A millimeter wave massive MIMO channel estimation method based on low-rank matrix recovery

Abstract

The application discloses a kind of millimeter wave massive MIMO channel estimation methods based on low-rank matrix recovery, belongs to 5G, 6G mobile communication technical field.The method includes the following steps: (1) construct millimeter wave massive MIMO system, construct millimeter wave channel model;(2) propose an improved θ-norm, utilize the low-rank characteristics of millimeter wave channel matrix, and the channel estimation problem is converted into the minimization problem based on improved θ-norm;(3) to the improved θ-norm is and formula decomposition, and the improved θ-norm is decomposed into the form of sum of kernel norm and singular value function;(4) the channel estimation problem is further converted into the double-target convex optimization problem based on joint kernel norm and singular value function;(5) using alternating optimization framework to solve double-target minimization problem.The application applies low-rank matrix recovery method to millimeter wave MIMO channel estimation, effectively reduces pilot overhead and improves estimation accuracy, and the calculation complexity is lower.

Description

Technical Field

[0001] This invention belongs to the field of 5G and 6G mobile communication technology, specifically relating to a millimeter-wave large-scale MIMO channel estimation method based on low-rank matrix recovery. Technical Background

[0002] With the rapid deployment of 5G and 6G, global mobile traffic is increasing rapidly, and capacity demand is expected to grow 1000-fold within the next 10 years. To meet this massive increase in data traffic and address the scarcity of spectrum resources below 6GHz, we need to develop higher frequency bands. Millimeter wave bands have become a primary focus for addressing these issues. Millimeter waves operate at frequencies between 30-300GHz, with corresponding wavelengths of 1-10mm. This band contains a large amount of unused bandwidth that wireless systems can utilize, making millimeter wave communication a crucial component of 5G and 6G. Massive MIMO, a core technology of 5G and 6G, allows for further expansion of antenna size in the millimeter wave band, enabling the use of spatial multiplexing to improve spectral efficiency and reduce transmission loss. However, with the increased antenna size, the pilot overhead required by traditional channel estimation algorithms increases dramatically, and low estimation accuracy remains a significant problem. Summary of the Invention

[0003] This invention addresses the problems of high channel overhead and low estimation accuracy in millimeter-wave massive MIMO systems by proposing a low-rank matrix recovery-based channel estimation method. Utilizing the low-rank characteristic of the millimeter-wave channel matrix, the channel estimation problem is transformed into a low-rank matrix recovery problem. An improved θ-norm-based matrix recovery method is employed, which improves accuracy by using the improved θ-norm to approximate the rank function more closely than the kernel norm. The improved θ-norm is decomposed into the sum of the kernel norm and singular value functions, constructing a new bi-objective low-rank matrix recovery model. Finally, the channel matrix H is solved using the Alternating Direction Method of Multipliers (ADMM).

[0004] A millimeter-wave large-scale MIMO channel estimation method based on low-rank matrix recovery is proposed, specifically including a dual-objective convex optimization channel estimation method based on an improved non-convex θ-norm and a joint kernel norm and singular value function. This method can effectively improve estimation accuracy, reduce training overhead and computational complexity, and includes the following steps:

[0005] Step 1: Construct a millimeter-wave massive MIMO system model and channel model. Both the transmitter and receiver use uniform linear arrays (ULA), and generate the channel matrix.

[0006] Step 2: Propose an improved θ-norm, which utilizes the low-rank property of the millimeter-wave channel matrix to transform the channel estimation problem into a minimization problem based on the improved θ-norm;

[0007] Step 3: Perform a summative decomposition on the improved θ-norm, decomposing the θ-norm into the form of the sum of the nuclear norm and the singular value function;

[0008] Step 4: Further transform the channel estimation problem into a bi-objective convex optimization problem based on the joint nuclear norm and singular value functions;

[0009] Step 5: Solve the bi-objective minimization problem using the alternating optimization framework.

[0010] Furthermore, step 1 is detailed as follows:

[0011] The base station is equipped with N T Root antenna, There are N radio frequency links, and the user end has N R Root antenna, A radio frequency link is established, and channel estimation is performed on the downlink of the FDD system. The base station sends pilot sequences, and the pilot sequences are orthogonal to each other to avoid mutual interference between pilots.

[0012] The channel matrix H generated according to the system model is represented as follows:

[0013]

[0014] Where ρ represents the transmission path loss, L represents the number of effective paths, and g l ∈C represents the path gain of the l-th path. and α R (θ l ) represent the antenna array response vectors of the transmitting and receiving ends, respectively.

[0015] Furthermore, step 2 is detailed as follows:

[0016] The traditional theta norm is represented as follows:

[0017]

[0018] Where r represents the rank of matrix H. For the traditional θ-norm, we can see that in the ideal case, when θ is 1, the value of the θ-norm is 1 when the singular values ​​of H are greater than θ, which can approximate the rank function well. However, for singular values ​​less than θ, the value of the θ-norm is equal to the singular value itself, not equal to 1, so it cannot approximate the rank function well.

[0019] Therefore, to address the problem that the traditional θ-norm cannot approximate the rank function well, an improved θ-norm is proposed, as follows:

[0020]

[0021] For small parameters θ that approach 0, we can see that... It must be greater than 1, therefore The improved θ-norm can accurately approximate the rank function, where ||H||0 denotes the 0 norm of matrix H. Let H be the estimated matrix.

[0022] Furthermore, the decomposition process in step 3 is as follows:

[0023]

[0024] Among them, ||H|| * The nuclear norm of matrix H is represented by H. Let H denote the singular value functions of matrix H. Since both the nuclear norm and the singular value functions are convex, the sum of two convex functions is also convex. Therefore, the non-convex minimization problem based on the improved θ-norm becomes a bi-objective convex minimization problem that is easier to solve.

[0025] Furthermore, step 4 is detailed as follows:

[0026]

[0027] st Y = HX

[0028] P Ω (H)=P Ω (M)

[0029] Y represents the received signal matrix, X represents the pilot matrix, Ω represents the sampling matrix, M is the observation matrix of the channel matrix H, and P Ω (*) indicates the projection operator.

[0030] Furthermore, step 5 is detailed as follows:

[0031] Step 1: Construct the dual problem and introduce auxiliary variable matrices H = D and H = W as constraints;

[0032] Step 2: Further transform the bi-objective convex optimization expression in Step 1 into the form of augmented Lagrangian function L(H,D,W,Z1,Z2,μ1,μ2), where Z1 and Z2 represent augmented Lagrangian multipliers and μ1 and μ2 represent penalty parameters;

[0033] Step 3: Update variable H alternately by minimizing the augmented Lagrangian function L(H,D,W,Z1,Z2,μ1,μ2) of each variable. k+1 D k+1 W k+1 ;

[0034] Step 4: Update Z1 k+1 Z2 k+1 μ1 k+1 and μ2 k+1 ;

[0035] Step 5: k = k + 1;

[0036] Step 6: Finally, output the channel estimation matrix.

[0037] Compared with the prior art, the present invention has the following significant advantages:

[0038] In the context of millimeter-wave massive MIMO system models, this invention applies the low-rank matrix recovery method to the estimation problem of channel state information in millimeter-wave massive MIMO systems. Leveraging the low-rank characteristic of the millimeter-wave massive MIMO channel matrix, the channel estimation problem is transformed into a low-rank matrix recovery problem. An improved θ-norm minimization-based channel estimation method is proposed, and further, the θ-norm non-convex optimization problem is transformed into a bi-objective convex optimization problem combining the kernel norm and singular value functions, solved using an alternating optimization framework. In this method, a sufficiently small θ allows the improved θ-norm to approximate the rank function better than the kernel norm. Furthermore, this method significantly reduces computational complexity in millimeter-wave massive MIMO channel estimation. Simulation results demonstrate that this method can reduce pilot overhead and effectively improve the accuracy of the channel matrix. Attached Figure Description

[0039] Figure 1 This is a comparison chart of the convergence of the present invention under different signal-to-noise ratios.

[0040] Figure 2 This is a schematic diagram comparing the performance of the present invention and the SVT algorithm under different iteration numbers.

[0041] Figure 3 This is a performance comparison chart of the present invention and the SVP-H and SVT algorithms under different channel overhead conditions.

[0042] Figure 4 This is a performance comparison chart of the present invention and the SVP-H and SVT algorithms under different signal-to-noise ratio conditions. Detailed Implementation

[0043] The present invention will now be described in further detail with reference to the accompanying drawings and specific implementation methods.

[0044] This invention employs uniform linear arrays (ULA) at both the transmitter and receiver to construct a millimeter-wave massive MIMO communication system. The base station of this system is equipped with N... T Root antenna, There are N radio frequency links, and the user end has N R Root antenna, A radio frequency link is established, and channel estimation is performed on the downlink of the FDD system. The base station sends pilot sequences, and the pilot sequences are orthogonal to each other to avoid mutual interference between pilots.

[0045] Generate pilot sequence X, channel matrix H, and received signal matrix Y according to the system model.

[0046] To obtain channel state information, the transmitted pilot signal sequence expression is as follows:

[0047]

[0048] Where T represents the number of pilot sequences.

[0049] The downlink channel matrix is ​​generated based on the system model. A geometric channel is used as the channel model for millimeter-wave massive MIMO, and a narrowband millimeter-wave channel model is adopted. For ease of study, the impact of multipath delay is ignored, and point-to-point communication is used between the base station and the user. The channel matrix is ​​then represented as follows:

[0050]

[0051] In the formula, N T N R Here, ρ represents the number of antennas at the system's transceiver end, L represents the transmission path loss, and L represents the number of paths between the system's transceiver end, with the number of paths being much smaller than the number of antennas at the transceiver end. l This represents the delay of the l-th path. l θ represents the path gain of the l-th path. l and Indicates the arrival angle and departure angle of the l-th path.

[0052] α T and α R The antenna array response vectors at the transmitting and receiving ends are represented as follows:

[0053]

[0054]

[0055] Where λ represents the millimeter-wave wavelength, D represents the element spacing of the antenna array, and θ l ,

[0056] In particular, we made

[0057]

[0058]

[0059] G = [g1, g2, ... g] L ]∈R L×L

[0060] The channel matrix H can then be expressed as:

[0061]

[0062] Where G represents the path gain matrix, A T and A R These represent the antenna array response matrices of the transmitting and receiving ends, respectively.

[0063] Based on the system model, the received signal matrix is ​​generated, and the received signal at the base station is:

[0064] Y = HX + E

[0065] Where E represents a mean of 0 and a variance of σ. 2 Additive white Gaussian noise (AWGN) is used. For ease of computation, the training sequence x is designed to satisfy E{xx}. T}=P t P t This represents the average transmit power of the base station antenna.

[0066] Low-rank matrix recovery theory operates directly on the matrix, recovering the original channel matrix from the observation matrix containing partial sampled values.

[0067] Traditional channel estimation methods use the nuclear norm to approximate the rank function, but since the nuclear norm is convex, it is not accurate enough. Therefore, we use an improved θ-norm to approximate the rank function, transforming the channel estimation problem into a minimization problem based on the improved θ-norm:

[0068]

[0069] st Y = HX

[0070] P Ω (H)=P Ω (M)

[0071] in, This represents the sampling matrix, where Ω contains m 1s and N... R ×N T-m zeros, M is the observation matrix of the channel matrix H, P Ω (H) represents the projection operator, specifically:

[0072]

[0073] The rank function is approximated by reducing the value of θ in the improved θ-norm:

[0074]

[0075] Where ||H||0 represents the 0 norm of matrix H. Let H be the estimated matrix.

[0076] Regularizing the constraints in the above equation further transforms it into an unconstrained minimization problem:

[0077]

[0078] in This represents the Frobenius norm.

[0079] The improved θ-norm is decomposed into a summation of the nuclear norm and singular value functions:

[0080]

[0081] Among them, ||H|| * The nuclear norm of matrix H is represented by H. Let H be the singular value function of matrix H.

[0082] Since both the nuclear norm and singular value functions are convex, the sum of two convex functions is also convex. Therefore, the non-convex minimization problem based on the improved θ-norm becomes a convex minimization problem that is easier to solve.

[0083] The problem of minimizing based on the improved θ-norm is further expressed as a bi-objective convex optimization problem based on the joint nuclear norm and singular value functions:

[0084]

[0085] The alternating optimization framework is used to solve the bi-objective convex optimization problem based on the joint nuclear norm and singular value functions.

[0086] To recover the matrix and construct the dual problem, we introduce auxiliary matrices H = D and H = W, further expressing the above equation in the following form:

[0087]

[0088] stH=D,H=W

[0089] The above equation can be further transformed into the form of an augmented Lagrange function:

[0090]

[0091] Where Z1 and Z2 represent augmented Lagrange multipliers, and μ1 and μ2 represent penalty parameters.

[0092] The update for each variable can be obtained by minimizing the augmented Lagrangian function L(H,D,W,Z1,Z2,μ1,μ2) for each variable, as follows:

[0093] enter: Z1 0 =0,Z2 0 =0, μ1>0, μ2>0, k=1,k max =100, ρ=1.1, v=0.1.

[0094] Update H k+1 :

[0095] Given W, D, Z1, Z2, calculate H. k+1 as follows:

[0096]

[0097] Taking the derivative of the above equation directly and then setting the derivative to 0, we can obtain the update formula for the channel matrix H:

[0098] H k+1 =(YX) T -P Ω (M)-μ1D-μ2W+Z1+Z2)(XX T -μ1I-μ2I) -1

[0099] +P Ω (YX T -P Ω (M)-μ1D-μ2W+Z1+Z2)(XX T -μ1I-μ2I-I) -1

[0100] Since the above equation is quite complex and requires inversion, we use gradient descent to solve it. The gradient of H is obtained as follows:

[0101]

[0102] At this point, the iterative update formula for H is:

[0103]

[0104] Where v∈(0,1) represents the iteration step size.

[0105] Update D k+1 :

[0106] Given W, Z1, and Z2, calculate D. k+1 as follows:

[0107]

[0108] Differentiating the above equation and setting the derivative to 0, we obtain the updated formula for D:

[0109]

[0110] in, U and V represent the left and right singular value vectors, respectively.

[0111] Update W k+1 :

[0112] Given Z1 and Z2, calculate W. k+1 as follows:

[0113]

[0114] Taking the derivative of the above equation and setting it to zero, we obtain the updated formula for W as follows:

[0115]

[0116] in, i = 1, 2, ..., rank(W), where A and B represent the left and right singular value vectors.

[0117] Update Z1 and Z2:

[0118] Z1 k+1 =Z1 k +μ1(H k+1 -D k+1 )

[0119] Z2 k+1 =Z2 k +μ2(H k+1 -W k+1 )

[0120] Update μ1, μ2:

[0121] μ1 k+1 =ρμ1 k μ2 k+1 =ρμ2 k

[0122] Update k = k + 1;

[0123] Finally, the channel estimation matrix is ​​output.

[0124] To verify that the improved θ-norm mentioned in this invention is superior to the kernel norm, the performance of the proposed channel estimation method based on the improved θ-norm in millimeter-wave massive MIMO systems was compared with that of the SVP-H and SVT algorithms in terms of signal-to-noise ratio (SNR), pilot overhead, and number of iterations. The normalized mean square error (NMSE) was used to evaluate the performance of each algorithm for CSI estimation; a smaller NMSE indicates better performance.

[0125] When setting the parameters required for CSI estimation, the millimeter wave frequency band is 90Hz, and the antenna AoD angle is... L=10 represents an effective transmission path, the rank of the channel matrix is ​​6, and the maximum number of iterations k is set. max =100, the CSI estimation error uses the normalized mean square error and Frobenuis norm error:

[0126]

[0127] Figure 1 The NMSE performance of the proposed algorithm under different signal-to-noise ratios with varying iteration numbers was compared. For ease of analysis, the channel overhead T = 50, N... R and N T Let's take 100. As shown in the figure, even under low signal-to-noise ratio (SNR) conditions, the channel estimation method proposed in this invention still exhibits fast convergence speed and reliable convergence. The comparison of NMSEs shows that as the SNR increases, the CSI estimation accuracy of the algorithm proposed in this invention increases with the number of iterations and eventually tends towards the convergence value. Therefore, the algorithm proposed in this invention has high estimation accuracy and reliable convergence.

[0128] Figure 2 The NMSE performance of SVT and the proposed algorithm of this invention as the number of iterations was compared. For ease of analysis, SNR was set to 20dB, channel overhead T = 50, and N R and N T Let's take 100. The channel estimation method proposed in this invention and the SVT algorithm both exhibit fast convergence speeds within the same number of iterations. A comparison of NMSEs shows that the CSI estimation accuracy of both the proposed algorithm and the SVT algorithm increases with the number of iterations. When the iteration results tend to converge, the algorithm proposed in this invention has higher estimation accuracy compared to the SVT algorithm.

[0129] Figure 3 The NMSE performance of SVP-H, SVT, and the algorithm proposed in this invention as channel overhead varies was compared. For ease of analysis, SNR was set to 20 dB, and N... R and N T The channel overhead was set to 50, with values ​​of 15, 20, 25, 30, and 35 respectively. As the channel overhead T increased, the estimation accuracy of all algorithms improved. When the training overhead was low, SVP-H exhibited poor CSI estimation performance, while the performance of SVT and ADMM algorithms improved significantly with increasing channel overhead T. The algorithm proposed in this invention demonstrates excellent CSI estimation performance with low training overhead, and its performance further improves with increasing training overhead. For the same pilot overhead, the algorithm proposed in this invention exhibits the best CSI estimation performance.

[0130] Figure 4 The NMSE performance of SVP-H, SVT, and the algorithm proposed in this invention as the signal-to-noise ratio (SNR) changes was compared. For ease of analysis, the channel overhead T = 50, N... R and N T The signal-to-noise ratio (SNR) was set to 50, and the values ​​were 5, 10, 15, 20, and 25. As the SNR increased, the performance of all algorithms improved significantly. This is because with a higher SNR, the signal power and energy in the system are greater, and the noise interference decreases, thus resulting in better NMSE estimation performance. At a lower SNR, the noise interference is greater, and the SVP-H algorithm has lower NMSE estimation performance. As the SNR increases, the estimation performance of all algorithms improves significantly. When the SNR is the same, the SVT algorithm and the algorithm proposed in this invention are significantly better than the SVP-H algorithm, with the algorithm proposed in this invention showing the best performance. Therefore, the algorithm proposed in this invention can achieve higher CSI estimation performance with a lower SNR.

[0131] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A millimeter-wave large-scale MIMO channel estimation method based on low-rank matrix recovery, characterized in that, Includes the following steps: Step 1: Construct a millimeter-wave massive MIMO system model and channel model. Both the transmitter and receiver use uniform linear arrays (ULA), and generate the channel matrix. Step 2 proposes an improved θ-norm. Utilizing the low-rank characteristic of the millimeter-wave channel matrix, the channel estimation problem is transformed into a minimization problem based on the improved θ-norm. The improved θ-norm is expressed as follows: Where r represents the rank of matrix H, ||H||0 represents the 0 norm of matrix H, and for small parameters θ that tend to 0, It must be greater than 1, therefore The improved θ-norm can accurately approximate the rank function; Step 3: Perform a summative decomposition on the improved θ-norm, decomposing it into the sum of the nuclear norm and the singular value function, as shown below: Among them, ||H|| * The nuclear norm of matrix H is represented by H. The singular value function representing matrix H; Step 4: The channel estimation problem is further transformed into a bi-objective convex optimization problem based on the joint nuclear norm and singular value functions; Step 5: Solve the bitarget convex minimization problem using the alternating direction multiplier method (ADMM) to obtain the channel estimation matrix.

2. The millimeter-wave massive MIMO channel estimation method based on low-rank matrix recovery according to claim 1, characterized in that, Step 1 is described in detail as follows: The base station is equipped with N T Root antenna, There are N radio frequency links, and the user end has N R Root antenna, A radio frequency link is established, and channel estimation is performed on the downlink of the FDD system. The base station sends pilot sequences, and the pilot sequences are orthogonal to each other to avoid mutual interference between pilots. The channel matrix H generated according to the system model is represented as follows: Where ρ represents the transmission path loss, L represents the number of effective paths, and g l ∈C represents the path gain of the l-th path. and α R (θ l ) represent the antenna array response vectors of the transmitting and receiving ends, respectively.

3. The millimeter-wave massive MIMO channel estimation method based on low-rank matrix recovery according to claim 1, characterized in that, The decomposition process in step 3 is as follows: Since the improved θ-norm is non-convex, it is difficult to solve. Therefore, the improved θ-norm is decomposed into a summation of the nuclear norm and the singular value function, as specifically expressed below: Among them, ||H|| * The nuclear norm of matrix H is represented by H. Let H denote the singular value functions of matrix H. Since both the nuclear norm and the singular value functions are convex, the sum of two convex functions is also convex. Therefore, the non-convex minimization problem based on the improved θ-norm becomes a bi-objective convex minimization problem that is easier to solve.

4. The millimeter-wave massive MIMO channel estimation method based on low-rank matrix recovery according to claim 1, characterized in that, Step 4 is as follows: Since both the nuclear norm and the singular value function are convex, the sum of two convex functions is also convex. Therefore, the non-convex minimization problem based on the improved θ-norm becomes a bi-objective convex minimization problem based on the joint nuclear norm and singular value function, which is easier to solve, as follows: st Y = HX P Ω (H)=P Ω (M) in, Let H be the estimation matrix, Y be the received signal matrix, X be the pilot matrix, Ω be the sampling matrix, M be the observation matrix of the channel matrix H, and P be the sampling matrix. Ω (*) indicates the projection operator.

5. The millimeter-wave massive MIMO channel estimation method based on low-rank matrix recovery according to claim 1, characterized in that, The specific steps of step 5 are as follows: Step 1: Construct the dual problem and introduce auxiliary variable matrices H = D and H = W as constraints; Step 2: Further transform the bi-objective convex optimization expression in Step 1 into the form of augmented Lagrangian function L(H,D,W,Z1,Z2,μ1,μ2), where Z1 and Z2 represent augmented Lagrangian multipliers and μ1 and μ2 represent penalty parameters; Step 3: Update variable H alternately by minimizing the augmented Lagrangian function L(H,D,W,Z1,Z2,μ1,μ2) of each variable. k+1 D k+1 W k+1 ; Step 4: Update Z1 k+1 Z2 k+1 μ1 k+1 and μ2 k+1 ; Step 5: k = k + 1; Step 6: Finally, output the channel estimation matrix.

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