With the further research in communication systems, especially in wireless communication systems, a statistical model called Nakagami
It is well known that radiowave propagation is complicated under the environment of wireless channels and various effects exert different impacts on the characteristics of radiowave propagation. Considerable research focusing on building the statistical models and characterizing these effects has been undertaken. Multipath fading is of great significance among the effects and is caused by the combination of various signal components containing randomly diffracted, reflected, delayed, and scattered components. There are different models used under different radio propagation environments, such as Rayleigh, Rice, Hoyt, and Weibull distributions [
Numerical calculations and theoretical analyses in wireless communications under the Nakagami
In this paper, an expression with lower computational complexity and better performance for approximating the Nakagami
The contributions of our work are summarized as follows:
We propose a briefer and more efficient expression to approximate the quantile function of the Nakagami
We propose the generalized oppositionbased learning strategy in quantum space, which is first proposed in the literature. Meanwhile, we apply this strategy to QSSA to improve its convergence speed.
We propose the quantum intelligent algorithm which is named GOQSSA. It is the first time that quantum swarm intelligence is combined with salp swarm algorithm and quantum evolutionary strategy, and the results prove the accuracy of the proposed expression and the efficiency of GOQSSA.
The remaining part is structured as follows: Section
Many researchers have focused on numerical calculations and theoretical analyses under the Nakagami
In recent years, a new method to approximate the quantile function of Nakagami
Concepts and properties have been interpreted in [
The Nakagami
According to the relationship between PDF and CDF in probability theory, the CDF is defined as
Hence, the quantile function of Nakagami
We have obtained the exact value of the Nakagami
So far, we have modelled the approximation problem. Next, we will expound the methodology on both expression and algorithm aspects.
There are two main aspects to be addressed when solving this problem. One is the expression designed to approximate Nakagami
It is well known that the foremost aspect is the mathematical expression for a curvefitting problem, and as mentioned above, we propose a new approximation model with higher accuracy as follows:
In order to compare clearly with available approximations previously reported in the literature, we introduce them briefly. Beaulieu and Cheng [
This is much simpler than (
There are only four coefficients in this expression, with an inverse hyperbolic tangent function and two exponential functions. However, in practice, equation (
Comparison between the inverse hyperbolic tangent function
SSA is inspired by salps’ swarming behavior [
According to the theory of SSA and quantum computing [
There is a salp swarm containing
The position of each salp denotes a
According to the theory of quantum computing, the quantum rotation angles are updated through quantum positions of the last generation, and then, we use the updated quantum rotation angles to update and get new quantum positions. Meanwhile, considering the structure of the salp chain, the leader should guide the swarm to move in the direction to the food source and the followers follow the one in front of themselves, we propose three strategies to update their quantum rotation angles. The first and second strategies are designed for the leader and can be respectively expressed as
In equations (
Here, we have obtained the quantum rotation angles of all salps, and the next step is to update their quantum positions with equation (
Oppositionbased learning was first proposed to improve algorithms in machine intelligence by Tizhoosh [
Let
Let
Let
In this section, we give the method to approximate Nakagami
Experiments under several combinations of
Here, we give the results optimized by total four algorithms—PSO, ABC, SSA, and GOQSSA. There are numerous heuristic algorithms in the literature, including cuckoo search algorithm [
In this paper, we conduct independent simulations 100 times for each algorithm under each combination of fading and scale parameters. Convergence curves are shown in Figures
Average of optimal fitness values computed by four algorithms in 100 runs under different fading parameters when
Average of optimal fitness values computed by four algorithms in 100 runs under different fading parameters when
Statistical comparisons between different algorithms when

Algorithm  Best  Worst  Mean  STD 

2.5  PSO  1.2409 
2.3049 
1.1032 
9.7759 
ABC  5.5411 
1.9640 
1.4567 
2.2771  
SSA  4.0482 
2.3033 
4.9134 
9.0627  
GOQSSA 







3.0  PSO  1.5375 
2.6207 
1.0526 
1.0562 
ABC  6.9084 
2.2046 
1.6213 
3.0613  
SSA  4.3616 
2.6344 
5.8407 
1.0535  
GOQSSA 







3.5  PSO  1.6453 
2.8989 
1.1871 
1.1737 
ABC  1.0771 
2.3312 
1.7890 
2.4986  
SSA  4.8817 
2.8878 
3.8861 
9.2479  
GOQSSA 







4.0  PSO  1.2613 
3.1806 
8.7276 
1.0394 
ABC  1.2184 
2.5891 
1.9198 
2.8857  
SSA  5.5371 
3.2140 
3.6417 
9.3109  
GOQSSA 




Statistical comparisons between different algorithms when

Algorithm  Best  Worst  Mean  STD 

7.0  PSO  1.3551 
4.1978 
2.5441 
1.9053 
ABC  7.5002 
3.1097 
2.1384 
5.2317  
SSA  1.1977 
4.1861 
4.1626 
1.0410  
GOQSSA 







8.0  PSO  2.1655 
4.4409 
1.8379 
1.9599 
ABC  9.0083 
3.2507 
2.4481 
4.7022  
SSA  1.3828 
4.4421 
2.8054 
7.3801  
GOQSSA 







9.0  PSO  2.0526 
4.6547 
2.1849 
2.1072 
ABC  7.7279 
3.4422 
2.4093 
5.7204  
SSA  1.5481 
4.6437 
2.4459 
6.3161  
GOQSSA 







10.0  PSO  2.2120 
4.8394 
2.4978 
2.2095 
ABC  1.2149 
3.8000 
2.5844 
5.5310  
SSA  1.6970 
4.8273 
3.6282 
9.1703  
GOQSSA 




As we can see in these figures, PSO and ABC quickly trap into local optimal solutions. There is some probability that SSA can escape from some local optimal solutions, but SSA fails to compute the global optimal solutions in the end. The statistical data in Tables
Another main aspect in the approximation problem is the expression for approximating Nakagami
Comparison between our proposed approximation and existing approximations based on curvefitting under different fading parameters when

Bilim and Develi [ 
Kabalci [ 
Kabalci [ 
Proposed 

0.5  0.0572  0.0213  0.0333  6.2606E − 4 
0.6  0.0554  0.0153  0.0258  1.2451E − 3 
0.7  0.0520  0.0102  0.0245  1.9087E − 3 
0.8  0.0512  0.0073  0.0156  2.4405E − 3 
0.9  0.0503  0.0076  0.0169  2.8529E − 3 
1.0  0.0489  0.0098  0.0161  3.1685E − 3 
1.5  0.0438  0.0189  0.0206  3.8556E − 3 
2.0  0.0398  0.0217  0.0226  3.9459E − 3 
2.5  0.0385  0.0228  0.0230  4.0482E − 3 
3.0  0.0355  0.0234  0.0240  4.3616E − 3 
3.5  0.0334  0.0237  0.0243  4.8817E − 3 
4.0  0.0322  0.0240  0.0254  5.5371E − 3 
5.0  0.0306  0.0243  0.0252  7.0110E − 3 
6.0  0.0294  0.0245  0.0250  8.4937E − 3 
7.0  0.0278  0.0247  0.0248  9.8905E − 3 
8.0  0.0261  0.0248  0.0250  1.1179E − 2 
9.0  0.0322  0.0250  0.0255  1.2359E − 2 
10.0  0.0289  0.0251  0.0256  1.3440E − 2 
Comparison between our proposed approximation and existing approximations based on curvefitting under different fading parameters when

Bilim and Develi [ 
Kabalci [ 
Kabalci [ 
Proposed 

0.5  0.0829  0.0301  0.0429  1.6811E − 4 
0.6  0.0805  0.0216  0.0274  7.1155E − 4 
0.7  0.0743  0.0145  0.0246  1.3029E − 3 
0.8  0.0719  0.0104  0.0278  1.7516E − 3 
0.9  0.0713  0.0108  0.0271  2.0812E − 3 
1.0  0.0716  0.0139  0.0294  2.3099E − 3 
1.5  0.0606  0.0268  0.0320  2.3751E − 3 
2.0  0.0575  0.0307  0.0320  1.5034E − 3 
2.5  0.0531  0.0323  0.0326  9.5815E − 4 
3.0  0.0505  0.0331  0.0343  1.9993E − 3 
3.5  0.0475  0.0336  0.0344  3.4285E − 3 
4.0  0.0456  0.0339  0.0355  4.8606E − 3 
5.0  0.0517  0.0343  0.0347  7.5286E − 3 
6.0  0.0407  0.0347  0.0354  9.8914E − 3 
7.0  0.0519  0.0349  0.0360  1.1977E − 2 
8.0  0.0475  0.0352  0.0362  1.3828E − 2 
9.0  0.0427  0.0354  0.0357  1.5481E − 2 
10.0  0.0404  0.0356  0.0358  1.6970E − 2 
Considering the simulation results and analyses above, we finally compute the coefficients of the proposed expression with the assistance of GOQSSA. The coefficients of the proposed expression computed by GOQSSA under various scaling parameters (
Coefficients of the proposed expression computed by GOQSSA when the fading parameter






0.5  0.8177  0.3688  0.9329  0.3072 
0.6  0.7495  0.4459  0.6658  0.2240 
0.7  0.6945  0.5333  0.5653  0.1213 
0.8  0.6469  0.6016  0.5027  0.0649 
0.9  0.6053  0.6529  0.4565  0.0367 
1.0  0.5690  0.6915  0.4197  0.0244 
1.5  0.4401  0.7884  0.3048  0.0388 
2.0  0.3618  0.8232  0.2423  0.0735 
2.5  0.3090  0.8389  0.2022  0.1027 
3.0  0.2708  0.8470  0.1741  0.1257 
3.5  0.2417  0.8516  0.1532  0.1439 
4.0  0.2187  0.8543  0.1370  0.1587 
5.0  0.1845  0.8571  0.1134  0.1810 
6.0  0.1602  0.8582  0.0970  0.1971 
7.0  0.1418  0.8587  0.0848  0.2093 
8.0  0.1274  0.8588  0.0755  0.2188 
9.0  0.1157  0.8588  0.0680  0.2265 
10.0  0.1061  0.8587  0.0620  0.2328 
Coefficients of the proposed expression computed by GOQSSA when the fading parameter






0.5  1.1043  0.5072  0.8819  0.4577 
0.6  0.9166  0.7319  0.5434  0.2766 
0.7  1.1486  0.7440  0.5821  0.2042 
0.8  0.8466  0.8224  0.4835  0.2209 
0.9  0.7872  0.8912  0.4393  0.1898 
1.0  0.7361  0.9436  0.4041  0.1730 
1.5  0.5602  1.0786  0.2943  0.1656 
2.0  0.4559  1.1293  0.2342  0.1860 
2.5  0.3861  1.1531  0.1957  0.2063 
3.0  0.3358  1.1660  0.1686  0.2234 
3.5  0.2977  1.1736  0.1484  0.2375 
4.0  0.2677  1.1785  0.1327  0.2491 
5.0  0.2233  1.1841  0.1099  0.2670 
6.0  0.1919  1.1870  0.0940  0.2799 
7.0  0.1685  1.1888  0.0823  0.2896 
8.0  0.1502  1.1901  0.0733  0.2971 
9.0  0.1357  1.1909  0.0661  0.3031 
10.0  0.1237  1.1916  0.0602  0.3080 
Comparisons between values of the proposed expression and the exact values under different fading and scaling parameters. (a) Comparisons when the fading parameter is set as 0.8, 1.5, and 3.0 and scaling parameter is 1. (b) Comparisons when the fading parameter is set as 0.9, 2.5, and 6.0 and scaling parameter is 1. (c) Comparisons when the fading parameter is set as 0.7, 1.5, and 3.5 and scaling parameter is 2. (d) Comparisons when the fading parameter is set as 1.0, 2.0, and 4.0 and scaling parameter is 2.
Nakagami
In this paper, a simple and efficient expression for approximating the Nakagami
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
This research was supported by the National Natural Science Foundation of China (Grant no. 61571149), China Postdoctoral Science Foundation (Grant no. 2015T80325), and Fundamental Research Funds for the Central Universities (HEUCFP201808).