Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. The DFT is the most important discre

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Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (

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In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo

Fourier series

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric s

EE-205: Signals and systems (for EL&IC)

This class teaches the theory of linear time-invariant (LTI) systems. These systems serve both as models of physical reality (such as the wireless channel) and as engineered systems (such as electrical circuits, filters and control strategies).

MATH-205: Analysis IV

Learn the basis of Lebesgue integration and Fourier analysis

MATH-203(c): Analysis III

Le cours étudie les concepts fondamentaux de l'analyse vectorielle et l'analyse de Fourier en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.

Fourier non-uniqueness sets from totally real number fields

Martin Peter Stoller

Let K be a totally real number field of degree n >= 2. The inverse different of K gives rise to a lattice in Rn. We prove that the space of Schwartz Fourier eigenfunctions on R-n which vanish on the "component-wise square root" of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres root mS(n-1) for integers m >= 0 and, as m -> infinity, there are similar to c(K)m(n-1) many points on the m-th sphere for some explicit constant c(K), proportional to the square root of the discriminant of K. This contrasts a recent Fourier uniqueness result by Stoller (2021) Using a different construction involving the codifferent of K, we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes "root Lambda" for general lattices Lambda subset of R-n. Using results about lattices in Lie groups of higher rank we prove that if n >= 2 and a certain group Gamma(Lambda) >= PSL2.(R)(n) is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n >= 5 and all real lambda >= 2, Fourier interpolation results for sequences of spheres root 2m/lambda Sn-1, where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincare type for Hecke groups of infinite covolume and is similar to the one in Stoller (2021).

EUROPEAN MATHEMATICAL SOC-EMS2022

On Local Features for Face Verification

{W}e compare four local feature extraction techniques for the task of face verification, namely (ordered in terms of complexity): raw pixels, raw pixels with mean removal, 2D Discrete Cosine Transform (DCT) and local Principal Component Analysis (PCA). The comparison is performed in terms of discrimination ability and robustness to illumination changes. We also evaluate the effectiveness of several approaches to modifying standard feature extraction methods in order to increase performance and robustness to illumination changes. Results on the XM2VTS database suggest that when using a Gaussian Mixture Model (GMM) based classifier, the raw pixel technique provides poor discrimination and is easily affected by illumination changes; the mean removed raw pixel technique provides performance that is fairly close to 2D DCT and local PCA, but is considerably affected by illumination changes. The performance of 2D DCT and local PCA techniques is quite similar, suggesting that the 2D DCT technique is to be preferred over the local PCA technique, due to the lower complexity of the 2D DCT. Both 2D DCT and local PCA techniques are considerably more robust to illumination changes compared to the raw pixel techniques. Modifying the 2D DCT and local PCA techniques by removing the first coefficient, which is deemed to be the most affected by illumination changes, clearly enhances robustness; removing more than the first coefficient causes a noticeable reduction in performance on clean images and provides no further gains in robustness. Compared to just throwing out the first coefficient, the use of deltas can achieve a small increase in performance and robustness. Lastly, we suggest that it is more appropriate to use analysis blocks of size 8x8 (as opposed to 16x16) with 2D DCT decomposition; out of the 64 resulting coefficients, the second through to 21-st (resulting in 20 dimensional feature vectors) are the most robust to illumination changes while providing good discriminatory information.

IDIAP2004

Fourier uniqueness and interpolation in Euclidean space

Martin Peter Stoller

We prove that every Schwartz function in Euclidean space can be completely recovered given only its restrictions and the restrictions of its Fourier transform to all origin-centered spheres whose radii are square roots of integers. In particular, the only Schwartz function which, together with its Fourier transform, vanishes on these spheres, is the zero function. We show that this remains true if we replace the spheres by surfaces or discrete sets of points which are sufficiently small perturbations of these spheres. In a complementary, opposite direction, we construct infinite dimensional spaces of Fourier eigenfunctions vanishing on on certain discrete subsets of those spheres. The proofs combine harmonic analysis, the theory of modular forms and algebraic number theory.

EPFL2022