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. 2020 Mar 17;10:4852. doi: 10.1038/s41598-020-60878-7

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Absolute numerical error as a function of the polar angle of the transform parameter W, plotted for chirp contours with 1024 points on the unit circle. The nine plots illustrate the variety of shapes that the error function can take depending on the choice of discretization. In all plots, the size of the transform is fixed at 1024. What varies is the number of angles, i.e., polar angles of W that are discretized using regularly-spaced intervals. Each point in each plot shows the average value of the absolute error, computed with the CZT–ICZT procedure over 10 random input vectors. The top row, i.e., plots (a–c), shows the results for the case when the number of regularly-spaced polar angles is close to the number of points on the chirp contour, i.e., 1024. The second and the third row show the results when the number of angles is approximately 2 times and 4 times greater than the size of the transform, respectively. The left column, i.e., plots (a,d,g), shows the results for the case when the number of angles is a power of two. The plots in the middle column are for discretizations with 1026, 2050, and 4098 angles, which are composite numbers that are not powers of two. The right column shows plots for the case when the number of points is a prime number.