改善非均勻傅利葉以及非均勻哈特利快速轉換演算法的精確度

Abstract

本論文研究非均勻的傅利葉快速轉換演算法中準確因子對運算結果準確性的改善，並提出非均勻哈特利快速轉換的演算法。在上述兩種非均勻快速的演算法中，證明其準確因子可使用餘弦函數的次方，在不增加計算複雜度的情況下，可將計算結果誤差最多降低為1/30。文中使用多組函數測試，以證明計算準確值的改善。本論文的第二部份，根據傅利葉的非均勻轉換演算法，以周圍均勻點展開任意點的方式，對哈特利的核心函數做非均勻展開，首度推導出非均勻哈特利快速轉換演算法。此外，本文也對此快速演算法所得結果做誤差的分析。

Based on the regular Fourier matrix, a new set of accuracy factors is proposed for the nonuniform fast Fourier transform (NUFFT) algorithm to improve the accuracy of transformed data. Numerical results show that the accuracy of the results can be improved up to 30 times better than that of the previous factors. The second part of this thesis proposes a nonuniform (unequally spaced) fast Hartley transform algorithm. Numerical results have good agreement with those of directional evaluation. The comparison of new sets of accuracy factors is also discussed. Numerical examples are shown for the applications in computational electromagnetics.