Fast Discrete Fourier Transform Research Articles

A New Method to Construct Algorithms for Fast Discrete Fourier Transform of Finite Complex and Real Signals Based on Thesecond Type Parametric Discrete Fourier Transforms

A New Method to Construct Algorithms for Fast Discrete Fourier Transform of Finite Complex and Real Signals Based on Thesecond Type Parametric Discrete Fourier Transforms

Speech Encryption with Fractional Watermark

Research on the feature of speech and image signals are carried out from two perspectives, the time domain and the frequency domain. The speech and image signals are a non-stationary signal, so FT is not used for the non-stationary characteristics of the signal. When short-term stable speech is obtained by windowing and framing the subsequent processing of the signal is completed by the Discrete Fourier Transform (DFT). The Fast Discrete Fourier Transform is a commonly used analysis method for speech and image signal processing in frequency domain. It has the problem of adjusting window size to a for desired resolution. But the Fractional Fourier Transform can have both time domain and frequency domain processing capabilities. This paper performs global processing speech encryption by combining speech with image of Fractional Fourier Transform. The speech signal is embedded watermark image that is processed by fractional transformation, and the embedded watermark has the effect of rotation and superposition, which improves the security of the speech. The paper results show that the proposed speech encryption method has a higher security level by Fractional Fourier Transform. The technology is easy to extend to practical applications.

Fast Discrete Fourier Transform on Local Fields of Zero Characteristic

For the discrete Fourier transform with respect to the system of characters of a local field with zero characteristic, we propose a fast algorithm. We find the complexity of the algorithm.

Parallel Lattice Structure for Dual Windows Computation in Multiwindow Gabor Transform

The multiwindow discrete Gabor transform (M-DGT) is a useful time-frequency analysis tool for non-stationary signal processing. Given an arbitrary Gabor frame, a parallel lattice structure of block time-recursive algorithm for fast and efficient computation of dual/analysis Gabor windows for M-DGT is presented. By using a multiple window dual Gabor frame, the dual Gabor windows can be expressed by synthesis and analysis windows with a block-circulant matrix. Then, a parallel lattice structure of block time-recursive can be derived to solve the dual Gabor windows by the block-circulant matrix computed by the fast discrete Fourier transform (FFT). When compared to three existing methods, the proposed algorithm can reduce computational complexity and save computational time. Experimental results indicate that the proposed algorithm is valid to compute the dual windows of the M-DGT, which make the algorithm attractive for fast time-frequency signal analysis and processing.

Effective Approach to Calculate Analysis Window in Infinite Discrete Gabor Transform

The long-periodic/infinite discrete Gabor transform (DGT) is more effective than the periodic/finite one in many applications. In this paper, a fast and effective approach is presented to efficiently compute the Gabor analysis window for arbitrary given synthesis window in DGT of long-periodic/infinite sequences, in which the new orthogonality constraint between analysis window and synthesis window in DGT for long-periodic/infinite sequences is derived and proved to be equivalent to the completeness condition of the long-periodic/infinite DGT. By using the property of delta function, the original orthogonality can be expressed as a certain number of linear equation sets in both the critical sampling case and the oversampling case, which can be fast and efficiently calculated by fast discrete Fourier transform (FFT). The computational complexity of the proposed approach is analyzed and compared with that of the existing canonical algorithms. The numerical results indicate that the proposed approach is efficient and fast for computing Gabor analysis window in both the critical sampling case and the oversampling case in comparison to existing algorithms.

Constructing spatial discretizations for sparse multivariate trigonometric polynomials that allow for a fast discrete Fourier transform

The paper discusses the construction of high dimensional spatial discretizations for arbitrary multivariate trigonometric polynomials, where the frequency support of the trigonometric polynomial is known. We suggest a construction based on the union of several rank-1 lattices as sampling scheme. We call such schemes multiple rank-1 lattices. This approach automatically makes available a fast discrete Fourier transform (FFT) on the data. The key objective of the construction of spatial discretizations is the unique reconstruction of the trigonometric polynomial using the sampling values at the sampling nodes. We develop different construction methods for multiple rank-1 lattices that allow for this unique reconstruction. The symbol M denotes the total number of sampling nodes within the multiple rank-1 lattice. In addition, we assume that the multivariate trigonometric polynomial is a linear combination of T trigonometric monomials. The ratio of the number M of sampling points that are sufficient for the unique reconstruction to the number T of distinct monomials is called oversampling factor in this context. The presented construction methods for multiple rank-1 lattices allow for estimates of this number M . Roughly speaking, the oversampling factor M / T is independent of the spatial dimension and, with high probability, only logarithmic in T , which is much better than the oversampling factor that is expected for a sampling method that uses one single rank-1 lattice. The newly developed approaches for the construction of spatial discretizations are probabilistic methods. The arithmetic complexity of these algorithms depend only linearly on the spatial dimension and, with high probability, only linearly on T up to some logarithmic factors. Furthermore, we analyze the computational complexities of the resulting FFT algorithms, that exploits the structure of the suggested multiple rank-1 lattice spatial discretizations, in detail and obtain upper bounds in O ( M log ⁡ M ) , where the constants depend only linearly on the spatial dimension. With high probability, we construct spatial discretizations where M / T ≤ C log ⁡ T holds, which implies that the complexity of the corresponding FFT converts to O ( T log 2 ⁡ T ) .

Fast Discrete Fourier Spectra Attacks on Stream Ciphers

In this paper, some new results are presented on the selective discrete Fourier spectra attack introduced first as the Ronjom-Helleseth attack and the modifications due to Ronjom, Gong, and Helleseth. The first part of this paper fills some gaps in the theory of analysis in terms of the discrete Fourier transform (DFT). The second part introduces the new fast selective DFT attacks, which are closely related to the fast algebraic attacks in the literature. However, in contrast to the classical view that successful algebraic cryptanalysis of LFSR-based stream cipher depends on the degree of certain annihilators, the analysis in terms of the DFT spectral properties of the sequences generated by these functions is far more refined. It is shown that the selective DFT attack is more efficient than known methods for the case when the number of observed consecutive bits of a filter generator is less than the linear complexity of the sequence. Thus, by utilizing the natural representation imposed by the underlying LFSRs, in certain cases, the analysis in terms of DFT spectra is more efficient and has more flexibility than classical and fast algebraic attacks. Consequently, the new attack imposes a new criterion for the design of cryptographic strong Boolean functions, which is defined as the spectral immunity of a sequence or a Boolean function.

On Some Applications of the Fast Discrete Fourier Transform

Motivated by the excellent work of Bill Davis and Jerry Uhl’s Differential Equations & Mathematica [1], we present in detail several little-known applications of the fast discrete Fourier transform (DFT), also known as FFT. Namely, we first examine the use of the FFT in multiplying univariate polynomials and integers and approximating polynomials with sines and cosines (also known as the fast Fourier fit or FFF). We then examine the use of the FFF in solving differential equations with Laplace transforms and rediscovering trigonometric identities.

Fast Fourier Transform – Calculation and Interpretation

The article is intended to the students, who make their first steps in the application of the Fourier transform to physics problems. We examine several elementary examples from the signal theory and classic optics to show relation between continuous and discrete Fourier transform. Recipes for correct interpretation of the results of FDFT (Fast Discrete Fourier Transform) obtained with the commonly used application programs (Matlab, Mathcad, Mathematica) are given.

Closed-Form Orthogonal DFT Eigenvectors Generated by Complete Generalized Legendre Sequence

In this paper, we propose a new method for deriving the closed-form orthogonal discrete Fourier transform (DFT) eigenvectors of arbitrary length using the complete generalized Legendre sequence (CGLS). From the eigenvectors, we then develop a novel method for computing the DFT. By taking a specific eigendecomposition to the DFT matrix, after proper arrangement, we can derive a new fast DFT algorithm with systematic construction of an arbitrary length that reduces the number of multiplications needed as compared with the existing fast algorithm. Moreover, we can also use the proposed CGLS-like DFT eigenvectors to define a new type of the discrete fractional Fourier transform, which is efficient in implementation and effective for encryption and OFDM.

A quantum search algorithm of two entangled registers to realize quantum discrete Fourier transform of signal processing

The discrete Fourier transform (DFT) is the base of modern signal processing. 1-dimensional fast Fourier transform (ID FFT) and 2D FFT have time complexity O (N log N) and O (N2 log N) respectively. Since 1965, there has been no more essential breakthrough for the design of fast DFT algorithm. DFT has two properties. One property is that DFT is energy conservation transform. The other property is that many DFT coefficients are close to zero. The basic idea of this paper is that the generalized Grover's iteration can perform the computation of DFT which acts on the entangled states to search the big DFT coefficients until these big coefficients contain nearly all energy. One-dimensional quantum DFT (ID QDFT) and two-dimensional quantum DFT (2D QDFT) are presented in this paper. The quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, ID and 2D QDFT have time complexity O(√N) and O (N) respectively. QDFT and quantum convolution demonstrate that quantum computation to process classical signal is possible.

Fast Discrete Fourier Transform Computations Using the Reduced Adder Graph Technique

It has recently been shown that the n-dimensional reduced adder graph (RAG-n) technique is beneficial for many DSP applications such as for FIR and IIR filters, where multipliers can be grouped in multiplier blocks. This paper highlights the importance of DFT and FFT as DSP objects and also explores how the RAG-n technique can be applied to these algorithms. This RAG-n DFT will be shown to be of low complexity and possess an attractively regular VLSI data flow when implemented with the Rader DFT algorithm or the Bluestein chirp-z algorithm. ASIC synthesis data are provided and demonstrate the low complexity and high speed of the design when compared to other alternatives.

A fast DFT planar array synthesis tool for generating contoured beams

An efficient procedure, based on a discrete Fourier transform (DFT) expansion of planar array distributions, is described for synthesizing contoured beams in satellite antenna applications. The DFT provides a global basis set that spans over the entire array, and thus produces highly directional beam patterns to serve as corresponding local basis functions for representing the desired far zone radiation field of the array. The unknown coefficients of the DFT expansion are then found via a minimum least square error (MLSE) technique. While the total number of DFT coefficients exactly equals the number of array elements, it is found that only a small fraction of the total unknown DFT coefficients generally remain dominant, and only these need to be solved, thus making the present approach very fast.

A Novel Approach to Fast Discrete Fourier Transform

Discrete Fourier transform (DFT) is an important tool in digital signal processing. In the present paper, we propose a novel approach to performing DFT. We transform DFT into a form expressed in discrete moments via a modular mapping and truncating Taylor series expansion. From this, we extend the use of our systolic array for fast computation of moments without any multiplications to one that computes DFT with only a few multiplications and without any evaluations of exponential functions. The multiplication number used in our method isO(Nlog2N/ log2log2N) superior toO(Nlog2N) in FFT. The execution time of the systolic array is onlyO(Nlog2N/ log2log2N) for 1-D DFT andO(Nk) fork-D DFT (k⩾2). The systolic implementation is a demonstration of the locality of dataflow in the algorithms and hence it implies an easy and potential hardware/VLSI realization. The approach is also applicable to DFT inverses.

A New Fast Discrete Fourier Transform

This paper presents a new fast Discrete Fourier Transform (DFT) algorithm. By rewriting the DFT, a new algorithm is obtained that uses 2n?2(3n?13)+4n?2 real multiplications and 2n?2(7n?29)+6n+2 real additions for a real data N=2n point DFT, comparable to the number of operations in the Split-Radix method, but with slightly fewer multiply and add operations in total. Because of the organization of multiplications as plane rotations in this DFT algorithm, it is possible to apply a pipelined CORDIC algorithm in a hardware implementation of a long-point DFT, e.g., at a 100 MHz input rate, a 1024-point transform can be realized with a 200 MHz clocking of a single CORDIC pipeline.

A pipeline design of a fast prime factor DFT on a finite field

A conventional prime factor discrete Fourier transform (DFT) algorithm of the Winograd type is used to realize a discrete Fourier-like transform on the finite field GF(q/sup /n). A pipeline structure is used to implement this prime-factor DFT over GF(q/sup /n). This algorithm is developed to compute cyclic convolutions of complex numbers and to aid in decoding the Reed-Solomon codes. Such a pipeline fast prime-factor DFT algorithm over GF(q/sup /n) is regular, simple, expandable, and naturally suitable for most implementation technologies. An example illustrating the pipeline aspect of a 30-point transform over GF(q/sup /n) is presented. >

Deconvolution of gravity data due to lateral density distributions

A filter technique for deconvolution of gravity data due to an infinite horizontal slab with laterally varying density is described. The inverse filtering is performed in the spectral domain. Numerical calculations are carried out by using fast discrete Fourier transform (FFT) algorithm. Synthetic examples prove the accuracy of the method. To demonstrate the practical use, the filter technique is applied to a gravity profile in the Grossvenediger area (Austria).

A fast pipelined DFT processor and its programming consideration

A new discrete Fourier transform (DFT) processor with a pipelined structure has been developed. This processor is designed to optimise computation of the pair of operationsAx0 ±Bx1, which is mostly encountered in various fast DFT algorithms. For real-valued data and coefficients, the processor needs only two machine cycles to calculate the pair of operations. A straightforward multiple-stage transform algorithm has been proposed to implement real-valued prime-factor or radix-type transforms. About half of the computation can be saved by taking into account the fact that transform outputs are conjugate pairs for real inputs. The short Winograd Fourier transform algorithm is suggested as a basic building block for large transforms because it is more efficient than the fast Fourier transform.

Hybrid time-delay/phase-shift digital beamforming for uniform collinear arrays

Digital phase-shift beamforming of narrow-band signals from uniform collinear hydrophone arrays may be accomplished by computationally efficient discrete Fourier transform (DFT) algorithms. However, phase-shift beamforming is a valid approximation to delay-and-sum steering only over a restricted bandwidth limited by array aperture and steering angle. A multiple-beam digital beamforming technique is described which is a hybrid of phase-shift and time-delay operations designed for bandwidths in excess of the phase-shift limit. The beamformer employs fast DFTs for phase-shifting, organized according to a particular decomposition of the array into subarrays, plus a limited amount of delay steering by interpolation. The result is a substantial reduction in the number of digital arithmetic operations compared with the alternative of delay-and-sum steering by interpolation. The design principles of the hybrid beamformer are described and an example presented to illustrate its computational advantages.

Addendum to "A New Hybrid Algorithm for Computing a Fast Discrete Fourier Transform"

Recently,1 the authors proposed a hybrid algorithm for computing the discrete Fourier transform (DFT) of certain long transform lengths. In that technique, a Winograd-type algorithm was used in conjunction with the Mersenne prime-number theoretic transform to perform a DFT. Even though this technique requires fewer multiplications than either the standard fast Fourier transform (FFT) or Winograd's more conventional algorithm, it increases the number of additions considerably. In this letter it is proposed to use Winograd's algorithm for computing the Mersenne prime-number theoretic transform in the transform portion of the hybrid algorithm. It is shown that this can reduce significantly the number of additions while still maintaining about the same number of multiplications.