Lagrangian dual relaxation (LDR) detector

The archive provided in this page contains ready-to-use binaries and MATLAB functions for the LDR detector. The binaries and functions can be freely distributed for academic or personal use. Please contact the authors if you intend to employ the binaries or functions in the archive for commercial purpose.

Source Code

The LDR detector is written using C MEX function and MATLAB.

Source codes (last updated on 25-11-2013)

Lagrangian dual relaxation detector

The Lagrangian dual relaxation (LDR) detector considers maximum-likelihood (ML) MIMO detection with PAM constellations

$begin{aligned} min_{mathbf s in 2mathbb Z^n + mathbf 1 } quad &|mathbf y - mathbf H mathbf s |_2^2 text{s.t.} quad & s_i^2 leq u^2, ~~i=1,hdots,n, end{aligned}$

where mathbf y in mathbb R^m , $mathbf H in mathbb R^{m times n}$ , and is the symbol bound. For example, we have $s_i in {pm 1}$ for u=1 , $s_i in {pm 1, pm 3}$ for u=3 , and $s_i in {pm 1, pm 3, pm 5, pm 7}$ for u=7 . Note that ML MIMO detection with QAM constellations can also be recast as the above optimization problem.

The LDR detector tackles the ML problem by its Lagrangian dual problem

$max_{bm lambda in mathbb R^n} min_{{mathbf s} in 2 mathbb Z^n + mathbf 1} | {mathbf y} - {mathbf H}{mathbf s} |^2 + mathbf s^T mathbf D(bm lambda) mathbf s - u^2 bm lambda^T mathbf 1,$

where bm lambda in mathbb R^n is the dual variable associated with the bound constraint, and mathbf D(bm lambda) is the diagonal operator. The Lagrangian dual problem is handled by the projected subgradient (PS) method

$bm lambda^{(k+1)}= [bm lambda^{(k)}+ alpha_k mathbf g^{(k)}]^+,$

where [cdot]^+ is the projection on to the nonnegative orthant, ${alpha_k}$ is a predefined stepsize sequence, and $mathbf g^{(k)}= (mathbf s^{(k)})^2 - u^2 bm 1$ with

$mathbf s^{(k)}= argmin_{{mathbf s} in 2 mathbb Z^n + mathbf 1} | {mathbf y} - {mathbf H}{mathbf s} |^2 + mathbf s^T mathbf D(bm lambda^{(k)}) mathbf s - u^2 { (bm lambda^{(k)}})^T mathbf 1.$

The above problem is an integer least-squares problem, which can be solved optimally by a lattice decoder (LD). Alternatively, we can adopt inexact LD and lattice reduction aided (LRA) methods for efficient approximations.

For more detailed description of the LDR detector, please see the following paper

Jiaxian Pan, Wing-Kin Ma and Joakim Jaldén “MIMO Detection by Lagrangian dual maximum-likelihood relaxation: Reinterpreting regularized lattice decoding,” IEEE Trans. Signal Process., vol. 62, no. 2, pp. 511-524, Jan. 2014. [PDF]

Simulations

We demonstrate the performance of the SDR detector in problem size m=n=32 .