One of the most important methods used to cope with multipath fading effects, which cause the symbol to be received incorrectly in wireless communication systems, is the use of multiple transceiver antenna structures. By combining the multi-input multi-output (MIMO) antenna structure with non-orthogonal multiple access (NOMA), which is a new multiplexing method, the fading effects of the channels are not only reduced but also high data rate transmission is ensured. However, when the maximum likelihood (ML) algorithm that has high performance on coherent detection, is used as a symbol detector in MIMO NOMA systems, the computational complexity of the system increases due to higher-order constellations and antenna sizes. As a result, the implementation of this algorithm will be impractical. In this study, the backtracking search algorithm (BSA) is proposed to reduce the computational complexity of the symbol detection and have a good bit error performance for MIMO-NOMA systems. To emphasize the efficiency of the proposed algorithm, simulations have been made for the system with various antenna sizes. As can be seen from the obtained results, a considerable reduction in complexity has occurred using BSA compared to the ML algorithm, also the bit error performance of the system is increased compared to other algorithms.

Insufficient frequency spectral resources are the biggest obstacle to high-rate data transmission. Interest in the use of the NOMA technique, which is a kind of multi-carrier system that provides high-rate data transmission by using spectral resources efficiently, has increased considerably in recent years. In addition to providing spectral efficiency of NOMA, its resistance to multipath damping is another important advantage of the system. Therefore, the NOMA technique also forms the basis of 5th generation communication systems [

The most significant drawback to being considered in the multiplexing methods is the destructive effects caused by multipath fading. One of the most important ways to deal with these destructive effects is to use a multi-antenna transceiver. By using NOMA combined with multiple antenna structures, not only high-rate data transmission is provided but also the destructive effects caused by fading are minimized [

Many difficult real-time engineering problems have been solved by using heuristic algorithms inspired by the behavior of many events or creatures in nature [

In this study, the BSA is proposed for symbol detection in MIMO-NOMA systems with various transmitter and receiver antennas. To highlight the complexity reduction and bit error rate (BER) efficiency of the proposal, the proposed BSA is compared to the classical algorithms such as ML, ZF and VBLAST and heuristic algorithms such as GA, PSO. The simulation results and complexity analysis demonstrate that our proposal is an effective solution for symbol detection.

This paper is organized as follows, Section 2 introduces the concept of the MIMO-NOMA system and ML scheme. The architecture of the backtracking search algorithm for data detection is described in Section 3. The simulation results and conclusion are drawn in Section 4 and Section 5 respectively.

NOMA combined with MIMO is a new multicarrier modulation system that provides orthogonal access to the users either in frequency, time, space, or code. For this scheme, each user uses the same time and frequency where they are selected considering the power levels. In NOMA, the successive interference cancellation (

When we consider a MIMO-NOMA system consists of

Where

Where

Where

Where

As can be seen from

Channel gains can be presented as follows,

Besides PA coefficients are ordered as

To detect symbol, an ML algorithm can be implemented by maximizing the (9) metric;

Data can be detected by using the

For ML detection

BSA, is a population-based evolutionary algorithm that provides to make the right choice from a variety of ways to reach a goal or search for a value.

BSA is distinct from other similar algorithms in that it has a memory for storing a population from a previous generation, which is then used to create the search-direction matrix for the next iteration. Besides, Since BSA has a basic structure, it is efficient, fast, and capable of solving multidimensional problems. The mix-rate, which is the only control parameter in BSA, decreases the sensitivity of the inceptive values to the algorithm’s parameters greatly. Initialization, selection I, mutation, crossover, and selection II are the five processing stages in BSA [

The processing steps of the BSA are given in

For the symbol detection problem, in the BSA population of the individuals corresponds to the symbols. At the first steps of BSA, the individuals

Where

The population is updated at the beginning of each epoch as follows

Where

BSA alters a randomly selected population from the previous generation as the old population and keeps this old population in memory until it changes.

Population members are shuffled using the permitting function as follows;

The mutation process is then performed using the

where F is a real number that is used for control of amplification in search space.

The purpose of mutation is to include the experiences of previous generations in the process. In the crossover step, the final version of the trial population ^{−2}.

The performance evaluation of the algorithms has been made by considering the systems with 10 MHz bandwidth and 16QAM modulation for various antennas size over the channel. The NOMA system and channel parameters are given in

Parameter | Value |
---|---|

Bandwidth | 10 MHz |

Number of Subcarriers | 64 |

Number of User | 2 |

Power Allocation Factor | 0.75 0.25 |

Modulation | 16QAM |

Path | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

0 | 0.3 | 1.0 | 1.6 | 5.0 | 6.6 | |

−2.5 | 0 | −3 | −5 | −2 | −4 |

Besides the parameters of the proposed heuristic algorithms for data detection are given in

BSA | PSO | GA |
---|---|---|

Number of Population = 30 | Swarm Size = 30 | Number of Population = 30 |

Step Size Amplification = 3.rnd |
Max Velocity = 20 | Crossover Rate = 0.8 |

Mix Rate = 1 | Inertia factor = 0.9 to 0.4 | Mutation Rate = 0.5 |

Learning factor = 2 |

A comparison between the results of the proposed method to other detectors for a NOMA system with a 2 × 2 antennas array is shown in

Although it can be observed from ^{−1} BER, the BSA algorithm has a gain of 14 dB better than the worst-performing ZF and 2 dB better than the PSO with the closest performance to it. Also, the difference between BSA and VBLAST at a value of 20 dB signal to noise ratio (SNR) is around 10^{−1}.

The BER performance of the proposed detector for the systems with 4 × 4 antennas is shown in ^{−2} GA requires 18 dB, PSO requires 16.5 dB whereas BSA requires 14 dB. Also, at fewer BER values the required SNR value of the proposal is less than the other algorithms.

Finally comparing the performance of the algorithms for the system with 8 × 8 antennas in ^{−3} while PSO and GA need around 23.5 dB and 26 dB of SNR respectively to reach the BER = 10^{−3}. As can be seen from all figures that our proposal offers large SNR improvements compared to other detectors.

Furthermore, to demonstrate the computational complexity benefits of BSA over the ML detector, we can analyze the computational complexity of the detectors in terms of number complex multiplication [

In the ZF algorithm, there are

In the VBLAST algorithm, after the calculation of

When we consider the ML algorithm, the number of multiplications required for matrix multiplication and squaring operations are,

Where C is constellation size, in heuristic algorithms, the number of multiplications required to calculate the fitness of each particle in the population are,

After applying the population updating parameters (µ) and _{itr} times repetition to convergence to an optimal value, the complexity becomes as

The complexity comparison analysis of the detectors can be found in terms of the number of multiplications in

DETECTOR | 2 × 2 NOMA | 4 × 4 NOMA | 8 × 8 NOMA |
---|---|---|---|

ML | 1536 | 1.320.720 | 309 B |

ZF | 48 | 384 | 3072 |

VBLAST | 70 | 712 | 8864 |

BSA | 1440_{itr} = 8, µ = 2 |
5400_{itr} = 10, µ = 2 |
23760_{itr} = 12, µ = 2 |

PSO | 1800_{itr} = 10, µ = 2 |
6480_{itr} = 12, µ = 2 |
29700_{itr} = 15, µ = 2 |

GA | 1980_{itr} = 11, µ = 2 |
7560_{itr} = 14, µ = 2 |
31680_{itr} = 16, µ = 2 |

As can be seen from this computational complexity analysis, the number of transceiver antennas increases the computational load of each algorithm. However, this amount of load is even higher for the ML detector when the number of antennas increases. For instance, the complexity value of the ML algorithm is 3250 for a 2 × 2 antenna, while the number of antennas is as high as 309 billion in the case of 8 × 8. It can be seen from this analysis that the increase in the number of antennas and modulation index makes the algorithm less practical. Although the complexity of the ZF and VBLAST algorithms is low, their error performance is worse than the heuristic algorithms. Although the value of complexity in heuristic algorithms is similar, the number of iterations required for convergence has a direct effect on the complexity of these algorithms.

To reach optimal solutions for the 4 × 4 NOMA, 8 iterations in the BSA, 10 iterations in the PSO, and 11 iterations in the GA are needed. As a result, BSA, PSO, and GA need 1440, 1800, and 1980 multiplications respectively. Among heuristic algorithms, BSA has the lowest complexity.

Furthermore, BSA provides a 6.25% complexity reduction compared to the ML algorithm 2 × 2 systems. However, it significantly reduces the complexity by 99.59% and 99.9% in 4 × 4 and 8 × 8 systems.

In this study, the use of the BSA algorithm is proposed for data detection in the MIMO-NOMA systems which provide high-rate and quality of service. With this proposed algorithm, computational complexity is reduced in systems where the number of antennas and modulation constellation is large, and BER performance is increased compared to ML algorithm. As can be seen from the simulation results, the BSA algorithm has better values than the other heuristic algorithms in any case; also, it has a close performance to the ML algorithm. Besides, it is better than the classic algorithms such as ZF and VBLAST used for data detection, in terms of BER. So, we can say that the proposed algorithm can be used in a remarkable way for data detection in MIMO-NOMA systems.