Data reduction using a generalized regressive discrete Fourier series

With the development of optical measurement techniques it is possible to obtain vast amounts of data. In vibrometry applications in particular operational deflection shapes are often obtained with very high spatial resolution. It has long been known that it is possible to reduce (approximate) the measurement data by means of a Fourier decomposition.One of the most common techniques for evaluating optical measurement data is by means of a Fourier analysis. It is well known that for periodic and band-limited sequences the Discrete Fourier Transform (DFT) returns the true Fourier coefficients when exactly 1 period (or a multiple) is processed. Leakage will occur when less than 1 period is considered. This gives rise to non-negligible errors, which can be resolved by using the Generalized Regressive Discrete Fourier Series (GRDFS), introduced in this article. The measured signal is represented by a model using sines and cosines. The coefficients of those sines and cosines are then estimated together with the phase and frequency on a global scale by means of a frequency domain system identification technique. By making use of the regressive technique proposed in this paper, it is possible to reduce the data in comparison to the classical Fourier decomposition by a sizeable factor.In this article the method will be applied in particular to the reduction of data for laser vibrometer measurements performed on an Inorganic Phosphate Cement (IPC) beam (1D), as well as on an aluminium plate (2D). The proposed technique will also be validated on both 1D and 2D simulations of varying complexity.