The nonuniform discrete Fourier transform (NDFT), used in many processing schemes, can be computed using a fast algorithm known as the non uniform fast Fourier transform (NFFT). The NFFT is not a new algorithm, but it is an approximation scheme that can be use to calculate an approximate spectrum. In one dimension, computational complexity of the NFFT is O(NlogN) which is a dramatic improvement...

The Sparse Fast Fourier Transform is a recent algorithm developed by Hassanieh et al. at MIT for Discrete Fourier Transforms on signals with a sparse frequency domain. A reference implementation of the algorithm exists and proves that the Sparse Fast Fourier Transform can be faster than modern FFT libraries. However, the reference implementation does not take advantage of modern hardware featur...

We present fast, spatially dispersionless and unconditionally stable high-order solvers for Partial Differential Equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain “Fourier continuation” (FC) method for the resolution of the Gibbs phenomenon, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value p...

Beamspace processing is an emerging paradigm to reduce hardware complexity in all-digital millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO) basestations. This approach exploits sparsity of mmWave channels but requires spatial discrete Fourier transforms (DFTs) across the antenna array, which must be performed at baseband sampling rate. To mitigate resulting DFT implementati...

The fractional Fourier transforms (FrFTs) is one-parametric family of unitary transformations {F} α=0. FrFTs found a lot of applications in signal and image processing. The identical and classical Fourier transformations are both the special cases of the FrFTs. They correspond to α = 0 (F = I) and α = π/2 (F = F), respectively. Up to now, the fractional Fourier spectra Fi = Fi {f} , i = 1, 2, ....

The symmetry and periodicity properties of the discrete Fourier transform (DFT) allow a variety of useful and interesting decompositions. In particular, by clever grouping and reordering of the complex exponential multiplications it is possible to achieve substantial computational savings while still obtaining the exact DFT solution (no approximation required). Many “fast” algorithms have been ...