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In order to apply sphere decoding algorithm in multiple-input multiple-output communication systems and to make it feasible for real-time applications, its computational complexity should be decreased. To achieve this goal, this paper provides some useful insights into the effect of initial and the final sphere radii and estimating them effortlessly. It also discusses practical ways of initiating the algorithm properly and terminating it before the normal end of the process as well as the cost of these methods. Besides, a novel algorithm is introduced which utilizes the presented techniques according to a threshold factor which is defined in terms of the number of transmit antennas and the noise variance. Simulation results show that the proposed algorithm offers a desirable performance and reasonable complexity satisfying practical constraints.

The Nondeterministic Polynomial-time hard (NP-hard) complexity of Maximum Likelihood (ML) decoding, the optimal decoder, generally prohibits its use in practical Multiple-Input Multiple-Output (MIMO) systems [

SD was first introduced in [

Various papers have analyzed the complexity of SD such as [

Stopping criteria can be used to reduce the complexity of SD, since it results in terminating the decoding process earlier and thus prevents a huge amount of extra calculations. Some researchers have worked on these criteria for some special scenarios. This paper also discusses the convergence radius and proposes a new stopping criterion.

SD searches a lattice through a given set of points that is bounded by the search sphere with the received point as center. Therefore, the method requires determining an initial search radius,

In this paper,

In the assumed MIMO system,

To obtain a lattice representation of this multiple antenna system, the complex matrix equation is transformed into the real matrix equation as

The reminder of this paper is organized as follows. Section

A finite lattice can be defined as

As mentioned earlier, SD searches the lattice through a given set of points bounded by a sphere with the received point as center and a specific initial radius. Whenever a point is found inside the sphere, the radius is reduced to the value of the distance between the new and the received point. Under assumption of

The first candidate for

By SE enumeration, the candidates are spanned in a zigzag order, starting from the midpoint. Hence, at each level

A full search can be depicted as a search tree, like Figure

The possible searching paths in a tree for

The work of [

If the candidate is valid and a leaf node is reached

If the candidate is not valid, the algorithm will go to step 4. If the algorithm is in the top level, it means that there is no valid symbol in the sphere. Thus, the algorithm terminates. Otherwise, the algorithm will go up to

This paper uses the SE-SD algorithm introduced in [

Several approaches have been proposed to find an appropriate initial radius. Because of the advantages of the Schnorr-Euchner enumeration, the conventional methods choose the positive infinity as the initial radius. Obviously, this approach avoids declaring an empty sphere. It is also clear that the first point found with

Another case for

Some works consider a small fixed number as

A useful approach is to choose

In such a way, a lattice point can be found inside the sphere with a high probability:

It is important to note that the radius is chosen based on the statistics of the noise and not

To investigate the behavior of the algorithm, we find the average number of flops, a measure for the complexity, and Bit Error Rate (BER) of the SE-SD algorithm for various ^{8} random symbols per any particular

The work of [

Although (

BER versus initial radius of SE-SD algorithm when

A huge number of figures that show the complexity versus

The average number of flops versus initial radius of SE-SD algorithm when

The average number of flops versus initial radius of SE-SD algorithm when

For any

The curves related to low

Moderate

High

From Figures

Simulation results that depict the average complexity and BER of SE-SD algorithm as a function of

While an approach is to choose

Figure

However, as it can be seen from Figure

Stopping criterion is a potential mean for saving computational complexity in iterative algorithms like SE-SD. The work of [

The idea of [

It was mentioned that when the algorithm finds a symbol in the sphere, calculates a new radius for the sphere and when the algorithm reaches its convergence radius, it still tries to find a symbol inside the new sphere, but it does not succeed, because there is no one. Of course, the attempts to find a new symbol after reaching the convergence radius cause significant extra calculations. Thus, if SE-SD algorithm terminates as soon as it reaches the convergence radius, the huge amount of unnecessary computations can be prevented. Consequently, the complexity of the algorithm will be reduced considerably.

Finding the convergence radius is itself really complicated. Through the computer simulations of SE-SD for each scenario we recorded the average

When a new sphere radius is calculated in step 5 of the algorithm, it should be compared to the estimated

Although the problem of finding suitable

SE-SD with

SE-SD which

early terminated PSE-SD (EPSE-SD).

Table

Reduction in complexity corresponding the number of transmit antennas.

Number of transmit antennas | SE-SD complexity reduction (%) when | PSE-SD complexity reduction (%) | EPSE-SD complexity reduction (%) |
---|---|---|---|

20 | 30 | 40 | 58 |

12 | 15 | 21 | 33 |

10 | 9 | 16 | 27 |

6 | −5 | −2 | 4 |

4 | −13 | −7 | 1 |

According to Table

However, if

Table

Reduction in complexity corresponding noise variance.

Noise variance | SE-SD Complexity reduction (%) when | PSE-SD complexity reduction (%) | EPSE-SD complexity reduction (%) |
---|---|---|---|

2.49 | 20 | 30 | 52 |

1.14 | 3 | 8 | 22 |

1.02 | −2 | 3 | 13 |

0.203 | −13 | −7 | 1 |

0.056 | −32 | −30 | −27 |

In fact the noticeable beneficial effect of allocating

As a result of the presented discussion, EPSE-SD seems not to be the efficient decoding algorithm for some cases; a criterion should be introduced to help us to choose one of the initiation and termination techniques of SE-SD. We propose the TF-based algorithm performing different decoding techniques via a Threshold Factor (TF). TF of this algorithm is defined as a function of

There are four major cases according to the value of TF. First, when TF is greater than 300, the TF-based algorithm only performs SIC decoding. For instance, if

In the second case when TF is between 3.2 and 300, the algorithm initially performs SIC decoding to find

In the third case, when TF is between 0.4 and

Finally, the fourth case is when TF is less than 0.4 and the algorithm performs SE-SD with

(1) Calculate TF

(2) If

(3) If

Calculate

(4) If

(5) If

In order to make SE-SD feasible for real applications, some techniques should be utilized to decrease the complexity of this algorithm. We presented new methods of initiation and termination of the SE-SD algorithm that can contribute to achieve the goal of having a reasonable complexity.

We showed that for a high number of transmit antennas, using Babai distance as initial radius leads to considerable performance degradation due to the big problem size. The suitable initial sphere radius which results in low complexity and desirable performance in the range of

To estimate the initial radius and the final one, Babai distance should be found through SIC decoding. Therefore, the presented technique sounds not to be useful for some cases in which the extra complexity of SIC decoding is comparable to that of SE-SD process. This investigation proposed an algorithm that utilizes different techniques according to a threshold factor defined in terms of the number of transmit antennas and noise variance. Using threshold factor, the novel algorithm offers a reasonable complexity without any performance degradation.

This work is supported by Developing Research and Strategic Planning Department of Mobile Communication Company of Iran (MCI) as a part of advanced communication systems project. The authors would like to thank Mr. Vahid Sadoughi, the CEO of MCI, for his financial support. They would like to thank Mrs. Samaneh Movassaghi (University of Technology, Sydney) and Mr. Mahyar Shirvani Moghaddam (University of Sydney) for the great help they provided. Also, the authors would like to thank the editor and anonymous reviewers of the International Journal of Antennas and Propagation (IJAP) for their very helpful comments and suggestions which have improved the presentation of the paper.