Generalized Adaptive Polynomial Window Function

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 nonlinear 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. INDEX TERMS Discrete Fourier transforms, signal processing, optimization methods, adaptive algorithms.

Comments on “Generalized Adaptive Polynomial Window Function”

“…Sidelobe suppression of a minimum four-term Nuttall window is −98.17 dB, as can be seen from Tbl. 2 in [2], and not −93.8 dB as reported in [1]. Also, reported maximum sidelobe −102.8 dB of ''Optimized Nuttall'' is obtained only a 64 point window, and degrades for windows with more points, as summarized in Tbl.…”

“…In the above article [1], the authors introduced a Generalized Adaptive Polynomial (GAP) window function and presented examples of how GAP-based window functions outperform existing window functions. Claims in [1] need clarification, and more importantly, do not hold in all cases.…”

“…In [1], the authors compare the Hann window with ''Optimized Hann'' in Fig. 3, and present that main lobe width has been reduced from 1.46 to 1.25 bins while keeping the sidelobe level at −31.5 dB.…”

“…Lastly, article [1] does not report the scalloping loss of optimized flat-top windows. As their name suggests, flat-top windows are designed to have a maximally flat response in the first bin.…”

New Class of Cosine-Sum Windows

“…For applications using the window function, we raise the suppression of co-channel interference [2], harmonic analysis in real-time systems [3], satellite altimeter waveforms [4], radar systems [5]- [7], sensor arrays [8], and audio systems [9], [10]. The window function has been studied in various ways [11]- [22].…”