Huang et al. Figure 1. System architecture of the proposed FALPINE waveform recognition. shown in Algorithm 1. After the decomposition process, the IMFs are extracted and regarded as the EMD features. Algorithm 1. Decompose signal sðtÞ into IMFs [22] 1) Identify all extrema of sðtÞ; 2) Interpolate the local maxima to form an upper envelop uðtÞ; 3) Interpolate the local minima to form an lower envelop lðtÞ; 4) Calculate the mean envelop mðtÞ¼ðuðtÞþ lðtÞÞ=2; 5) Extract the mean mðtÞ from the signal and obtain hðtÞ¼ sðtÞmðtÞ; 6) Check whether hðtÞ satisfies the IMF property. If hðtÞ is an IMF, iterate all the above steps on the residue rðtÞ¼ sðtÞ hðtÞ. Otherwise, keep iteration on hðtÞ as sðtÞ. (b) VMD feature. Similar to EMD, the VMD method is also able to adaptively decompose an arbitrary signal into a number of principle modes or IMFs [23]. Unlike the EMD, which is limited by its algorithmic ad hoc nature lacking mathematical theory and thus reducing the algorithm's robustness of the decomposition, the VMD provides a promising solution to the decomposition that is theoretically well founded in the frequency domain to extract frequency band components of the input signals. The fundamental ofVMD is solving the constrained variational problem as the following equation: min uk;wk s.t. f ZX j@t k X k uk ¼ f dðtÞþ j pt ukðtÞ ejwktj2dt (3) Figure 2. Algorithmic flow ofVMD [23]. APRIL 2023 IEEE A&E SYSTEMS MAGAZINE 17