Swift Convergent Weighted Quadrigeminal Beamformer Using Smart Antenna System

The Section 2 demonstrates the thorough and methodical literature review that was carried out. This background information serves as the basis for the proposed research project.

When the received signals are noncircular polarized widely linear minimum variance distortionless response beamformer is better than traditional minimum variance distortionless response beamformer method which may be subjected to the direction of arrival errors due to mismatch of extended steering vectors; thus nonconvex optimization problem is converted into a semidefinite programming problem and then they are solved iteratively [1].

For short distance and indoor mobile communication, leaky wave antenna arrays are designed which create bilateral beamforming. The antenna is capable of multiport feeding network built on dual layer substrate integrated waveguides. These arrays have one and four ports on lower and upper layer respectively. When two individual upper ports are fed, they generate beams in the direction of ±25° and ±45° [2].

The main agenda of beamforming is to give the maximum gain in desired direction and nulling in the direction of interference. Beamforming is modeled based on neural networks which can be trained well with the help of array excitations and covariance matrix of the steering vector, computation time is reduced with a well trained neural network which aids in quick array excitation [3].

Circular antenna array with minimum variance distortionless response beamforming method is used to steer the beam patterns in the desired directional space, this technique reduces the side lobe level and increases the resolution and accuracy. Different performance parameters that were considered are as follows: propagation time delay, signal to interference ratio, antenna efficiency and spatial correlation function [4].

The novel techniques for DoA and Beamforming were implemented, which were capable of performing the detection and estimation of the desired user from the three dimensional spatial field. Equally spaced uniform linear antenna array was used to deploy spatial division multiple access in real time with the help of smart antenna system. The DoA algorithm displayed the optimal bias rate with acme resolution values. The very sharp and narrow pencil beams were generated by the Beamforming algorithm, which had very less mean square error values with high directivity when compared to the existing methods [5].

Downlink multiuser beamforming optimization techniques for stochastic systems with imperfect channel state information are trained using a Deep Neural Network (DNN). This deep neural network model provides beamforming based on the imperfect observations received at the base station by the channel state information; thus step size, error signals, and antenna weights are adaptively computed to form a directional beam toward the main user and side lobes toward interfering users [6].

Energy harvesting nodes in the IoT devices are enabled with the use of multiantenna wireless power transmission. Hybrid energy beamforming architecture with single RF chain and analog phase shifter impairments are adopted because large antenna arrays operating at the radio frequency are bulky, expensive and space consuming. Analytical approximation solution for nonlinear RF energy harvesting model is considered using least squares estimator with the time allocation and transmit power for channel estimation and hybrid energy beamforming phases are computed [7].

The spectral efficiency gets doubled by using a full duplex communication system which operates on smart antenna architecture that allows every antenna to be shared between downlink transmissions and uplink transmissions where signal separation at the base station plays a vital role. Antenna beamforming is utilized to maximize the spectral efficiency by the weighted sum technique, which uses block coordinate descent to decompose the given problem into assignment problems; thus total spectral efficiency gains increase with the number of antennas in the full duplex communication system [8].

Adaptive filters and genetic algorithm are combined to optimize linear antenna array, which has (n) number of antenna elements in it. Genetic algorithm is used to find the distance and the progressive changes in the phase shifts to achieve the desired half power beamwidth values; similarly adaptive filters are used to maximize the radiation pattern in desired direction and eliminate interference in the other spatial directions [9].

The covariance matrix for adaptive beamforming is estimated from the data snapshots, which attenuates the uncorrelated noise components and mitigates the directional interference. The minimum variance distortionless response algorithm acts as the basis for the dominant mode rejection algorithm which replaces the minuscule eigenvalues in the covariance matrix with their respective average values without even altering the large eigenvalues, thus increases the white noise gain and signal to interference plus noise ratio while suppressing the loud interferers [10].

A novel gaussian triangular factor method is proposed which is based on vector subspaces and it performs the detection of the desired mobile users from the 3-D spatial domain by using a smart antenna system. The entire eigenspace is decomposed into the upper element and lower element factor then the computation of the power spectrum gives the peaks at the angle of the detection of the desired user. This method provided the optimal values for different performance parameters like disturbance error, time complexity, detection error, and resolution when compared to the existing methods [11].

The power consumption and hardware cost can be reduced by employing an antenna array with adaptive phase shifters. The continuous and discrete phase shifters are examined for interference suppression where the continuous phase shifters are concerned only with the turning phase problem, which is a quadratically constrained quadratic program by nature that can be solved by convex optimization technique by using semidefinite relaxation. Similarly, binary quadratic programming problem for the optimization of the adaptive beamformer is achieved through discrete phase shifters, thus discrete and continuous phase shifters can be mixed into a single scheme for beamforming, which optimizes the implementation cost and SINR [12].

Software defined radios are used in digital beamforming with adaptive cancellation of the interference and effects of multipath environments where the carrier offset frequency is a big issue and conventional algorithms fail to achieve their mark. This method improves the overall system efficiency by providing very directed beams toward the desired user without degrading the system performance factors and SINR without the need for synchronization at the receiver or transmitter [13].

The phase-only constraint is also known as the constant modulus constraint which is an NP-hard nonconvex optimization problem. Many existing methods solve it indirectly by its more tractable form by relaxing the constraints but the price is paid in terms of degrading the performance and huge computational cost. However, there is a need to solve the problem using the direct method without any relaxation using the Riemannian manifold optimization which is based on the conjugate gradient algorithm where the problem is first transformed into an unconstrained problem placed on a complex circle manifold and then the step size and the gradient descent directions are computed iteratively to achieve a reduction in one unit magnitude complexity and 4 dB null gain [14].

Multiantenna beamforming for the receivers and transmitters operating in a Rayleigh fading environment with impaired phase noise uses the reconfigurable intelligent surface. The SNR distribution is approximated as either gamma random variables or as the chi-squared non-central random variables with two or three independent random variables associated with it; thus the amount of fading experienced by the channel decreases linearly with N ≫ 1 [15].

The SNR of the received signal is improved by the use of acoustic beamforming which can be made adaptive by using spectral domain noise covariance matrix to form the beam pattern in each bin of the frequency and responds to any temporal changes. Nonstationary acoustic environments estimate the noise covariance matrix very challenging task, which is further used to construct a beamformer mechanism. A spherical microphone array is evaluated by measuring the room impulse responses and simulation method [16].

Mobile parameter roots lies on the unit circle lies for the MVDR beamformer which are not utilized very effectively, the proposed method intends to optimize the roots on the unit circle through a non-iterative process to obtain a solution in a closed form. Multiple one-dimension optimization problems are generated from (N) dimensional MVDR problem and perform excellently even when a limited number of snapshots are available to the base station [17].

Smart antenna technology gets an edge by utilizing the intelligent reflecting surface technique which aids in remote communication between the base station and a single user via connecting to an line of sight (LoS) link of the nearby intelligent reflecting surface system through signal reflection by multi-hopping technique. For a given beam route, set of active and passive beamformers at the base station (BS) is generated, which is solved using the optimization technique of a simple shortest path problem utilizing the graph theory [18].

Massive MIMO beamforming systems benefit greatly from the new addition of antennas to a conventional MIMO system, but they also suffer due to the contamination of the pilot because of the interference coming from the user in the nearby cell. This issue can be mitigated through a pilot allocation scheme by training the Convolutional Neural Networks (CNNs) to identify which users share the same pilot sequences to avoid the contamination of the pilot [19].

Using Capon’s algorithm signal of interest, coefficients of the beamformer, interferences, and steering vectors are computed for the nested subarray from the set of the large uniform linear array. Interference plus Noise Covariance Matrix (INCM) and augmented (INCM) are constructed through spatial smoothing operations and vectorization techniques, thus increasing the degrees of freedom of the nested array and reducing the complexity of implementation [20].

Multiuser downlink optimization of the beamforming problem arises due to the stochastic nature of the imperfect channel state information available at the base station. This issue can be resolved with the use of deep neural networks which only accept the links with perfect channel state information, thus training of DNN will give an efficient beamforming solution [21].

Existing methods are segregated as conjugate gradient and unigradient based methods; they are compared with the proposed method on different performance parameters as shown in the results section. Unigradient based techniques can perform elementary beamforming under non-erratic conditions, one such example is Linear Weight Beamformer (LWBF) method. But when operating under erratic conditions gradient based methods perform optimally when compared to their contrary; Unconstrained Gradient Beamformer (UGBF) and Low Complexity Beamformer (LCBF) methods are indicators of the gradient based beamforming techniques.

4.1 Linear Weight Beamformer (LWBF)

The Linear Weight Beamformer (LWBF) method is an unigradient based technique, the computation of weights is simple and very much straight forward, for non-erratic channel conditions this method performs elementary beamforming; the computation of the phase shifts are as follows.

$\begin{aligned} w_{L W B F}(n+1) & =w_{L W B F}(n) +\frac{A^{-1} \cdot l\left(\theta_0\right)}{l^H\left(\theta_0\right) \cdot A^{-1} \cdot l\left(\theta_0\right)} +\sum_{i=1}^{N_{J a m}} \frac{A^{-1} \cdot l\left(\theta_i\right)}{l^H\left(\theta_i\right) A^{-1} \cdot l\left(\theta_i\right)}\end{aligned}$              (12)

where,

A = Autocorrelation Matrix, $l\left(\theta_0\right)$ = Steering vector for desired user angle θ, $l\left(\theta_i\right)$ = Steering vector for jammer sequence, N_Jam = Number of Jammers.

4.2 Unconstrained Gradient Beamformer (UGBF)

The shortcomings of the LWBF method can be compensated with the utilization of unconstrained gradient optimization techniques served by Unconstrained Gradient Beamformer (UGBF) method, which uses the augmented covariance matrix of the input data to compute the phase shifts and next estimate values in an iterative way.

The step size for new angular orientation is very vital in finding the next angular orientation path which is computed as a ratio of current and previous gradient values, it is defined as following.

$\phi_k(n)=\frac{\Re\left[\zeta_k^H(n) \zeta_k(n)\right]}{\Re\left[\zeta_k^H(n-1) \cdot \zeta_k(n-1)\right]}$         (13)

where, ϕ_k(n) = step size for new angular orientation; $\zeta_{\mathrm{k}}, \zeta_{\mathrm{k}}^{\mathrm{H}}$ = gradient and its Hermitian transpose.

The next transmit direction s_k(n) values are computed based on the Eq. (14), which comprises the calculation of gradient and previous signal direction of transmission.

$s_k(n)=\zeta_k(n)+\phi_k(n) \cdot s_k(n-1)$              (14)

The previous and current signal transmit direction values along with the autocorrelation and cross correlation matrix aid in computation of the gradient pathway, which is represented in Eq. (15).

$\Gamma_k(n)=A(n) \cdot s_k(n)+K(n) \cdot s_k^*(n-1)$              (15)

The secondary weight updating function for the new estimate values are:

$\psi_k(n)=\frac{\Re\left[\zeta_k^H(n-1) \cdot s_k(n-1)\right]}{\Re\left[s_k^H(n) \cdot \Gamma_k(n)\right]}$           (16)

where, ψ_k(n) = step size for next estimate value; $\zeta_{\mathrm{k}}^{\mathrm{H}}$ = Hermitian transpose of the gradient; s_k, s_k^H = signal transmit direction, Γ_k(n) = gradient path.

The next update estimate value is an amalgamation of previous estimate value and transmit direction it is shown as:

$\xi_k(n+1)=\xi_k(n)+\psi_k(n) \cdot s_k(n)$              (17)

The Weight update equation is given as

$w_{U G B F}(n)=\frac{\xi_k(n)}{2 \cdot \mathfrak{R}\left[l^H\left(\theta_0\right) \cdot \xi_k(n)\right]}$             (18)

4.3 Low Complexity Beamformer (LCBF)

The Low Complexity Beamformer (LCBF) method is an adaptive beamforming method, which builds on top of the Unconstrained Gradient Beamformer (UGBF) method. A plethora of modifications was proposed to improve the performance based on misadjustment in the steady state. However, very few works were dedicated to improving the convergence rate and MSE values, which are dependent on the autocorrelation matrix.

The amount of previous and current gradient values that need to be added in the computation of the next angular orientation is judged by the step size.

$\phi_k(n)=\frac{\Re\left[\zeta_k^H(n) \cdot \zeta_k(n)\right]}{\Re\left[\zeta_k^H(n-1) \cdot \zeta_k(n-1)\right]}$           (19)

where, $\mathfrak{R}[\cdot]$ ← correlation operator, $\zeta_{\mathrm{k}}, \zeta_{\mathrm{k}}^{\mathrm{H}}$ = gradient and its Hermitian transpose.

Next beam transmit direction is computed using gradient and the product of step size with the previous signal transmit direction.

$s_k(n)=\zeta_k(n)+\phi_k(n) \cdot s_k(n-1)$             (20)

Pathway of the gradient is consolidation of correlation matrix and autocorrelation with present and previous beam transmit directions.

$\Gamma_k(n)=A(n) \cdot s_k(n)+K(n) \cdot s_k^*(n-1)$            (21)

Step size to compute the next new estimate values takes tolerance-value into consideration, which are ranging from 0 ≤ $\tau$ ≤ 0.823; this acts as a constraint to optimize the cost function to produce main beam in desired direction.

$\begin{aligned} & \psi_k(n) =\frac{\Re\left[s_k^H(n) \cdot \zeta_k(n-1)\right] \cdot(\tau)-\lambda \cdot \Re\left[s_k^H(n) \cdot l\left(\theta_0\right)\right]}{\Re\left[s_k^H(n) \cdot \Gamma_k(n-1)\right]}\end{aligned}$              (22)

where, $\tau$ = Tolerance value 0 ≤ $\tau$ ≤ 0.823.

New estimate values depend on largely on the step size, which will indicate the amount of previous transmit directions to be considered and the previous estimate value that needs to be added.

$\xi_k(n+1)=\xi_k(n)+\psi_k(n) \cdot s_k(n)$             (23)

Weight update equation plays a vital role in computing phase shifts, which has a direct effect on beam steering.

$w_{L C B F}(n)=\frac{\xi_k(n)}{2 \cdot \Re\left[l^H\left(\theta_0\right) \cdot \xi_k(n)\right]}$             (24)

where, $l^H\left(\theta_0\right)$ = Hermitian Transpose of steering vector for desired user angle θ, ξ_k(n) = next estimate value.

The design and implementation of the proposed Weighted Quadrigeminal Beamformer (WQBF) method are done from the ground up; and it is compared with the existing methods on different performance parameters such as robustness, accuracy, rate of convergence, System Error for Fixed Angle, mean and root mean squared errors.

The proposed method is capable to provide very accurate beamforming under the presence of heavy fading and noise. The antenna array weights were computed adaptively, the main beam was given to the desired user and nulls were presented to the interfering users in the 3-Dimensional spatial field.

Furthermore, it aided in the reduction of grating lobes and side lobes, which are the primary causes of signal leakage. The existing beamforming methods were outmatched by the superior performance delivered by the proposed WQBF method; the different performance parameters are MSE (0.2501), RMSE (1.4078), Beam Directivity (very high), Time Complexity (0.7316 sec) and Rate of Convergence (14 Iterations, MSE = 0.2501).