### Table 1: Fast Fourier and other transform factorizations.

1997

"... In PAGE 2: ... 2 The Language of Factorizations A fast transform algorithm can be seen as a sparse factorization of the transform matrix. Table1 displays di erent factorizations for the DFT matrix as well as other transforms that arise in signal processing. The abbreviation Bn;m = (Fn Im)Dn;m is used in the FFT, with D a diagonal matrix of weights, Pn;p is the stride permutation matrix.... In PAGE 10: ... In general, the same transformation strategy can han- dle all problems of the same dimension, di erent transformation strategies are needed for multidimensional problems. bbb We successfully transform all fast Fourier and other transform and transposition algo- rithms shown on Table1 using the above rules for determining the loop structure and high level assignments. We generated the radix code, the straight-line code that computes an FFT in the loop nest by symbolically capturing the assignments and operations of an FFT calculation of appropriate length using the prime factor FFTs as the base cases.... ..."

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### Table 3. Detection Results on Fast Fourier Transform of Different Ball Bearings

2004

"... In PAGE 10: ...18% Broken cage with one loose element 2892 14 0.48% Damage cage, four loose element 2892 0 0% No evident damage; badly worn 2892 0 0% Table3 shows the results using Fourier transform. Table 4 is the corresponding results using statistical moments.... ..."

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### Table XI: Timings (seconds) for fast Fourier transform using Express (and NX) on the iPSC/860.

### Table1. Some Properties of the Fourier Transformation Signal Fourier Transform

"... In PAGE 2: ... The discrete Fourier trans- formation (FT) F(u; v) of a 2D discrete image f[x; y] 2 IRN N is de ned as F(u; v) = 1 N N?1 X j=0 N?1 X k=0 f[j; k] exp(?2 i(uj + vk) N ) (1) with i = p?1 and u; v = 0; 1; :::; N ? 1. Using the FT properties shown in Table1 , the following characteristics of the amplitude spectrum A of F(u; v) can be derived: it is invariant with respect to translation, inverse-variant with respect to scaling and variant with respect to rotation. Thus, features based on the amplitude spectrum of an image are translation invariant.... ..."

### Table 1: A list of three-dimensional Fourier transforms of various integrable functions used in meshless methods. with the exception of that of the Gaussian, the inverse multiquadric, and the Sobolev spline [28, Theorems 6.10, 6.13, and Page 133], the Fourier transform of each function was computed using (6).

"... In PAGE 6: ...Examples of integrable radial basis functions are given in Table1 . Non-integrable radial basis functions and polynomial terms are sometimes used, examples of which include the multiquadric and the thin-plate spline.... ..."

### Table 1: Some properties of the Fourier transform

"... In PAGE 16: ...imension of u is 1=time, i.e., frequency). In the case of image processing, the signal is a function of space rather than time, and in that case the domain of the Fourier transform is called spatial frequency. Table1 summarizes some important properties of the one-dimensional Fourier transform. The symbol is used to denote convolution:... ..."

### Table 2: Running times in milliseconds for direct evaluation, fast Gauss transform and improved fast Gauss transform in three di- mensions.

2003

"... In PAGE 6: ...etween 0 and 1. The bandwidth of the Gaussian is h =0.2. We set the relative error bound to 2% which is reasonable for most kernel density estimation, because the estimated density function itself is an approximation. Table2 re- ports the CPU times using direct evaluation, the original fast Gauss transform (FGT) and the improved fast Gauss trans- form (IFGT). All the algorithms are programmed in C++ and were run on a 900MHz PIII PC.... ..."

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### Table 3 shows the results using Fourier transform. Table 4 is the corresponding results using statistical moments. Throughout all the different conditions of ball bearing, Fourier transform seems to be more sensitive to detect any anomaly than statistical moments. Both methods detect better when the damage is more severe.

2004

"... In PAGE 10: ... The moments of first (mean), second (variance), third, four, and fifth order are used, so the resulted data points become 5-dimensional. Table3 . Detection Results on Fast Fourier Transform of Different Ball Bearings Ball bearing conditions Total number of data points Number of detected anomalies Percentage detected New bearing (normal) 2739 0 0% Outer race completely broken 2241 2182 97.... ..."

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### Table 6.2: Comparisons of runtimes in seconds of the Fast Fourier Transform benchmark on various targets of the testbed

1999

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