## 2 The Principle of Extracting Transmission Characteristics  
For the near-field emission analysis, a wideband microstrip line electric probe is used [23]. As shown in Figure 1, a common near-field probe measurement system is illustrated. The printed trace, including straight lines, corners, and vias, is excited by a signal generator and matched by a load of 50 \(\Omega\). The probe is placed above the trace and connected to an SA to obtain the near-field emission spectrum. The microstrip line electric probes, similar to the coaxial probes, utilize capacitive coupled probing [24]. The coupling capacitance between the printed trace and the probe tip is \(C_{p}\). The voltage generates a time-varying electric field that induces a current signal \(i_{c}\) in \(C_{p}\) flowing to the detecting tip. The current can be calculated as  
\[i_{c}=C_{p}A\frac{d\overset{\rightarrow}{E}(t)}{dt}, \tag{1}\]  
where \(\overset{\rightarrow}{E}(t)\) is the electric field strength, \(t\) denotes time variable, and \(A\) is the system factor. Based on the theory [25], the system consisting of the electric probe and the printed trace can be presumed to be a linear and time invariant (LTI) system. The principle of the extraction of transmission characteristics from a spectrum has two important components. Firstly, as shown in Figure 1, the input voltage \(v_{in}(t)\) of the LTI system on port 1 is a wideband signal, \(v_{l}(t)\) is the voltage on the connector of the printed trace (port 2), and the output is the induced voltage \(v_{p}(t)\) on the probe tip (port 3). Through convolution, the relationships between the voltages \(v_{in}(t)\), \(v_{l}(t)\), and \(v_{p}(t)\) can be written as  
\[v_{l}(t)=\int_{-\infty}^{\infty}v_{in}(t)(t-\tau)h_{l}(\tau)d\tau, \tag{2}\]and  
\[v_{p}(t)=\int_{-\infty}^{\infty}v_{l}(t)(t-\tau)h_{t}(\tau)d\tau, \tag{3}\]  
where \(h_{l}(\tau)\) is the response of the printed trace and \(h_{t}(\tau)\) is the response of the induction process. By converting them to the frequency domain, we have  
\[V_{L}(f)=V_{IN}(f)H_{L}(f), \tag{4}\]  
and  
\[V_{P}(f)=V_{L}(f)H_{T}(f), \tag{5}\]  
where \(V_{IN}(f)\), \(V_{L}(f)\), and \(V_{P}(f)\) denote the Fourier transforms of \(v_{in}(t)\), \(v_{l}(t)\), and \(v_{p}(t)\), respectively. \(H_{L}(f)\) is the transmission coefficient of the printed trace, and \(H_{T}(f)\) is the transfer function of the induction process shown in Figure 1. When one is considering the whole system, according to Equations (3) and (4), \(H_{L}(f)\) can be derived as  
\[H_{L}(f)=\frac{V_{P}(f)}{V_{IN}(f)H_{T}(f)}. \tag{6}\]  
In Equation (5), \(V_{P}(f)\) can be measured by an SA; \(H_{T}(f)\) can be measured by a VNA or a simulation or a model. Once \(V_{IN}(f)\) is determined, the transmission characteristics of the interconnects can be calculated from Equation (6). Therefore, next, we discuss how to reconstruct the input signal \(V_{IN}(f)\) from the output \(V_{P}(f)\) and how to extract the transmission characteristics from the near-field emission. Secondly, for printed traces, considering the quality of signal transmission, they are commonly low-pass linear systems. The transmission characteristic function \(H_{L}(f)\) of the interconnects can be written as  
\[H_{L}(f)=\left\{ \begin{array}{l}H_{P}(f),f<f_{c}\\ H_{C}(f),f\geq f_{c}\end{array} \right., \tag{7}\]  
where \(H_{P}(f)\) is the transmission coefficient in the pass band, \(H_{C}(f)\) is the transmission coefficient in the stop band, and \(f_{c}\) is the cut-off frequency. In the pass band, there is virtually no attenuation. Hence, after normalization, we have  
\[\left|H_{P}(f)\right|\approx 1\ \text{or}\ \left|H_{P}(f)\right|_{dB}\approx 0, \tag{8}\]  
Figure 1: Analysis model of a microstrip line electric probe and printed trace system. where \(|H_{P}(f)|_{dB}\) is the decibel form of the transmission coefficient in the pass band.