## 3 The Key Technologies of Extracting Transmission Characteristics
### The Input Signal Estimation  
Trapezoidal signals were chosen as the wideband input of the printed trace because they can generate a significant number of electromagnetic emissions in most digital circuits and switched-mode power supplies. A trapezoidal signal in the time-domain is illustrated in Figure 5. These parameters, including amplitude \(A\), period \(T\), rise time \(\tau_{r}\), high-voltage time \(t_{A}\), and fall time \(\tau_{f}\), are used to express the trapezoidal signal in the time domain. Based on the positive frequency Fourier series [27], the trapezoidal signal can be written as  
\[v(t)=c_{0}+\sum_{n=1}^{+\infty}|c_{n}|\cos(2n\pi f_{0}t+\angle c_{n}),n\in N, \tag{12}\]  
\[c_{n}=-j\frac{A}{\pi n}e^{-jn\pi(1.5D_{1}+D_{2}+0.5D_{3})}\]  
\[[Sa(n\pi D_{1})e^{jn\pi(0.5D_{1}+D_{2}+0.5D_{3})}-Sa(n\pi D_{3})e^{-jn\pi(0.5D_ {1}+D_{2}+0.5D_{3})}], \tag{13}\]  
where \(f_{0}=1/T\) denotes the fundamental frequency of the signal, \(nf_{0}\) denotes harmonics of that fundamental frequency, \(c_{n}\) represents the expansion coefficients (they may have a complex value, denoted by \(c_{n}=|c_{n}|\angle c_{n}\)), and \(D_{1}\), \(D_{2}\), and \(D_{3}\) denote the proportions of the rise time, high-voltage time, and fall time, respectively.  
When the amplitudes of the emission spectrum are obtained for \(f<f_{c}\), according to Equation (8), the spectrum of the input signal for \(f<f_{c}\) is also obtained. Then, we extract the time parameters from the partial spectrum of the input signal. Since the expansion coefficients are in terms of cyclic frequency \(f_{0}\), when the fundamental frequency \(f_{0}\) is extracted, the amplitude of the spectral components \(|c_{n}|\) can be obtained. Then, according to Equation (12), the parameters \(D_{1}\), \(D_{2}\), and \(D_{3}\) are extracted. To guarantee the high accuracy of the reconstruction, \(f_{c}\) must be at least greater than \(3f_{0}\).  
The fundamental frequency \(f_{0}\) can be calculated using the method in [8], which is based on the autocorrelation function. For periodic signal \(f(t)=f(t+T)\), the autocorrelation function \(R(\tau)\) can be written as  
\[R(\tau)=\int_{-\infty}^{+\infty}f(t)f(t-\tau)dt=\int_{-\infty}^{+\infty}f(t) f(t-(\tau+T))dt=R(\tau+T). \tag{14}\]  
Therefore, the autocorrelation function of the periodic signal is also periodic with the same repetition period. For the superposition of several periodic signals, when \(\tau=nT\), \(n\in N\), \(R(\tau)\) will reach maximal values, the period \(T\) can thus be extracted by calculating the autocorrelation function. Then, the parameters \(D_{1}\), \(D_{2}\), and \(D_{3}\) can be obtained by fitting the practical measured harmonic amplitudes \(|A_{n}|\) into the AVDO algorithm [27]. When the initial values of the parameters \(D_{1}\), \(D_{2}\), and \(D_{3}\) are obtained, the first calculated harmonic amplitudes \(|c_{i}|\) according to Equation (12) are compared with the practical measured \(|A_{i}|\). If the root-mean-square error (RMSE) disagrees with the setting condition, the parameters must be changed and compared repeatedly. The iteration will be finished until it agrees with the RMSE setting, such as  
Figure 5: The periodic trapezoidal signal and its time domain parameters.  
\[\text{Minimize}\;error_{RMSE}=\sqrt{\frac{1}{N}{\sum_{i=1}^{n}\left[20\log 10(\left| \overline{c_{i}}\right|-\left|\overline{A_{i}}\right|)\right]^{2}}}, \tag{15}\]  
where \(\overline{A_{i}}\) denotes the normalization amplitudes of the \(i\)th practical measured harmonic amplitude with decibel form and \(\overline{c_{i}}\) is the normalization amplitude of the \(i\)th predicted harmonic amplitude. Finally, the parameters \(D_{1}\), \(D_{2}\), and \(D_{3}\) are exported. Table 1 shows the results. The error is no more than 3%, indicating that the method can extract the time domain parameters from spectrum amplitudes accurately. When \(D_{1}\), \(D_{2}\), and \(D_{3}\) are extracted, the amplitudes of harmonic amplitudes in the whole measured band can be obtained according to Equation (12).