CN108365874B - FDD-based large-scale MIMO Bayesian compressed sensing channel estimation method - Google Patents

Abstract

The invention belongs to the technical field of wireless communication, and provides a Bayesian compressed sensing channel estimation method based on FDD (frequency division duplex) large-scale MIMO (multiple input multiple output), which is used for obtaining accurate channel state information. The invention designs a mode-coupled Gaussian prior model for describing the common sparsity among different antennas, wherein coefficients in a channel vector are divided into groups with equal length, and each group has a common hyper-parameter, so that the coefficients of each group have the same sparsity; furthermore, Bayesian inference is carried out based on an iterative method through an expectation maximization step, wherein channel coefficients are used as hidden variables, and hyperparameters are used as unknown parameters; and finally, taking the posterior mean value of the obtained channel vector as the estimation of the channel. Simulation shows that the BCS method provided by the invention is superior to similar methods to a great extent, and can reach a performance boundary with an ideal least square algorithm as a baseline.

Description

FDD-based large-scale MIMO Bayesian compressed sensing channel estimation method

Technical Field

The invention belongs to the technical field of wireless communication, and particularly relates to a compressed sensing channel estimation method based on a sparse Bayesian model.

Background

Due to the great spatial freedom degree brought by the large-scale antenna, the large-scale MIMO (Multiple-Input Multiple-Output) system can improve the frequency spectrum and energy efficiency by several orders of magnitude; and thus is widely recognized as one of the key technologies of the next generation wireless system.

In order to fully exert the characteristics of massive MIMO, a base station needs to accurately obtain uplink and downlink CSI (Channel State Information); in a Time Division Duplex (TDD) mode, a channel has reciprocity, and uplink CSI and downlink CSI are considered to be the same; therefore, only the terminal is required to transmit the uplink pilot, and the base station estimates the corresponding Channel State Information (CSI); when the pilot sequence is reasonably designed, the base station can accurately estimate the channel. However, in a frequency Division multiplexing (FDD) mode, reciprocity of a channel is not established, and a traditional way of obtaining downlink CSI is to perform channel estimation at a user end and feed back the channel estimation to a base station; due to the large-scale number of antennas of the base station, the estimated unknown coefficients in the downlink channel are more; therefore, directly performing downlink channel estimation would cause excessive training and computation overhead.

In order to avoid the adverse factor of obtaining downlink CSI in a large-scale MIMO system, most researchers so far employ a TDD mode; however, FDD and TDD have many advantagesPoints, especially for applications where communication symmetry and delay are sensitive; therefore, it is also important to make the acquisition of FDD downlink CSI for massive MIMO challenging. To alleviate the training and computational burden, some methods based on compressed sensing are proposed; one scheme is that a compressed measurement value of downlink CSI is fed back to a base station from each user, and then the base station jointly estimates channel matrixes of a plurality of users by adopting an algorithm based on Orthogonal Matching Pursuit (OMP); however, this algorithm only considers flat fading channels, resulting in applications in narrowband systems only. In order to process a frequency selective channel in a broadband system, an Adaptive Structured Subspace tracking Algorithm (ASSP) is proposed, and the ASSP algorithm can reach the performance boundary of the algorithm and has moderate training overhead due to the characteristic that different antennas have common sparsity; but the performance of the A SSP algorithm is limited by a threshold p_thA decision that this needs to be carefully adjusted according to the signal-to-noise ratio; the signal-to-noise ratio is not a priori available, thus greatly limiting the practical application of the ASSP algorithm.

Based on the method, the invention provides an improved Bayesian compressed sensing channel estimation method based on FDD large-scale MIMO.

Disclosure of Invention

The invention aims to provide an FDD large-scale MIMO improved Bayesian compressed Sensing channel estimation method, which is based on a BCS (Bayesian compressed Sensing) method and is used for estimating FDD large-scale MIMO broadband CSI.

In order to achieve the purpose, the invention adopts the technical scheme that:

a Bayesian compressed sensing channel estimation method based on FDD large-scale MIMO is characterized by comprising the following steps:

setting the number of antennas of the base station as M, adopting the orthogonal frequency division multiplexing technology, and selecting N by the base station_pThe training symbols are sent by pilot frequency sub-carriers, the training symbol vector sent by the m antenna is p_mExpressing the position set of pilot subcarriers as

When the base station communicates with a single user, the channel between the mth antenna of the base station and the user is represented as:

h_m＝[h_m,1,h_m,2,...,h_m,L]^T，m＝1,2,...,M

wherein h is_m,lRepresents h_mThe L-th multipath, L1, 2, L represents the channel length;

step 1, initializing channel variance

Variance of noise beta⁽⁰⁾Upper limit of iteration number N_TPresetting an error eta;

step 2, adopting an EM method to carry out iterative computation:

e, step E:

Γ⁽ⁿ⁾＝(D^(n-1))^-1-β^(n-1)(D^(n-1))^-1A^H(I+β^(n-1)A(D^(n-1))^-1A^H)^-1A(D^(n-1))^-1

μ⁽ⁿ⁾＝β^(n-1)Γ⁽ⁿ⁾A^Hy

where D represents the diagonal variance matrix:

D＝diag([α₁ ^(n-1)I_1×M,…,α_l ^(n-1)I_1×M,…,α_L ^(n-1)I_1×M]^T)，I_1×Mis a unit vector;

a denotes the measurement matrix:

A＝[A₁,A₂,...,A_L]，A_l＝[Φ₁(l),Φ₂(l),...,Φ_M(l)]，Φ_m(l) Is phi_mThe first column of (a) is,

from F_LAccording to location set

Selected N_pColumn composition, F_LThe first L columns of the discrete Fourier transform matrix F are formed; ●^HRepresenting a conjugate transpose operation;

y is a received signal;

and M:

wherein m is_l ⁽ⁿ⁾＝[μ⁽ⁿ⁾ _(l-1)M+1,…,μ⁽ⁿ⁾ _(l-1)M+M]^T、μ⁽ⁿ⁾ _tIs mu⁽ⁿ⁾The t-th element of (V)_lIs x_lA posterior variance matrix of, V_lThe (p, t) -th element V of_lΓ ((l-1) M + p, (l-1) M + t), Γ (i, j) is the (i, j) th element of Γ,

after the position of the channel adjustment element is represented, the sparse vector of the block is obtained, wherein x_l,m＝h_m,l，

And (4) termination judgment:

if the number of the first-time-series terminal,

or N ═ N_TThen the iteration is terminated;

outputting the channel estimation value

The invention has the beneficial effects that:

the invention provides an improved Bayesian compressed sensing channel estimation method based on FDD large-scale MIMO, which is used for obtaining accurate channel state information, thereby fully exerting the potential of a large-scale multiple-input multiple-output (MIMO) system. The invention provides an improved Bayesian Compressive Sensing (BCS) method for estimating Channel State Information (CSI) in frequency division multiplexing massive MIMO; designing a mode-coupled Gaussian prior model for describing the common sparsity among different antennas, wherein coefficients in a channel vector are divided into groups with equal length, and each group has a common hyper-parameter, so that the coefficients of each group have the same sparsity; furthermore, bayesian inference is performed based on an iterative approach, with channel coefficients as hidden variables and hyper-parameters as unknown parameters, by an Expectation Maximization (EM) step; and finally, taking the posterior mean value of the obtained channel vector as the estimation of the channel. Simulation shows that the BCS method provided by the invention is superior to similar methods to a great extent, and can reach a performance boundary with an ideal Least Square (LS) algorithm as a baseline.

Drawings

Fig. 1 shows a sparse mode for a massive MIMO channel.

FIG. 2 is a graphical representation of a prior model, where a double circle is the observed data, a single circle is the hidden variable, and a box is the unknown parameter.

FIG. 3 is a graph of MSE performance versus SNR for various algorithms in an embodiment, N_p/N＝0.22，k＝6。

FIG. 4 shows MSE performance of various algorithms as a function of pilot ratio N in an embodiment_pAnd in the/N change graph, SNR is 20dB, and k is 6.

FIG. 5 is a graph showing the MSE performance of various algorithms according to the channel sparsity k in the embodiment, where SNR is 20dB, and N is_p/N＝0.22。

Detailed Description

The invention is described in detail below with reference to the following figures and examples:

the embodiment provides a Bayesian compressed sensing channel estimation method based on FDD large-scale MIMO, which specifically comprises the following steps:

and (3) channel model:

assuming that a base station with M antennas communicates with a single user, in the case of frequency selective fading, the channel between the mth antenna of the base station and the user can be represented as:

h_m＝[h_m,1,h_m,2,...,h_m,L]^T，m＝1,2,...,M (1)

wherein h is_m,lRepresents h_mThe L-th multipath in (1), L represents the channel length; due to the high resolution brought by the broadband system, L is usually large, however, due to the limited dispersion in the physical propagation environment, only a few number of multipaths are meaningful, and other channel coefficients are approximately 0; in other words, the channel vector

Are generally sparse; moreover, channels of different antennas at the massive MIMO base station end usually have correlation; thus, the channels of different transmission antennas have a common sparse pattern; the support set of the sparse channel is Ω_m＝{l:|h_m,l| ≠ 0 }; fig. 1 shows a sparse mode of a massive MIMO channel, where L is 12, Ω is {1,4,10}, and | Ω | } is 3, and it can be seen that

Receiving a signal model:

the base station selects N by using Orthogonal Frequency Division Multiplexing (OFDM) technology_pThe pilot frequency sub-carrier wave is used for sending a training symbol, and the user side carries out channel estimation after receiving a signal; between a total of N subcarriers, N_pUniformly placing the pilot frequency sub-carriers; expressing the position set of pilot subcarriers as

Wherein,

Is a rounding-down operation; the m-th antenna sends a training symbol vector of

Wherein, random variable

Uniform distribution subject to independent same distribution

At a user end, removing a cyclic prefix from a received signal, converting the received signal into a frequency domain, and obtaining a vector expression of the processed received signal as follows:

wherein the content of the first and second substances,

for the processed received signal,

Represents N_pComplex vector of row 1 and column, N_pFor the number of pilots, diag (-) denotes a diagonal matrix with the elements in it on its diagonal,

from F_LAccording to location set

Selected N_pColumn composition, F_LBy Discrete Fourier Transform (DFT) matrix

The first L columns of (a) make up,

is a 0-mean complex gaussian noise vector;

for the sake of analysis, expression (2) is as

Wherein

And is

In order to reflect the correlation of sparse patterns among different antennas and to facilitate calculation, the columns of phi and the elements in h are rearranged to make the channel have block sparsity, the rearranged measurement matrix is represented by A, and the block sparse channel is represented by x, and the following results are obtained:

y＝Ax+w(4)

wherein the content of the first and second substances,

is phi_mColumn l;

x_l＝[h_1,l,h_2,l,...,h_M,l]∈C^M×1i.e. vector x_lBy

The ith element of each vector; according to the correlation between different antenna channels, x_lHave the same sparsity; let the support set of x be S, we get:

y＝A_Sx_S+w (5)

wherein A is_SComposed of a columns selected from the support set S, x_SX is selected from the group of non-0S;

assuming that the user terminal S is known, x in an ideal state is obtained_SIs estimatedThe method comprises the following steps:

wherein the content of the first and second substances,

is the Moore-Penrose generalized inverse thereof; this estimation scheme is called the ideal Least Squares (L S) method, which the present invention uses as a performance baseline for the simulation.

The improved BCS method of the invention:

for the compressed sensing model in equation (4), y, a, x and w are real values while the conventional BCS method processes, and the channel is complex; moreover, the elements in x are independent of each other, that is, different elements in x have different superparameters; the traditional BCS method is not suitable for the current problem and needs more hyper-parameters; therefore, the present invention improves on this.

To describe x_lThe mutual sparsity of the middle elements is introduced into a variable alpha_lAs x_lA hyper-parameter shared by all elements in;

wherein x_l,m＝h_m,lLet x be_l,mIs a complex Gaussian distribution with a mean of 0 and a variance of 1/alpha_l(ii) a The probability density function is:

as seen from the above formula, α_lWhen having a large value, x_l,mTends to be 0 and alpha_lWhen it is small, x_l,mSince the probability of being a large value is high, a is appropriately adjusted according to y_lCan adaptively control x_l,mSparsity of (a);

because of alpha_lBy all x_lThe elements in (1) are shared with each other,they therefore have the same sparsity, based on the above equation, the probability density function of the prior information of x is:

that is, x is a complex gaussian vector with a mean value of 0, and the diagonal variance matrix D ═ diag ([ α [ ])₁I_1×M,…,α_LI_1×M]^T)，I_1×MIs a unit vector;

according to a signal model (4) and prior information (8), a channel x is estimated from a received signal y, the channel x is put into an EM framework to be solved, y is regarded as an observation vector, x is regarded as a hidden variable, and

as an unknown parameter, where 1/β is the variance of the 0-mean gaussian noise vector w, the model diagram is as in fig. 2; the EM algorithm works in an iterative manner, during the nth iteration, involving an Expectation (E) step, in which the posterior probability of x is updated, and a Maximization (M) step, in which the parameter θ that maximizes the Expectation is estimated; in summary, the EM method iterates between the following two steps:

e, step E: calculating p (x | y; theta)^(n-1))(9)

And M: estimating

The posterior probability of x in step E is calculated by:

wherein p (x; theta)^(n-1)) Is a prior probability density function of x, p (y; theta^(n-1)) The probability density of y is expressed, and can be treated as a constant in the process of calculating the mean value and the variance of the posterior probability; theta^(n-1) Is the n-1 th timeAfter iteration, a set of parameters is estimated, and in step M, it is expected to be calculated by:

wherein lnp (x, y; theta)^(n-1))＝ln(p(y|x；θ^(n-1))p(x；θ^(n-1)) Symbol) of a character<·>Expression with respect to the posterior probability p (x | y; theta)^(n-1)) (iii) a desire;

for ease of illustration, without ambiguity, θ or its superscript will be ignored below;

in step E, the signal model (4) and Gaussian noise w are synthesized to easily obtain:

taking the logarithm of equation (11) to obtain:

lnp(x|y；θ)＝lnp(y|x；θ)+lnp(x；θ)+const＝-(x-μ)^HΓ^-1(x-μ)+const (14)

wherein const is a constant unrelated to x;

substituting (8) and (13) into (11) to calculate the posterior probability p (x | y; theta) of x, combining the formula (14), obtaining the mean value mu and the variance gamma of the posterior probability of x as follows:

as can be seen from (14), the posterior probability of x follows a complex gaussian distribution, the mean vector is μ, and the variance matrix is Γ;

it should be noted that, in the step E, the parameter θ is calculated and known in the last iteration, and here, the expression is concise, and the iteration mark is omitted;

in the step M, the objective function Q (x, θ) ═ ln (p (y | x; θ) p (x; θ)) >, and the values (8) and (13) are substituted into Q (x, θ), and the following are calculated:

it is understood that the expectation is calculated with respect to the posterior probability p (x | y; θ), so the magnitude of Q (x, θ) depends on θ;

solving for Q (x, theta) with respect to a_lIs a solution of 0, to obtain

Wherein the content of the first and second substances,

1,2, L, μ t is the tth element of μ;

is x_lA posterior variance matrix of, V_lThe (p, t) -th element V of_lΓ ((l-1) M + p, (l-1) M + t), Γ (i, j) being the (i, j) -th element of Γ; a is shown by formula (17)_lInversely proportional to | | x_l||²:<||x_l||²>A posterior mean of (i) when<||x_l||²>When larger, α_lSmaller, which causes elements in xl to have large values; on the other hand, when<||x_l||²>Smaller, α_lLarger, which will force x to_lThe smaller of the middle elements is close to the average value 0; moreover, since a common hyper-parameter α is used_l，x_lHave a common sparsity;

similarly, by making the derivative of Q (x, θ) with respect to β 0, it is derived

Wherein tr (-) is a trace of matrix; it is easy to see that the above calculation completes one iteration;

in the step M, μ and Γ are known from the calculation in the step E of the current iteration, and the iteration mark is omitted for the sake of simplicity;

in the above EM algorithm, initialization parameters are required for iteration start

Maximum number of iterations N_TError η; running iteration until the number of iterations reaches N_TOr reaching the respective iteration end condition, i.e.

Wherein, mu⁽n) is the posterior mean value after the nth iteration;

in each iteration, the computational complexity depends mainly on the computation of Γ in (15); if t is calculated directly, it results in higher complexity

The method uses Woodbury identity equation to calculate gamma, and the calculation of gamma is expanded as follows:

Γ＝D^-1-βD^-1A^H(I+βAD^-1A^H)^-1AD^-1 (19)

with a complexity of

Thus, the overall complexity of the proposed BCS method is

Wherein N is_iThe number of iterations actually used in the iteration; in the present embodiment, an ASSP algorithm is used as a comparative example 1, and the complexity of the ASSP algorithm is

Under typical settings N_p＝410，N_p0.2, 64, 6 (used in the simulations below), both methods have similar complexity;

if { x is not utilized_lCommon sparsity in given xEach element of (1) sets independent hyper-parameters, namely an original BCS algorithm, as a comparative example 2; the hyper-parameter set is combined into

Using a similar derivation:

1,2, r, ML, wherein r_nIs the nth element on the diagonal of Γ; the posterior probability of x and the calculation of β still use the original expression.

During simulation, the system parameters are set to antenna number M64, channel length L64, pilot number N2048, and sparsity | Ω | k. Now, according to a complex Gaussian distribution of 0 mean and unit variance, k elements are randomly selected as channels

M is more than or equal to 1 and less than or equal to M is the serial number of the antenna, k is the sparsity of the channel, h_mThe other L-k elements in (a) are set to 0. Normalizing the obtained channel to enable | | | x | | non-woven phosphor²And x is the adjusted channel in (4). The simulation compares the original BCS method, the ASSP algorithm, the improved BCS algorithm and the ideal LS algorithm. Wherein, the initial hyper-parameter in the original BCS algorithm is set as 100, the threshold in the ASSP algorithm is set as p_th0.06, the initial hyper-parameter setting in the improved BCS method of the present invention is

The number of iterations and the error are set to N_T＝20,η＝10-³。

Briefly describing the simulation process as follows:

step 1. generating corresponding original sparse channel according to the above description

Step 2. according to

x_l＝[h_1,l,h_2,l,...,h_M,l]∈CM×¹The sparse channels generated in the step 1 are arranged into x in the formula (4) to describe the common sparsity of the channels, so that the x has block sparsity to facilitate algorithm calculation;

step 3, generating corresponding 0 mean value Gaussian random noise w according to the set SNR value;

in this case, the signal normalization signal-to-noise ratio is calculated by the formula

The variance of the noise can be obtained according to the formula, and the variances of the real part and the imaginary part of the noise are all

Generating a corresponding 0-mean complex Gaussian noise w;

step 4. according to

Constructing the measurement matrix in (4), wherein

Is phi_mThe first column of (a) is,

the DFT matrix is of the form

Wherein

1≤i≤N_p，0≤k≤L-1。

Step 5 generates a corresponding reception signal y according to equation (4) y ═ Ax + w.

Step 6, inputting y and A into the above four algorithms for channel estimation, and calculating corresponding MSE, wherein the formula is as follows

Wherein | x | purple₂The operator is the two-norm of x. Adjusting SNR and N separately_pN and k are plotted in fig. 3, 4, 5.

The improved BCS method pseudo code is as follows

Input y, A

Initialization

Iterative computation

The method comprises the following steps:

the method comprises the following steps:

and (5) termination judgment:

if it is not

Or N ═ N_TThe iteration is terminated.

Output of

Description of simulation results

The channel estimation Mean Square Error (MSE) performance as a function of signal-to-noise ratio is described as shown in FIG. 3, where N is_p/N＝0.22And k is 6. It can be seen that the curve of the proposed improved BCS almost coincides with the curve of the ideal LS in the full SNR region displayed. In contrast, the gap between the ASSP algorithm and the ideal LS algorithm is large when SNR < 25 dB. When the SNR is larger than or equal to 25dB, the curves of the ASSP algorithm, the improved BCS algorithm and the ideal LS algorithm are relatively close. Compared with the ASSP algorithm and the improved BCS algorithm, the original BCS algorithm does not utilize the inherent common sparsity of the massive MIMO channel, so that compared with an ideal LS baseline, the performance is lost to a certain extent.

The impact of the training resources on performance is depicted in fig. 4, where the SNR is 20dB and k is 6. As expected, with the number of pilots N_pAll methods exhibit superior performance. Similar to fig. 3, the improved BCS algorithm reaches the performance boundary of the ideal LS. However, the ASSP algorithm requires more pilot overhead to achieve the desired LS performance. As can be seen, the ASSP algorithm is at N_p0.21 to N_pThe MSE drop between 0.23 and/N is large, which means that the algorithm requires an empirical choice of N in order to achieve reliable channel estimation_p。

The effect of the channel sparsity k on the estimation performance is depicted in fig. 5, where SNR is 20dB, N_pand/N is 0.22. As k increases, more meaningful elements in x need to be estimated. Thus, for all methods, the larger k the worse the performance is estimated. Although the performance of the improved BCS suffers somewhat as sparsity increases, it still closely approaches the ideal LS baseline.

Combining the results of fig. 3, 4, and 5, it can be concluded that the improved BCS algorithm can achieve performance close to the ideal baseline. Compared with the ASSP algorithm, the channel estimation method can process channel estimation with larger sparsity level with better performance under smaller SNR and training overhead.

While the invention has been described with reference to specific embodiments, any feature disclosed in this specification may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise; all of the disclosed features, or all of the method or process steps, may be combined in any combination, except mutually exclusive features and/or steps.

Claims (1)

1. FDD-based massive MIMO Bayes compressed sensing channel estimation method is characterized by comprising the following steps:

setting the number of antennas of the base station as M, adopting the orthogonal frequency division multiplexing technology, and selecting N by the base station_pThe training symbols are sent by pilot frequency sub-carriers, the training symbol vector sent by the m antenna is p_mExpressing the position set of pilot subcarriers as

When the base station communicates with a single user, the channel between the mth antenna of the base station and the user is represented as:

h_m＝[h_m,1,h_m,2,...,h_m,L]^T，m＝1,2,...,M

wherein h is_m,lRepresents h_mThe L-th multipath, L1, 2, L represents the channel length;

step 1, initializing channel variance

Sum noise variance β⁽⁰⁾Upper limit of iteration number N_TPresetting an error eta;

step 2, adopting an EM method to carry out iterative computation:

e, step E:

Γ⁽ⁿ⁾＝(D^(n-1))^-1-β^(n-1)(D^(n-1))^-1A^H(I+β^(n-1)A(D^(n-1))^-1A^H)^-1A(D^(n-1))^-1

μ⁽ⁿ⁾＝β^(n-1)Γ⁽ⁿ⁾A^Hy

where n represents the number of iterations and D represents the diagonal variance matrix:

D＝diag([α₁ ^(n-1)I_1×M,…,α_l ^(n-1)I_1×M,…,α_L ^(n-1)I_1×M]^T)，I_1×Mis a unit vector;

a denotes the measurement matrix:

A＝[A₁,A₂,...,A_L]，A_l＝[Φ₁(l),Φ₂(l),...,Φ_M(l)]，Φ_m(l) Is phi_mThe first column of (a) is,

from F_LAccording to location set

Selected N_pColumn composition, F_LThe first L columns of the discrete Fourier transform matrix F are formed; a^HRepresenting a conjugate transpose operation;

y is a received signal;

and M:

wherein m is_l ⁽ⁿ⁾＝[μ⁽ⁿ⁾ _(l-1)M+1,…,μ⁽ⁿ⁾ _(l-1)M+M]^T、μ⁽ⁿ⁾ _tIs mu⁽ⁿ⁾The t-th element of (V)_lIs x_lA posterior variance matrix of, V_lThe (p, t) -th element V of_lΓ ((l-1) M + p, (l-1) M + t), Γ (i, j) is the (i, j) th element of Γ,

representing post-block sparseness of channel-adjustment element positionsVector of where x_l,m＝h_m,l；

And (4) termination judgment:

if the number of the first-time-series terminal,

or N ═ N_TThen the iteration is terminated;

outputting the channel estimation value

Images