Abstract
We present a novel method to design and optimize window functions based on combinations of linearly independent functions. These combinations can be performed using different strategies, such a sums of sines/cosines, series, or conveniently using a polynomial expansion. To demonstrate the flexibility of this implementation, we propose the Generalized Adaptive Polynomial (GAP) window function, a non-linear polynomial form in which all the current window functions could be considered as special cases. Its functional flexibility allows fitting the expansion coefficients to optimize a certain desirable property in time or frequency domains, such as the main lobe width, sidelobe attenuation, and sidelobe falloff rate. The window optimization can be performed by iterative techniques, starting with a set of expansion coefficients that mimics a currently known window function and considering a certain figure of merit target to optimize those coefficients. The proposed GAP window has been implemented and several sets of optimized coefficients have been obtained. The results using the GAP exemplify the potentiality of this method to obtain window functions with superior properties according to the requirements of a certain application. Other optimization algorithms can be applied within this strategy to further improve the window functions.
Highlights
The Discrete Fourier Transform (DFT) is a powerful tool to perform Fourier analysis in discrete data, with widespread uses in several modern applications, such as in astronomy, chemistry, acoustic signals, geophysics, and digital processing [1]–[3]
The expansion of (2), with the appropriate coefficients presented in Table 1, allows to mimic any of the well-established window functions, one can call the function as a generalized window function with flexibility to allow searching for sets of expansion coefficients that could provide highly-optimized results for signal analysis
The Generalized Adaptive Polynomial (GAP) windows present the same spectral characteristics when compared to those traditional implementations
Summary
The Discrete Fourier Transform (DFT) is a powerful tool to perform Fourier analysis in discrete data, with widespread uses in several modern applications, such as in astronomy, chemistry, acoustic signals, geophysics (seismic data), and digital processing [1]–[3]. This functional form is very flexible, which allows searching by a systematic method to obtain sets of expansion coefficients that could provide superior and optimized properties, considering a reference figure of merit that takes into account the property that is intended to be improved.
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