Adaptive Preconditioned Iterative Linear Detection and Architecture for Massive MU-MIMO Uplink

Abstract

Being the enabling technique for 5G wireless communications, massive multiple-input multiple-output (MIMO) system can drastically increase the capacity efficiency. However, a few hundreds of antennas will inevitably introduce notable complexity and therefore hinders its direct adoption. Though the state-of-the-art (SOA) iterative methods such as conjugate gradient (CG) detection show complexity advantage over the conventional ones such as MMSE detection, their convergence rates slow down if the antenna configurations become more complicated. To this end, first this paper devotes itself in exploring the convergence properties of iterative linear solvers and then leverages the proposed adaptive precondition technique to improve the convergence rate. This adaptive precondition technique is general and has been incorporated with steepest descent (SD) detection as a show case. An approximated calculation for log-likelihood ratios (LLRs) is proposed for further complexity reduction. Analytical and numerical results have shown that with the same iteration number, the adaptive preconditioned SD (APSD) detector outperforms the CG one around 1 dB when BER = 10⁻³. Hardware architecture for the APSD detector is proposed based on iteration bound analysis and architectural optimization for the first time. Architectures for other adaptive preconditioned iterative linear detectors can be easily derived by following similar design flow. Compared with the SOA designs, FPGA implementations have verified the APSD detector’s advantage in balancing throughput and complexity, and guaranteed its application feasibility for 5G wireless.

Appendices

Appendix A: Proof of Corollary 1

The SCN of a given matrix A, which is defined as κ(A), is the ratio of the largest and smallest eigenvalues. Since the MMSE equalization matrix A can be decomposed as:

$$ \begin{array}{ll} \mathbf{A} &= \left( \mathbf{H}^{H}\mathbf{H}+N_{0}E_{s}^{-1}\mathbf{I}\right)\\ &=\mathbf{U}\mathbf{\Lambda}\mathbf{ U}^{H}+N_{0}E_{s}^{-1}\mathbf{U}\mathbf{I}\mathbf{U}^{H}\\ &=\mathbf{U}(\mathbf{\Lambda}+N_{0}E_{s}^{-1}\mathbf{I})\mathbf{U}^{H}, \end{array} $$

(22)

where U is the unitary matrix and Λ is the diagonal matrix with eigenvalues as its entries.

To obtain the abstract of the distribution of κ(A), the joint probability density function (PDF) of the ordered eigenvalues of A is required. Fortunately, for each eigenvalue of matrix A, we have

$$ \lambda\mathbf{A}=\lambda\mathbf{H}^{H}\mathbf{H}+N_{0}E_{s}^{-1}. $$

(23)

Here, H ^H H is a complex central Wishart distribution with B degree of freedom and U eigenvalues. The joint PDF of the ordered eigenvalues of H ^H H for Cases 1, 2, and 3 in Section 2.1 is given by [59] and the corresponding analysis of the SCN distribution is given by [55]. Thus, the joint PDF of the ordered eigenvalues, $\mathbf {\lambda }\triangleq [\lambda _{1}, \lambda _{2}\ldots \lambda _{U}]$ with λ ₁ ≥ λ ₂ ≥… ≥ λ _U ≥ 0 of A is

$$ f_{\mathbf{\lambda}\mathbf{A}}(\mathbf{\lambda})=f_{\mathbf{\lambda}\mathbf{H}^{H}\mathbf{H}}(\mathbf{\lambda})|\frac{d({\mathbf{\lambda}\mathbf{H}^{H}\mathbf{H}})}{d({\mathbf{\lambda}\mathbf{A}})}|. $$

(24)

According to Eq. 23, we have

$$\begin{array}{@{}rcl@{}} f_{\mathbf{\lambda}\mathbf{A}}(\mathbf{\lambda})=f_{\mathbf{\lambda}\mathbf{H}^{H}\mathbf{H}}(\mathbf{\lambda}-N_{0}E_{s}^{-1}). \end{array} $$

(25)

This indicates that λ A and λ H ^H H have the same distribution when only the influence of system loading factor α and channel correlation coefficient ζ are considered. Finally, Corollary 1 can be proved by the result in [55] (See Statement 1 and 2 in Section 3.1).

Appendix B: Derivation of APSD Detection Algorithm

In classical SD method, $\hat {\mathbf {s}}$ is iteratively calculated [54] by:

$$ \hat{\mathbf{s}}^{(j)}=\hat{\mathbf{s}}^{(j-1)}+\alpha^{(j-1)} \mathbf{z}^{(j-1)}. $$

(26)

Since z ^(j− 1) is the iterative residual, −z ^(j− 1) is the iterative research direction and α ^(j− 1) is the iterative research length in the (j − 1)-th iteration.

In the proposed APSD algorithm, the system is preconditioned from $\mathbf {A}\hat {\mathbf {s}}=\tilde {\mathbf {y}}$ to $\mathbf {(LDL^H)}^{-1}\mathbf {A}\hat {\mathbf {s}}=\mathbf {(LDL^H)}^{-1}\tilde {\mathbf {y}}$. The iterative residual in the j-th iteration changes from $\tilde {\mathbf y}^{(j)}$ in

$$ \hat{\mathbf{s}}^{(j)} = \hat{\mathbf{s}}^{(j -1)} + \frac{\left( \tilde{\mathbf y}^{(j -1)} \cdot \tilde{\mathbf y}^{(j -1)} \right)} {\left( \mathbf{A} \tilde{\mathbf y}^{(j -1)} \cdot \tilde{\mathbf y}^{(j -1)} \right)} \tilde{\mathbf y}^{(j -1)} $$

(27)

to

$$ \begin{array}{ll} \mathbf{z}^{(j)}&=\mathbf{L}^{-1}\mathbf{D}^{-1}\mathbf{L}^{-H}\mathbf A\hat{\mathbf{s}}^{(j)} -\mathbf{L}^{-1}\mathbf{D}^{-1}\mathbf{L}^{-H}\tilde{\mathbf{y}}^{(0)}\\ &=\mathbf{L}^{-1}\mathbf{D}^{-1}\mathbf{L}^{-H}\tilde{\mathbf{y}}^{(j)}, \end{array} $$

(28)

where

$$ \tilde{\mathbf{y}}^{(j)}= {\tilde{\mathbf{y}}}^{(0)}-\mathbf{A}\mathbf{s}^{(j)}. $$

(29)

Substitute s ^(j) to Eq. 26, then $\tilde {\mathbf {y}}^{(j)}$ is calculated by:

$$ \tilde{\mathbf{y}}^{(j)}=\tilde{\mathbf{y}}^{(j-1)}-\alpha^{(j-1)} \mathbf A\mathbf{z}^{(j-1)}. $$

(30)

To derive α ^(j− 1), we consider the basic principle of SD method that, the set of search direction z should satisfy

$$ \mathbf{z}^{(k)T}\mathbf{z}^{(j)}= 0 (k\neq j). $$

(31)

Then α ^(j− 1) is obtained based on Eqs. 28, 30, and 31,

$$ \alpha^{(j-1)}=\frac{(\mathbf{z}^{(j-1)}, \mathbf{y}^{(j-1)})}{(\mathbf{A}\mathbf{z}^{(j-1)}, \mathbf{z}^{(j-1)})}. $$

(32)

Summarize Eqs. 26, 28, 29, and 32, the proposed algorithm is obtained

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{llll} \hat{\mathbf{s}}^{(j)} = & \hat{\mathbf{s}}^{(j-1)} + \frac{\left( \mathbf{z}^{(j - 1)} \cdot \tilde{\mathbf{y}}^{(j-1)}\right)}{\left( \mathbf A \mathbf{z}^{(j - 1)} \cdot \mathbf{z}^{(j - 1)} \right)} \mathbf{z}^{(j-1)},\\ \tilde{\mathbf{y}}^{(j)} = & \tilde{\mathbf{y}}^{(j-1)} - \frac{\left( \mathbf{z}^{(j - 1)} \cdot \tilde{\mathbf{y}}^{(j-1)}\right)}{\left( \mathbf A \mathbf{z}^{(j - 1)} \cdot \mathbf{z}^{(j - 1)} \right)} \mathbf{A} \mathbf{z}^{(j-1)}, \\ \mathbf{z}^{(j)} =& (\mathbf{L}\mathbf{D}\mathbf{L}^{H})^{-1} \tilde{\mathbf{y}}^{(j)}. \end{array}\right. \end{array} $$

(33)

Combining with Algorith1, we can obtain Algorithm 2.