Non-uniform discrete Fourier transform

About: Non-uniform discrete Fourier transform is a research topic. Over the lifetime, 4067 publications have been published within this topic receiving 123952 citations.

Fast algorithms for signal reconstruction without phase

Abstract: We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important insignal processing, especially speech recognition technology, and has relevance for state tomography in quantumtheory. We show that a generic frame gives reconstruction from the absolute value of the frame coecients inpolynomial time. An improved ec iency of reconstruction is obtained with a family of sparse frames or framesassociated with complex projective 2-designs.Keywords: Frames, mutually unbiased bases, equiangular frames, projective 2-designs, discrete chirps 1. INTRODUCTION For years engineers believed that if a signal was represented with a sucient degree of redundancy, then itsreconstruction should be possible without using the recorded phase information. Typical examples are thereconstruction of an unknown square-integrable function from the magnitude of its windowed Fourier Transform ,also known as the short-time Fourier transform ,orofanundecimated wavelet transform in audio and imageprocessing. Onemotivation for studyingthis problemarosefromspeech recognitiontechnologywhere informationabout the phase of a signal is lost in the course of processing using cepstral analysis (see [3,6,29,30,32,35]).The problem of reconstruction without phase appears also in the context of quantum theory, where a quantumstate is determined by measu ring correlations with other states [28]. More precisely, a pure quantum state isgiven by a ray in a complex Hilbert space, and is usually re presented by normalized v ectors in this ray. Quantummeasurements only give access to the magnitudes of the inner product between thes e vectors with other statevectors. Therefore, reconstructing a pure quantum state from measurements is the same as “nding a vector, upto a unimodular constant, from the magnitudes of linear t ransform coeents. This is also referred to as quantumstate tomography [31].Moreover, there is a closely connected problem in optics with applications to X-ray, crystallography, electronmicroscopy, and coherence theory (see [5,18,19,27]). This problem is to reconstruct a discrete signal from themodulus of its Fourier transform under constraints in both the original and the Fourier domain. For “nite signalsthe approach uses the Fourier transform with redundancy 2. All signals with the same modulus of the Fouriertransform satisfy a polynomial factorization equation. In dimension one this factorization has an exponentialnumber of possible solutions. In higher dimensions the factorization is shown to have genericallya unique solution(see [22]).Recently, the problem of r econstruction without phase was solve d for a large class of frames in “nite-dimensional Hilbert spaces [3]. It is easily seen that, in the case of a real Hilbert space, this problem is equivalent

Calculation of geometric moments using Fourier plane intensities.

Abstract: A new system for calculating the geometric moments of an input image is presented. The system is based on a mathematical derivation that relates the geometric moments of the input image to the intensity of the Fourier transform of the image. Since optical systems are very efficient at obtaining Fourier transforms, an optical system is used to provide the transform of the input image in this design. An array of detectors is then used to sample the Fourier plane, and a digital postprocessor performs a differentiation process on these Fourier magnitude samples to obtain a vector of values which are combined in a predetermined fashion to provide the geometric moments of the original input function.