# Numerical Laplace Inversion Method for Through-Silicon Via (TSV) Noise Coupling in 3D-IC Design
## 2 Calculation of Time-Domain TSV Noise Coupling in 3D-IC Design with NILT  
The use of the Laplace transform method has simplified the solution of transients on transmission lines (TL), of transients of dynamic systems, and other problems in electrical engineering. However, some difficulties appear when transforming solutions to the time domain. This makes researchers concerned to find accurate and precise numerical methods. One of these numerical methods is the numerical inverse Laplace transform (NILT) method, which can be used in cases when, for instance, the transform is a transcendental, irrational or some other complex function; then finding the solution in its analytical form is difficult and sometimes impossible [26; 27]. The NILT method has been used in several works. In [28], NILT methods were selected to evaluate their performance for dealing with solution transportation in the subsurface under uniform or radial flow conditions. The authors of [29] evaluate and compare some numerical algorithms of the NILT method for the inversion accuracy of some fractional order differential equation solutions. In [30; 31; 32; 33; 34; 35] the multidimensional NILT method has been explained in detail for electrical circuits. In this paper, we were interested in 1D-NILT. Thus, a one-dimensional Laplace transform of a function \(f(t)\), with; \(t\geq 0\), is defined as:  
\[F(s)=\int\limits_{0}^{\infty}f(t)e^{-st}dt \tag{1}\]  
Under the assumption \(\left|f(t)\right|\leq Me^{st}\), M is real positive, \(\alpha\) is a minimal abscissa of convergence, and \(F(s)\) is defined on a region \(\left\{s\in C\,:\mathrm{Re}[s]>\alpha\right\}\), with \(s=c+j\Omega\), c is defined as an abscissa of convergence, \(\Omega=\frac{2\pi}{\tau}\) as the generalized frequency step, and \(\tau\) forms a region of the solutions \(t\in[0\,\tau]\). The original function can be given using the Bromwich integral [36]:  
\[f(t)=\frac{1}{2\pi j}\int\limits_{c-j\infty}^{c+j\infty}F(s).e^{st}ds \tag{2}\]  
By using a rectangular rule of integration as mentioned in [30], Equation (3) is found. \[\tilde{f}(t)=\frac{\exp(ct)}{\tau}\ \sum\limits_{n=0}^{\infty}F(s)\exp(jn \Omega t) \tag{3}\]  
As explained in [30], by substituting \(s=c+jn\Omega\) into Equation (1), if the obtained function has integration ranges split into infinite numbers of steps of the length \(\tau\), \(F(s)\) could be written as:  
\[F_{n}=F(c+jn\Omega)=\sum\limits_{l=0}^{\infty}\ \int\limits_{l\tau}^{(l+1)\tau} g(t)\ \exp(-jn\Omega t)\ dt \tag{4}\]  
\(g(t)\) is an exponentially damped object function. Then for \(t\in[l\tau,\tau(l+1)]\), the functions \(g_{l}(t)\) and \(F(s)\) are given by:  
\[g_{l}(t)=f(t)\exp(-ct) \tag{5}\]\[F(c+jn\Omega)=\tau\sum_{l=0}^{\infty}C_{l,n} \tag{6}\]  
where:  
\[C_{l,n}=\frac{1}{\tau}\int\limits_{l\tau}^{(l+1)\tau}g_{l}(t)\ \exp(-jn\Omega t)dt \tag{7}\]  
Applying complex Fourier series to Equation (5), \(g_{l}(t)\) could be found as:  
\[g_{l}(t)=\sum_{n=-\infty}^{+\infty}C_{l,n}\exp(jn\Omega t) \tag{8}\]  
Moreover, by substituting Equation (6) into Equation (3) and considering Equation (8), it is found that the approximate original function exponentially damped could be expressed as the infinite sum of the newly defined periodical function, Equation (5). By exploiting all the previous equations, \(\widetilde{f}(t)\) is obtained and the absolute error \(\varepsilon(t)=\widetilde{f}(t)-f(t)\) can be computed. \[\widetilde{f}(t)=f(t)+\sum_{l=1}^{\infty}f(l\tau+t).\exp(-cl\tau) \tag{9}\]  
A limiting absolute error is determined as \(\varepsilon_{M}(t)\geq\varepsilon(t)\), then \(\left|f(t)\right|\leq Me^{at}\), so a limiting relative error \(\delta_{M}\) could also be controlled, and a path of integration from a required limit relative error could be chosen using Equation (10).