## 1 Introduction  
In this paper, we propose two architectures based on field-programmable gate arrays (FPGAs) [1] for implementing an inverse source identification algorithm. The inverse source identification problem that we address involves determining the source from measurements produced by it on the exterior boundary. The inverse source problem has many applications in practical problems such as inverse electroencephalography and inverse electrocardiography, for which study has been conducted using operational equations defined in Hilbert spaces [2; 3]. These operational equations are ill-posed in the sense of Hadamard [4]. Here, we solved the operational equation, from which, the algorithm was obtained using Fourier series. More precisely, we solved the normal equations obtained using the Tikhonov regularization method. Some inverse source problems lead us to algebraic systems of linear equations, which are obtained from the operational equations by discretization [5]. The matrices in these systems of equations are ill-conditioned, which must be considered when we solve the system with error on the right side because the systems are unstable, which is a consequence of the ill-posedness of the operational equations. At this point, it is important to consider that handling precision inadequately might worsen the results, as the solution found for data with errors might be too far from the one for data without errors in addition to increasing other costs via hardware resources and critical paths. In [6], the authors pointed out that using \(2\times 2\) matrices to show how ill-conditioning and precision can affect system design (resources, cost, etc. ), and they illustrated the effect generated in the calculation of the inverse of an ill-conditioned matrix when its elements were approximated by truncation. We consider a circular, non-homogeneous medium and that the sources are located on the separation interface of two homogeneous media, which make up the non-homogeneous media. Sources and measurements are correlated by a boundary value problem, which allows us to make an operational statement from where the ill-posedness (in the sense of Hadamard) is analyzed, and the source identification algorithm is obtained. We consider the algorithm given in [7], in which the inverse source problem was developed for sources located on the interface of two homogeneous media. The authors developed the algorithm using the technique of Fourier series for circular geometry and the finite element method for a complex geometry. Since we are considering a circular geometry, we implemented the algorithm in an FPGA using the Fourier series technique, i.e., we used the trigonometric base to express the solution to the inverse problem. Ill-posedness is related to numerical instability, which can produce significant changes in the solution due to small measurement changes. The Tikhonov regularization method is employed to handle this numerical instability; it depends on a parameter called the Tikhonov regularization parameter, which must be properly chosen in terms of the measurement error [5]. We chose the Tikhonov regularization parameter numerically. Associated with the trigonometric base is an arithmetic kernel, which allows hardware resources to be reused in a repetitive type system to reduce the number of operations [8]. The proposed architectures, labeled Mode 1 and Mode 2, were tested on four FPGAs of the 7 Series of Xilinx [1]: Spartan-7 xc7s100 fgga484, Virtex-7 xc7v585 fffg1157, Kintex-7 xc7k70 ffbg484, and Artix-7 xc7a35 tcpg236. We report on the performance of architectures Mode 1 and Mode 2 in terms of power and resource consumption. The two architectures can also be replicated, leading to a multichannel structure. This offers the possibility of applying these architectures to problems for which their nature is multichannel, such as identifying bioelectrical sources from electroencephalographic or electrocardiographic signals.