### I. Introduction

### II. Comparison between Sufficient and Insufficient Jem Signals

*f*

*, which is a reciprocal of the spool rate (one rotation period). Thus, to estimate the blade number exactly, the spool rate and JEM spectrum should be clearly obtained from the measured JEM signal.*

_{R}*F*,

*F*

^{−1}, and

*x*are the Fourier transform operator, inverse Fourier transform operator, and the JEM signal in the time domain, respectively [11]. Since the spool rate determines the fundamental periodicity of the JEM signal, the JEM spectrum comprises spool line spectra, and the cepstrum has an outstanding peak. Then, a certain threshold is determined to designate the time samples as the peak, with cepstrum values above the threshold. Finally, the spool rate is estimated using the detected peak.

### III. Proposed Algorithm for Estimating the Blade Number

Step 1. Extraction of the FCF using the harmonic selection rule.

Step 2. Selection of the cutoff of the preprocessing LPF as the FCF.

Step 3. Decomposition of the signal filtered by the LPF into a finite number of IMFs using EMD.

Step 4. Application of the ACF to the desired combination of extracted IMFs to obtain a refined AC waveform.

Step 5. Acquisition of the blade number using the detected peaks of the ACF.

*z*is the combined complex JEM data in the time domain, and

*N*is the total length of the JEM data [11]. The ACF is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise. In this study, the JEM ACF yields a refined AC waveform by applying it to the combined signal with IMF 1 and the last IMF, which denotes the first harmonic component and the zero-Doppler component, respectively. This refined AC waveform makes it easy to automatically extract the blade number on the jet engine via detected peaks within the outstanding peaks, which will be presented in Step 5. Fig. 6 shows the AC waveform of the measured raw JEM signal with insufficient dwell time and the reconstructed JEM signal with IMF 1, respectively. The AC waveform of the insufficient JEM signal has ambiguous peaks from the complicated waveform. Unlike the AC waveform shown in Fig. 6, useful information can now be obtained from the well-presented AC waveform shown in Fig. 5(d).