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.

The discrete time wavelet transform: its discrete time Fourier transform and filter bank implementation

Abstract: Viewing the discrete time wavelet transform DTWT[m,n] of a sampled signal s(nT) as a sequence in n, a closed-form expression is derived for its Discrete Time Fourier Transform (DTFT) D/sup (m)/(e/sup j/spl omega//) in terms of the DTFTs of the sampled mother wavelet /spl psi/(nT) and sampled signal s(nT). Next, an expression is derived for the output Y/sup (m)/(e/sup j/spl omega//) of a filter bank defined by the digital filters a[n] and b[n] and excited by s(nT). The filter a[n] emerged as an ideal low-pass filter, and the filter b[n] turned out to be a time-reversed and complex conjugated version of the sampled mother wavelet /spl psi/(nT).

The Nonuniform Discrete Fourier Transform And Its Applications In Signal Processing

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Sparse representation of two- and three-dimensional images with fractional Fourier, Hartley, linear canonical, and Haar wavelet transforms

Abstract: Fractional Fourier Transform are introduced as sparsifying transforms.Linear Canonical Transforms are introduced as sparsifying transforms.Various approaches for compressing three-dimensional images are suggested. Display Omitted Sparse recovery aims to reconstruct signals that are sparse in a linear transform domain from a heavily underdetermined set of measurements. The success of sparse recovery relies critically on the knowledge of transform domains that give compressible representations of the signal of interest. Here we consider two- and three-dimensional images, and investigate various multi-dimensional transforms in terms of the compressibility of the resultant coefficients. Specifically, we compare the fractional Fourier (FRT) and linear canonical transforms (LCT), which are generalized versions of the Fourier transform (FT), as well as Hartley and simplified fractional Hartley transforms, which differ from corresponding Fourier transforms in that they produce real outputs for real inputs. We also examine a cascade approach to improve transform-domain sparsity, where the Haar wavelet transform is applied following an initial Hartley transform. To compare the various methods, images are recovered from a subset of coefficients in the respective transform domains. The number of coefficients that are retained in the subset are varied systematically to examine the level of signal sparsity in each transform domain. Recovery performance is assessed via the structural similarity index (SSIM) and mean squared error (MSE) in reference to original images. Our analyses show that FRT and LCT transform yield the most sparse representations among the tested transforms as dictated by the improved quality of the recovered images. Furthermore, the cascade approach improves transform-domain sparsity among techniques applied on small image patches.

Inverse transform narrow band/broad band sound synthesis

Abstract: An additive sound synthesis process for generating complex, realistic sounds is realized in a computationally efficient manner. In accordance with one aspect of the invention, polyphony is efficiently achieved by dosing the energy of a given partial between separate transform sums corresponding to different channels. In accordance with another aspect of the invention, noise (87) is injected by randomly perturbing the phase of the sound, either on a per-partial basis or on a transform-sum basis. In the latter instance, the phase is perturbed in different regions of the spectrum to a degree determined by the amount of energy present in the respective regions of the spectrum. In accordance with yet another aspect of the invention, a transform sum (83) representing a sound is processed in the transform domain to achieve with great economy effects achievable only at much greater expense outside the transform domain. Other transforms besides the Fourier transform may be used to advantage. For example, use of the Hartley transform produces comparable results but allows transforms to be computed at approximately twice the speed as the Fourier transform.

Comparing Power Spectra Computed by Blackman-Tukey and Fast Fourier Transform

Abstract: Since controversy has arisen as to whether the Blackman-Tukey or the fast Fourier transform (FFT) technique should be used to compute power spectra, single and cross spectra have been computed by each approach for artificial data and real data to provide an empirical means for determining which technique should be used. The spectra were computed for five time series, two sets of which were actual field data. The results show that in general the two approaches give similar estimates. For a spectrum with a large slope, the FFT approach allowed more window leakage than the Blackman-Tukey approach. On the other hand, the Blackman-Tukey approach demonstrated a better window closing capability. From these empirical results it is concluded that the Blackman-Tukey technique is more effective than the FFT approach in computing power spectra of short historic time series, but for long records the fast Fourier transform is the only feasible approach.