CN106453163A - Massive MIMO (Multiple Input Multiple Output) channel estimation method - Google Patents
Massive MIMO (Multiple Input Multiple Output) channel estimation method Download PDFInfo
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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
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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
- H04B7/0452—Multi-user MIMO systems
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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
- H04L25/0242—Channel estimation channel estimation algorithms using matrix methods
Abstract
The invention belongs to the technical field of wireless communication, and particularly relates to a channel estimation algorithm for a multi-user massive MIMO (Multiple Input Multiple Output) system under a frequency division duplex (FDD) mode. In the multi-user massive MIMO system, channel estimation is realized by a sparse signal recovering technology through an inference method based on Bayes compressed sensing, and channel estimation overhead of the FDD massive MIMO system can be lowered greatly.
Description
Technical field
The invention belongs to the multi-user under wireless communication technology field, more particularly to a kind of FDD (FDD) pattern is big
The channel estimation method of scale multiple-input and multiple-output (Multiple Input Multiple Outpu, MIMO) system.
Background technology
Extensive mimo system is one of key technology of the 5th Generation Mobile Communication System, and its main advantage is:System
Capacity increases as antenna amount increases;Reduce sending signal power;Simply existing precoder and detector i.e. up to
To optimal performance;Region orthogonalization between channel, eliminates co-channel interference in cell.
The premise for realizing these advantages is base station (BS) channel knowledge status information (CSIT).In time division duplex (TDD) system
In system, channel estimation is carried out using the reciprocity of up-downgoing channel in user side (MS).Then channel estimation expense is independently of base
Stand the extensive antenna array antenna number at end, only relevant with number of users.Therefore, in TDD system, the expense of channel estimation will not be made
Become the burden of system.And for the extensive mimo system of FDD, the flow process of its channel estimation is:Base station is to each user's broadcast pilot
Signal, then mobile subscriber feeds back to base station using Signal estimation CSIT is received.In this case, pilot signal number and base station day
Line number is directly proportional, and as, in extensive mimo system, antenna amount is huge, conventional channel estimation methods are (as least square
Method) huge training expense will be faced so that the training time is elongated, even more than channel want do the time so that channel estimation
Lose meaning.
In multi-user MIMO system, due to base station end and the greatest differences of user side antenna number, base station end and user side
Reaction for scattering effect is also completely different, present base station end propagation path openness and user side propagation path rich
Fu Xing.Meanwhile, as the scattered signal of identical scattering object is partially received between different user, between its channel, there is portion
The characteristic that split-phase is closed, here it is the joint sparse of the extensive mimo system channel of multi-user.
Compressed sensing is a kind of brand-new signal sampling theory, and it is openness using signal, is being much smaller than Nyquist
The discrete sample of signal in the case of speed, is obtained with stochastical sampling, sparse below some bases using primary signal
Characteristic, is projected under little measurement, recovers original signal by nonlinear algorithm, and therefore compressive sensing theory can
Maximum signal message is retained by minimum measurement.Management loading (Sparse Bayesian Learning, SBL)
Proposed in calendar year 2001 by the Tipping of Microsoft Research initially as a kind of machine learning algorithm, be subsequently introduced into sparse letter
Number recover field (BCS).
Bayes's compressed sensing is the method using probability, increases sparse prior to signal, is inferred by Bayesian statistic
Method, derive signal recover algorithm.Due to Bayes's flexibility ratio height, can pass through to change the form of probability priori, with
Adapt to multiple different signal priori.Bayesian frame provides multiple useful estimating methods, for example:Expectation
Maximization(EM)、Variational Expectation Maximization(VEM)、Maximal Likelihood
(ML).EM algorithm is a kind of method for seeking parameter Maximum-likelihood estimation that Dempster, Laind, Rubin were proposed in 1977,
It can estimate to parameter from incomplete data set, be a kind of very simple and practical learning algorithm.The master of EM algorithm
Syllabus is to provide a simple iterative algorithm to calculate posterior density function, and its great advantage is simple and stable, but appearance
Local optimum is easily absorbed in.VEM algorithm is the EM algorithm of generalization, be by BEAL M J. earliest in its paper《Variational
Algorithms for Approximate Bayesian Inference》In proposed, be mainly used in Bayesian Estimation
In complicated statistical model in machine learning field.
Content of the invention
In the extensive mimo system of multi-user, estimating method of the present invention based on Bayes's compressed sensing, using sparse
Signal recovery technology realizes channel estimation, can reduce the expense of the extensive mimo system channel estimation of FDD in a large number.
Understand for convenience, introduce model and function that the present invention is used first:
The extensive mimo channel estimating system model of FDD multi-user:
Assume that it is flat block decline to need the channel that estimates, i.e., channel status is constant within certain time.System has one
Individual BS, K MS, the BS are configured with the extensive antenna array with N number of antenna, and each MS has M antenna, then FDD is many
The mathematical model that the extensive mimo channel of user is estimated can be expressed as Yj=HjX+Nj, wherein, YjRepresent the reception of j-th MS
Signal matrix, HjRepresent the channel matrix between BS and j-th MS, X is pilot signal, NjFor receiving noise signal matrix.
Standard compression sensing mathematics model:
Y=Α x+n, wherein, Α is calculation matrix of the size for m × n, and y is that compressed signal is tieed up in m × 1, and x is n × 1 dimension
Sparse signal, its degree of rarefication is only s < < n element non-zero in s, i.e. x, remaining element all 0, n be m × 1 tie up be
System noise and its element obey average be 0, variance be σ2Gauss distribution, m < < n.
Bayes's compressed sensing model is by maximizing marginal likelihood function:
Pass throughStudy first α and β is obtained, i.e.,Wherein, μ i represents sparse signal xiAverage, ViiRepresent sparse signal xiVariance; Represent sparse signal<||Yi-φXi||2>'s
Average,Represent sparse signal<||Yi-φXi||2>Variance.
By become integration Bayes's expectation maximization method obtain parameter k, formula is as follows:
Wherein, parameter k clothes
It is distributed from Bernoulli Jacob.
BS end and MS end are all configured with uniform straight line array (ULA), are converted according to the virtual angle domain of mimo channel, by each MS
Corresponding channel decomposing is:Wherein, UR∈CM×MAnd UT∈CN×NIt is the angle domain change at MS end and BS end respectively
Change unitary matrice, unitary matrice UT(p, q) unit be:P, q ∈ [0, N-1],
Unitary matrice UR(a, the b) unit of (a, b) isA, b ∈ [0, M-1],It is the channel matrix of angle domain.In extensive antenna array,Row vector have identical sparse support set,
It is the multiple Gauss distribution of the position identical and non-zero entry obedience zero mean unit variance of their nonzero element.Different MS pairs
There is common factor in the sparse support set that part related relation, i.e. each MS is there is also between the different channels matrix that answers.By jth
The sparse support set expression of individual MS channel is Ωj, thenIt is the jointly sparse support set of each MS.
A kind of Masssive mimo channel method of estimation, comprises the steps:
S1, initialization, specially:
S11, BS broadcast T pilot signal X with T time slot to K MSP=[x(1),x(2),...,x(T)]∈CN×T, wherein,
N is the antenna number of BS, and mimo channel is estimated pilot signal X in mathematical modelPIt is converted into the compressed sensing measurement of angle domain
Matrix, represents have with symbol ΦUnitary matricep,q∈
[0, N-1], ΦH∈CN×TElement i.e. from setIn with equiprobability extract, P is the pilot signal of each time slot
Power, *HRepresent * conjugate transpose;
The receipt signal matrix of S12, K MS are { Rj, wherein, RjRepresent the receipt signal matrix of j-th MS, j=1,
2,...,K;
S13, symbol conversion is carried out, orderWherein,
Unitary matriceA, b ∈ [0, M-1], M represent compression for the antenna number of MS, Φ
Perceive calculation matrix, XjFor the conjugate transpose of angle domain channel matrix, angle domain channel matrix isΕjFor etc.
Effect Gaussian noise matrix, NjFor receiving noise signal matrix;
S2, the sparse support collection Combined estimator of each user, i.e., using sparse of multitask BCS algorithm K MS of Combined estimator
Hold set, obtain K estimation channel sparse support set, described K estimation channel sparse support set expression be Ω1,
Ω2,...ΩK;
S3, the sparse support set iterative estimate of each user, specially:
The sparse iteration control variable N for supporting set common of S31, each MS of settingiterWith maximum iteration time Nset;
S32, given initial value:
Receipt signal matrix Y;The prior probability matrix of X, matrix element is obeyed average and isVariance isMultiple Gauss divide
ClothJointly sparse support and non-jointly sparse support parameterThe multiple Gauss distribution variance of noiseMeet
Bernoulli Jacob is distributed, and probability isEmpirical valueAnd it is initialized as 0 intermediate variable
The reception signal Z=Φ X of S33, not Noise, it is assumed that the prior probability of Z is obeyed average and isVariance isAnswer
Gauss distributionThe posterior probability of Z is obtained by the prior probability of Z, probability is obeyed average and isVariance isMultiple height
This distributionMore new regulation is(ForInverse);
S34, by Z posterior probability combine intermediate variablePush away X prior probability
S35, by Z posterior probability, common Sparse parameter and non-common Sparse parameterNoise variance parameter
Selection parameterThrough intermediate variableJoin operation, obtain update X posterior probability
S36, renewal initial parameter valueIteration S33-S35 is until meeting iteration control variable NiterMost
Big iterationses NsetRequirement, can obtain comprising jointly sparse support and non-jointly sparse support each user channel
Status information
S4, the extensive mimo channel of multi-user estimate, the channel estimation results of each user areIts
In, the sparse support section of X is obtained when meeting and necessarily imposing a condition by Bayes's compressed sensing alternative manner proposed by the present invention
Arrive, remainder all 0, described impose a condition as empirical condition.
Further, the sparse support set omega of channel of K estimation is obtained described in S21,Ω2,...ΩKConcrete steps are such as
Under;
S21, assume the M reception antenna of each user have identical sparse support set, sparse of K MS
Degree of holding is S, and for different user K, the number of common sparse support position is Sc, noncomitant sparse support position
Number is S-Sc, and wherein, Sc is the symbol for representing sparse support position number, and S-Sc represents exclusive non-jointly sparse of each user
Support position number;
S22, according to Bayes's compressed sensing algorithm, if it is α that the condition of sparse channel of j-th MS obeys parameterj=[αj1,
αj2,...,αji,...,αjN]TMultivariate multiple Gauss distribution, wherein, element αjiSparse support is had for multi-userOr be exclusive sparse supportI.e. joint probability is close
Degree functionWherein, m=1,2 ..., K, HjRepresent the
Channel between j MS and BS,Represent i-th element of channel vector, empirical valueBernoulli Jacob's distribution obeyed by initial value, per
Individual element kiProbability isI=1,2 ... N;
S23, joint K MS of consideration, according toDerive the parameter of sparse support position jointly
SetMore new regulationNon- jointly sparse support
The parameter sets of positionMore new regulationWherein, Represent m-th sparse signalAverage,Represent m-th sparse signalVariance, m=
1,2,…K;
It is 0 that S24, the noise of different user obey average, and variance isMultiple Gauss distribution, then parameter betamMore new formula
For
S25, joint consider K MS, and the more new formula of co-localization parameter k of different user isWherein,Wherein, π is ki=0 probability;
Compressed sensing calculation matrix Φ and angle domain receipt signal matrix Y described in S26, input S13, to S23-S25's
Parameter carries out Joint iteration estimation, and the sparse support set omega of different user is obtained.
Further, initial parameter value is updated described in S36Detailed process is:The X's for obtaining in S34
Posterior probability is substituted in the parameter more new formula in S23-S25.
The invention has the beneficial effects as follows:
Inventive process avoids the process of the posterior probability direct matrix in verse to the X for arriving of EM algorithm, greatly simplifies
Operand, improves arithmetic speed and operational precision.Meanwhile, with the channel estimation methods phase such as OMP, ST-BCS, SOMP, JOMP
Than the present invention improves the accuracy of channel estimation, under certain condition, channel estimation errors can be caused to reach 10-3, using this
Inventive method causes extensive mimo channel to estimate that realization in practice becomes possibility.
Description of the drawings
Fig. 1 is the extensive mimo channel joint sparse schematic diagram of multi-user and its physics view.
Fig. 2 is inventive algorithm flow chart.
Fig. 3 is that inventive algorithm and remaining sparse signal reconfiguring algorithm are implemented on the extensive mimo system of multi-user in difference
Performance comparison figure under training expense.
Fig. 4 is that inventive algorithm and remaining sparse signal reconfiguring algorithm are implemented on the extensive mimo system of multi-user in difference
Performance comparison figure under signal to noise ratio.
Specific embodiment
With reference to specific embodiment, the present invention is described in further detail.
Fig. 1 is the extensive mimo channel schematic diagram of multi-user.
Assume that number of users K=20, base station end and user side are all each configured with uniform linear array (ULA), and base station
Antenna number N=160, each user antenna number identical and be M=2.Assume sparse support number (degree of rarefication) phase of each subscriber channel
With and be S=15, common sparse support number Sc=8.
Fig. 2 is that the extensive mimo channel of multi-user estimates flow chart, according to flow chart, algorithm can be entered using above-mentioned parameter
Row emulation.
S1, initialization, specially:
S11, BS broadcast T pilot signal X with T time slot to 20 MSP=[x(1),x(2),...,x(T)]∈CN×T, its
In, antenna number N of BS takes 160, and mimo channel is estimated pilot signal X in mathematical modelPIt is converted into the compression sense of angle domain
Know calculation matrix, represent have with symbol Φ(Represent XPConjugate transpose), unitary matriceP, q ∈ [0, N-1], ΦH∈CN×T(ΦHRepresent Φ conjugate transpose)
Element is i.e. from setIn with equiprobability extract, P is the pilot signal power of each time slot;
S12, the receipt signal matrix of 20 MS are { Rj},RjRepresent j-th MS receipt signal matrix (j=1,2 ...,
20);
S13, symbol conversion is carried out, orderWherein,
Unitary matriceAntenna number M of a, b ∈ [0, M-1], MS represents pressure for 2, Φ
Contracting perceives calculation matrix, XjFor the conjugate transpose of angle domain channel matrix, angle domain channel matrix isΕjFor
Equivalent Gaussian noise matrix, NjFor receiving noise signal matrix;
S2, the sparse support collection Combined estimator of each user, i.e., using sparse of multitask BCS algorithm 20 MS of Combined estimator
Hold set, obtain 20 estimation channel sparse support set, described 20 estimation channel sparse support set expression be
Ω1,Ω2,...Ω20, specially:
S21, assume 2 reception antennas of each user have identical sparse support set, 20 MS's is sparse
Support is 15, and for different user, the number of common sparse support position is 8, noncomitant sparse support position
Number is 7;
S22, according to Bayes's compressed sensing algorithm, if it is α that the condition of sparse channel of j-th MS obeys parameterj=[αj1,
αj2,...,αji,...,αjN]TMultivariate multiple Gauss distribution, wherein, element αjiSparse support is had for multi-userOr be exclusive sparse supportI.e. joint probability is close
Degree functionWherein, m=1,2 ..., 20, HjRepresent
Channel between j-th MS and BS,Represent i-th (i=1,2 ... 160) individual element of channel vector, empirical valueInitial value
Obey Bernoulli Jacob's distribution, each element kiProbability is(i=1,2 ... 160);
S23, joint 20 MS of consideration, according toDerive the ginseng of sparse support position jointly
Manifold is closedMore new regulation(Represent m-th
Sparse signalAverage,Represent m-th sparse signalVariance, m=1,2 ... 20), non-jointly sparse support position
Parameter setsMore new regulationWherein,
It is 0 that S24, the noise of different user obey average, and variance isThe multiple Gauss distribution of (m=1,2 ..., 20), then
Parameter betamMore new formula be
S25, joint consider 20 MS, and the more new formula of co-localization parameter k of different user isWherein,
Compressed sensing calculation matrix Φ and angle domain receipt signal matrix Y described in S26, input S13, to S23-S25's
Parameter carries out Joint iteration estimation, and the sparse support set omega of different user is obtained.
S3, the sparse support set iterative estimate of each user, specially:
The sparse iteration control variable N for supporting set common of S31, each MS of settingiter=10-3, maximum iteration time NsetFor
200;
S32, given initial value:
Receipt signal matrix Y;The prior probability matrix of X, matrix element is obeyed average and isVariance isMultiple Gauss divide
ClothJointly sparse support and non-jointly sparse support parameterThe multiple Gauss distribution variance of noiseMeet
Bernoulli Jacob is distributed, and probability isEmpirical valueAnd it is initialized as 0 intermediate variable
The reception signal Z=Φ X of S33, not Noise, it is assumed that the prior probability of Z is obeyed average and isVariance isAnswer
Gauss distributionThe posterior probability of Z is obtained by the prior probability of Z, probability is obeyed average and isVariance isMultiple height
This distributionMore new regulation is(ForInverse);
S34, by Z posterior probability combine intermediate variablePush away X prior probability
S35, by Z posterior probability, common Sparse parameter and non-common Sparse parameterNoise variance parameter
Selection parameterThrough intermediate variableJoin operation, obtain update X posterior probability
S36, renewal initial parameter valueIteration S33-S35 is until meeting iteration control variable NiterMost
Big iterationses NsetRequirement, can obtain comprising jointly sparse support and non-jointly sparse support each user channel
Status information
S4, the extensive mimo channel of multi-user estimate, the channel estimation results of each user areIts
In, the sparse support section of X is obtained when meeting and necessarily imposing a condition by Bayes's compressed sensing alternative manner proposed by the present invention
Arrive, remainder all 0, described impose a condition as empirical condition.
Fig. 3 is that the performance that the inventive method is applied to when the extensive mimo channel of multi-user is estimated is extensive with other sparse signals
Double calculation method is applied to performance comparison figure when same channel is estimated for different expenses.It can be seen that the calculation of the present invention
Method base station end send 45 pilot signals when just reached optimal performance, with LS (method of least square), OMP, SOMP,
ST-BCS, JOMP (Joint OMP) algorithm is compared, and estimation difference also significantly reduces.Remaining algorithm will reach optimal performance needs
Base station sends more pilot signals.By contrast, illustrate that the algorithm of the present invention is reducing the extensive mimo channel of multi-user
With obvious advantage in terms of estimation expense so that extensive mimo channel estimates that realization in practice becomes possibility.
Fig. 4 is that the performance that inventive algorithm is applied to when the extensive mimo channel of multi-user is estimated is extensive with other sparse signals
Double calculation method is applied to performance comparison figure when same channel is estimated for different signal to noise ratios.Illustrate the present invention in different noises
More consistent than performance under environment.Can draw under different signal to noise ratios and Fig. 3 identical conclusion.
Claims (3)
1. a kind of Masssive mimo channel method of estimation, it is characterised in that comprise the steps:
S1, initialization, specially:
S11, BS broadcast T pilot signal X with T time slot to K MSP=[x(1),x(2),...,x(T)]∈CN×T, wherein, N is
The antenna number of BS, mimo channel is estimated pilot signal X in mathematical modelPIt is converted into the compressed sensing measurement square of angle domain
Battle array, represents have with symbol ΦUnitary matricep,q∈[0,
N-1], ΦH∈CN×TElement i.e. from setIn with equiprobability extract, P is the pilot signal work(of each time slot
Rate, *HRepresent * conjugate transpose;
The receipt signal matrix of S12, K MS are { Rj, wherein, RjRepresent the receipt signal matrix of j-th MS, j=1,2 ...,
K;
S13, symbol conversion is carried out, orderWherein, unitary matriceA, b ∈ [0, M-1], M are the antenna number of MS, and Φ represents that compressed sensing is surveyed
Moment matrix, XjFor the conjugate transpose of angle domain channel matrix, angle domain channel matrix isΕjFor equivalent Gauss
Noise matrix, NjFor receiving noise signal matrix;
S2, the sparse support collection Combined estimator of each user, i.e., using the sparse support collection of multitask BCS algorithm K MS of Combined estimator
Close, obtain K estimation channel sparse support set, described K estimation channel sparse support set expression be Ω1,
Ω2,...ΩK;
S3, the sparse support set iterative estimate of each user, specially:
The sparse iteration control variable N for supporting set common of S31, each MS of settingiterWith maximum iteration time Nset;
S32, given initial value:
Receipt signal matrix Y;The prior probability matrix of X, matrix element is obeyed average and isVariance isMultiple Gauss distributionJointly sparse support and non-jointly sparse support parameterThe multiple Gauss distribution variance of noiseMeet primary
Nu Li is distributed, and probability isEmpirical valueAnd it is initialized as 0 intermediate variable
The reception signal Z=Φ X of S33, not Noise, it is assumed that the prior probability of Z is obeyed average and isVariance isMultiple Gauss
DistributionThe posterior probability of Z is obtained by the prior probability of Z, probability is obeyed average and isVariance isMultiple Gauss divide
ClothMore new regulation is(ForInverse);
S34, by Z posterior probability combine intermediate variablePush away X prior probability
S35, by Z posterior probability, common Sparse parameter and non-common Sparse parameterNoise variance parameterSelect ginseng
NumberThrough intermediate variableJoin operation, obtain update X posterior probability
S36, renewal initial parameter valueIteration S33-S35 is until meeting iteration control variable NiterChange with maximum
For times NsetRequirement, can obtain comprising jointly sparse support and non-jointly sparse support each user channel status
Information
S4, the extensive mimo channel of multi-user estimate, the channel estimation results of each user areWherein, X
Sparse support section is obtained when meeting and necessarily imposing a condition by Bayes's compressed sensing alternative manner proposed by the present invention, remaining
Part all 0, described imposes a condition as empirical condition.
2. a kind of Masssive mimo channel method of estimation according to claim 1, it is characterised in that:K is obtained described in S2
The sparse support set omega of the channel of individual estimation1,Ω2,...ΩKComprise the following steps that;
S21, assume the M reception antenna of each user have identical sparse support set, the sparse support of K MS
S is, for different user K, the number of common sparse support position is Sc, and the number of noncomitant sparse support position is
S-Sc, wherein, Sc is the symbol for representing sparse support position number, and S-Sc represents the exclusive non-jointly sparse support of each user
Position number;
S22, according to Bayes's compressed sensing algorithm, if it is α that the condition of sparse channel of j-th MS obeys parameterj=[αj1,αj2,...,
αji,...,αjN]TMultivariate multiple Gauss distribution, wherein, element αjiSparse support is had for multi-userOr be exclusive sparse supportI.e. joint probability is close
Degree functionWherein, m=1,2 ..., K, HjRepresent the
Channel between j MS and BS,Represent i-th element of channel vector, empirical valueBernoulli Jacob's distribution obeyed by initial value, per
Individual element kiProbability isI=1,2 ... N;
S23, joint K MS of consideration, according toDerive the parameter sets of sparse support position jointlyMore new regulationNon- jointly sparse support position
Parameter setsMore new regulationWherein, Represent m-th sparse signalAverage,Represent m-th sparse signalVariance, m=
1,2,…K;
It is 0 that S24, the noise of different user obey average, and variance isMultiple Gauss distribution, then parameter betamMore new formula be
S25, joint consider K MS, and the more new formula of co-localization parameter k of different user isWherein,Wherein, π is ki=0 probability;
Compressed sensing calculation matrix Φ and angle domain receipt signal matrix Y described in S26, input S13, the parameter to S23-S25
Joint iteration estimation is carried out, the sparse support set omega of different user is obtained.
3. a kind of Masssive mimo channel method of estimation according to claim 1, it is characterised in that:Update described in S36
Initial parameter valueDetailed process is:The parameter in the posterior probability substitution S23-S25 of the X for obtaining in S34 more
In new formula.
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