Fractional order algorithms have shown promising results in various signal processing applications due to their ability to improve performance without significantly increasing complexity. The goal of this work is to investigate the use of fractional order algorithm in the field of adaptive beamforming, with a focus on improving performance while keeping complexity lower. The effectiveness of the algorithm will be studied and evaluated in this context. In this paper, a fractional order least mean square (FLMS) algorithm is proposed for adaptive beamforming in wireless applications for effective utilization of resources. This algorithm aims to improve upon existing beamforming algorithms, which are inefficient in performance, by offering faster convergence, better accuracy, and comparable computational complexity. The FLMS algorithm uses fractional order gradient in addition to the standard ordered gradient in weight adaptation. The derivation of the algorithm is provided and supported by mathematical convergence analysis. Performance is evaluated through simulations using mean square error (MSE) minimization as a metric and compared with the standard LMS algorithm for various parameters. The results, obtained through Matlab simulations, show that the FLMS algorithm outperforms the standard LMS in terms of convergence speed, beampattern accuracy and scatter plots. FLMS outperforms LMS in terms of convergence speed by 34%. From this, it can be concluded that FLMS is a better candidate for adaptive beamforming and other signal processing applications.

Smart antennas use the spatial domain to improve wireless communication systems by providing greater coverage and higher data rates through spatial diversity in a multipath channel environment. They are widely used in wireless communication, radar, sonar, navigation, and tracking systems [

Smart antenna systems rely on adaptive algorithms to make the necessary adjustments for system efficiency. These algorithms are classified as either non-blind or blind. Training signals are available for weight adaptation in non-blind algorithms. In the literature, adaptive algorithms such as least mean square (LMS) [

Fractional calculus is a relatively new area of research that has been gaining attention for its applications in various fields of engineering [

This paper’s organization describes that in Section 2, a system model for FLMS-based adaptive beamforming and the convergence analysis of the FLMS are presented. Section 3 includes results from Matlab simulations and discussions. The final section, Section 4, provides conclusions and suggestions for future work.

This section of the paper models the adaptive array beamforming system using the proposed FLMS algorithm. The beamformer in the model receives desired and interfering signals from various directions and impinges on

The FLMS algorithm is a non-blind algorithm used for beamforming that aims to estimate the optimal weights for the received signals (

For the proposed algorithm, some assumptions are made, such as the system environment being stationary and the input signals being statistically independent. Further signals are assumed to be uncorrelated and to have a zero mean.

In this setup of adaptive array beamforming, the signal array vector is composed of binary phase shift keying (BPSK) signal received at array elements and is represented by the transposed vector:

The desired and interfering signals are random and drawn from uniform, independent, and identical distributions. The signal array vector may be represented as a combination of desired and interfering signals in the element form given by:

Here

Taking the first array element as a reference, the steering vectors are represented as follows:

In this equation,

Weight

The array factor for ULA is given as:

In this equation,

The cost function

By putting

The equation for the updated weight vector is given using an iterative approach by differentiating the cost function for LMS

or

Here

It is a modified LMS that employs fractional order derivatives along with standard order derivatives in the weight adaptation of LMS. The cost function for LMS is the mean square error minimization in

The weight update equation for the FLMS has both the standard order and fractional order derivatives, given as:

Here

By definition of fractional order derivative of

By taking the cost function’s fractional order derivative with respect to

Putting

By taking the standard order derivative of the cost function:

Using

Here

In the weight vector adaptation, FLMS employs fractional in addition to standard order derivatives. The cost function

The error signal is estimated as:

The algorithm’s output is given by the equation below:

The weight vector given in

The weight update equation in FLMS contains both the standard and fractional order derivatives. It’s because the incoming signal is a Gaussian process. The weight update equation is:

The fractional order

Once a fractional order derivative is applied, the weight update equation becomes:

Here

The received signal consists of desired, interfering, and noise signals. When weights are applied to this input signal, an output signal is produced. Input

LMS weight updates equation as given in

The weight updates equation for FLMS as given in

Some other frame of reference is employed to reveal the properties of convergence of FLMS. The error vector

Here

Error vector in the next instant (

For LMS it becomes:

After putting

After substitutions and rearranging

In simplified form, it can be written as:

After applying the orthogonality principle, this vector simplifies to:

After pre-multiplying by a unitary matrix

After putting

Here

Suppose

For FLMS weight vectors consider

For column vector

When the statistical independence expectation and orthogonality principle are applied to both sides, the equation becomes:

Replacing

At a stable state, the mean power of the weight difference should decrease as the number of iterations increases, we have:

It is only possible for

It can also be represented in terms of eigenvalues

The simulations of the proposed algorithm use the setup and parameters listed in

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

Number of array antenna elements | 20 |

Number of samples | 1000 |

Number of runs | 500 |

Step size ( |
0.4, 0.1, 0.09, 0.05 |

Fractional order ( |
0.9, 0.7, 0.5 |

Modulation scheme | bpsk |

Elements spacing | 0.5 |

SNR | 10, 15, 20 |

Angel of desired signal ( |
10° |

Angel of interferer ( |
45° |

There are 20 isotropic array elements in ULA.

Element spacing is half the wavelength of the signal (

The desired BPSK modulated signal arrives at an angle of 10°.

The interfering BPSK modulated signal arrives at an angle of 45°.

Both desired and interfering signals are contaminated with Gaussian noise.

The simulations evaluate a fractional order adaptive beamforming algorithm using Monte Carlo simulations. The simulations test the algorithm’s performance under different signal-to-noise ratio (SNR), fractional order, and step size conditions. The evaluation is based on the comparison of mean square error (MSE) learning curves and convergence rates for 1000 samples and 500 independent runs. The desired signal has a direction of arrival (DOA) of 10°, while the interfering signal has a DOA of 45°.

The results in ^{st} iteration, while LMS reaches the same value at the 78^{th} iteration. This corresponds to 34% improvement. Similarly for ^{rd} iterations while LMS converges at the 45^{th} iteration. For SNR value of 15 dB, ^{st} iteration whereas LMS at the 62^{nd} iteration. The results establish FLMS provides better performance than the LMS algorithm.

The learning curves in ^{th} iteration, while with ^{th} iteration. The same pattern can be observed in

The results shown in

The mean squared error (MSE) performance for a signal-to-noise ratio (SNR) of 20 dB is evaluated and presented in

The MSE performance is evaluated in

The beampatterns in

The scatter plot in

Beamforming is an important feature of modern communication and radar systems. A fractional order adaptive algorithm is proposed for beamforming applications. The fractional-order adaptive beamforming (FLMS) algorithm was derived and simulated in this research work. A mathematical convergence analysis provided shows that the use of fractional derivatives in a system can improve its convergence and steady state response by reducing the spread of the autocorrelation matrix’s eigenvalues. This leads to better performance even with larger step sizes. The simulation results illustrate that FLMS outperforms the standard least mean square (LMS) algorithm in terms of MSE convergence speed by 34%. FLMS also showed better directivity and gain, resulting in improved coverage while reducing power consumption. The results indicate that FLMS has the potential to be applied in various categories of adaptive signal processing applications, i.e., system identification, inverse modeling, prediction, etc, and offer improved performance over traditional algorithms. In future work, the FLMS can be evaluated by changing various parameters, such as the number of array elements, the number of interferers, and different modulation schemes. Additionally, the fractional order approach can be extended to other adaptive filtering algorithms, such as RLS and NLMS, to evaluate their performance in adaptive beamforming applications. These lines of inquiry can provide new insights and potential avenues for improving the performance of adaptive beamforming algorithms.

This work is supported by the Office of Research and Innovation (IRG project # 23207) at Alfaisal University, Riyadh, KSA.

The authors declare that they have no conflicts of interest to report regarding the present study.