be written in matrix form:
Figure BDA0002861253990000031
Figure BDA0002861253990000032
and
Figure BDA0002861253990000033
respectively representing array response matrixes of a receiving end and a transmitting end;
Figure BDA0002861253990000034
a path complex gain matrix is represented.
In a ULA, the general form of the array response vector is:
Figure BDA0002861253990000035
λ is the signal wavelength and d is the spacing of adjacent antenna elements.
And sparse modeling is carried out by utilizing the sparsity of the millimeter wave channel, and the channel estimation problem is converted into a sparse signal recovery problem. The angle parameters are quantized into an angle grid by the following definitions:
Figure BDA0002861253990000036
Figure BDA0002861253990000037
g is the number of grids, and G is more than or equal to max (N)T,NR)。
Figure BDA0002861253990000038
Satisfies the following conditions:
Figure BDA0002861253990000039
obtaining a corresponding array response matrix according to the divided angle grids as follows:
Figure BDA00028612539900000310
Figure BDA00028612539900000311
the channel matrix based on the angle grid is represented as:
Figure BDA00028612539900000312
Figure BDA00028612539900000313
referred to as an approximate channel matrix defined in a discrete angular domain, representing a quantized channel;
Figure BDA00028612539900000314
a matrix with sparsity L (not a diagonal matrix), i.e. there are L non-zero entries corresponding to AoA/AoD;
Figure BDA00028612539900000315
is a quantization error matrix generated by quantizing the angle.
By vectorization of formulae
Figure BDA00028612539900000316
Vectorizing the channel matrix may result in:
Figure BDA00028612539900000317
by having a fine quantization angle grid
Figure BDA00028612539900000318
Forming a redundant dictionary as the sparse transformation base of CS, wherein the sparse representation form of vec (H) under the dictionary is
Figure BDA00028612539900000319
After the angle domain is quantized, in the whole signal transmission stage, the received signal at the receiving end can be represented as:
Figure BDA0002861253990000041
Figure BDA0002861253990000042
order to
Figure BDA0002861253990000043
At this time
Figure BDA0002861253990000044
Which may be referred to as an equivalent sensing matrix, equation 11 reduces to:
Figure BDA0002861253990000045
the method comprises the following specific steps:
step 1, estimating AoA and AoD according to the received signals and the beam forming matrix, and optimizing an angle grid:
(1.1) estimation of AoA and AoD: known received signal matrix
Figure BDA0002861253990000046
Wherein WHIs the conjugate transpose of W, which represents the precoding matrix at the receiving end,
Figure BDA0002861253990000047
in order to be a matrix of channels,
Figure BDA0002861253990000048
representing the path complex gain matrix, NTNumber of antennas at transmitting end, NRL is the number of channel paths for the number of antennas at the receiving end, F represents the precoding matrix at the transmitting end,
Figure BDA0002861253990000049
is as { xpIs a matrix of diagonal elements, xpWhich is indicative of the pilot symbols, is,
Figure BDA00028612539900000410
in order to transmit the end pilot beam(s),
Figure BDA00028612539900000411
in order to receive the pilot beams at the receiving end,
Figure BDA00028612539900000412
is a noise matrix, ARAnd ATArray response matrices corresponding to AoA and AoD respectively, and when subscripts are removed, the matrix response matrices have
Figure BDA00028612539900000413
A (N) represents the nth column of the array response matrix, theta corresponds to AoA or AoD, lambda represents signal wavelength, d represents antenna spacing, and N is the number of antennas at a transmitting end or a receiving end; the beamforming matrixes of the transmitting end and the receiving end are respectively F ═ FRFFBBAnd W ═ WRFWBBWherein the RF beamforming matrix FRFAnd WRFDefined as DFT matrix, base band precoding matrix FBBAnd WBBDefined as a unit matrix, the beamforming matrices F and W are DFT matrices. The ith row and jth column elements of the DFT matrix are expressed as:
Figure BDA00028612539900000414
therefore, when the received signal matrix and the beamforming matrix are known, AoA and AoD of the L paths are estimated from the relation between the received signal and the angle and beamforming matrix. First, find the index of the first L elements with the largest modulus value in Y, where the index includes the information of the row number row and the column number col of the L elements. Then, AoA corresponding to the L paths is estimated according to the L rows and the receiving end beamforming matrix W, and AoD corresponding to the L paths is estimated according to the L columns and the transmitting end beamforming matrix F, specifically:
Figure BDA0002861253990000051
wherein, N is the number of antennas at the transmitting end or the receiving end. When it is row, N is NRTheta corresponds to thetarI.e., AoA; when col, N corresponds to NTTheta corresponds to thetatI.e., AoD.
(1.2) simplifying the array response matrix according to the estimated values of AoA and AoD: obtaining cos (theta) corresponding to the estimated values AoA and AoD, and obtaining the cos (theta) by dividing an angle grid
Figure BDA0002861253990000052
Find the L grids closest to cos (theta), get the position index of the L angle grids, and then will find the position index of the L angle grids
Figure BDA0002861253990000053
And
Figure BDA0002861253990000054
the column corresponding to the L position index is reserved, the elements of other irrelevant columns are set to be 0, and a new array response matrix is obtained
Figure BDA0002861253990000055
And
Figure BDA0002861253990000056
simplifying the array response matrix after grid division; wherein
Figure BDA0002861253990000057
Is the angle of the grid division and,
Figure BDA0002861253990000058
and
Figure BDA0002861253990000059
respectively dividing the grids to obtain array response matrixes corresponding to AoA and AoD;
step (2), obtaining the angle grid according to the optimized angle
Figure BDA00028612539900000510
And
Figure BDA00028612539900000511
then the equivalent sensing matrix
Figure BDA00028612539900000512
Is rewritten as:
Figure BDA00028612539900000513
wherein
Figure BDA00028612539900000514
To represent
Figure BDA00028612539900000515
The companion matrix of (a);
step (3), solving the optimal solution of the channel estimation optimization problem:
(3.1) channel estimation optimization problem: according to the above steps, the received signal at the receiving end is rewritten as:
Figure BDA00028612539900000516
Figure BDA00028612539900000517
is composed of
Figure BDA00028612539900000518
In the form of a vectorization of (c),
Figure BDA00028612539900000519
to represent
Figure BDA00028612539900000520
And
Figure BDA00028612539900000521
the path complex gain matrix for the corresponding lower path,
Figure BDA00028612539900000522
n is a noise vector, so the method is based on the compressed sensing theory and simultaneously carries out the channel parameter matching
Figure BDA00028612539900000523
Introduction of l2The norm constraint obtains the optimization problem of the millimeter wave channel estimation as follows:
Figure BDA00028612539900000524
Figure BDA00028612539900000525
for the channel parameter to be estimated, y is the known receiving end signal, and μ is the penalty factor of the constraint term. And solving the optimal solution of the optimization problem by a design algorithm, thereby realizing the estimation of the millimeter wave channel.
(3.2) solving the optimal solution: the channel estimation optimization problem obtained in the above step is a convex optimization problem, so that a first derivative of the parameter to be estimated is directly obtained for the objective function, and the value of the corresponding parameter to be estimated is the global optimal solution of the convex optimization problem when the first derivative is equal to 0.
For the objective function
Figure BDA0002861253990000061
Concerning the number to be estimated
Figure BDA0002861253990000062
The first derivative is calculated as:
Figure BDA0002861253990000063
let a derivative equal to 0, get the expression:
Figure BDA0002861253990000064
wherein
Figure BDA0002861253990000065
Is a matrix with mu as diagonal element and the other elements as 0;
Figure BDA0002861253990000066
then the inverse can obtain the channel parameters
Figure BDA0002861253990000067
The estimated values of (c) are:
Figure BDA0002861253990000068
step (4), reconstructing a channel matrix: will estimate the value
Figure BDA0002861253990000069
Inverse quantization can be obtained
Figure BDA00028612539900000610
Thereby based on grid-based
Figure BDA00028612539900000611
And
Figure BDA00028612539900000612
and
Figure BDA00028612539900000613
obtaining an estimated value of a channel matrix
Figure BDA00028612539900000614
Comprises the following steps:
Figure BDA00028612539900000615
compared with the prior art, the invention has the beneficial effects that: the invention utilizes the sparsity of a millimeter wave channel to quantize angle parameters into angle grids, optimizes the angle grids by receiving signals and a beam forming matrix, and simultaneously adopts l2The norm constrains sparsity of channel parameters, so that a channel estimation problem is converted into a convex optimization problem, a global optimal solution of the optimization problem can be directly obtained by solving a minimum value, and a millimeter wave channel estimation result is obtained. The method has the advantages of good estimation precision, low calculation complexity and good realizability.
Drawings
Fig. 1 is a single-user millimeter wave massive MIMO communication system employing a hybrid beamforming structure.
Fig. 2 is a simulation plot of Normalized Mean Square Error (NMSE) versus Pilot-to-Noise Ratio (PNR) for an estimated channel in an example of the present invention. The figure has two curves in common, one is the simulation curve of the invention, and the other is the simulation curve adopting the OMP algorithm.
The specific implementation mode is as follows:
in order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to specific examples below.
As shown in FIG. 1, the number N of antennas at the transmitting end and the receiving end is definedT＝NR32; pilot beam
Figure BDA0002861253990000071
Radio frequency chain N RF8; the channel path number L is 5; adjacent antenna element spacing
Figure BDA0002861253990000072
λ is the signal wavelength; baseband precoding matrix FBB/WBBIs a unit array; RF beamforming matrix FRF/WRFSetting as a DFT matrix; the angle grid number G is 32.
The method comprises the following specific steps:
step 1, estimating AoA and AoD according to the received signals and the beam forming matrix, and optimizing an angle grid:
1.1. estimation of AoA and AoD: known received signal matrix
Figure BDA0002861253990000073
Wherein A isRAnd ATArray response matrices corresponding to AoA, AoD, respectively, with subscripts removed, then:
Figure BDA0002861253990000074
a (n) represents the nth column of the array response matrix, and θ corresponds to AoA or AoD. The beamforming matrixes of the transmitting end and the receiving end are respectively F ═ FRFFBBAnd W ═ WRFWBBWherein the RF beamforming matrix FRFAnd WRFDefined as DFT matrix, base band precoding matrix FBBAnd WBBDefined as a unit matrix, the beamforming matrices F and W are DFT matrices. The ith row and jth column elements of the DFT matrix can be expressed as:
Figure BDA0002861253990000075
therefore, when the received signal and the beamforming matrix are known, AoA and AoD of the L paths can be estimated from the relation between the received signal and the angle and beamforming matrix. First, find the index of the first L elements with the largest modulus value in Y, where the index includes the information of the row number row and the column number col of the L elements. Then, AoA corresponding to the L paths can be estimated according to the L rows and the receiving end beamforming matrix W, AoD corresponding to the L paths can be estimated according to the L columns and the transmitting end beamforming matrix F, specifically:
Figure BDA0002861253990000076
wherein, N is the number of antennas at the transmitting end or the receiving end. When it is row, N is NRTheta corresponds to thetarI.e., AoA; when col, N corresponds to NTTheta corresponds to thetatI.e., AoD.
1.2. According to AEstimates of oA and AoD simplify the array response matrix: obtaining cos (theta) corresponding to the estimated values AoA and AoD, and obtaining the cos (theta) by dividing an angle grid
Figure BDA0002861253990000081
Find the L grids closest to cos (theta), get the position index of the L angle grids, and then will find the position index of the L angle grids
Figure BDA0002861253990000082
And
Figure BDA0002861253990000083
the column corresponding to the L position index is reserved, the elements of other irrelevant columns are set to be 0, and a new array response matrix is obtained
Figure BDA0002861253990000084
And
Figure BDA0002861253990000085
and simplifying the array response matrix after the grid division is finished.
Step 2, obtaining the angle grid according to the optimized angle
Figure BDA0002861253990000086
And
Figure BDA0002861253990000087
then the equivalent sensing matrix
Figure BDA0002861253990000088
Is rewritten as:
Figure BDA0002861253990000089
and 3, solving the optimal solution of the channel estimation optimization problem:
3.1. channel estimation optimization problem: according to the above steps, the received signal of the receiving end can be rewritten as
Figure BDA00028612539900000810
Therefore, based on the compressed sensing theory, the channel parameters are simultaneously processed
Figure BDA00028612539900000811
Introduction of l2Norm constraint to obtain the optimization problem of millimeter wave channel estimation as
Figure BDA00028612539900000812
Wherein
Figure BDA00028612539900000813
Is represented by
Figure BDA00028612539900000814
Y is the known receiver signal, and μ is the penalty factor of the constraint term, which is set to 5 in this example.
3.2. Solving an optimal solution: the channel estimation optimization problem obtained by the invention is a convex optimization problem, so that a first derivative related to the parameter to be estimated can be directly solved for the objective function, and the value of the corresponding parameter to be estimated is the global optimal solution of the convex optimization problem when the first derivative is equal to 0. For the objective function
Figure BDA00028612539900000815
Concerning the number to be estimated
Figure BDA00028612539900000816
The first derivative is calculated as:
Figure BDA00028612539900000817
let this first derivative equal 0, the expression can be derived:
Figure BDA00028612539900000818
wherein
Figure BDA00028612539900000819
Is a matrix with mu as diagonal element and the other elements as 0;
Figure BDA00028612539900000820
then the inverse can obtain the channel parameters
Figure BDA00028612539900000821
Is estimated as
Figure BDA00028612539900000822
Step 4, reconstructing a channel matrix: will estimate the value
Figure BDA00028612539900000823
Inverse quantization can be obtained
Figure BDA00028612539900000824
Obtained from an angle-based mesh optimization
Figure BDA00028612539900000825
And
Figure BDA00028612539900000826
and
Figure BDA00028612539900000827
obtaining an estimated value of a channel matrix of
Figure BDA00028612539900000828
Fig. 2 is a simulation test chart of NMSE performance of an estimated channel according to PNR in the present invention, which includes a simulation curve of the present invention and a simulation curve based on an OMP algorithm. As can be seen from fig. 2, the performance of the channel estimation method NMSE of the present invention is substantially close to the performance of the channel estimation method NMSE based on the OMP algorithm. However, the calculation amount of the OMP algorithm is proportional to the number G of grids, and the algorithm complexity is high. In contrast, the channel estimation method of the present invention optimizes the angle grid to make the equivalent sensing matrix
Figure BDA0002861253990000091
The medium and large number of elements are all 0, and the channel estimation is solved without adopting an iterative mode of an OMP algorithmThe optimization problem is calculated, the optimal solution of the problem is solved directly through matrix inversion, and therefore the calculation complexity is greatly reduced, and the method has good realizability.

Claims (1)

1. A millimeter wave channel estimation method based on angle grid optimization and norm constraint is applied in the following scenes: a single-user millimeter wave large-scale MIMO communication system adopting a hybrid beam forming structure; the antenna array is a uniform linear array, and is characterized in that the method comprises the following specific steps:
estimating an arrival angle AoA and a departure angle AoD according to the received signals and the beam forming matrix, and optimizing an angle grid:
(1.1) estimation of AoA and AoD: known received signal matrix
Figure FDA0002861253980000011
Wherein WHIs the conjugate transpose of W, which represents the precoding matrix at the receiving end,
Figure FDA0002861253980000012
in order to be a matrix of channels,
Figure FDA0002861253980000013
representing the path complex gain matrix, NTNumber of antennas at transmitting end, NRL is the number of channel paths for the number of antennas at the receiving end, F represents the precoding matrix at the transmitting end,
Figure FDA0002861253980000014
is as { xpIs a matrix of diagonal elements, xpWhich is indicative of the pilot symbols, is,
Figure FDA0002861253980000015
in order to transmit the end pilot beam(s),
Figure FDA0002861253980000016
in order to receive the pilot beams at the receiving end,
Figure FDA0002861253980000017
is a noise matrix, ARAnd ATArray response matrices corresponding to AoA and AoD respectively, and when subscripts are removed, the matrix response matrices have
Figure FDA0002861253980000018
A (n) represents the nth column of the array response matrix, θ corresponds to AoA or AoD; λ represents signal wavelength, d represents antenna spacing, and N is the number of antennas at the transmitting end or the receiving end; the beamforming matrixes of the transmitting end and the receiving end are respectively F ═ FRFFBBAnd W ═ WRFWBBWherein the RF beamforming matrix FRFAnd WRFDefined as DFT matrix, base band precoding matrix FBBAnd WBBDefining as a unit matrix, so that the beam forming matrixes F and W are DFT matrixes; the ith row and jth column elements of the DFT matrix are expressed as:
Figure FDA0002861253980000019
therefore, under the condition that a received signal matrix and a beam forming matrix are known, the AoA and the AoD of the L paths are estimated according to the relation between the received signals and the angle and the beam forming matrix; firstly, finding the indexes of the first L elements with the maximum module value in Y, wherein the indexes comprise the information of the row number row and the column number col of the L elements; then, AoA corresponding to the L paths is estimated according to the L rows and the receiving end beamforming matrix W, and AoD corresponding to the L paths is estimated according to the L columns and the transmitting end beamforming matrix F, specifically:
Figure FDA00028612539800000110
wherein, N is the number of antennas of the transmitting end or the receiving end; when it is row, N is NRTheta corresponds to thetarI.e., AoA; when col, N corresponds to NTTheta corresponds to thetatI.e., AoD;
(1.2) simplifying array response based on estimated value of AoA and AoDMatrix: obtaining cos (theta) corresponding to the estimated values AoA and AoD, and obtaining the cos (theta) by dividing an angle grid
Figure FDA00028612539800000224
Find the L grids closest to cos (theta), get the position index of the L angle grids, and then will find the position index of the L angle grids
Figure FDA0002861253980000021
And
Figure FDA0002861253980000022
the column corresponding to the L position index is reserved, the elements of other irrelevant columns are set to be 0, and a new array response matrix is obtained
Figure FDA0002861253980000023
And
Figure FDA0002861253980000024
simplifying the array response matrix after grid division; (ii) a Wherein
Figure FDA0002861253980000025
Is the angle of the grid division and,
Figure FDA0002861253980000026
and
Figure FDA0002861253980000027
respectively dividing the grids to obtain array response matrixes corresponding to AoA and AoD;
step (2), obtaining the angle grid according to the optimized angle
Figure FDA0002861253980000028
And
Figure FDA0002861253980000029
then the equivalent sensing matrix
Figure FDA00028612539800000210
Is rewritten as:
Figure FDA00028612539800000211
wherein
Figure FDA00028612539800000212
To represent
Figure FDA00028612539800000213
The companion matrix of (a);
step (3), solving the optimal solution of the channel estimation optimization problem:
(3.1) channel estimation optimization problem: according to the above steps, the received signal at the receiving end is rewritten as:
Figure FDA00028612539800000214
Figure FDA00028612539800000215
is composed of
Figure FDA00028612539800000216
In the form of a vectorization of (c),
Figure FDA00028612539800000217
to represent
Figure FDA00028612539800000218
And
Figure FDA00028612539800000219
the path complex gain matrix for the corresponding lower path,
Figure FDA00028612539800000220
n is a noise vector, so the method is based on the compressed sensing theory and simultaneously carries out the channel parameter matching
Figure FDA00028612539800000221
Introduction of l2The norm constraint obtains the optimization problem of the millimeter wave channel estimation as follows:
Figure FDA00028612539800000222
Figure FDA00028612539800000223
the channel parameter to be estimated is y a known receiving end signal, and mu is a penalty factor of a constraint term; solving the optimal solution of the optimization problem by a design algorithm, thereby realizing the estimation of the millimeter wave channel;
(3.2) solving the optimal solution: the channel estimation optimization problem obtained in the above step is a convex optimization problem, so that a first derivative of the parameter to be estimated is directly solved for the objective function, and the value of the corresponding parameter to be estimated is the global optimal solution of the convex optimization problem when the first derivative is equal to 0;
for the objective function
Figure FDA0002861253980000031
Concerning the number to be estimated
Figure FDA0002861253980000032
The first derivative is calculated as:
Figure FDA0002861253980000033
let a derivative equal to 0, get the expression:
Figure FDA0002861253980000034
wherein
Figure FDA0002861253980000035
Is a matrix with mu as diagonal element and the other elements as 0;
Figure FDA0002861253980000036
then the inverse can obtain the channel parameters
Figure FDA0002861253980000037
The estimated values of (c) are:
Figure FDA0002861253980000038
step (4), reconstructing a channel matrix: will estimate the value
Figure FDA0002861253980000039
Inverse quantization can be obtained
Figure FDA00028612539800000310
Thereby based on grid-based
Figure FDA00028612539800000311
And
Figure FDA00028612539800000312
and
Figure FDA00028612539800000313
obtaining an estimated value of a channel matrix
Figure FDA00028612539800000314
Comprises the following steps:
Figure FDA00028612539800000315
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