Spectral Processing of Self-Mixing Interferometric Signal Phase for Improved Vibration Sensing Under Weak- and Moderate-Feedback Regime

Abstract

Technology of Pakistan. His current research interests include magnetic field based measurement and monitoring for power systems, magnetic energy harvesters and innovative instrumentation problems. Thierry Bosch (M'93-SM'06) is a Professor with Institut National Polytechnique Toulouse-ENSEEIHT and member of the Optoelectronics for Embedded Systems Research Group, LAAS-CNRS. His research interests include laser industrial instrumentation development, including range finding techniques, vibration and velocity measurements. He is presently Vice President of IEEE IM France.

On nature of determining displacement from sole knowledge of self interferometric perturbed phase

“…Ofcourse, left hand side of (4) is equal to the right hand side,if the derivative exists and if derivative exists then by chain rule dxo dx f also exists, and we can say that continuous mapping between x f and x o exits. On plotting the left and right hand sides, for a given SMI signal, it is obvious from figure(4) that equation 15 Therefore, a simple denoising process,as mentioned in TFSP method [10], carried out by an averaging filter [4] at 100 points, creates an error of less than 20µm, which is quite not enough, since our range is in µm.This is shown in figure(6). Another way to use approach of exploiting the noisy nature of perturbed phase is to use Savitzky-Golay filter [8] which is a smooting filter that tends to fit polynomials on subsets of adjacent data points.…”

“…Another way to use approach of exploiting the noisy nature of perturbed phase is to use Savitzky-Golay filter [8] which is a smooting filter that tends to fit polynomials on subsets of adjacent data points. Given the structure of perturbed phase, as shown in figure (5) as triangular structures over unperturbed phase, makes Savitzky-Golay filter, the best method to exploit the TFSP method [10].As shown in figure (7),Savitzky-Golay filter is carried out, with frame length of 201 points, for a displacement of 5Hz sampled at 10kHz. Fig (7), shows that error between reconstructed phase (x or ) and x o is bounded Fig.…”

“…Thus,the extraction of x o from x f in frequency domain, becomes a spectral filtering process followed by minimizing contribution of T (ω) superimposed on F [x o ](ω), due to which extracted unperturbed phase would be slighty defornmed by just spectral filtering. The later part can be achieved by following some perturbative or homotopical deformation methods such that dynamics of equation 1become preserved, inorder to tackle the problem of superposition.It is to be noted that this is not simple as ordinary filtering process and error is not random and has structure.This approach has been used in TFSP method [10] to estimate x o from x f without knowledge of C and α by simply putting a low pass filter which is an incomplete procedure.…”

“…Laser diode(LD) based Self-Mixing (SM) or optical feedback interferometry is an attractive sensing scheme due to the compact, self-aligned, and low-cost nature of the SM sensor [10]. A basis process of self mixing phenomenon for displacement measurement is shown in figure (1).…”

“…The proposed method increases the accuracy of the SM sensor to λ 16 . The root of the this research was tackeled in [7], where behavioral model of a self-mixing laser diode sensor was presented to simplify the solutions to non-linear self interferometric phase equation and thus modelling the basis of all algorithms by means of piecewise analysis on cases of C. Moreover, in [10], TFSP method was introduced where perturbed phase was estimated from unperturbed phase thus removing the need for estimation of parameters of phase equation, by means of Fourier series but to my understanding, the logic of generally connecting Fourier coefficents of perturbed phase and unperturbed phase remained unproven. Therefore analysis has been performed on this paper in order to rigorously deduce the spectral property of perturbed phase as compared to unperturbed phase, along with providing theoretical background on deriving them in time domain.…”