1. Introduction
Massive multiple-input multiple-output (MIMO) technology dramatically expands the capacity of wireless communication systems without increasing the system bandwidth and transmit power and effectively resolves the contradiction between the limited spectrum resource and the rapid growth in capacity demand. Therefore, it has become one of the most promising solutions of 5G systems [
1,
2,
3,
4]. Equipped with up to hundreds of antennas at the base station (BS), massive MIMO system simultaneously serves multiple single-antenna users in the network, resulting in an order of magnitude improvement in spectrum utilization and energy efficiency of wireless systems [
5,
6].
Apart from the salient technical merits, massive MIMO system encounters various challenging problems in practice, one of which is the multi-user signal detection in the uplink. Due to the large number of antennas, the multi-user interference is remarkably intensified in system uplink and the implementation complexity is highly accrued, compared with the conventional MIMO systems. Theoretically, the maximum likelihood (ML) algorithm serves as the optimal solution for signal detection in MIMO systems and it becomes stringently burdensome to be implemented effectively in practical applications. Since the computational complexity of ML algorithm rises exponentially with the increase of the number of antennas and the modulation order of the baseband signal [
6,
7], it is actually infeasible to be employed in massive MIMO systems. When the reduced or fixed complexity is considered as the first priority design objective, the tabu search (TS) algorithm [
8] and the fixed-complexity sphere decoding (FSD) algorithms [
9] were proposed to obtain the close optimal ML detection performance, but their complexity still becomes not practically affordable for a large scale configuration of MIMO system with high modulation order.
Benefiting from the large number of antennas in massive MIMO systems, linear signal detection algorithms, such as zero forcing (ZF) and minimum mean square error (MMSE) algorithms, have been shown and verified to achieve nearly optimal detection performance, at the cost of involving high-dimensional matrix inversion with high complexity [
6] (
, where
K is the number of users simultaneously transmitting over the uplink). In recent years, various low complexity signal detection algorithms based on the MMSE criterion have been proposed for massive MIMO systems in the literature. In our previous work, we have investigated a variety of low complexity signal detection algorithms for massive MIMO systems under the MMSE criterion-based signal detection, where the key idea of achieving simplified complexity in signal detection is to find a solution that manages to evade the high-dimension matrix inverse operation, and a comparative study has been presented [
10].
To the authors’ best knowledge, the MMSE criterion-based low complexity signal detection algorithms can be basically categorized into three typical types, namely the approximate matrix inversion algorithms (AMIA), the iterative approaches for solving linear equations (IASLE), and the matrix gradient search methods (MGSM). Firstly, the AMIA algorithms deal with the matrix inversion operation, required by the MMSE signal detection, in an approximation manner where the Neumann series expansion and Newton iteration are evoked to estimate the matrix inversion approximately [
11,
12,
13]. The approximation accuracy depends on the number of Neumann items or the Newton iterations and may result in a high complexity when the number of items or iterations is set large for achieving a satisfactory performance. Secondly, in an entirely different mechanism, the IASLE tackle the problem of matrix inversion by finding solution to the system equation [
14,
15,
16,
17], where the transmitted multi-user signal vector is directly estimated and thus the high dimensional matrix inversion is purposely circumvented. Thirdly, based on the same idea of the IASLE, the MGSM methods are proposed to acquire the equation solution by matrix gradient search and hence the direct matrix inversion operations are bypassed, saving a lot of computations [
18,
19]. From the perspective of system performance, the AMIA algorithms are usually inferior to the IASLE and MGSM algorithms and when the number of items or iterations is large, the algorithm complexity is approaching
again. As for the IASLE and MGSM algorithms, in case that some special properties of the weighting matrix is not guaranteed, for instance, if the weighting matrix is not symmetric positive and strictly diagonal dominant, they may encounter serious performance degradation or even fail to operate properly. Drawbacks of these algorithms need to be overcome by means of finding new type of algorithms. The aforementioned typical MMSE criterion-based multi-user signal detection algorithms are compared in
Table 1.
In this paper, in order to obtain an easy-to-implement multi-user signal detection scheme for the uplink massive MIMO system, we propose a soft decision algorithm based on the Kaczmarz iteration [
20,
21,
22]. In our previous work, the Kaczmarz algorithm was proposed to serve as a matrix-inverse approximation method for implementing the MMSE criterion-based signal detection with reduced complexity [
22]. By circumventing the high-dimensional matrix inversion computations, we effectuated a simplified detection scheme for acquiring the transmitted signal vector in linear equation solving manner. To further improve the system performance and accelerate the converging speed in iterations, an optimal relaxation parameter is introduced to accelerate the convergence of the proposed Kaczmarz algorithm. To be more specific, the proposed improved Kaczmarz algorithm falls in one category of the low complexity MMSE criterion-based signal detection algorithms, that is, the IASLE, and it is actually a combination of the iterative approach for solving linear equations and the conventional Kaczmarz algorithm. Based on the output of the proposed improved Kaczmarz algorithm, theoretical log-likelihood ratios (LLRs) of the user bit streams are derived and one approximate method of estimating the LLRs for channel decoding is presented as well. Simulation results verify that the proposed algorithm outperforms the typical algorithms mentioned in
Table 1 in terms of bit error rate (BER) with significantly relieved computational complexity. In comparison with the Kaczmarz algorithm, the proposed improved Kaczmarz algorithm yields a much better BER performance, given the same number of iterations. Additionally, the proposed improved Kaczmarz algorithm converges rapidly in operation and achieves its performance quite close to that of the MMSE algorithm with only a small number of iterations.
The rest of this paper is organized as follows. In
Section 2, we describe the system model. In
Section 3, the Kaczmarz iteration based signal detection is introduced. In
Section 4, we propose the improved Kaczmarz algorithm based soft output signal detection. In
Section 5, simulation and analysis are presented. Finally,
Section 6 concludes the paper.
Notation: Lower-case and upper-case boldface letters are used to represent column vectors and matrices, respectively. The superscripts , , and stand for the transpose, conjugate-transpose, and inverse of matrix, separately. The operator , , and denote the vector/matrix norm, the statistical expectation of a given argument, and the inner product of two vectors, respectively. is the K dimensional unit diagonal matrix.
2. System Model
An uplink massive MIMO system is considered, where the BS is equipped with N antennas and totally K single-antenna users are located within the coverage area of the BS (). The bit stream of each user is first encoded using a channel encoder and then mapped to the constellation points in the set . The constellation symbol vector , , contains the transmit signals from the K users and it is assumed that , where represents the average power of the user signals.
Based on the system configuration, the received signal at the BS can be expressed as:
where
is the received signal vector at the BS,
is the white Gaussian noise vector with zero mean and variance
for each entry, and
is the
channel matrix with its entry
denoting the channel coefficient between the
k-th user and the
n-th BS antenna.
2.1. MMSE Detection
If linear detection is utilized under the MMSE criterion, an estimate of the transmitted signal,
, at the BS is expressed as:
where
indicates the weighting matrix,
is the matched filter output, and
is the
Gram matrix.
2.2. LLRs Generation
Based on the MMSE weighting, the transmitted signal can be presented as:
where
represents the channel matrix after equalization.
The received signal of the
k-th user can be expressed as:
where
is the equivalent channel gain after equalization and
denotes the noise-plus interference (NPI) for the
k-th user, with its variance calculated as:
where
is the
k-th diagonal element of
, and
.
The LLR of the
b-th bit of the
k-th user’s symbol is hence obtained as:
where the coefficient
is equivalently the signal-to-interference-plus-noise ratio (SINR) for the
k-th user,
and
are the symbol subsets of
where the
b-th bit of the constellation symbols is 0 and 1, respectively.
From the above analysis, it is easy to observe that the calculations of , , and all require computation of first, leading to a high complexity as . In order to circumvent the high complexity computations in matrix inversion, a low complexity MMSE soft detection scheme is proposed in this paper.
3. Kaczmarz Iteration Based Signal Detection
Since the IASLE methods usually outperform the AMIA algorithms, we also consider another typical IASLE algorithm, known as the Kaczmarz algorithm, in handling the signal detection task as the one of system equation solving. The Kaczmarz algorithm is widely used in various fields, among which it is also known as the algebraic reconstruction technique (ART) [
23] in computed tomography. It provides an iterative method for solving the large scale over-determined linear equation
, where
is an
matrix
,
represents a
vector to be determined, and
is an
measurement vector. In the iterative process of the Kaczmarz algorithm,
, the
k-th row of the matrix
, is traversed in a periodic manner. In each step, the solution of the last inner iteration is
within the
t-th outer iteration and it is orthogonally projected, as
, onto the hyperplane associated with the row vector
. Given an initial solution
for solving
, the
t-th iteration’s solution of the Kaczmarz algorithm can be expressed as:
where
t and
k respectively represent the index of outer and inner iterations,
is the predetermined largest number of iterations, and
is the vector norm of
.
Applying the Kaczmarz algorithm to detecting the transmitted signal
in the linear equation
for massive MIMO systems [
22], we can obtain an estimate of the transmitted signal as:
where
is the
k-th row vector of
and the initial solution is denoted as
that is usually set as a zero vector. Details of the Kaczmarz algorithm based signal detection are given in Algorithm 1.
Algorithm 1 Kaczmarz algorithm based signal detection |
Input:, , , . |
Output:. |
1: , , |
2: , , , |
3: for do |
4: for do |
5: |
6: , |
7: end for |
8: . |
9: end for |
4. Improved Kaczmarz Algorithm Based Soft Output Signal Detection
As a non-optimal solution compared to the MMSE signal detection, the performance of the Kaczmarz algorithm can be ameliorated. We propose an improved Kaczmarz algorithm, utilizing a traversal scheme based on norm ordering to enhance the signal detection performance. Specifically, a traversal scheme based on norm ordering consists of the following three steps.
:
Norm ordering. The norms of
are sorted in descending order, resulting in the corresponding subscript set
permutated from the index set
, with
, and then the entry in the subscript set
is sequentially selected for traversing. Equation (
8) can be expressed as:
:
Introducing the relaxation parameter in iterations. An optimal relaxation parameter is introduced to accelerate convergence. Equation (
9) can be modified as:
where
denotes the relaxation parameter for the
-th iteration.
:
Finding the optimal relaxation parameter. The convergence speed of the Kaczmarz Algorithm merely depends on the condition number of
and the algorithm converges for any
t and
k when
is set as a constant
[
24]. Equation (
10) can be simply expressed as:
Usually, the relaxation parameter
is set as 1 for simplicity, as shown in Equation (
9). However, it is heuristically found in experiments that the optimal relaxation parameter
can be set as
, where
and
is the minimum and maximum eigenvalue of
, respectively.
In massive MIMO uplink system, as the number of BS antennas and the number of users increase, the eigenvalues of
obey the Marchenko-Pastur distribution, where
and
asymptotically converge as [
2]:
where
. The optimal relaxation parameter, as a function of
, can be therefore determined as:
where it is easy to observe the fact that
. Based on the above analysis, we further demonstrate via simulations that the optimal relaxation parameter
exists in a very narrow range and it can be set as a constant determined by the system configuration only. Detailed description of the improved Kaczmarz algorithm based soft output signal detection is presented in Algorithm 2.
Algorithm 2 Improved Kaczmarz algorithm based soft output signal detection |
Input:, , , . |
Output:. |
1:% Initialization |
2: , , |
3: , |
4: , |
5: % Setting the optimal relaxation parameter |
6: , |
7: sort in descending order, obtaining the subscript set , , |
8: % Solving the system equation in iterations |
9: for do |
10: for do |
11: |
12: |
13:
|
14: , |
15: end for |
16: . |
17: end for |
18: % Computing the approximate LLRs |
19:
|
20: , |
21: , |
22:
|
23:
|
4.1. Initial Estimation
For massive MIMO systems, the columns of
are asymptotically orthogonal, meaning that
is positive definite and diagonally dominant. According to the channel hardening phenomenon [
25], there is
. This special property enables us to utilize
to approximate
with trivial error and the initial solution of Equation (
11) can be set as:
where
is the diagonal matrix corresponding to
.
4.2. Two Methods to Compute LLRs
4.2.1. Exact Method
Through iterative operation, the solution vector
gradually converges to the solution for the equation
, and hence there should be a corresponding matrix
for each iteration. The most straightforward method of computing LLR is to use the Kaczmarz algorithm to estimate
after each iteration. Combining Equations (
2) and (
11), an estimate of the transmit vector at each iteration can then be computed as:
where
represents the
-th unit row vector that only has the
-th element being non-zero.
For the
-th element in
, we have
where the corresponding
-th row vector of
can be expressed as:
where
is the
-th diagonal element of
. Therefore,
can be obtained from the above discussions, with an initial solution
. The corresponding channel gain and NPI variance can be derived as:
where
and
Then, the LLRs for channel decoding can be obtained by substituting Equations (
18) and (
19) into Equation (
6).
4.2.2. Approximated Method
The exact method precisely computes the LLRs and yields theoretically optimal BER performance. However, from Equation (
17), updating
involves multiplication and addition between the matrix and the vector for each iteration, causing the final complexity order to rise to
again. To solve this problem, an approximated method to calculate the LLRs is proposed, which completely avoids the complicated matrix inversion. Since
is diagonal dominant for uplink massive MIMO systems, it can be replaced by the diagonal matrix
with tolerable error. Then, the approximated channel gain and NPI variance are obtained in a non-iterative manner as:
where
,
Then, the LLRs for channel decoding can be obtained by substituting Equations (
20) and (
21) into Equation (
6) with much lower complexity.