Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions

We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of L-bases, a class of bases that include the Bernstein-Bézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for L-bases or by examining change of basis algorithms for L-bases. For polynomials of degree n n in s s variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity O ( n s + 1 ) O(n^{s+1}) , a nested multiplication algorithm with computational complexity O ( n s log ⁡ n ) O(n^s \log n) and a ladder recurrence algorithm with computational complexity O ( n s ) O(n^s) . These algorithms also generate a new generalization of the Aitken-Neville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange L-bases. The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity O ( n s ) O(n^s) , a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities O ( n s ) O(n^s) , O ( n s ) O(n^s) and O ( n ) O(n) per point respectively for the evaluation of multivariate polynomials at several points.

The Fast Fourier Transform (FFT) is an efficient algorithm to calculate the Discrete Fourier Transform (DFT), which is often employed in Digital Signal Processing (DSP) and communication. In FFT, complex multiplication and addition units in the butterfly module consume most of the hardware resources. Compared to the addition operation, multiplication is more complicated. In this paper, the multiplier in the complex multiplication unit of the FFT is approximated. Four Radix-4 Booth multipliers with different approximation levels are proposed to reduce the hardware complexity. The pipeline FFT and the parallel FFT based on the proposed approximate multipliers are implemented and extensively evaluated. Compared with the state-of-the-art FFT designs, the LUTs amount is reduced up to 20.3% and 29.1% for pipeline and parallel FFTs, respectively. The power is reduced up to 69.9% for pipeline FFT, and the delay is reduced up to 45.7%. Moreover, the PSNR is reduced by less than 1dB in both pipeline FFT and parallel FFT. Proved by experiment results, the overall performance of the proposed designs is better than FFT designs using other approximate multipliers.

The discrete Fourier transform in d dimensions with equispaced knots in space and frequency domain can be computed by the fast Fourier transform (FFT) in $${\cal O}(N^d \log N)$$ arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multivariate approximation, interpolations on sparse grids were introduced. In particular, for frequencies chosen from an hyperbolic cross and spatial knots on a sparse grid fast Fourier transforms that need only $${\cal O}(N \log^d N)$$ arithmetic operations were developed. Recently, the FFT was generalised to nonequispaced spatial knots by the so-called NFFT. In this paper, we propose an algorithm for the fast Fourier transform on hyperbolic cross points for nonequispaced spatial knots in two and three dimensions. We call this algorithm sparse NFFT (SNFFT). Our new algorithm is based on the NFFT and an appropriate partitioning of the hyperbolic cross. Numerical examples confirm our theoretical results.

Signal source separation is an important aspect of both measurement and signature intelligence (MASINT) as well as modern commercial smart grid applications. In some MASINT scenarios, real-time signal processing is required in a small physical form factor, which can become prohibitive with the high sample rates required of modern signal identification algorithms. We develop a new Fast Fourier Transform (FFT) algorithm, the Aliased FFT (AFFT), that exploits spectral redundancy and sparsity in the signal to drastically reduce the number of computations required from O(N logN), for the normal FFT, to one that approaches the idealized O(K logK) for a K-sparse signal. We also develop a signal source separation algorithm using harmonically aligned signal projections (HASP) that is useful for transforming a one-dimensional signal of interest (SOI) into a two-dimensional image in such a way to facilitate automated feature extraction. An automated feature extractor is described which can accurately achieve super-resolution far beyond that of the traditional FFT by again taking advantage of the harmonic structure present in the SOI. We demonstrate HASP and this feature extractor for the application of load disaggregation-identifying the presence of certain types of devices connected to a power system.

We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an oracle function $$f :L^m \rightarrow {\mathbb {F}}_q$$ , where $$L\subset {\mathbb {F}}_q$$ , a verifier distinguishes whether f is the evaluation of a low-degree polynomial or is far (in relative Hamming distance) from being one, by making only a few queries to f. This topic has been studied in the context of locally testable codes, interactive proofs, probalistically checkable proofs, and interactive oracle proofs. We present the first interactive oracle proofs of proximity (IOPP) for tensor products of Reed–Solomon codes (evaluation of polynomials with bounds on individual degrees) and for Reed–Muller codes (evaluation of polynomials with a bound on the total degree) that simultaneously achieve logarithmic query complexity, logarithmic verification time, linear oracle proof length and linear prover running time. Such low-degree polynomials play a central role in constructions of probabilistic proof systems and succinct non-interactive arguments of knowledge with zero-knowledge. For these applications, highly-efficient multivariate low-degree tests are desired, but prior probabilistic proofs of proximity required super-linear proving time. In contrast, for multivariate codes of length N, our constructions admit a prover running in time linear in N and a verifier which is logarithmic in N. Our constructions are directly inspired by the IOPP for Reed–Solomon codes of [Ben-Sasson et al., ICALP 2018] named “FRI protocol”. Compared to the FRI protocol, our IOPP for tensor products of Reed–Solomon codes achieves the same efficiency parameters. As for Reed–Muller codes, for fixed constant number of variables m, the concrete efficiency of our IOPP for Reed–Muller codes compares well, all things equal.

Fast Fourier Transform (FFT) is computationally an efficient algorithm that transforms a function in time domain to function in frequency domain. All 3G and 4G wireless technologies use OFDM and FFT is an integral block of transceiver loop. In this paper three FFT architectures are designed that fall within the Cooley-Tukey class of algorithms. These FFT designs are targeted for OFDM applications especially in future generations of Software Defined Radio (SDR) systems. The FFT is performed on 64-point complex valued input samples with 16-bit precision. Each core is optimized for a combination of area, power, and speed. The design is targeted for a partially reconfigurable FPGA platform such as Xilinx Virtex-4 or above. The FFT cores have been simulated in Model Sim and synthesized using Xilinx ISE software. The cores can be readily downloaded on to target FPGA board to check for run time configurability. The performance evaluation of these cores show that they run at clock frequency of about 80MHz.

Group Invariant Fourier Transform Algorinthms

Compressive sensing (CS) theory has been applied to sparse microwave imaging in many ways that provide better performance and significantly reduce the sampling rate. However, the computational complexity of reconstruction puts strict constraint on some practical applications with large-scale problems in radar imaging. In this paper, we propose a novel fast reconstruction scheme by realizing the traditional matched filtering with CS technique, which maintains the good performance of matched filtering with a reduced number of observations. Meanwhile, a new sparse basis is formed which offers excellent potential for reducing the computational complexity in reconstruction with fast Fourier transform (FFT) and nonuniform FFT, i.e. NUFFT, where the computational complexity can decrease from to . The feasibility and efficiency of the proposed scheme are validated as well through both numerical simulations and raw data processing results.

Chapter 6 - Fast Fourier transform methods

Interpolation lattices for hyperbolic cross trigonometric polynomials

The structure of the various forms of the fast Fourier transform (FFT) is well described by patterns of “butterfly” operations, each involving only an individual pair of inputs or intermediate results, but ultimately yielding one of the most elegant, efficient, and ubiquitous computational algorithms known to mathematics. Here, the structure underlying the FFT is developed in a more general context of a decomposition applicable to any arbitrary complex unitary matrix. By developing this multi-layer decomposition (MLD), it becomes clear that: Any unitary matrix may be decomposed into a butterfly flow graph. Each individual butterfly may be interpreted as an equivalent 2 × 2 standard complex Givens rotation. Identification of the butterfly flow graph for any particular unitary matrix is intimately related to the singular value decomposition (SVD) of the different block sub-matrices of the original unitary matrix. The resulting butterfly flow graph may be interpreted as a prescription of the individual planar coordinate rotations necessary to precisely implement the general multi-dimensional coordinate rotation defined by the unitary matrix. It may also be interpreted as the equivalent polar representation for the matrix. The general form of the butterfly flow graph is not “fast.” The FFT achieves its computational efficiency due to certain specific mathematical symmetries, permitting removal of extraneous butterflies. The form of the butterfly flow graph is sensitive to the order in which the inputs and outputs are specified. A distinct advantage of the MLD is that any approximations or modifications made within this structure continue to result in a full unitary transformation.

Chapter 6 - Continuous, Discrete, and Fast Fourier Transform

This paper presents a new optoelectronic hybrid numerical system for the parallel processing of algebrapolynomial evaluation. Free—space optical perfect shuffle inter—connection network is adopted in our sys-tems. Optical interconnection networks can make the operating time the minimum ,which is proportional tolog 2N.1. INTRODUCTIONThe problem of evaluating polynomials in minimum time as well as Fast Fourier Transform is one im-portant problem of real—time processing. The most effective way to reduce the operating time is the way ofparallel processing. This paper, a new optoelectronic hybrid system of polynomial evaluation using opticalperfect shuffle interconnection networks is reported, in our system, data line can be pipelined in stages and

The fast Fourier transform (FFT) is a highly efficient algorithm for computing the discrete Fourier transform (DFT). It is widely employed in various applications, including digital communication, image processing, and signal analysis. Recently, in-memory computing architectures based on emerging technologies, such as resistive RAM (RRAM), have demonstrated promising performance with low hardware cost for data-intensive applications. However, directly mapping FFT onto RRAM crossbars is challenging because the algorithm relies on many small, sequential butterfly operations, while cross-bars are optimized for large-scale, highly parallel vector–matrix multiplications (VMMs). In this paper, we introduce ReFFT, a system architecture that reformulates FFT computations for efficient execution on RRAM crossbars. ReFFT combines the reduced computational complexity of FFT with the parallel VMM capability of RRAM. We incorporate measured device data into our framework to analyze the effect of variability and develop an adaptive mapping scheme that improves twiddle-factor programming accuracy, leading to a 9.9 dB peak signal-to-noise ratio (PSNR) improvement for a 256-point FFT. Compared with prior RRAM-based DFT designs, ReFFT achieves up to 4.6× and 19.5× higher energy efficiency for 256- and 2048-point FFTs, respectively. The system is further validated in digital communication and satellite image compression tasks.

A new technique that significantly minimizes the aliasing error encountered in the conventional use of the fast Fourier transform (FFT) algorithms for the efficient evaluation of Fourier transforms of spatially limited functions (such as those that occur in the radiation pattern analysis of reflector antennas and planar near field to far field (NF-FF) transformation) is presented and illustrated through a typical example. Employing this technique and a discrete Fourier series (DFS) expansion for the integrand, a method for computing the radiation integrals of reflector antennas and planar NF-FF transformation integrals at arbitrary observation angles with optimum use of computer memory and time is also described.