CN113242195B - Narrow-band millimeter wave MIMO channel estimation method under low-precision all-digital architecture - Google Patents

Narrow-band millimeter wave MIMO channel estimation method under low-precision all-digital architecture Download PDF

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CN113242195B
CN113242195B CN202110734629.9A CN202110734629A CN113242195B CN 113242195 B CN113242195 B CN 113242195B CN 202110734629 A CN202110734629 A CN 202110734629A CN 113242195 B CN113242195 B CN 113242195B
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CN113242195A (en
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罗雷
李心安
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Chongqing University of Post and Telecommunications
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/0202—Channel estimation
    • H04L25/024—Channel estimation channel estimation algorithms
    • 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

Abstract

The invention relates to a method for estimating a narrowband millimeter wave MIMO channel under a low-precision all-digital architecture, which belongs to the field of wireless communication signal processing and comprises the following steps: a transmitting end transmits mutually orthogonal pilot signals; the receiving end radio frequency link carries out low-precision quantization on the analog receiving signal to obtain a digital receiving signal; converting the millimeter wave MIMO channel estimation problem into a noise-containing quantization sparse signal reconstruction problem by using the existing angle domain millimeter wave MIMO channel model; firstly, constructing an optimization problem by using a consistency reconstruction criterion, and estimating a channel vector support set; then, the reconstruction problem is subjected to dimension reduction processing, and the calculation complexity is reduced; further calculating the condition expectation of the sampled received signal to obtain the maximum likelihood estimation; and finally, performing least square estimation on the millimeter wave MIMO channel. Compared with the traditional channel estimation method, the method carries out customized design according to the characteristic of sparsity of the millimeter wave MIMO channel in the angle domain, and has higher estimation precision and lower calculation complexity.

Description

Narrow-band millimeter wave MIMO channel estimation method under low-precision all-digital architecture
Technical Field
The invention belongs to the field of wireless communication signal processing, and relates to a narrowband millimeter wave MIMO channel estimation method under a low-precision all-digital architecture.
Background
With the rapid increase of user data traffic and the shortage of available spectrum resources, millimeter wave (mmWave) communication becomes a key technology of next-generation mobile communication. However, millimeter waves have severe path loss, atmospheric attenuation and rain attenuation, and are not conducive to outdoor long-distance transmission. Millimeter wave systems are often combined with large-scale multiple-input multiple-output (MIMO) architectures for implementing spatial multiplexing and beamforming. The large-scale MIMO architecture has larger antenna array gain for compensating the serious path loss of the millimeter wave channel; meanwhile, the shorter wavelength of the millimeter wave enables the same size of space to be equipped with more antenna elements.
The number of the receiving end radio frequency links in the all-digital architecture is the same as that of the receiving antennas, and more radio frequency links bring more power consumption. Analog-to-digital converters (ADCs) are the most power consuming electronic components in radio frequency links, and the power consumption of ADCs increases dramatically with increased resolution. Therefore, the use of a low resolution ADC at the receiving end can greatly reduce the system power consumption. However, the dimensionality of a channel matrix is increased by a large-scale MIMO architecture, and the traditional channel estimation method is too high in calculation complexity and difficult to be practically applied; and the receiving end only adopts the low-precision received signal to estimate the channel matrix, and the estimation precision is lower.
Disclosure of Invention
In view of this, the present invention provides a method for estimating a narrowband millimeter wave MIMO channel under a low-precision all-digital architecture. The method divides the channel estimation problem into two stages, wherein the first stage determines the upper and lower boundaries of the value range of the sampled received signal by using the quantized received signal according to the consistency criterion, constructs an optimization problem and estimates the support set of the channel vector; and in the second stage, the support set is utilized to carry out dimensionality reduction processing on the channel estimation problem, an expected maximum algorithm (EM) is further used for estimating the sampled received signal, and finally least square estimation of a channel vector is obtained.
In order to achieve the purpose, the invention provides the following technical scheme:
1. a narrow-band millimeter wave MIMO channel estimation method under a low-precision all-digital architecture is characterized in that: the method comprises the following steps:
s1: a transmitting end transmits mutually orthogonal pilot signals;
s2: the receiving end radio frequency link carries out low-precision quantization on the analog receiving signal to obtain a digital receiving signal;
s3: the receiving end constructs an optimization problem according to the quantized received signal r and the perception matrix phi, and estimates a support set of an equivalent channel vector h
Figure BDA0003141146720000011
S4: according to the estimated support set, the dimension of the sparse signal reconstruction problem is reduced, and the original problem is changed into that:
Figure BDA0003141146720000021
wherein the content of the first and second substances,
Figure BDA0003141146720000022
representing a selection set
Figure BDA0003141146720000023
A low-dimensional channel vector formed by the corresponding elements in the middle,
Figure BDA0003141146720000024
representing a selection set
Figure BDA0003141146720000025
A low-dimensional sensing matrix formed by the middle corresponding column vectors; obtaining a maximum likelihood estimate of a sampled received signal y using an expectation maximization algorithm EM
Figure BDA0003141146720000026
Then calculating a low-dimensional channel vector
Figure BDA0003141146720000027
Least squares estimation of
Figure BDA00031411467200000219
Further, in the step S1, the transmitting end transmits a pilot signal
Figure BDA0003141146720000028
Pilot signal selection Nt×NtFirst P columns of the dimensional Hadamard matrix, where NtThe number of antennas at a sending end is represented, and P represents the pilot frequency length; sending end digital precoding matrix
Figure BDA0003141146720000029
Calculated as follows:
Figure BDA00031411467200000210
wherein]m,nThe mth row and nth column elements of the matrix are represented, j represents an imaginary unit, w represents a wavelength, pi represents a circumferential rate, and d represents a spacing between adjacent antenna elements.
Further, in step S2, the receiving end obtains a quantized received signal R ═ Q (y) ═ Q (HU) at the receiving endTZ + N), wherein Q (-) represents the element-by-element quantization process, and the Saleh-Vallenzuela channel model is adopted to obtain the sparse representation of the millimeter wave MIMO channel
Figure BDA00031411467200000211
The millimeter wave MIMO channel estimation problem is converted into a sparse signal reconstruction problem by utilizing column vectorization:
Figure BDA00031411467200000212
wherein
Figure BDA00031411467200000213
Which represents the received signal after the quantization,
Figure BDA00031411467200000214
which represents the received signal after sampling and which is,
Figure BDA00031411467200000215
represents the equivalent channel matrix to be estimated,
Figure BDA00031411467200000216
representing a noise matrix whose elements obey a complex Gaussian distribution
Figure BDA00031411467200000217
And are independent of each other, URRepresents Nr×NrA matrix of the dimensional dictionary is used,
Figure BDA00031411467200000218
represents Nt×NtAnd (3) maintaining a dictionary matrix, converting the complex number field reconstruction problem into a real number field reconstruction problem, and solving the problem:
r＝Q(y)＝Q(Φh+n)
wherein:
Figure BDA0003141146720000031
Figure BDA0003141146720000032
y represents the sampled received signal; the above process converts the millimeter wave channel estimation problem into a sparse signal reconstruction problem to solve: knowing the quantized received signal r and the sensing matrix phi, solving an equivalent channel vector h.
Further, the optimization problem constructed in the step S3 is:
Figure BDA00031411467200000311
s.t.l≤Φh≤u
||h||2＝1
wherein h represents an optimization variable, u and l respectively represent an upper boundary and a lower boundary determined by the quantized received signal, and l is more than or equal to y and less than or equal to u; the specific solving process comprises the following steps:
s31: introducing relaxation factor lambda and penalty function
Figure BDA0003141146720000033
Wherein mu (x) represents a step function, and changes the original optimization problem into the following relaxation optimization problem:
Figure BDA0003141146720000034
s.t.||h||2＝1
s32: initializing channel vectors
Figure BDA0003141146720000035
The iteration counter k is 0, and the gradient is decreased by a step delta;
s33: updating an iteration counter k to k + 1;
s34: gradient descent process on penalty function component:
calculating a penalty function term gradient:
Figure BDA0003141146720000036
projecting the gradient onto a unit sphere:
Figure BDA0003141146720000037
gradient descent:
Figure BDA0003141146720000038
S35：l1gradient descent process on norm component: introducing a contraction function
Figure BDA0003141146720000039
Figure BDA00031411467200000310
l1Gradient decrease in norm component:
Figure BDA0003141146720000041
s36: normalization:
Figure BDA0003141146720000042
s37: re-executing step S33 until l of channel vector difference obtained by two adjacent iterations2The norm is less than a given threshold;
s38: a supporting set of the resulting channel vectors is calculated.
Further, the specific step of solving the reconstruction problem after the dimension reduction in step S4 includes:
s41: initializing a channel vector:
Figure BDA0003141146720000043
s42: calculating the conditional expectation of the sampled received signal y:
Figure BDA0003141146720000044
the closed expression of its elements is represented as:
Figure BDA0003141146720000045
wherein e isiA unit vector representing that the ith element is 1, and the other elements are 0, wherein erfc (·) represents a complementary error function;
s43: according to calculated expected value
Figure BDA0003141146720000046
Obtaining least squares estimates of channel vectors
Figure BDA0003141146720000047
Figure BDA0003141146720000048
S44: the estimated value obtained in step S43
Figure BDA0003141146720000049
Substituting into step S42, repeating the calculation process until the difference l of the channel vectors obtained from two adjacent iterations2The norm is less than some given threshold.
The invention has the beneficial effects that:
(1) the optimization problem constructed in the step S3 is a convex optimization problem, and complex matrix pseudo-inverse operation is not needed in the solution; step S4 performs dimensionality reduction on the sparse signal reconstruction problem using the channel vector support set estimated in step S3, thereby significantly reducing the computational complexity.
(2) The invention obtains the maximum likelihood estimation of the sampled received signal, thereby reducing the influence of the amplitude distortion of the quantized received signal on the channel estimation performance and further improving the channel estimation performance.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.
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For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a block diagram of a system employed by the present invention;
FIG. 2 is a graph of the contraction function used in step S3 of the method of the present invention;
FIG. 3 is a graph of normalized mean square error of channel estimation as a function of signal-to-noise ratio (SNR) for a line-of-sight propagation millimeter wave MIMO channel when 1-bit, 2-bit, 3-bit, 4-bit ADC quantization is employed at the receiving end, respectively;
fig. 4 is a graph of channel estimation normalized mean square error as a function of signal-to-noise ratio (SNR) for a two-path non-line-of-sight propagation millimeter wave MIMO channel when 1-bit, 2-bit, 3-bit, and 4-bit ADCs are used for quantization at the receiving end, respectively.
Detailed Description
The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and examples may be combined with each other without conflict.
Wherein the showings are for the purpose of illustrating the invention only and not for the purpose of limiting the same, and in which there is shown by way of illustration only and not in the drawings in which there is no intention to limit the invention thereto; to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product; it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The same or similar reference numerals in the drawings of the embodiments of the present invention correspond to the same or similar components; in the description of the present invention, it should be understood that if there is an orientation or positional relationship indicated by terms such as "upper", "lower", "left", "right", "front", "rear", etc., based on the orientation or positional relationship shown in the drawings, it is only for convenience of description and simplification of description, but it is not an indication or suggestion that the referred device or element must have a specific orientation, be constructed in a specific orientation, and be operated, and therefore, the terms describing the positional relationship in the drawings are only used for illustrative purposes, and are not to be construed as limiting the present invention, and the specific meaning of the terms may be understood by those skilled in the art according to specific situations.
Referring to fig. 1 to 4, the present invention provides a method for estimating a narrowband millimeter wave MIMO channel under a low-precision all-digital architecture. As shown in fig. 1, a transmitting end transmits mutually orthogonal pilot sequences, and a receiving end obtains an analog received signal by using an antenna array and performs channel estimation by performing low-resolution ADC quantization.
In order to improve the channel estimation precision and reduce the calculation complexity, the invention converts the millimeter wave channel estimation problem into the noise-containing quantization sparse signal reconstruction problem and provides a dimension reduction-based method for solving the reconstruction problem. Firstly, the value range of the sampled received signal is determined by utilizing the quantized received signal, the value range is used as a constraint condition, an optimization problem is constructed by combining sparsity priori knowledge of a channel vector, and a channel vector support set is solved. And then, the dimension of the noise-containing quantized sparse signal reconstruction problem is reduced by utilizing the estimated support set, and the calculation complexity is reduced. And further, the prior knowledge that the sampled received signals obey Gaussian distribution is utilized to solve the conditional expectation of the quantized received signals to obtain the maximum likelihood estimation of the sampled received signals. And finally, calculating least square estimation of a channel vector according to the estimated sampled received signal.
The channel estimation method provided by the invention comprises the following steps:
s1 transmitting terminal sends pilot signal
Figure BDA0003141146720000061
Pilot signal selection Nt×NtFirst P columns of the dimensional Hadamard matrix, where NtDenotes the number of antennas at the transmitting end and P denotes the pilot length. Sending end digital precoding matrix
Figure BDA0003141146720000062
Calculated as follows:
Figure BDA0003141146720000063
wherein]m,nThe mth row and nth column elements of the matrix are represented, j represents an imaginary unit, w represents a wavelength, pi represents a circumferential rate, and d represents a spacing between adjacent antenna elements.
S2 the receiving end obtains the quantized received signal R ═ Q (y) ═ Q (HU)TZ + N), wherein Q (-) represents the element-by-element quantization process, and the Saleh-Vallenzuela channel model is adopted to obtain the sparse representation of the millimeter wave MIMO channel
Figure BDA0003141146720000064
The millimeter wave MIMO channel estimation problem is converted into a sparse signal reconstruction problem by utilizing column vectorization:
Figure BDA0003141146720000065
wherein
Figure BDA0003141146720000066
Which represents the received signal after the quantization,
Figure BDA0003141146720000067
which represents the received signal after sampling and which is,
Figure BDA0003141146720000068
represents the equivalent channel matrix to be estimated,
Figure BDA0003141146720000069
representing a noise matrix whose elements obey a complex Gaussian distribution
Figure BDA00031411467200000610
And are independent of each other. The above complex field reconstruction problem can be converted into a real field reconstruction problem solution:
r＝Q(y)＝Q(Φh+n)
wherein:
Figure BDA0003141146720000071
Figure BDA0003141146720000072
y represents the sampled received signal. The above process converts the millimeter wave channel estimation problem into a sparse signal reconstruction problem to solve: knowing the quantized received signal r and the sensing matrix phi, solving an equivalent channel vector h.
S3 receiving end constructs optimization problem according to quantized received signal r and sensing matrix phi, estimates support set of equivalent channel vector h
Figure BDA0003141146720000073
The optimization problem of the structure is as follows:
Figure BDA0003141146720000074
s.t.l≤Φh≤u
||h||2＝1
wherein h represents an optimization variable, u and l respectively represent an upper boundary and a lower boundary determined by the quantized received signal, and l is more than or equal to y and less than or equal to u. The main idea is to sequentially add the penalty function component to l1Gradient descent is performed on the norm component. The specific solving process comprises the following steps:
s31 introduces relaxation factor lambda and penalty function
Figure BDA0003141146720000075
Wherein mu (x) represents a step function, and changes the original optimization problem into the following relaxation optimization problem
Figure BDA0003141146720000076
s.t.||h||2＝1
S32 initializing channel vector
Figure BDA0003141146720000077
The iteration counter k is equal to 0 and the gradient is decreased by a step delta
S33 updates iteration counter k-k +1
S34 penalty function component gradient descent procedure:
calculating a penalty function term gradient:
Figure BDA0003141146720000078
projecting the gradient onto a unit sphere:
Figure BDA0003141146720000079
gradient reduction:
Figure BDA00031411467200000710
S35 l1gradient descent process on norm component: introducing a contraction function
Figure BDA00031411467200000711
The function graph is shown in fig. 2. Is prepared from1The derivative of the norm is obtained and the contraction function represents l1Gradient descent process on norm component. The function expression is as follows:
Figure BDA0003141146720000081
l1gradient decrease in norm component:
Figure BDA0003141146720000082
s36 normalization:
Figure BDA0003141146720000083
s37 re-executes step S33 until l of channel vector difference obtained by two adjacent iterations2Norm less than a given threshold
S38 support set of channel vector obtained by calculation
S4, using the channel vector support set estimated in the step S3 to perform dimensionality reduction processing on the sparse signal reconstruction problem, calculating the maximum likelihood estimation of the sampled received signal, and further obtaining the least square estimation of the channel vector. The specific calculation process is as follows:
s41 initializing channel vector
Figure BDA0003141146720000084
S42 calculating conditional expectation of sampled received signal y
Figure BDA0003141146720000085
The closed expression of its elements can be expressed as:
Figure BDA0003141146720000086
wherein e isiThe unit vector representing the ith element as 1 and the remaining elements as 0, and erfc (·) represents the complementary error function. S43 expected value according to calculation
Figure BDA0003141146720000087
Obtaining least squares estimates of channel vectors
Figure BDA0003141146720000088
Figure BDA0003141146720000089
S44 estimating the value obtained in step S43
Figure BDA00031411467200000810
Substituting into step S42, repeating the calculation process until the channel vector difference obtained from two adjacent iterationsL of2The norm is less than some given threshold.
As shown in fig. 3, under the line-of-sight propagation millimeter wave MIMO channel, the channel estimation method proposed by the present invention performs better than the most advanced expected maximum and approximate message transfer joint algorithm (EM-VAMP), especially when the receiving end adopts 3-bit and 4-bit quantization.
As shown in fig. 4, under the two-path sparse millimeter wave MIMO channel, the channel estimation method proposed by the present invention has the same conclusion as that under the line-of-sight propagation millimeter wave MIMO channel.
It is especially noted that when the receiving end adopts a 3-bit ADC, the channel estimation performance is greatly improved, and the performance gap between the 3-bit ADC and the 4-bit ADC is small, so the proposed algorithm is especially suitable for channel estimation when the receiving end adopts a 3-bit ADC.
Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.

Claims (1)

1. A narrow-band millimeter wave MIMO channel estimation method under a low-precision all-digital architecture is characterized in that: the method comprises the following steps:
s1: a transmitting end transmits mutually orthogonal pilot signals; transmitting terminal transmits pilot signal
Figure FDA0003589321130000011
Pilot signal selection Nt×NtFirst P columns of dimension Hadamard matrix, where NtThe number of antennas at a sending end is represented, and P represents the pilot frequency length; sending end digital precoding matrix
Figure FDA0003589321130000012
Calculated as follows:
Figure FDA0003589321130000013
wherein]m,nRepresenting the mth row and nth column elements of the matrix, j representing an imaginary number unit, w representing a wavelength, pi representing a circumferential rate, and d representing the interval between adjacent antenna array elements;
s2: the receiving end radio frequency link carries out low-precision quantization on the analog receiving signal to obtain a digital receiving signal; the receiving end obtains quantized received signals R ═ Q (y) ═ Q (HU)TZ + N), wherein Q (-) represents the element-by-element quantization process, and the Saleh-Vallenzuela channel model is adopted to obtain the sparse representation of the millimeter wave MIMO channel
Figure FDA0003589321130000014
The millimeter wave MIMO channel estimation problem is converted into a sparse signal reconstruction problem by utilizing column vectorization:
Figure FDA0003589321130000015
wherein
Figure FDA0003589321130000016
Which represents the received signal after the quantization,
Figure FDA0003589321130000017
which represents the received signal after sampling and which is,
Figure FDA0003589321130000018
represents the equivalent channel matrix to be estimated,
Figure FDA0003589321130000019
representing a noise matrix whose elements obey a complex Gaussian distribution
Figure FDA00035893211300000110
And are independent of each other, URTo representNr×NrA matrix of the dimensional dictionary is used,
Figure FDA00035893211300000111
represents Nt×NtAnd (3) maintaining a dictionary matrix, converting the complex number field reconstruction problem into a real number field reconstruction problem, and solving:
r＝Q(y)＝Q(Φh+n)
wherein:
Figure FDA00035893211300000112
Figure FDA00035893211300000113
y represents the sampled received signal; the above process converts the millimeter wave channel estimation problem into a sparse signal reconstruction problem to solve: after the quantization is known, receiving a signal r and sensing a matrix phi, and solving an equivalent channel vector h;
s3: the receiving end constructs an optimization problem according to the quantized received signal r and the sensing matrix phi, and estimates a support set of the equivalent channel vector h
Figure FDA0003589321130000021
The optimization problem of the structure is as follows:
Figure FDA0003589321130000022
s.t.l≤Φh≤u
||h||2＝1
wherein h represents an optimization variable, u and l respectively represent an upper boundary and a lower boundary determined by the quantized received signal, and l is more than or equal to y and less than or equal to u; the specific solving process comprises the following steps:
s31: introducing relaxation factor lambda and penalty function
Figure FDA0003589321130000023
Wherein mu (x) represents a step function, and changes the original optimization problem into the following relaxation optimization problem:
Figure FDA0003589321130000024
s.t.||h||2＝1
s32: initializing channel vectors
Figure FDA0003589321130000025
The iteration counter k is 0, the gradient step size δ;
s33: updating an iteration counter k to k + 1;
s34: gradient descent process on penalty function component:
calculating a penalty function term gradient:
Figure FDA0003589321130000026
projecting the gradient onto a unit sphere:
Figure FDA0003589321130000027
gradient descent:
Figure FDA0003589321130000028
S35：l1gradient descent process on norm component: introducing a contraction function
Figure FDA0003589321130000029
Figure FDA00035893211300000210
l1Gradient decrease in norm component:
Figure FDA00035893211300000211
s36: normalization:
Figure FDA00035893211300000212
s37: re-executing step S33 until l of channel vector difference obtained from two adjacent iterations2The norm is less than a given threshold;
s38: calculating a support set of the obtained channel vectors;
s4: according to the estimated support set, the dimension of the sparse signal reconstruction problem is reduced, and the original problem is changed into that:
Figure FDA0003589321130000031
wherein Q (-) represents an element-by-element quantization process,
Figure FDA0003589321130000032
representing a selection set
Figure FDA0003589321130000033
A low-dimensional channel vector formed by the corresponding elements in the middle,
Figure FDA0003589321130000034
representing a selection set
Figure FDA0003589321130000035
A low-dimensional sensing matrix formed by the middle corresponding column vectors; obtaining a maximum likelihood estimate of a sampled received signal y using an expectation maximization algorithm EM
Figure FDA0003589321130000036
Then, a low-dimensional channel vector is calculated
Figure FDA0003589321130000037
Least squares estimation of
Figure FDA0003589321130000038
The concrete steps for solving the reconstruction problem after dimensionality reduction comprise:
s41: initializing a channel vector:
Figure FDA0003589321130000039
s42: calculating the conditional expectation of the sampled received signal y:
Figure FDA00035893211300000310
the closed expression of its elements is represented as:
Figure FDA00035893211300000311
wherein e isiA unit vector representing that the ith element is 1, the rest elements are 0, and erfc (-) represents a complementary error function;
s43: according to calculated expected value
Figure FDA00035893211300000312
Obtaining least squares estimates of channel vectors
Figure FDA00035893211300000313
Figure FDA00035893211300000314
S44: the estimated value obtained in step S43
Figure FDA00035893211300000315
Substituting into step S42, repeating the calculation process until adjacentL of channel vector difference obtained by two iterations2The norm is less than some given threshold.
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