# 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
\[c=\alpha-\frac{1}{\tau}\ln(1-\frac{1}{1+\delta_{M}})\approx\alpha-\frac{1}{ \tau}\ln(\delta_{M}) \tag{10}\]  
This formula is valid, with a relative error achieved by the NILT \(\widetilde{f}(t)\), if infinite numbers of terms are used in series, and is a suitable technique for accelerating a convergence and for achieving the convergence of infinite series in a suitable way. Equation (3) can be rewritten using FFT and IFFT algorithms for an effective computation. Based on the experience of the authors of [31], the quotient-difference (q-d) algorithm of [PERSON] seems to give errors rather close to \(\delta_{M}\) predicted by Equation (10), while considering a relatively small number of additional terms. While considering a discrete variable in the original domain, \(t_{k}=kT\), where T is a sampling period, \(\widetilde{f}(t)\) could be expressed as:  
\[f(t)=\frac{1}{2\pi j}\int\limits_{c-j\infty}^{c+j\infty}F(s).e^{st}ds \tag{2}\]0  
The above stated formula could be decomposed as:  
\[f(t)=\frac{1}{2\pi j}\int\limits_{c-j\infty}^{c+j\infty}F(s).e^{st}ds \tag{2}\]1  
where \(N=2^{k}\), \(k\) integer, \(\widetilde{F}^{(\pm n)}=\widetilde{F}(c-jn\Omega)\), \(\widetilde{G}^{(\pm n)}=\widetilde{F}^{(\pm N\pm n)}\), \(z_{\pm k}=\exp(\pm j\frac{2 nkT}{\tau})\), and \(C_{k}=\frac{\exp(ckT)}{\tau}\), while \(\tau=N\tau\), \(\forall k\), and \(z_{kk}^{N}=\exp(\pm j2\pi k)=1\). In Equation (12), the first and the third sum are evaluated using the FFT and IFFT algorithms, respectively, while other parts, which present the infinite sum, are used as the input data in the q-d algorithm that uses a very small number of necessary additional terms, as explained in [24]. The computing region should be chosen as: \(O_{cal}=(0,t_{cal})\), where \(t_{cal}=\left(\frac{N}{2}-1\right)\). Time-domain noise coupling could be easily obtained by the explained method in 3D technology based on TSVs. In order to compute the noise coupling, different circuits were treated. The first structure is illustrated in \[f(t)=\frac{1}{2\pi j}\int\limits_{c-j\infty}^{c+j\infty}F(s).e^{st}ds \tag{2}\]2 This figure represents a basic structure of the TSV-TSV noise coupling [3]. It is composed of two signal TSVs, two ground TSVs, and is terminated by I/O drivers. The simplified lumped circuit model of this structure is given in Figure 2, where \(C_{TSV-equiv}\) is the total equivalent TSV capacitance, \(R_{sub-equiv}\) is the substrate resistance, and \(C_{sub-equiv}\) is the substrate capacitance. In this simplified model, proposed in [3], the TSV resistance (\(R_{TSV}\)), the TSV inductance (\(L_{TSV}\)), and the depletion region are neglected, but in our work \(R_{TSV}\) and \(L_{TSV}\) are kept. In the study just mentioned, the authors assume that their effects appear in frequencies above 12 GHz. To consider the effect of the depletion region, which is modeled by a capacitance, it is enough to add its value to the TSV capacitance. The I/O drivers can be modeled as a resistor for the output driver and as a capacitor for the input driver that represents the MOS gate capacitance. The I/O drivers are presented by the impedances \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), and \(Z_{4}\). To apply the NILT method, the conceptual structure can be modeled with a T-matrix, as illustrated in the figure. The entire matrix of the circuit is the product of \(T_{1}\), \(T_{2}\), and \(T_{3}\), as defined below. \[\left( \begin{array}{c}V_{1}\\ I_{1}\end{array} \right)=\left[T\right]\;\left( \begin{array}{c}V_{2}\\ -I_{2}\end{array} \right) \tag{13}\]  
where:  
\[\left[T\right]=\left[T_{4}\right]\left[T_{1}\right]\left[T_{2}\right]\left[T_{ 3}\right]\left[T_{4}\right] \tag{14}\]  
\[\left[T_{1}\right]=\left[ \begin{array}{cc}1&0\\ 1/\left(Z_{2}+\frac{R_{tsv}}{2}+s\frac{L_{tsv}}{2}\right)&1\end{array} \right] \tag{15}\]  
\[\left[T_{2}\right]=\left[ \begin{array}{cc}1&Z_{eq}\\ 0&1\end{array} \right] \tag{16}\]  
\[\left[T_{3}\right]=\left[ \begin{array}{cc}1&0\\ 1/\left(Z_{3}+\frac{R_{tsv}}{2}\right)+s\frac{L_{tsv}}{2}&1\end{array} \right] \tag{17}\]  
\[\left[T_{4}\right]=\left[