## 7 Discussion  
The source identification problem arises in many applications, such as inverse electroencephalography, inverse electrocardiography, and inverse geophysics. An FPGA is an integrated circuit that allows the implementation of algorithms and other complex mathematical operations. In this work, we implemented a source identification algorithm considering a non-homogeneous medium made up of two homogeneous media and with sources located on the interface separating the homogeneous media. The algorithm is expressed as a Fourier series expansion that considers the circular geometry.  
Since the source identification problem presents numerical instability, we used the Tikhonov regularization method and a cut-off of the Fourier expansion to handle the numerical instability. The Tikhonov regularization parameter and the truncation term were chosen through numerical testing.  
This work presents a hardware architecture with two designs to implement the source identification algorithm. These architectures were tested on four FPGAs from the 7 Series of Xilinx: Spartan-7 xc7s100 fgga484, Virtex-7 xc7v585 fffg1157, Kintex-7 xc7k70 tfbg484, and Artix-7 xc7a35 tcpg236. Each architecture contains two channels, each of which is composed of two sections: one for the arithmetic operations and one for the control of the operations. We report the resource consumption of the 7 Series of Xilinx FPGAs. These resources include LUTs, RAM blocks, FFs, DSP blocks, and power consumption. One of the developed architectures can be used for problems with a larger number of channels. One example of this case is inverse electroencephalography, for which many inverse source problems must be solved per second.  
To validate the hardware implementation, we constructed three synthetic examples in MATLAB. The results of both the hardware and software implementations were then compared in terms of error. These synthetic examples correspond to three types of functions that encompass a wide range of sources.  
The first example is a trigonometric polynomial, while the second is a smooth function with rapidly decaying Fourier coefficients. The third example is a jump function, which represents a different class of functions. In this last case, the Fourier series exhibits the Gibbs phenomenon at the jump point. However, the regularization technique applied here acts as a smoothing mechanism, eliminating the Gibbs phenomenon in the recovered source.  
As mentioned in the article, we conducted additional software and hardware implementations using similar functions and observed comparable results.  
Given that we are considering two regularization parameters, we performed numerical tests to identify the pair that most effectively recovers the sources. We found that when the Tikhonov regularization parameter is fixed and the truncation parameter increases, the error between the exact source and the recovered source also increases. Similarly, when the truncation parameter is fixed, the results obtained for other regularization parameters are less accurate. Moreover, the precision of the hardware implementation does not improve when considering other pairs of parameters close to the chosen pair.