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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 ().
[2]Paolo Prandoni, [3]LCAV, EPFL
In this notebook we will build a dual-tone multi-frequency decoder
based on the short-time Fourier transform.
In [1]:
[2]Paolo Prandoni, [3]LCAV, EPFL
In this notebook we will look at the practical issues associated to
plotting the DFT and in particular the DFT of real-world signals. We
will examine how to map the DFT coefficients to real-world frequencies
and we will investigate the frequency resolution of the DFT and the
effects of zero padding.
In [1]:
[2]Paolo Prandoni, [3]LCAV, EPFL
In this notebook we will explore various ways to implement a system
that changes the voice of a speaker. Typical applications for these
techniques, strictly sorted by plausibility, include international spy
missions and prank telephone calls, and devices such as the one
displayed here can be easily bought on the web... or in a novelty shop.
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