For the last few decades, the parameter estimation of electromagnetic plane waves i.e., far field sources, impinging on antenna array geometries has attracted a lot of researchers due to their use in radar, sonar and under water acoustic environments. In this work, nature inspired heuristics based on the flower pollination algorithm (FPA) is designed for the estimation problem of amplitude and direction of arrival of far field sources impinging on uniform linear array (ULA). Using the approximation in mean squared error sense, a fitness function of the problem is developed and the strength of the FPA is utilized for optimization of the cost function representing scenarios for various number of sources non-coherent located in the far field. The worth of the proposed FPA based nature inspired computing heuristic is established through assessment studies on fitness, histograms, cumulative distribution function and box plots analysis. The other worthy perks of the proposed scheme include simplicity of concept, ease in the implementation, extendibility and wide range of applicability to solve complex optimization problems. These salient features make the proposed approach as an attractive alternative to be exploited for solving different parameter estimation problems arising in nonlinear systems, power signal modelling, image processing and fault diagnosis.

Parameter estimation specially direction of arrival (DOA) estimation of plane waves plays a vital role in the areas of wireless communication, earthquake, medicine, tracking, navigation, and radio astronomy [

To overcome these problems adaptive algorithms were used and methods of maximum likelihood were developed. Stochastic maximum likelihood and deterministic maximum likelihood i.e., SML and DML methods were a few to mention [

Techniques, being metaheuristic, have been exploited for the determination of DOA unlike adaptive techniques namely Least Mean Square, MUSIC, ESPRIT and Recursive Least Square etc. due to their effective strength in optimization [

In this paper, an effective optimization mechanism of flower pollination algorithm (FPA) is employed as a newly introduced algorithm for the parameter estimation of electromagnetic waves of the far field. FPA mimics the process of pollination in flowering plants. FPA is proposed by Yang [

Exploitation of pollination based optimization technique FPA for the novel study of DOA estimation

Augmented power of FPA is built for the parameter estimation (effectively) of plane waves of sources.

The design mechanism is validated for different scenarios of far field sources.

The accuracy, robustness, and reliability of the algorithm are proven via results of the statistics in terms of parameters fitness.

Ease of implementation, simple in concept, extendibility, handling complex models and wide range of applicability are further advantages of the scheme.

The paper is arranged as follows: In Section 2, plane waves incident on a ULA is given as general data model for parameter estimation, while details about proposed scheme that has foundation on FPA are given in Section 3. Section 4 provides results and discussion on the results. The last section presents the conclusion and future work.

For model development, consider narrow band sources of EM plane waves P in number. The plane waves are falling on ULA. The ULA has “N” elements. The inter-element spacing is “d” which is uniform between any two consecutive elements. It is portrayed in

for

In

Here

S is the symbol used here for the steering matrix. It has got steering vectors of P sources. AWGN introduced in each antenna element is symbolized here as

A new meta heuristic technique called FPA was originally proposed by Yang [

In

In this work, FPA is developed for parameter estimation of electromagnetic plane waves. The flow chart of FPA is portrayed in

Step 1 Population Initialization

“

For the current optimization problem, the constraints associated with are:

Amplitude bounds (lower and upper) are symbolized here as lb and ub respectively, and

Step 2 Computation of Fitness

The fitness function is expressed in terms of mean squared error. For noiseless environment it is given as:

Fitness is computed for each individual of population F using

Step 3 Determine g^{*}, the initial best solution

Step 4 Defines a probability switch

Step 5 Compute fitness value of all n members/solution/flowers.

Step 6 if

Step 7 Draw a step vector L (d-dimensional) obeying Levy Distribution

Step 8 Carry out global pollination via

Else

Step 9 Draw a uniform distribution

Step 10 Randomly choose

Step 11 Do local pollination via

Step 12 Evaluate the new best solution

Step 13 If the new

Step 14 ^{t} = ^{t+1}

Step 15 Find the current best solution

Step 16 For global best solution, store the parameters, and its fitness for each run.

Step 17 For reliability, Steps 1–16 are repeated for sufficiently huge runs to have a huge set of data.

Step 18 For FPA performance evaluation, the fitness as in

where N is index of element in vector v.

In this section, work is presented in terms of simulations for three scenarios. In each scenario we have two sources model (2SM), three sources model (3SM) and four sources model (4SM). The EM plane wave sources are P where

The scheme designed (based on FPA) is utilized for parameter estimation. It is employed for both situations (noisy as well as noiseless) as given in the section of methodology. For each scenario, five cases are worked out as: Case 1: 2 SM with no noise and given noise is added for rest of the four cases namely Case 2: 2 SM having 65 dB, Case 3: 2 SM having 55 dB, Case 4: 2 SM having 45dB and Case 5: 2 SM having 35 dB. The results are obtained for 100 independent runs of the FPA. Objective function for any of the scenario is formulated as:

For this data, the algorithm was run 100 times independently. The best estimated parameters are given in ^{−29} in 85 runs. Three sources reach to a fitness of about 10^{−28} in 96 runs. Likewise, four sources have about 10^{−7} fitness in 100 runs. Remaining graphs of the 1st case can also be shown. Likewise, ^{−31} in about 88 runs. The same three sources get a fitness of about 10^{−30} in about 97 runs and the same four sources get a fitness of about 10^{−8} in 100 runs.

Noise | Amplitudes | DOAs | Fitness | NAE/error values | ||
---|---|---|---|---|---|---|

i_{1} |
i_{2} |
|||||

0 | 0.5 | 1 | 0.87266 | 1.4835 | 0.E+00 | 0.E+00 |

65 dB | 0.4993 | 0.99953 | 0.8724 | 1.4834 | 0.E+00 | 8.93E −04 |

55 dB | 0.49909 | 1.0035 | 0.87375 | 1.4824 | 0.E+00 | 0.E+00 |

45 dB | 0.49351 | 1.005 | 0.87786 | 1.4878 | 0.E+00 | 0.E+00 |

35 dB | 0.51367 | 0.99537 | 0.84826 | 1.4984 | 0.E+00 | 1.E −16 |

Noise | Amplitudes | DOAs | Fitness | NAE/error values | ||||
---|---|---|---|---|---|---|---|---|

i_{1} |
i_{2} |
i_{3} |
||||||

0 | 3 | 1 | 5 | 1.8326 | 2.7053 | 0.87266 | 0.E+00 | 0.E+00 |

65 dB | 3 | 0.99983 | 4.9999 | 1.8325 | 2.7046 | 0.87231 | 0.E+00 | 0.E+00 |

55 dB | 2.9998 | 1.001 | 5.0018 | 1.8342 | 2.7077 | 0.8722 | 0.E+00 | 0.E+00 |

45 dB | 3.0004 | 1.0046 | 4.9965 | 1.8413 | 2.703 | 0.87028 | 0.E+00 | 0.E+00 |

35 dB | 3.0179 | 0.98696 | 5.0066 | 1.8064 | 2.6689 | 0.87465 | 0.E+00 | 0.E+00 |

Noise | Amplitudes | DOAs | Fitness | NAE/error values | ||||||
---|---|---|---|---|---|---|---|---|---|---|

i_{1} |
i_{2} |
i_{3} |
i_{4} |
|||||||

0 | 0.99973 | 1.5001 | 2 | 2.5 | 8.9012 | 0.34907 | 1.0472 | 1.9199 | 1.E −07 | 6.E+00 |

65dB | 1.0004 | 1.5002 | 1.9993 | 2.5014 | 2.6173 | 0.34871 | 1.0466 | 1.9201 | 1.E −08 | 9.E −05 |

55dB | 0.99744 | 1.497 | 2.0016 | 2.4983 | 3.6609 | 0.34731 | 1.0483 | 1.9175 | 7.E −11 | 1.E+00 |

45dB | 0.99791 | 1.506 | 1.9866 | 2.5054 | 8.897 | 0.35053 | 1.0449 | 1.921 | 6.E −09 | 6.E+00 |

35dB | 1.0033 | 1.4633 | 2.0245 | 2.4645 | 9.9539 | 0.37401 | 1.0512 | 1.9301 | 1.E −09 | 7.E+00 |

^{−30} to 10^{−25} for two sources, and about 2 runs give the same fitness for three sources while the same two runs give a fitness in the range 10^{−20} to 10^{−15} for four sources. ^{−35} to 10^{−30} for two sources, 4 runs give a fitness in the range of 10^{−30} to 10^{−25} for three sources, and about 2 runs give a fitness of 10^{−6} to 10^{−5} for four sources in the presence of 65 dB noise respectively. ^{−29} for two sources, about 5% of runs give a fitness of more than 10^{−27} for three sources and about a fraction of one run gives a fitness of about 10^{−7} for four sources. ^{−31} for about 12% of the runs for two sources, a fitness of more than 10^{−29} for 4% of runs for three sources, and a fitness of about 10^{−8} for about a fraction of 1% runs for four sources in presence of 65dB noise. ^{−6} for two sources, more than 10^{−2} for three sources and more than 10^{−1} for four sources. Likewise, the best fitness is more than 10^{−28} for two sources, about 10^{−16} for three sources and about 10^{−7} for four sources. 75% of fitness is about 10^{−6} for two sources, less than 10^{−3} for three sources and about 10^{−1} for four sources. Exactly half of the fitness is about 10^{−8} for two sources, 10^{−4} for three sources and less than 10^{−1} for four sources. ^{−4} for two sources, less than 10^{−2} for three sources, and less than 10^{−1} for four sources. Likewise, the best fitness is more than 10^{−28} for two sources, about 10^{−7} for three sources, and 10^{−8} for four sources. 75% of the fitness is about 10^{−6} for two sources, 10^{−2} for three sources and 10^{−1} for four sources. Exactly half of the fitness is about 10^{−8} for two sources, 10^{−3} for three sources, and less than 10^{−1} for four sources in the presence of 65 dB noise.

Likewise, all

An innovative application of flower pollination heuristic is introduced for reliable parameter estimation of electromagnetic plane waves impinging on antenna array geometries. The accuracy, stability and robustness of the proposed flower pollination heuristic is verified from actual value of system parameter for single and multiple autonomous runs. The worth of the proposed FPA is further established through statistical assessments based on fitness, histograms, cumulative distribution function and box plots analysis for two, three and four source model of DOA parameter estimation in noisy and noiseless environments.

In future, one may exploit the proposed methodology for different optimization problems including power signal estimation [