This paper is part 2 in a series of papers about the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). The focus of this paper is on a fast implementation of the DFT, called the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). The implementation is based on a well-known algorithm, called the Radix 2 FFT, and requires that its' input d...

Legendre coefficients of an integrable function f(x) are proved to coincide with the Fourier a nonnegative index suitable Abel-type transform itself. The numerical computation N can thus be carried out efficiently in O(NlogN) operations by means single fast f(x). Symmetries associated exploited further reduce computational complexity. dual problem calculating sum expansions at prescribed set po...

This paper is part 3 in a series of papers about the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). The focus of this paper is on computing the Power Spectral Density (PSD) of the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). The implementation is based on a well-known algorithm, called the decimation in time Radix 2 FFT, and re...

In this paper, we generalize the windowed Fourier transform to the windowed linear canonical transform by substituting the Fourier transform kernel with the linear canonical transform kernel in the windowed Fourier transform definition. It offers local contents, enjoys high resolution, and eliminates cross terms. Some useful properties of the windowed linear canonical transform are derived. Tho...

Dirichlet series and Fourier series can both be used to encode sequences of complex numbers an , n ∈ N. Dirichlet series do so in a manner adapted to the multiplicative structure of N, whereas Fourier series reflect the additive structure of N. Formally at least, the Mellin transform relates these two ways of representing sequences. In this paper, we make sense of the Mellin transform of period...

The Fourier Series, the founding principle behind the field of Fourier Analysis, is an infinite expansion of a function in terms of sines and cosines. In physics and engineering, expanding functions in terms of sines and cosines is useful because it allows one to more easily manipulate functions that are, for example, discontinuous or simply difficult to represent analytically. In particular, t...

These notes provide an introduction to the mathematics used in medical imaging. In Section 6 we review the basics of calculus and multivariable calculus. In Section 2 we introduce Fourier Series, which is a premonition for the introduction of the Fourier Transform in Section 3. Finally, we will treat the mathematics of CT-Scans with the introduction of the Radon Transform in Section 4. Section ...

Spherical Harmonics arise on the sphere S in the same way that the (Fourier) exponential functions {e}k∈Z arise on the circle. Spherical Harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform. Without a fast transform, evaluating (or expanding in) Spherical Har...