The Application of Adaptive Linear and N on-Linear Filters to Fringe Order Identification in White-Light Interferometry Systems

Conventional optical interferometry systems driven by highly coherent light sources have a very short unambiguous operating range, a direct consequence of the flatness of the interference fringes visibility profile at the output of the system. The range can be extended by using a white-light interferometer (WU), which is driven by a low-coherence source and produces a Gaussian visibility profile with a unique maximum in correspondence of the central fringe. Due to system and/or measurement noise, however, the position of the maximum (from which an accurate measurement of the measurand - displacement, temperature, pressure, flow, etc. - can be derived) is not easily detectable, and can lead to large measurement errors. This is especially true in a multiplexing scheme, where the source power is distributed evenly among various sensors, with a corresponding drop in the overall signal-to-noise ratio. The inclusion of a signal processing scheme at the receiver end is thus a necessity. As the fringe pattern at the output of a WLI system is basically a noisy sine wave amplitude modulated by a Gaussian envelope, it can be classified as a non-stationary, narrow-band, linear but non-Gaussian signa\. So far, no attempt has been made to apply digital filtering techniques, as understood in the signal processing community, to the output signal of a WLI system. This thesis constitutes a first step in that direction. Since the only measurable information given by the system is contained in the output signal, the system is modelled as a "black box" driven by the system and measurement noise processes and containing an unknown set of parameters. Standard least squares techniques can then be applied to estimate the parameters of the model, as is usually done in the field of system identification when only noisy output measurements are available. It is shown that identification of the model parameters is equivalent to finding a set of coefficients for an inverse filter which takes the WU signal at its input and delivers the unknown noise process at the output. The non-stationarity of the signal is accounted for by allowing for time variations of the model parameters; this justifies the use of adaptive filters with time-varying coefficients. A new central fringe identification scheme is proposed, based on a modification of the standard least mean square (LMS) adaptive filtering algorithm in combination with amplitude thresholding of the fringe pattern. The new scheme is shown to offer considerable improvement in the identification rate when tested against current schemes over comparable operating ranges, while retaining the computational simplicity and operational speed of the standard LMS. Its performance is also shown to be largely independent of the step-size parameter controlling the rate of convergence and tracking in the standard LMS, which is known to be the main obstacle for a successful application of the algorithm in a practical setting. The non-Gaussianity of the signal is explored and an attempt is made to apply higher-order statistics (HOS) algorithms to central fringe identification. The effectiveness of Gaussianity tests on pilot Gaussian data is seen to depend not only on the number and length of records available but, perhaps more importantly, on the bandwidth of the process. Violation of the stationarity assumption is shown to lead to mis-classification of a seemingly non-Gaussian signal into a Gaussian one, as the visibility profile may alter the distribution of the underlying sinusoid making it appear Gaussian, even when beam diffraction and wavefront aberrations combine to produce a nonGaussian profile. HOS-based adaptive algorithms may still be of some benefit, however, if processing is confined to that region of the fringe pattern where sufficient non-Gaussianity is allowed to develop. Non-linear adaptive filters based on the Volterra theories are finally applied to compensate for possible non-linearities introduced by mismatches in optical components, chromatic aberrations, and analogue-to-digital converters. It is shown that although a Volterra filter is able to reproduce the low-amplitude distortions of the fringe pattern better than a linear filter does, the identification rate does not improve. Reasons are given for such behaviour.