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We develop a canonical dual approach for solving the MIMO problem. First, a special linear transformation is introduced to reformulate the original problem into a

In the recent decade, multiple antennae communication systems have developed very fast since it could provide substantial performance gain over their single antenna counterparts [

Note that, in the communication scenarios, the signal model is always written in the following form:

Lattice decoding is an important research direction for the MIMO detection. It has received a lot of attention for its good tradeoff between detection accuracy and complexity [

Another big family of MIMO detection algorithms is based on semidefinite relaxation (SDR). The SDR method relaxes the ML detection problem into a convex semidefinite programming (SDP) problem which leads to a polynomial-time complexity in the problem dimension. The SDR detector was first developed for the binary phase-shift keying (BPSK) constellation [

Besides, there are some other algorithms for the MIMO problem. Sphere decoder method is a classical one [

In this paper, we present a canonical duality approach to the MIMO problem. The canonical duality theory is originally proposed for handling general nonconvex and/or nonsmooth systems [

The paper is arranged as follows. In Section

The MIMO problem can be written as follows:

Let

The transformation

We first show that, for any

Then, we show that, for an arbitrary integer

Therefore, the transformation is a full mapping from

It is worth pointing out that this transformation is a linear mapping. Moreover, since we use the advantage of the special structure of the original problem, the size of the reformulated problem is smaller than the problems derived by some traditional transformation methods [

Now, replacing

Let

Note that problem

Let

Moreover, we define a set as follows:

Then following the work of [

Moreover, for two vectors

If

Note that since

Moreover, the Hessian matrix

Theorem

The canonical dual problem

Note that problem

Moreover, we have

KKT conditions provide necessary conditions for local minimizers in a nonconvex programming problem. Next, we show that the canonical dual problem is a concave maximization dual problem over a convex feasible domain under certain conditions. First, we define a subset of set

Assume that

Note that if

On the other hand, since

Above all, we transform the original problem which is a multi-integer quadratic programming problem into a piecewise continuous canonical dual problem by using the canonical duality theory. It is worth pointing out that the original problem and the canonical dual problem have no duality gap over

Note that if

The key issue for solving the problem

Let

Let

Let

If

Return

In this section, we use simulations to compare the canonical dual method (CDM) with some other benchmarked approximating methods, such as inexact ML sphere decoding (ML-SD) [

The channel matrix

We use two different problem sizes as

Figures

Symbol error rate comparison for different methods,

16-QAM

64-QAM

Symbol error rate comparison for different methods,

16-QAM

64-QAM

Table

Average computational time for different methods under different cases.

Methods | Cases | |||
---|---|---|---|---|

8 16-QAM | 16 16-QAM | 8 64-QAM | 16 64-QAM | |

ML-SD | 8.04 s | 31.2 s | 15.39 s | 38.44 s |

SDR | 6.21 s | 28.17 s | 10.32 s | 36.45 s |

MMSE-LD | 10.42 s | 35.76 s | 18.47 s | 54.53 s |

CDM | 7.27 s | 36.89 s | 14.23 s | 45.17 s |

From the simulation results, we can see that the CDM method outperforms all other methods in all situations. And the SER gaps between the CDM method and other methods are significantly wide. Moreover, the computation time indicates that the CDM method is quite efficient compared with other methods. It is worth pointing out that the CDM method is much more effective for the high order QAM problem.

In this paper, we have developed a canonical dual method to solve the MIMO problem. By introducing a tricky linear transformation, the original problem can be reformulated as a

The canonical dual approach offers a different angle to study the MIMO problem. It sheds some light on designing a new but more efficient algorithm. For the future study, some searching methods can be combined in the algorithm to improve the efficiency for finding the stationary points. And we aim to figure out which subclass of the MIMO problem can be exactly solved by the canonical duality theory.

The authors declare that there is no conflict of interests related to this paper.

Ye Tian’s research has been supported by the Chinese National Science Foundation no. 11401485.