key: cord-0756355-cokrp45d
authors: Bhardwaj, Vibhor Kumar; Maini, Surita
title: Measurement of micro-harmonic vibration from optical feedback interferometry using wavelet trend analysis
date: 2020-07-30
journal: Opt Commun
DOI: 10.1016/j.optcom.2020.126330
sha: a7680bbc29079cd9156a9ccd45dd60c5bbde7641
doc_id: 756355
cord_uid: cokrp45d

Self-mixed optical feedback interferometry based laser sensors show promising results in the measurement of the vibration frequency. To date several measurement methods have been developed to extract the vibration information from the self-mixed (SM) signal; however, the complexity and accuracy of the methods still need improvement. The presented work tries to fulfill the gap by realizing a novel method using maximal overlap discrete wavelet transformation (MODWT) and multi-resolution analysis (MRA). The proposed method can reconstruct the micro-harmonic (< [Formula: see text]) vibration up to 1 kHz even under weak feedback conditions. The mean squared error and the maximum relative error of the proposed method for this range remained below 1.89 [Formula: see text] & 8.79%, respectively. Although, above 1 kHz, the proposed method turns out to be futile to reconstruct the vibration signal but still capable to measure vibration frequency up to 10 kHz with an accuracy of [Formula: see text] 0.0001. The method also found suitable to measure non-sinusoidal vibration frequency with reasonable accuracy even for the moderate feedback conditions. The authors envision that the proposed method will provide a compact, non-contact, and low-cost alternative for the vibration frequency measurement hence useful in early fault detection schemes and lung abnormality diagnosis.

Vibration is one of the critical parameters of modern non-contact metrological and medical 

α (linewidth enhancement factor), C (Acket's parameter or feedback level), and m (modulation 6 index). Equation (1-4) are widely solved numerically to produce the synthetic SMI signals for 7 the analysis of distance and vibration measurement schemes [26] , [27] . 8 The set of equations described above explains the self-mixing effect through the change in the 9 effective-phase of the laser beam due to the optical events occurred in the optical path. 26 The vibrations in a subject can also be defined as a continuous physical variation in the 27 effective distance between the subject and the point of observation. In context to the SMI, the 28 point of the observation is the front facet of the laser diode, while the distance considers as the 29   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 The decomposition process involves circularly high-pass filtering to yield wavelet coefficients 20 and low pass filtering to obtain scaling coefficients to a level j as follows:

Where , & , are the transfer function of the high pass filter and low pass filter, 24 respectively that circularly filter the signal X= {Xt, t=0, 1… n-1}. Figure 3 illustrates a 25 MODWT operation perform on signal X to decompose the signal into three levels of detail and 26 approximation. Here it is worth noting that the MODWT does not require the down-sampling 27 as both the coefficients are of the same length 'N'. J o u r n a l P r e -p r o o f Journal Pre-proof the trend of the SMI signal obtained from the vibrating object is extracted using the MODWT 1 method.

In the second step, a local signal is generated having the same frequency and amplitude as 3 of the extracted trend signal. In the third step, a cross-correlation between the trend and the 4 local signal is evaluated to determine the similarity index and phase difference between them.

In the last step, a time scale shifting in the locally generated signal is performed based on the 6 cross-correlation analysis. The time-shifted or reconstructed signal is a processed copy of the 7 trend signal, hence contains the same characteristical nature as followed by the vibration signal 8 as per the hypothesis. Figure i.e. 50mV peak to peak in the frequency range of 0 to 10 kHz. The laser diode was operated 25 above the threshold condition at a constant voltage & current of 2.6V and 26mA, respectively. 26 The monitor photodiode was connected to a customized trans-impedance amplifier (TIA). The

TIA was designed using OPA857IRGTT to convert the photodiode current to an appropriate 28 voltage level (peak to peak 3V). The output of the TIA was fed to a bandpass filter (BPF) 29 having a lower and upper cutoff frequency of 8 Hz and 20 kHz, respectively for first-level 30 electronic filtering. The filtered SMI signal was acquired using a data acquisition system 31 (DAQ, Rigol-M300). 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 The acquired signal was digitally preprocessed through Matlab to improve the quality of 2 the signal. The preprocessing included smoothing of signal followed by noise removal steps 3 such as median filtering and moving average filtering. The preprocessing step removes the 4 common noise from the SMI signal and provides a smooth signal for further analysis. In the 5 next step, the MODWT was performed on the preprocessed signal to an appropriate 6 decomposition level. The evaluated coefficients were then analyzed using MRA. The post-7 processing involves the signal characterization, i.e., estimation of the amplitude and frequency 8 of the decomposed signal using Z-transformation followed by the normalization process. A 9 local signal was generated based on the post-processing result as a pure sinusoid. In the next 10 step, a cross-correlation was performed between the decomposed signal and the locally 11 generated signal. The cross-correlation process evaluates the similarity index between the 12 decomposed & locally generates signals. Lastly, by analyzing the decomposed or reconstructed 13 signal (locally generated), the vibration information such as frequency and amplitude were 14 evaluated. Except for the measurement, a statistical operation was also performed to evaluate 15 the relative and mean squared error. The extraction of the vibration information from the SMI signal was performed through 30 the step by step process as per the flowchart illustrated in figure 4 using Matlab. In the first 31 step, an SMI setup was employed to obtain the signal, as discussed in the previous section and 32   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f Journal Pre-proof illustrated in figure 5. In the next step, median and moving average filtering were performed 1 on the experimentally obtained SMI signal to obtain a smooth signal. Figure 7 shows a pre-2 processed SMI signal obtained at a distance of 20cm from a source vibrating at a 10Hz 3 frequency with a peak to peak amplitude of 5µm. The pre-processed signal was decomposed 4 to n levels using MODWT. The MODWT operation was performed using Symlet wavelets 5 with four vanishing moments. The output of this step is an MRA of the transformation matrix. The parameters of the resultant signal were evaluated in the fourth step to generate a local 9 vibration signal. The evaluation process includes the measurement of frequency, amplitude, 10 and phase of the decomposed signal. The locally generated signal has the same peak to peak 11 amplitude and the frequency as of the decomposed signal. In the proceeded step, a cross-12 correlation between the locally generated signal and the vibration signal was evaluated. The 13 cross-correlation examines the similarity between the two signals and provides information 14 about the time-shift between the two signals. Figure 9 shows the result of the cross-correlation 15 between the locally generated signal and the decomposed signal. Based on the cross-16 correlation, the locally generated signal was shifted to produce a local signal having the same 17 phase as of the decomposed signal.

The comprehensive study concludes that the phase-shift between the decomposed 19 signal and the locally generated signal is equal to the feedback phase ( ) of the SMI. 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 The study also concludes that the decomposition level needs to be supervised based on 17 the SNR/ shape of the decomposed signal and should be processed for more than three cycles 18 of the SMI signals. The inherent advantage of the method is that it doesn't require a complex 19 filtering process for de-noising as the wavelet transform inherently filters the signal. The 20 second significant advantage of the proposed method is logged for the peak deteriorated SMI Thus, during the execution of the algorithm, there is no need for extraordinary computational 32   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f Journal Pre-proof averaged electronic speckle pattern interferometry methods," Appl. Opt., vol. 35, no. 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 7   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f 1 Figure 7 SMI signal obtained at a distance of 20cm from a source vibrating at a 10Hz frequency 2 with a peak to peak amplitude of 3µm. 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 

Solving self-mixing equations for arbitrary feedback levels: a concise 1 algorithm

 Figure 8 Different levels of SMI signal decomposition.We as the author of the manuscript entitled "Measurement of Micro-Harmonic Vibration from Optical Feedback Interferometry Using Wavelet Trend Analysis" hereby declared that 1. There are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.2. No funding was received for this work.3. We have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property.