Non-uniform discrete Fourier transform

In applied mathematics, the nonuniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the shifted DFT. It has important applications in signal processing, magnetic resonance imaging, and the numerical solution of partial differential equations. As a generalized approach for nonuniform sampling, the NUDFT allows one to obtain frequency domain information of a finite length signal at any frequency. One of the reasons to adopt the NUDFT is that many signals have their energy distributed nonuniformly in the frequency domain. Therefore, a nonuniform sampling scheme could be more convenient and useful in many digital signal processing applications. For example, the NUDFT provides a variable spectral resolution controlled by the user. The nonuniform discrete Fourier transform transforms a sequence of complex numbers into another sequence of complex numbers defined by where are sample points and are frequencies. Note that if and , then equation () reduces to the discrete Fourier transform. There are three types of NUDFTs. The nonuniform discrete Fourier transform of type I (NUDFT-I) uses uniform sample points but nonuniform (i.e. non-integer) frequencies . This corresponds to evaluating a generalized Fourier series at equispaced points. It is also known as NDFT. The nonuniform discrete Fourier transform of type II (NUDFT-II) uses uniform (i.e. integer) frequencies but nonuniform sample points . This corresponds to evaluating a Fourier series at nonequispaced points. It is also known as adjoint NDFT. The nonuniform discrete Fourier transform of type III (NUDFT-III) uses both nonuniform sample points and nonuniform frequencies . This corresponds to evaluating a generalized Fourier series at nonequispaced points. It is also known as NNDFT. A similar set of NUDFTs can be defined by substituting for in equation ().

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (9)

Related courses (68)

Related lectures (468)

Related MOOCs (12)

Joint Demosaicing and Super-Resolution Imaging from a Set of Unregistered Aliased Images

Sabine Süsstrunk, Patrick Vandewalle, Karim Krichane

We present a new algorithm that performs demosaicing and super-resolution jointly from a set of raw images sampled with a color filter array. Such a combined approach allows us to compute the alignment parameters between the images on the raw camera data before interpolation artifacts are introduced. After image registration, a high resolution color image is reconstructed at once using the full set of images. For this, we use normalized convolution, an image interpolation method from a nonuniform set of samples. Our algorithm is tested and compared to other approaches in simulations and practical experiments.

2007

Least-squares spectral analysis

Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum based on a least-squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in the long and gapped records; LSSA mitigates such problems. Unlike in Fourier analysis, data need not be equally spaced to use LSSA.

Non-uniform discrete Fourier transform

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.

Digital Signal Processing I

Basic signal processing concepts, Fourier analysis and filters. This module can be used as a starting point or a basic refresher in elementary DSP

Digital Signal Processing II

Adaptive signal processing, A/D and D/A. This module provides the basic tools for adaptive filtering and a solid mathematical framework for sampling and quantization

Digital Signal Processing III

Advanced topics: this module covers real-time audio processing (with examples on a hardware board), image processing and communication system design.

EE-205: Signals and systems (for EL)

Ce cours pose les bases d'un concept essentiel en ingénierie : la notion de système. Plus spécifiquement, le cours présente la théorie des systèmes linéaires invariants dans le temps (SLIT), qui sont

COM-514: Mathematical foundations of signal processing

A theoretical and computational framework for signal sampling and approximation is presented from an intuitive geometric point of view. This lecture covers both mathematical and practical aspects of

COM-202: Signal processing

Signal processing theory and applications: discrete and continuous time signals; Fourier analysis, DFT, DTFT, CTFT, FFT, STFT; linear time invariant systems; filter design and adaptive filtering; samp

Fourier Series: Understanding Coefficients and Periodicity MATH-203(c): Analysis III

Explores Fourier series coefficients, periodicity, and series relations.

Discrete Fourier Transform: Main Properties EE-406: Fundamentals of electrical circuits and systems I

Covers the main properties of the Discrete Fourier Transform and convolution methods.

Frequency Estimation: Deterministic Signals with Low Noise Level ME-301: Measurement techniques

Explores frequency estimation in deterministic signals using the DFT and spectral resolution.