## 2 Proposed Methodology
### Waveform Imaging Using a Modified Recurrence Plot  
Recurrence plot (RP) is a method that can visualize the periodic nature of dynamical systems. To visualize a trajectory of the dynamical system in \(m\)-dimensional space (\(m\): integer), the trajectory should be projected on a 2D or 3D sub-space. For example, a trajectory of the dynamical system in a one-dimensional (1D) space can be projected to a 2D or 3D space. RPs are advantageous over other signal processing techniques (e.g., FT, WT, and WVD) since RPs can capture various patterns such as stationary, periodic/quasi-periodic, cyclostationary, and fluctuations. FT fails to capture the characteristic of cyclostationary waveforms. In this study, the trajectory in phase space corresponds to the dynamic resistance waveform acquired from a resistance spot welding system. The RP method enables conversion of waveforms into 2D images. Figure 2: Flowchart of the proposed methodology. Let \(x(i)\) be the \(i^{\text{th}}\) data point of the 1D waveform where \(i=1,\ldots,K;K\) is the number of data points. In the same manner, \(x(j)\) is the \(j^{\text{th}}\) data point of the 1D waveform, where \(j=1,\ldots,K\). The RP matrix at the \(i^{\text{th}}\) row and the \(j^{\text{th}}\) column is defined as [18]:  
\[R_{\text{o}}(i,j)= \begin{cases}1&\|x(i)-x(j)\|\leq\varepsilon\\ 0&\text{otherwise}\end{cases}  \tag{3}\]  
where \(\varepsilon\) is the predefined threshold (\(\varepsilon>0\)). When \(x(j)\) is sufficiently close to \(x(i)\), one is placed at the \(i^{\text{th}}\) row and the \(j^{\text{th}}\) column of the matrix. Otherwise, zero is placed. The original RP (\(R_{\text{o}}\left(i,j\right)\)) was introduced to present the closeness of two data points whose output is binary. However, the original RP cannot present the evolution of states between two data points, since the outcome from the original RP is either one or zero. In reality, dynamic resistance waveforms from spot welding systems show gradual changes due to the dynamic nature of the systems, as well as due to external noise. To incorporate the evolution of the system behavior, a modified RP is proposed:  
\[R_{\text{p}}(i,j)= \begin{cases}N&\|x(i)-x(j)\|\geq N\\ \|x(i)-x(j)\|&\text{otherwise}\end{cases}  \tag{4}\]  
where \(i\) and \(j\) are the vectors of the waveform data points, respectively; \(N\) is the predefined threshold (\(N>0\)); and \(R_{\text{p}}\left(i,j\right)\)) is the proposed RP matrix at the \(i^{\text{th}}\) row and the \(j^{\text{th}}\) column. The proposed RP is devised to give a fixed value of \(N\) when the difference between \(x(i)\) and \(x(j)\) is larger than the threshold \(N\). Otherwise, the difference is the output value of the modified RP. To compare the performance of the original RP and the proposed RP, simulation studies were conducted with two different types of waveform signals. The signals examined included (1) the positive half cycle of the sinusoidal waveforms and (2) the waveforms that emulate the evolution of dynamic resistance during resistance spot welding. The positive half cycles of the sinusoidal waveforms are shown with different amplitudes \(f(x)\), \(2f(x)\), and \(4f(x)\) in Figure 3. As shown in Figure 4, both the original and proposed RP show that the yellow area increases as the amplitude of the simulated waveforms increases. However, the rate of increase in the yellow area for the proposed RP is larger than that for the original RP. Therefore, it was verified that the proposed RP is more sensitive to the amplitude change of waveforms than the original RP. Figure 3: Simulated aperiodic signals. To understand the effect of the random noise from external sources on the proposed RP, random noise was added to the simulated waveforms. Three cases are shown in Figure 5: (1) \(2f(x)\), \((2)\,2f(x)+w(x)\), and (3) \(2f(x)+2w(x)\). The yellow area from the original RP fluctuated as the amplitude of the random noise increased, as is shown in Figure 6a-c.