1. Introduction
Millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) has been generally recognized as a strong potential key technique for enhancing quality in the future wireless communication field [
1]. The demand for higher speed data transmission, shorter delays, better user experience, and denser networks is increasing with the rapid development of wireless services such as virtual reality, multimedia, and the Internet of Things [
2]. The mmWave band has huge unexploited spectrum resources, which can overcome the spectrum congestion of standard wireless frequency bands and achieve an orders of magnitude increase in spectral efficiency to meet these requirements [
3,
4]. The work in [
5] expounded that the mmWave band can greatly speed up the data transmission rate. Additionally, massive MIMO refers to adding a number of antennas at the base station (BS), which can overcome the effects of fading, average out the dominant path-loss, and make up for the short wavelength of mmWave [
6,
7,
8], and the isolation between radiating elements is a very important issue [
9,
10]. The combination of mmWave and massive MIMO can be used to meet increasing demand in data traffic [
11,
12].
Many studies have been devoted to the research of mmWave massive MIMO multi-user systems in recent years. Compared with conventional point-to-point MIMO, multi-user MIMO with simplified resource allocation has greater advantages [
13,
14]. The work [
15] developed blind multi-user detection based on the model of [
16], which can approach the lower bound of performance under certain conditions. The work in [
17] summarized the benefits, challenges, and potential solutions of mmWave massive MIMO in cellular networks.
In order to achieve the theoretical performance gain and potential benefits of mmWave massive MIMO systems, accurate channel state information (CSI) is essential, which is also regarded as one of the main difficulties in mmWave communication [
18,
19]. However, due to the difference in channel characteristics compared to traditional sub-6 GHz, the smaller number of RF chains, and overwhelming overhead in downlink training and uplink channel feedback, accurate acquisition of CSI is challenging [
20,
21,
22]. Various novel channel estimation approaches have been recently proposed for mmWave massive MIMO [
23,
24,
25,
26,
27]. In particular, under the assumption of the BS antennas being highly correlated, the work in [
23] proposed an antenna grouping method to reduce feedback overhead. By utilizing the correlation among users, the work in [
24] provided a joint CSI acquisition proposal based on a low-rank matrix. To reduce errors caused by discretization and solve the line spectral estimation problem, the work in [
25] designed an iterative reweighted approach for joint dictionary parameter training and sparse signal recovery, but it had the limitation that only two parameters could be estimated. The work in [
26] extended the suggestion in [
25] to two dimensions and represented a super-resolution channel estimation scheme based on iterative reweighting in a point-to-point scenario. It is regrettable that such a solution in [
26] had the problem of slow estimation. Considering the problem of channel estimation for mmWave MIMO systems under a transmitter impairment model, a new algorithm based on Bayesian compressive sensing (BCS) was introduced in [
27]. However, the assumption of BCS was relatively harsh, and the accuracy of estimation was limited.
To obtain CSI more accurately and faster in multi-user mmWave massive MIMO, we develop a step-length optimization-based joint iterative channel estimation scheme in this paper. Our main contributions are outlined as follows:
1. For multi-user mmWave massive MIMO systems, we propose a novel joint estimation scheme, which can accurately estimate the azimuth angles of departure or arrival (AoDs/AoAs) and joint gain at the BS. Specifically, the proposed algorithm does not need to estimate the channel on the user side, and all channel parameters can be estimated at the BS, which is different from the traditional channel estimation scheme.
2. We combine the advantages of the Newton method and gradient descent method and derive an iterative step-length optimization approach to reduce computational complexity. It is confirmed that the proposed scheme has low complexity without losing the accuracy of channel estimation.
3. Compared with the traditional singular-value decomposition (SVD) algorithm and BCS algorithm under different situations, the simulation results show that the proposed algorithm achieves better channel estimation performance with fewer training data blocks.
The rest of this paper is arranged as follows.
Section 2 introduces the considered multi-user mmWave massive MIMO model.
Section 3 formulates the proposed joint iterative channel estimation algorithm, which accounts for the preprocessing, the choice of descent direction in the iteration, the optimization of the step-length, and the method for separating the downlink channel. Some simulation results are provided to illustrate the performance of estimation in
Section 4, and
Section 5 concludes this paper.
Notation: In this paper, vectors and matrices are denoted by boldface lowercase and uppercase symbols, respectively; , , , and correspond to the transpose, conjugate transpose, inverse, and Moore–Penrose pseudo-inverse of the matrix ; is the vector stacked by the columns of the matrix ; ⊗ is the Kronecker product; denotes the diagonal matrix with the vector on its diagonal; is the rank of ; , , and represent the Frobenius norm of , the -norm, and Euclidean norm of the vector , respectively; is the identity matrix; the sets of real and complex numbers are represented by and , respectively.
2. System Model
We considered a common mmWave massive MIMO system with hybrid precoding as illustrated in
Figure 1, where the number of antennas at the BS is
; but, only
RF chains with
to support
K user equipment (UE), and each UE has a single receive antenna [
18,
28]. In the communication process, it is assumed that there is only one transmission path via
data streams between BS and each UE.
Let
and
be the time slot of the transmitter and receiver, respectively. The BS pilots
represent the signal transmitted in the
time slot, where
L is the number of different pilot sequences, and its design meets the orthogonality condition
[
29,
30]. Then, the signal
received by the
user can be given by:
where
is the transmitted hybrid precoding matrix in the
time slot,
is the baseband beamformer followed by the RF beamformer
,
is the received independent and identically distributed additive white Gaussian noise elements having zero mean and the variance
, and
is the downlink channel vector, and by defining
as the normalized spatial angle [
31], it can be expressed as:
where
,
d, and
denote the propagation gain of the
downlink path and the antenna spacing at the BS and carrier wavelength, respectively.
is widely adopted here [
32].
is the AoD of the
path.
is the steering vector at the transmitter of the
path. In this work, we considered the typical uniform linear arrays (ULAs) [
33];
is given by:
Denote
where
is the AoA of the
path.
is the normalized spatial angle the same as
. The uplink channel vector
can be expressed as:
where
is the propagation gain of the
uplink path and
is the steering vector at the receiver of the
path. For typical ULAs,
is given by:
To reduce the overhead of downlink channel training and uplink channel feedback, we propose a joint estimation scheme, that is the user does not process the received signal and directly feeds it back to the BS [
24]. For all
K users, the comprehensive signal
received by the BS is:
where
is the number of complex paths,
is the received hybrid combining matrix in the
time slot,
and
are baseband combiners, and
is the noise received by the BS after combining. By defining
and synthesizing all
K users, we can further obtain:
where
,
,
, the steering matrix
and
are given by:
We further process the signal
in (
7) to make it multiply right by
; we have:
Next, we start to collect and arrange the signal
in (
10). By collecting the pilots in the
time slots of the transmitter, we can get:
where
,
. By dealing with the signal
in the
time slots of the receiver, we have after arranging:
where
and
are expressed as:
Now, the signal received by the BS in (
10) is written in a more compact form as:
where
; the combined channel matrix
can be written as:
3. Joint Iterative Channel Estimation Scheme
In this section, we firstly transform the above channel estimation problem into a new algorithm form. Then, we adopt the SVD preconditioning proposal to reduce the computational complexity. To improve algorithm performance, we give an angle estimation optimization scheme by optimizing the step-length and using a combination of the gradient descent algorithm and Newton method. Finally, we propose a scheme for successfully separating the downlink channel from the combined channel matrix.
3.1. Algorithm Formulation
The estimation of the combined channel matrix
in (
15) is equivalent to the estimation of combined path gain
and the two normalized spatial angles
and
for all paths. Due to the sparseness of the combined channel matrix
in the angle domain [
34], such a problem can be easily expressed as:
where
is the number of non-zero components of
, which also represents the number of estimated paths
K, and
is the error tolerance parameter related to noise.
Considering the low computational efficiency of finding the optimal solution in (
16), we propose to replace the
-norm with the use of the logarithmic sum function [
25,
35]:
where
is the
element of the gain
and
is a positive parameter to ensure that the definition of the log-sum function is valid. By adding a data fitting parameter
, the difficulty in (
17) can be formulated as an unconstrained optimization problem, which yields the following optimization:
Due to the need to ensure the monotonically decreasing characteristic, we introduce a suitable iterative function instead of the log-sum function [
26,
36]:
where
t is the number of iterations and
is a diagonal matrix, which is defined as:
where
,
is the path gain estimate of the
user at the
iteration. The minimization of the log-sum function
in (
18) is equivalent to the minimization of the surrogate function
in (
19), which is proven as follows. To get better estimates
,
,
in the
iteration and ensure the convergence of the function, the following inequality can be shown:
Since the maximum value is reached at
, we have:
Combining (
21) and (
22), we can finally get:
which explains the correctness of the above assumption.
By optimizing the path gain
in (
19) through the derivative method [
26], we can obtain the optimal point of
and the corresponding optimal value of
, that is,
where
is the column vector of the signal
received by the BS,
, and
is the column vector of the hybrid precoding matrix
. After that, we only need to consider the optimization of the normalized spatial angles
and
in (
26), which will be discussed in the next subsection.
3.2. Initial Angle Preprocessing
To optimize the spatial angle, we propose a preconditioning algorithm based on SVD [
37,
38] to select the initial value of the spatial angle effectively. Compared with using all
angle domain grids as the initial candidate value, the proposed algorithm can rapidly find the angle domain grids closest to the true orientation angle, which can significantly reduce the computational complexity of the subsequent algorithm. Specifically, we apply SVD to the matrix
in (
14) and get
, where
and
are unitary matrices that satisfy
and
, respectively,
is a diagonal matrix whose value on the diagonal is the singular value of
, and
. The signal is expressed in a more compact form as:
Suppose
the larger the
K singular value and their homologous singular vectors are roughly determined by
K paths, we can infer:
where
and
are the
column of
and
, respectively; the singular value can be written as:
Then, the spatial angles are normalized by
and assumed to lie in the quantized points; we have:
where
and
are the discrete Fourier transform matrices; we can further obtain the coarse estimates of angle, which can be formulated as:
where:
At this point, we get the initial candidates for the spatial angle. Furthermore, due to the uncertainty of the number of paths of the estimated channel at the beginning, we set , to ensure the accuracy of our estimated path, and some incorrect paths will be cut off in the proposed algorithm. The iterative search scheme of angles is explained in the next subsection.
3.3. Step-Length Optimization
In the previous section, it can be seen that the objective function
in (
19) is the weighted sum of two parts, where
is the regularization parameter [
39] that affects the tradeoff to some extent. In the proposed algorithm,
is simplified as:
where
e is a constant scale factor, and the squared residual at the
iteration
is expressed as:
The main task is to search for new estimates
and
from the neighborhood of
and
at the
iteration, so that the objective function
becomes smaller and eventually stabilizes. Considering the characteristics of the fast convergence speed of the Newton descent method and the high accuracy of the steepest descent method, we try to suggest combining the two for better performance; this searching can be accomplished by:
where
is the step-length at the
iteration and
is the corresponding descent direction. To balance the high computational complexity of the Newton iterative algorithm, we consider using the Newton direction as the descending direction to achieve fast convergence in the first iteration, that is denote
when
; the expression
is:
After that, we choose the gradient direction to achieve high accuracy and set
; the expression
is:
it can be seen that the search direction involves the gradient calculation of the iterative function
. Taking the normalized angle
as an example, the calculation method is explained as follows.
Define
,
, then
can be expressed as:
Take the partial derivative with respect to
, we can obtain:
and
; we have:
where:
Since there is only a certain transposition relationship between the gradient and the partial derivative, we have the gradient of
, and we can further obtain the second derivative:
where:
Next, we think about whether we can optimize the iteration step-length in (
35) to improve the speed of iteration. Considering that if the optimal step-length of
and
is directly solved, the algorithm complexity is very high, and the channel is only related to the path gain
and the spatial angles
and
, so when the gain has been optimized, we choose to use the optimal step-length of the channel to approximate the optimal step-length of the angles. The optimization method is as follows.
Denote
,
, and given the spatial angles
and
at the
iteration, we have:
Then, the cost function transformed from (
34) is:
Therefore, such an optimization can be reformulated as:
The key to this problem is how to get the suitable
; define
, and take the derivative with respect to
; we can get:
where:
(
52) can be simplified as:
Considering the complexity of the algorithm, the descent direction here is the gradient direction. Set
; we can finally get:
From this, we get the approximate optimal step-length for the angle update. During the iterative searching, the spatial angle and path gain estimates can converge quickly, until the latest estimates are almost the same as the previous ones. The proposed channel estimation scheme is shown in Algorithm 1. We can achieve the purpose of accurate and fast channel estimation, which will be verified in subsequent simulations.
Since the Newton direction is used only once, the computational complexity of each iteration mainly lies in the calculation of the gradient direction and the optimization of the step-length. The complexity to compute the gradient direction is , and the complexity of the step calculation is . As a result, we can conclude that the proposed algorithm has the complexity .
In the above algorithm, we successfully estimated the space angle , and the combined gain , where . Next, the solution to the problem of how to separate the downlink gain from the combined gain is revealed.
Specifically, we adopt the unit symbol pilot
sent by the UEs to the BS via the uplink channel, and the signal
received by the BS is written as:
where
; when
, we have
, where
, we obtain uplink gain
, and the downlink gain
can be separated by utilizing
. Since
can be easily designed as a matrix whose columns are full rank, the above conditions can be achieved.
Algorithm 1: The optimized iterative algorithm. |
Input: Received signals with noise , orthogonal pilot signals , coding matrix and , the number of detection paths , threshold and . Output: Estimated angles and gains of all paths. First stage • Perform ; • Compute , ; • Initialize , ; • Estimate gain by ( 25), and trim path if . Second stage 1. Repeat 2. Update ; 3. Build the function by ( 26); 4. If , then 5. , ; 6. Else 7. , ; 8. End if 9. Step optimization calculation: 9.1 , ; 9.2 Compute ; 9.3 Compute . 10. , ; 11. Estimate the path gain , and streamline path if ; 12. Until , and . |
4. Simulation Results
In this section, we demonstrate the simulation comparisons under different parameter scenarios, which can prove the effectiveness of the proposed step-length optimization-based joint iterative scheme. In particular, simulation results are provided to verify that the proposed algorithm can outperform the previous methods in certain aspects, as in [
26,
27]. Some default parameters are set to
,
, and
. Moreover, the number of antennas, UEs, and time slots is set to
,
, and
, respectively, which may change in the form of variables in subsequent simulations, and we set the variable
to represent the time slot for better presentation. As in [
2,
40], the signal-to-noise ratio (SNR) is defined as
, where
is the signal power.
I denotes the number of Monte Carlo runs, and the normalized mean squared error (NMSE) is defined as:
Figure 2 shows the convergence performance of the optimized scheme compared with the conventional super-resolution (SR) algorithm, which is reflected by the relationship between NMSE and the number of iterations for three SNR values. It is obvious that the proposed scheme can achieve convergence with a small number of iterations for each SNR value. For instance, the proposed algorithm completes convergence in about 20 iterations at an SNR of 20 dB, which also fully proves the acceleration brought by optimization.
Figure 3 compares the NMSE performance against the SNR. We assume the system design parameters
and
, as shown in
Figure 3, and the accuracy of the proposed solution is similar to that of the SR algorithm in different time slots, which proves that the proposed optimization does not sacrifice performance indicators. We can see that there is a certain proportional relationship between the time slot and the training pilot, so the NMSE performance becomes better as the number of time slots increases. In addition, the NMSE of the proposed and SR schemes is also shown in
Table 1 for further comparison. It is clearly observed that accurate channel estimation can be achieved.
Figure 4 depicts the iteration time of the proposed algorithm on the basis of the hypothetical scenario in
Figure 3, which can more intuitively reflect that the speed of the proposed scheme is significantly improved and the computational complexity is reduced. For example, when the SNR is 20 and the time slot is 10, the time taken by the proposed algorithm is 2.0534 s and the SR is 5.0885 s; it can be fully seen that our method achieves fast estimation. We can also see that as the SNR and time slot increase, the time consumed increase, and that is because the accuracy is also improved.
Figure 5 displays the NMSE performance comparison with BCS in the case of different numbers of UEs. Since it optimizes all the spatialization angles simultaneously and eliminates some false paths, the proposed scheme can obtain better NMSE performance, and the processing time of the BCS method is significantly higher than the proposed scheme under the same conditions. In addition, we can see that when the number of users increases, the channel estimation accuracy of the proposed algorithm decreases; but the decrease is small, and the proposed algorithm can still effectively estimate the channel.
Figure 6 compares the angular mean squared error (RMSE) performance under various UEs and antennas. The RMSE is given by:
As SNR increases, we observe that the RMSE of SVD remains constant, and its computational complexity is lower; but our algorithm yields the best performance. This is because the SNR is related to the entire angle domain grids, and our SVD method finds the best matching grid to get coarse estimates. Similarly to
Figure 5, the RMSE performance of channel estimation will decrease with the increase of users, and in contrast, the number of antennas is directly proportional to the estimated accuracy.
Figure 7 shows the performance of the gain mean squared error (GMSE) of the proposed algorithm in different time slots, and GMSE is expressed as
. Obviously, the recovery of the channel can gradually improve with the increase of time slots and SNR.