Efficient Fast Fourier Transform Algorithm Research Articles

REVISED FAST FOURIER TRANSFORM

The problem of realisation of the Discrete Fourier Transform in on-line is analysed because of non-efficient consuming a time for a newrecalculation of spectrum samples if one discrete-time signal sample or even some small portion of samples in period are replaced by newsample or by new samples, respectively. Using Fast Fourier Transform (FFT) procedure it is assumed that some signal samples in the respective period available for processing digitally are updated by a sensor in real time. It is urgent for every new sample that emerges to have a new spectrum. The ordinary recalculation of spectrum samples even with highly efficient Cooley-Tukey FFT algorithm is not suitable due to speedy varying in time real process to be observed. The idea is that FFT procedure should not be recalculated with every new sample, it is needed just to modify it when the new sample emerges and replaces the old one. We retrieve the recursive formulas for FFT algorithms that refer to the spectrum samples modification. In a case of appearing one new sample, the recursive algorithm calculates a new spectrum samples by simple addition of a residual between an old and new samples, multiplied on respective row of Fourier ‘code’ matrix, to a vector of old spectrum samples. An example of 8-point FFT is presented.

A Novel Architecture of Area Efficient FFT Algorithm for FPGA Implementation

Fast Fourier transform (FFT) of large number of samples requires huge hardware resources of field programmable gate arrays (FPGA), which needs more area and power. In this paper, we have presented an area efficient architecture of a FFT processor that reuses the butterfly elements several times. The FFT processor is simulated using VHDL and the results are validated on Virtex-6 FPGA. The proposed architecture out performs the conventional architecture of a N-point FFT processor in terms of area which is reduced by a factor of logN 2with negligible increase in processing time.

On-Chip Implementation of High Resolution High Speed Floating Point Adder/Subtractor with Reducing Mean Latency for OFDM

Problem statement: Fast Fourier Transform (FFT) is widely applied in OFDM trance-receiver communications system. Hence efficient FFT algorithm is always considered. Approach: This study proposed FPGA realization of high resolution high speed low latency floating point adder/subtractor for FFT in OFDM trance-receiver. The design was implemented for 32 bit pipelined adder/subtractor which satisfied IEEE-754 standard for floating-point arithmetic. The design was focused on the trade-off between the latency and speed improvement as well as resolution and silicon area for the chip implementation. In order to reduce the critical path and decrease the latency, the novel structure was designed and investigated. Results: Consequently, synthesis report indicated the latency of 4 clock cycles due to each stage operated within just one clock cycle. The unique structure of designed adder well thought out resulted 6691 equivalent gate count and lead us to obtain low area on chip. Conclusion: The synthesis Xilinx ISE software provided results representing the estimated area and delay for design when it is pipelined to various depths. The report showed the minimum delay of 3.592 ns or maximum frequency of 278.42 MHz.

Non-Gaussian Random Wave Simulation by Two-Dimensional Fourier Transform and Linear Oscillator Response to Morison Force

The one-dimensional fast Fourier transform (FFT) has been applied extensively to simulate Gaussian random wave elevations and water particle kinematics. The actual sea elevations/kinematics exhibit non-Gaussian characteristics that can be represented mathematically by a second-order random wave theory. The elevations/kinematics formulations contain frequency sum and difference terms that usually lead to expensive time-domain dynamic analyses of offshore structural responses. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency-difference terms, inverse FFT and forward FFT are implemented, respectively, across the two dimensions of the wave interaction matrix. Given specified wave conditions, the statistics of simulated elevations/kinematics compare well with not only the empirical fits but also the analytical solutions based on a modified eigenvalue/eigenvector approach, while the computational effort of simulation is very economical. In addition, the stochastic analyses in both time domain and frequency domain show that, attributable to the second-order nonlinear wave effects, the near-surface Morison force and induced linear oscillator response are more non-Gaussian than those subjected to Gaussian random waves.

Sequential detection of almost-harmonic signals

To detect a purely harmonic signal, it is difficult to beat a fast Fourier transform (FFT). However, when the signal is very long and weak, Parker and White (1999) have shown that a sequential probability ratio test (SPRT) operating on magnitude-square FFT data is far more efficient. Indeed, both from a numerical-error perspective and in terms of robustness against a deviation from a precisely tonal signal, the block-FFT/SPRT idea is very appealing. Here, the approach is extended to the case that the frequency is unknown, and expressions are developed for performance both in terms of detection and of sample number. The approach is applicable to a large number of practical problems, but particular attention is paid to the continuous gravitational wave (GW) example. The computational savings as compared with a fixed test vary as a function of signal strength, block length, bandwidth and operating point; however, gains of a factor of two are easy. That these gains are not more exciting relates mostly to the underlying FFT structure; although many SPRTs "end early," it is difficult to take advantage of that with an efficient FFT algorithm. However, the progressive reduction of the number of working SPRTs implies a substantial reduction of the ensemble of the candidate frequencies with time, which is an appealing feature, particularly in the GW case.

An efficient FFT algorithm based on the discrete sine transform

The authors earlier developed a fast recursive algorithm for the discrete sine transform (see IEEE Trans. Acoust. Speech Signal Process., vol.38, no.3, p.553-7, 1990). This algorithm is used as the basic building block for developing the real valued fast Fourier transform (FFT). It is assumed that the input sequence is real and of length N, an integer power of 2. The N-point discrete Fourier transform (DFT) of a real sequence can be implemented via the real (cos DFT) and imaginary (sin DFT) components. The N-point cos DFT in turn can be developed from N/2-point cos DFT and N/4-point discrete sine transform (DST). Similarly, the N-point sin DFT can be developed from N2-point sin DFT and N/4-point DST. Using this approach, an efficient algorithm (involving real arithmetic only) for N-point DFT is developed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The FFT as a Multigrid Algorithm

Apart from the fact that they are both ingenious and remarkably efficient, there would appear to be little kinship between fast Fourier transform (FFT) algorithms and basic multigrid methods. The fast Fourier transform is a powerful direct method for computing the discrete Fourier transform (DFT) of a sequence of complex numbers, whereas multigrid methods are highly refined iterative methods that can be used to approximate the solution of systems of equations. However, on closer examination, there are features of both algorithms that suggest at least a conceptual similarity. This note is written in a heuristic spirit and is intended to offer a motivation for the FFT that allows it to be viewed as a multigrid process. The outcome of this exercise is certainly not a more efficient FFT algorithm. But hopefully, it will serve to demonstrate the way in which multilevel thinking can be used to interpret algorithms and it may lead the way to new algorithms.

Efficient sinc function interpolation technique for center padded data

An efficient zooming FFT (fast Fourier transform) algorithm that allows center padding sinc function interpolation of 2-D images is presented. This algorithm avoids the phase shifts that would be introduced if the efficient Skinner interpolation method is used. Output pruning is incorporated to allow efficient determination of a zoomed subimage. Time savings of more than 50% can be achieved. Example images illustrating the use of the algorithm in conjunction with zooming and ARMA (autoregressive moving average) modeling of data are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Use of the fast fourier transform to improve direct numerical integration schemes for. transient structural response

A hybrid method is presented that combines a Laplace transform technique with a direct integration scheme. The Laplace transform is numerically inverted with an efficient Fast Fourier Transform algorithm. The response determined with the Laplace transform is used to reinitialize a direct integration scheme. The coupling of these solution techniques into a hybrid procedure results in accuracies unattainable with direct integration schemes alone at only a small additional computational cost. The hybrid method was applied to an illustrative problem, the longitudinal response of a rod due to a step load. It was found that the hybrid method retained its accuracy for extremely long response times.

Derivation and implementation of an efficient fast fourier transform algorithm (EFFT)

Abstract An analysis of the sequence of calculations during a standard radix-2 Fast Fourier Transform (FFT) computer algorithm reveals that additional multiplications beyond those normally eliminated by the conventional algorithms are also found to be unnecessary. In particular, the arguments for the trigonometric functions are always used in a well-defined “modified” bit-inverted order. The first four arguments within any pass yield reductions of multiplications via trigonometric identities. Further, by placing the trigonometric parameters in a look-up table in the same modified order, the minimum number of multiplications for a radix-2 FFT is performed by a simplified algorithm termed the EFFT (Efficient Fast Fourier Transform). This algorithm requires no reference to a bit-inversion procedure or look-up table until the output points are “unscrambled” at the conclusion. The feasibility of performing reasonably-large sized EFFT's on personal computers is examined. It is shown that using standard programming practices and currently available hardware, the execution time for the EFFT may he reduced to few seconds for complex transforms up to 2048 double-precision points. On an IBM Personal Computer, this transform may be performed in as little as 4.4 s.

Reduced Data Re-Order Complexity Properties of Polynomial Transform 2D Convolution and Fourier Transform Methods

This paper presents new results concerning the matrix data re-order requirements of polynomial-transform-based 2D convolution and 2D Fourier Transform methods which can be employed in digital processing of images and other 2D problems. The results indicate that several power-of-2 length-modified ring polynomial transform methods developed by Nussbaumer allow the total avoidance of the row-column matrix transpose commonly encountered in other algorithmic approaches, while also providing a number of other computational advantages. It is demonstrated that this property can be the source of significantly improved throughput on a number of existing data processing structures. An execution time comparison with an efficient Fast Fourier Transform algorithm base is made assuming the use of general register architecture and array processor units. It is also assumed that one makes use of recently developed efficient matrix transpose methods by Eklundh and Ari to support 2D FFT data re-order requirements. These comparisons demonstrate a two to four times throughput improvement for the use of the polynomial transform method in place of the 2D FFT approach to circularly convolve or generate 2D Fourier transforms for large 2D fields in the range 1024 × 1024 to 8192 × 8192.