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Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform.

ABSTRACT: The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

SUBMITTER: Yao X

PROVIDER: S-EPMC7924731 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:To optimally assess oscillatory phenomena within physiological variables, spectral domain transforms are used. A discrete Fourier transform (DFT) is one of the most common methods used to attain this spectral change. In traumatic brain injury (TBI), a DFT is used to derive more complicated methods of physiological assessment, particularly that of cerebrovascular reactivity (CVR). However, a practical application of a DFT will introduce various errors that need to be considered. This study will evaluate the pulse amplitude DFT derivation of intracranial pressure (AMP) to highlight how slight differences in DFT methodologies can impact calculations. Utilizing a high-frequency prospectively maintained data set of TBI patients with recorded arterial and intracranial blood pressure, various cerebral physiological aspects of interest were assessed using the DFT windowing methods of rectangular, Hanning, and Chebyshev. These included AMP, CVR indices (including the pressure reactivity and pulse amplitude index), and the optimal cerebral perfusion pressure (with all methods of CVR). The results of the different DFT-derived windowing methods were compared using the Wilcoxon signed-ranked test and histogram plots between individual patients and over the whole 100-patient cohort. The results for this analysis demonstrate that, overall and for grand average values, there were limited differences between the different DFT windowing techniques. However, there were individual patient outliers to whom the different methods resulted in noticeably different overall values. From this information, for derived indices utilizing a DFT in the assessment of AMP, there are limited differences within the resulting calculations for larger aggregates of data. However, when the amplitude of spectrally resolved response is important and needs to be robust in smaller moments in time, it is recommended to use a window that has amplitude accuracy (such as Chebyshev or flat-top).

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Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform.

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