# Weak Signal Processing Methods Based on Improved HHT and Filtering Techniques for Steel Wire Rope
## 4 Comparison and Results  
### Lowpass Filtering and Wavelet Denoising  
First, according to the results of the time and frequency analysis aforementioned, most of the defect signals are distributed within the low frequency band. Thus, the lowpass filtering techniques based on Butterworth is presented, and three typical wire rope defect signals with weak characterizations and low SNR are processed and compared. The transfer function of the Butterworth lowpass filter could be expressed as,  
\[|\mathrm{H(j\omega)}|=\frac{1}{\sqrt{1+\omega^{2n}}} \tag{5}\]  
where j is the imaginary unit, n is the order of the lowpass filter, \(\omega\) is the normalized cut off frequency in Equation (5). Besides, to further denoise the original wire rope detection signals, wavelet decomposition and reconstruction techniques were also applied in the signal denoising after the lowpass filtering. The continuous wavelet transform is defined as,  
\[\mathrm{W_{f}(a,b)}=\int_{-\infty}^{\infty}x(t)\Psi(\frac{t-a}{b})dt \tag{6}\]  
where x(t) is the time series signal, a and b represent the scale parameter and time center parameter in Equation (6), and the mother wavelet function is,  
\[\Psi(a,\tau)=\frac{1}{\sqrt{|a|}}\psi(\frac{t}{a}-\tau) \tag{7}\]  
where \(\psi\) is a wavelet function and \(\tau\) is the shifting parameter of the window function in Equation (7). As illustrated in Figure 5, three kinds of typical weak signals were randomly chosen from the experiments. Explanatorily, the S1 signal shown in Figure 5a means that the original signals were obtained from steel wire rope sample 1 with three-wire broken defect under the detection condition of the big liftoff distance of 20 mm by the coil sensor. Similarly, the signals of S2 and S3 expressed in Figure 5b,c represent that the signals were acquired from the one-wire broken defect of wire rope sensed by the inductive transducer under the liftoff distance of 20 mm and 13 mm, respectively. Specifically, S1 to S3 are the expanded part for signals 2, 3, and 5 as mentioned before. The original signals in the first line box of Figure 5a suggest that the weak wire rope inspection signals are mixed with strand signal features, while the FFT spectrum in the second line box implies that the original testing signals are distributed within the low frequency region smaller than 10 Hz, which also validates the time and frequency analysis results mentioned in Figure 3. Consequently, when the lowpass cutoff frequency was set as 5 Hz, the processed signals were shown in the third line box. Obviously, some interference signals were highlighted and became more prominent, while the defect signals were also filtered and abridged. After the further wavelet denoising by the wavelet of db6 (Daubechies 6 function) under the decomposition level of 16, the final processed signals were presented in the last line box. Obviously, the interferences of the noise signals are still prominent and difficult to distinguish from the real defect signals; in other words, the lowpass filtering and wavelet combined denoising methods are incapable of processing the weak wire rope signals to some extent. Figure 5: _Cont._  
Additionally, another two groups of wire rope weak signal processing results presented in Figure 5b,c indicate that both the original defect signals are difficult to identify due to the big liftoff distance and the interference of the strand as well as various noise signals in the environment. However, the weak magnetic flux leakage signals could still be caught by the inductive transducer. We observed by the frequency spectrum expressed in the second line box that the lowpass filtering and wavelet denoising parameters keep consistent with that described for these signals shown in Figure 5a. Namely, the lowpass cutoff frequency was 5 Hz, and db6 wavelet with 16 decomposition level was also chosen.