# Implementation of the Digital QS-SVM-Based Beamformer on an FPGA Platform

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## Abstract

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## 1. Introduction

- Recent FPGA technology has programmable logic and the capability of algorithm parallelization for further enhancing power consumption, flexibility, and accuracy.
- Recent advances in the FPGA architectures include a higher storage density, a drastic reduction in power consumption and cost, a large number of gates, and a high-performance processor.
- Recent FPGA software and high-level optimizations have to be accompanied by architectural changes in the FPGA board in order to satisfy drastic computations of SVM-based applications. Advances in FPGA technology have rigorously presented high-level software tools to be easily adjusted to the FPGA hardware.

- For the first time, the QS-SVM-based beamformer has been implemented using the hybrid antenna array with bowtie elements on an FPGA board.
- For the first time, this work presents an implementation of the proposed digital beamformer in both the real environment and hardware board.
- The implementation of the QS-SVM optimization method for the DoA estimation on an FPGA board has been rigorously demonstrated for the first time.
- We have achieved a superior performance of the digital QS-SVM-based beamformer in terms of beamforming, nullsteering, and beamsteering.
- A performance evaluation of the QS-SVM-based beamformer has been fulfilled in terms of throughput, latency, and performance efficiency.

## 2. Literature Review and Related Work

## 3. Proposed Methodology and Techniques for the Spatial Signal Processing

#### 3.1. Hybrid Antenna Array

#### 3.2. Methodology and Theoretical Framework

## 4. Methods of Modeling and Producing Data

#### 4.1. The Proposed Beamforming Technique

#### 4.2. The QS-SVM Technique for the DoA Estimation

**The training procedure:**The supervised learning algorithms are designed to construct a model within the training phase. The machine learning algorithm analyzes the training dataset to generate parameters for providing quadratic classifier surfaces such that a maximum margin engenders between each of the two different classes of data. Therefore, the output of the training procedure is a model trained by the training dataset. Suppose a non-linearly separable training dataset with correct values of outputs are in pair with correct values of inputs in the following form,

**Condition #1:**${\mathbf{x}}^{i},{\mathbf{y}}^{i}=-1,$

**Condition #2:**${\mathbf{x}}^{i},{\mathbf{y}}^{i}=+1,$

**The testing procedure:**The testing dataset, or the remainder of the dataset which was not employed for the training dataset, is fed to the model. As soon as the testing dataset is fed to the model, the model becomes fixed such that it cannot change anymore. Then, the quadratic classifier surfaces are used to separate data points of the testing dataset. In other words, to assess how well the model can process the real-world data and generate accurate predictions, we employ the unseen dataset, or testing dataset. In the mathematical sense, the decision function of ${D}_{ij}\left(x\right)$ can be expressed by,

## 5. Implementation Setup of the QS-SVM-Based Beamformer on the FPGA Board

#### 5.1. Real Environment and Software Implementation

#### 5.2. Hardware Environment and FPGA Implementation

**Quadrature Programming Solver:**The SVM algorithm allows for estimating a function, which maps the input data to a finite set of output labels [5,6] as discussed in detail in the previous section. Hence, the QS-SVM-based beamformer can overcome all constraints of the spatial reference techniques and achieve superior performance in terms of SINR. There is no information about the probability distribution of the input data. Therefore, we have to minimize errors between the array output and the reference signals (or here desired signals) for the number of available data in the left-hand side of Equation (22),

**Weight Updating:**Parameters of a wireless communication channel have to vary quickly, thereby weight updating should be performed at a higher rate than a statistic scenario. An alternative solution for updating the weight vector consists of directly computing the inverse of the correlation matrix using the MVDR algorithm. The primary goal of the MVDR beamforming technique is to minimize SINR expressed by the following formula [21,22],

**SVM Inner Product:**Hadamard multiplication (or element-wise multiplication) refers to the multiplication of array objects, which includes both of the Hadamard product of $a\ast b$ and the matrix product of $a\phantom{\rule{1.42262pt}{0ex}}@\phantom{\rule{1.42262pt}{0ex}}b$, as demonstrated in Figure 9 and Figure 11.

## 6. Results of the QS-SVM-Based Beamformer on the FPGA Board

- Null steering for undesired signals by replacing nulls of the radiation pattern of the proposed hybrid antenna array in the detected directions of undesired signals. Hence, we can weaken significantly or eliminate undesired signals.
- Keeping the desired signal unchanged by exerting power with the 0dB level in the detected direction of the desired signal. We should neither strengthen nor weaken the desired signal, due to the following two reasons: (1) since the desired signals may include noise, jamming, interference, and other unwanted signals, any amplification in the desired signal results in magnifying noise and other unwanted signals, and (2) any reduction in the desired signal is not of practical interest.

## 7. Performance Evaluation of the FPGA-Based Beamformer

#### 7.1. Throughput Evaluation

#### 7.2. Latency Evaluation

#### 7.3. Performance Efficiency

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of Open Access Journals |

TLA | Three Letter Acronym |

LD | Linear Dichroism |

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**Figure 1.**A demonstration of the 3D configuration of the hybrid antenna array with bowtie elements at the frequency of the operation of 10 GHz with bowtie elements. (

**a**) The assigned design parameters of the hybrid antenna array are listed in detail in Table 1. In this work, each hybrid antenna array consists of three cylindrical antenna arrays [15], and one circular antenna array [16], and (

**b**) a top view of the bowtie antenna; l: arm length, $\alpha $: flare angle, g: feed gap, w: bar width, b: bar length. In this work, the design parameters of each bowtie element are assumed to be l = 6 mm, $\alpha ={60}^{\circ}$, g = 0.02 mm, w = 0.02 mm, and its thickness is equal to 0.01 mm [7].

**Figure 2.**Procedures for the DoA estimation using the QS-SVM-based digital beamformer. It is assumed that there are M numbers of the proposed hybrid array antennas in Figure 1b of [7]. Each hybrid antenna array consists of 140 elements whose outputs have been measured. The proposed QS-SVM algorithm for the DoA estimation measures outputs of N array elements and predicts the directions of L signals impinging on the array elements. In this figure, position vectors are supposed to be in a two-dimensional plane of $\theta $ and $\varphi $. The process of the DoA estimation is to monitor the N outputs of antenna elements and predict the angle of arrival of L signals, $l=1,2,\dots ,L$. $N=M\times 140$. M = 5 in this work. ${K}^{\prime}$ denotes a number of sources that we can obtain after the DoA estimator.

**Figure 3.**(

**a**) A demonstration of the proposed QS-SVM-based digital beamformer. Here, position vectors are assumed to be in a plane of $\theta $ variable and $\varphi $ constant. An angle of ${45}^{\circ}$ is the direction of a desired signal, and the steering angle should be identical to it. However, angles of ${30}^{\circ}$, and ${50}^{\circ}$ represent the directions of undesired signals, and (

**b**) a demonstration of the proposed QS-SVM-based digital beamformer in the real and hardware environments.

**Figure 4.**A demonstration of the spatial LCMV beamforming technique in the real environment integrated with the FPGA blocks. The direction of the desired source is ${45}^{\circ}$. However, two other directions of ${50}^{\circ}$ and ${30}^{\circ}$ impinge on the hybrid array antennas as unwanted signals. The LCMV beamforming technique is capable of spatially filtering unwanted signals.

**Figure 5.**A demonstration of the complex Q-less QR forward-backward substitute on the FPGA board. In this work, Quadratic programming (QP) has been employed for solving the QS-SVM algorithm. The QP solver refers to the mathematical problem of finding weight vectors $\mathbf{w}$ in Equation (11) with respect to $f\left(\mathbf{x}\right)=0$ and under the aforementioned constraints of Equation (17) and (18).

**Figure 6.**The aforementioned subsystem in Figure 4, the LCMV beamforming technique, and other mathematical operations in the real environment. The Simulink demonstration of all mathematical operations in the MATLAB platform with detail.

**Figure 7.**Hardware implementation of the proposed digital SVM-based beamformer on the FPGA platform.

**Figure 8.**The quadrature programming solver for solving the QS-SVM problem in the HDL implementation.

**Figure 9.**(

**a**) A hardware demonstration for the inner products in the QS-SVM-based beamformer using HDL, (

**b**) QS-SVM inner product, and (

**c**) HDL complex multiplication.

**Figure 10.**The hardware implementation of HDL complex multiplications of the QS-SVM optimization method.

**Figure 11.**The spatial filtering performance of the QS-SVM-based digital beamformer on the FPGA board. The obtained results are associated with the part of the pattern plot in Figure 3.

**Figure 12.**Variation of throughputs of the classification performance of the QS-SVM-based beamformer in terms of SNRs consistent with Figure 3.

**Figure 13.**Latency analysis for different batch sizes consistent with Figure 3. Latency is presented in milliseconds and averaged over the test set of the network.

**Table 1.**The design parameters for the proposed hybrid antenna array of Figure 1.

Parameters | Definition | Value |
---|---|---|

${N}_{h}$ | Number of elements of any circular loop | ${N}_{h}=20$ |

${Q}_{h}$ | Number of elements of any cylinder | ${Q}_{h}=40$ |

${M}_{h}$ | Total number of cylinders in the proposed array | ${M}_{h}=3$ |

${P}_{h}$ | Number of circular loops in the cylinder | ${P}_{h}=2$ |

${d}_{v}$ | Vertical spacing between two consecutive circular loops | ${d}_{v}=0.5\lambda $ |

${d}_{r}$ | Horizontal spacing between two consecutive circular loops | ${d}_{r}=0.5\lambda $ |

$\varphi ,\theta $ | Maximum scanning angles | $\varphi ={45}^{\circ},\theta ={45}^{\circ}$ |

Antenna array with bowtie elements | Antenna array with dipole elements | |

Performance efficiency of the proposed QS-SVM beamformer | 96% | 75% |

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**MDPI and ACS Style**

Komeylian, S.; Paolini, C.
Implementation of the Digital QS-SVM-Based Beamformer on an FPGA Platform. *Sensors* **2023**, *23*, 1742.
https://doi.org/10.3390/s23031742

**AMA Style**

Komeylian S, Paolini C.
Implementation of the Digital QS-SVM-Based Beamformer on an FPGA Platform. *Sensors*. 2023; 23(3):1742.
https://doi.org/10.3390/s23031742

**Chicago/Turabian Style**

Komeylian, Somayeh, and Christopher Paolini.
2023. "Implementation of the Digital QS-SVM-Based Beamformer on an FPGA Platform" *Sensors* 23, no. 3: 1742.
https://doi.org/10.3390/s23031742