Fast Fourier Transform Computation Research Articles

New continuous-flow mixed-radix (CFMR) FFT Processor using novel in-place strategy

The paper proposes a new continuous-flow mixed-radix (CFMR) fast Fourier transform (FFT) processor that uses the MR (radix-4/2) algorithm and a novel in-place strategy. The existing in-place strategy supports only a fixed-radix FFT algorithm. In contrast, the proposed in-place strategy can support the MR algorithm, which allows CF FFT computations regardless of the length of FFT. The novel in-place strategy is made by interchanging storage locations of butterfly outputs. The CFMR FFT processor provides the MR algorithm, the in-place strategy, and the CF FFT computations at the same time. The CFMR FFT processor requires only two N-word memories due to the proposed in-place strategy. In addition, it uses one butterfly unit that can perform either one radix-4 butterfly or two radix-2 butterflies. The CFMR FFT processor using the 0.18 /spl mu/m SEC cell library consists of 37,000 gates excluding memories, requires only 640 clock cycles for a 512-point FFT and runs at 100 MHz. Therefore, the CFMR FFT processor can reduce hardware complexity and computation cycles compared with existing FFT processors.

Efficient 2D FFT implementation on mediaprocessors

We have developed an efficient implementation to compute the 2D fast Fourier transform (FFT) on a new very long instruction word programmable mediaprocessor. Using instruction-level parallelism and a multimedia instruction set, our radix-4 Cooley–Tukey algorithm optimally maps the FFT computation to the processing resources of the Hitachi/Equator’s MAP mediaprocessor. We have also achieved more efficient data I/O and lower data transfer time compared to traditional implementations by processing several columns in parallel during the column-wise stage of row–column decomposition. We used a programmable direct memory access engine and a double-buffering scheme in the data cache to perform the computation and the data transfer in parallel. Our implementation resulted in 22.4 ms total execution time for a 512 × 512-point 2D complex FFT, which is faster than previous single-chip programmable or dedicated solutions. The implementations on two other mediaprocessors, the TriMedia TM1100 and the BOPS ManArray, illustrate the importance of the instruction set architecture for achieving high performance and the trend of data I/O becoming the limitation on the 2D FFT performance in newer mediaprocessors.

Parallel Numerics and Applications

Parallel Numerics and Applications

Matrix Algorithms of Accelerated Computation of Fast Fourier Transform and Fast Hartley Transform

Matrix Algorithms of Accelerated Computation of Fast Fourier Transform and Fast Hartley Transform

Low‐power Application‐specific Parallel Array Multiplier Design for DSP Applications

Digital Signal Processing (DSP) often involves multiplications with a fixed set of coefficients. This paper presents a novel multiplier design methodology for performing these coefficient multiplications with very low power dissipation. Given bounds on the throughput and the quantization error of the computation, our approach scales the original coefficients to enable the partitioning of each multiplication into a collection of smaller multiplications with shorter critical paths. Significant energy savings are achieved by performing these multiplications in parallel with a scaled supply voltage. Dissipation is further reduced when conventional array multiplier is modified disabling the multiplier rows that do not affect the multiplication′s outcome. We have used our methodology to design low‐power parallel array multipliers for the Fast Fourier Transform (FFT). Simulation results show that our approach can result in significant up to 76% power savings over conventional array multipliers on 64‐coefficient FFT computation.

Optical Digital Fast Fourier Transform System

A system for optical digital fast Fourier transform (FFT) is proposed. Data exchange, adder, and multiplier circuits required for the optical FFT system are designed. Each circuit contains the same basic element of an optical beam switch composed of a polarization rotator and a birefringent plate. In the FFT algorithm, butterfly switching for data exchange is important and the method of optical switching networks is very suitable for such operations. FFT calculation for one-dimensional data of a sinusoidal signal is experimentally demonstrated using the optical system. The throughput of the proposed system is estimated and its future perspective is discussed.

Real-time FFT algorithm applied to on-line spectral analysis

On-line running spectral analysis is of considerable interest in many electrophysiological signals, such as the EEG (electroencephalograph). This paper presents a new method of implementing the fast Fourier transform (FFT) algorithm. Our "real-time FFT algorithm" efficiently utilizes computer time to perform the FFT computation while data acquisition proceeds so that local butterfly modules are built using the data points that are already available. The real-time FFT algorithm is developed using the decimation-in-time split-radix FFT (DIT sr-FFT) butterfly structure. In order to demonstate the synchronization ability of the proposed algorithm, the authors develop a method of evaluating the number of arithmetic operations that it requires. Both the derivation and the experimental result show that the real-time FFT algorithm is superior to the conventional whole-block FFT algorithm in synchronizing with the data acquisition process. Given that the FFT sizeN=2 r , real-time implementation of the FFT algorithm requires only 2/r the computational time required by the whole-block FFT algorithm.

Details of the integral equation method applied to the analysis of an adhesive layer crack

The mathematical model for a crack in an elastic adhesive layer sandwiched between two adherends proposed by Fleck, Hutchinson and Suo ((1991) Crack path selection in a brittle adhesive layer. International Journal of Solids and Structures27(13), 1683–1703) was considered. The elastic mismatch between the adhesive and adherend materials modifies the far-field values of the stress intensity factors and of the T-stress in a manner that depends on the position of the crack inside the layer and on the Dundurs parameters. A complex-potential stress-function formulation, using dislocation distributions represented by truncated Chebyshev series, yielded an integral equation that was solved numerically by the method of collocations. The symbolic derivation of the integral equations was checked, and a few differences from Fleck, Hutchinson and Suo's expressions were identified and reconciled. The computational aspects of the solution were studied in detail using two programming languages, MATHCAD and C++, run on standard PC hardware. A palette of numerical techniques were utilized to study and control the consistency and accuracy of the solution. Symmetry and anti-symmetry arguments were used to identify numerically sensitive regions. Fast Fourier Transform calculation of sine and cosine Fourier integrals was used to increase speed. Convergence of intermediate and final results was examined. It was found that the method is very sensitive to the details of numerical computation, especially for combinations of parameters that lead to nearly singular matrices.

Equiripple FIR filter design by the FFT algorithm

The fast Fourier transform (FFT) algorithm has been used in a variety of applications in signal and image processing. In this article, a simple procedure for designing finite-extent impulse response (FIR) discrete-time filters using the FFT algorithm is described. The zero-phase (or linear phase) FIR filter design problem is formulated to alternately satisfy the frequency domain constraints on the magnitude response bounds and time domain constraints on the impulse response support. The design scheme is iterative in which each iteration requires two FFT computations. The resultant filter is an equiripple approximation to the desired frequency response. The main advantage of the FFT-based design method is its implementational simplicity and versatility. Furthermore, the way the algorithm works is intuitive and any additional constraint can be incorporated in the iterations, as long as the convexity property of the overall operations is preserved. In one-dimensional cases, the most widely used equiripple FIR filter design algorithm is the Parks-McClellan algorithm (1972). This algorithm is based on linear programming, and it is computationally efficient. However, it cannot be generalized to higher dimensions. Extension of our design method to higher dimensions is straightforward. In this case two multidimensional FFT computations are needed in each iteration.

Signal processing methods for pulse oximetry

Current signal processing technology has driven many advances in almost every aspect of life, including medical applications. It follows that applying signal processing techniques to pulse oximetry could also provide major improvements. This research was designed to identify and implement one or more techniques that could improve pulse oximetry oxygen saturation (SpO 2) measurements. The hypothesis was that frequency domain analysis could more easily extract the cardiac rate and amplitude of interest from the time domain signal. The focus was on the digital signal processing algorithms that had potential to improve pulse oximetry readings, and then test those algorithms. This was accomplished using the Fast Fourier Transform (FFT) and the Discrete Cosine Transform (DCT). The results indicate that the FFT and DCT computation of oxygen saturation were as accurate without averaging, as weighted moving average (WMA) algorithms currently being used, and directly indicate when erroneous calculations occur.

Parallel FFT algorithms using radix 4 butterfly computation on an eight-neighbor processor array

Fast Fourier transform (FFT), which has wide and variety application areas, requires very high speed computation. Since parallel processing of FFT is very attractive for high speed FFT computation, many processor arrays and multiprocessor systems have been proposed with efficient FFT algorithms. As a result of the recent development of VLSI technology, several massively parallel computers have been implemented on commercial basis. The MasPar, which is one of the SIMD type massively parallel computers, consists of an eight-neighbor processor array. This paper discusses parallel 1-D FFT algorithms on an eight-neighbor processor array. We propose three algorithms according to various data allocation methods. Then we estimate and evaluate their processing time. With the number of processors N = N r × N r , processing time is estimated to be 2( N r − 2) t c + ( log 2 N r ) t b , where t c is the communication time between neighbor processors, and t b is the execution time for the radix 4 butterfly computation. We also compare these algorithms with the conventional radix 2 FFT algorithm implemented on a mesh processor array. It is shown that the radix 4 FFT algorithms are faster than the radix 2 algorithms. These algorithms get high speed FFT computation by combining the radix 4 FFT algorithm with the characteristics of the eight-neighbor processor array.

Hypercube concurrent processor implementation of a position invariant object classifier

A position-invariant fast object classification scheme is described. The grey-level image of objects is converted to binary form and a parallel region growing technique is employed to detect objects. A 2-D fast Fourier transform (FFT) is applied to each object region after translating the origin of the image co-ordinate system to the object centre and aligning the image co-ordinate axes with the object principal axes. The first five components from the principal lobe of the Fourier spectrum of each object are selected as characteristic features for minimum-distance object classification. For time efficiency, region growing and 2-D FFT computations were performed on a 16-node hypercube processor.

Improved FFT-based numerical inversion of Laplace transforms via fast Hartley transform algorithm

The disadvantages of numerical inversion of the Laplace transform via the conventional fast Fourier transform (FFT) are identified and an improved method is presented to remedy them. The improved method is based on introducing a new integration step length Δω = π/ mT for trapezoidal-rule approximation of the Bromwich integral, in which a new parameter m, is introduced for controlling the accuracy of the numerical integration. Naturally, this method leads to multiple sets of complex FFT computations. A new inversion formula is derived such that N equally-spaced samples of the inverse Laplace transform function can be obtained by [ m 2 ] + 1 sets of N-point complex FFT computations or by m sets of real fast Hartley transform (FHT) computations.

Diffraction tomography of strongly scattering infinite cylindrical objects of arbitrary cross-sectional shape

A new method for imaging an infinite cylindrical object with arbitrary cross-sectional shape, without any small-perturbation approximations, is proposed. In this method, the well-known integral equation equivalent to the inhomogeneous Helmholtz wave equation is discretized. The discretized equation expresses the scattered pressure field at any point in terms of the samples of the product of the unknown object function and the unknown pressure field present inside the object. Based on this relation, simultaneous equations are set up, relating these samples to the samples of the pressure field measured outside the object. By requiring the measurement points to lie on circles around the object, these equations are solved efficiently using the fast Fourier transform (FFT). From the product samples thus obtained, the samples of the unknown object function are extracted by means of a second set of FFT calculations. Details of illustrative computer simulations for the case of an elliptic cylinder are described.

Expandable Platform for Dynamic Signal Analysis

SciTech MicroSystems, who specialise in novel signal processing solutions, has announced the availability of the SPI Collection of array processor cards to perform the high speed, Fast Fourier Transform calculations required in signal processing applications. The modular design approach enables engineers to have the capabilities of a purpose built instrument within a PC at a considerably reduced cost.

Implementation and performance of the fast Hartley transform

The authors investigated the implementation aspects of the fast Hartley transform (FHT) in both software and hardware. They describe the modifications required to convert existing fast Fourier transform (FFT) programs to execute FHTs, showing the ease with which these modifications can be implemented. They compare execution time and memory storage requirements of both transforms and present power spectrum calculation and convolution as illustrative examples to compare the performances of the two transform techniques. They also give a comparative survey of the performances of various microprocessors and digital signal processors in FFT and FHT computation. >

Implementing fast Fourier transforms using the Am29500 family

The paper discusses the implementation of fast Fourier transform (FFT) algorithms using members of the Am29500 family of microprocessors and peripherals. First the suitability of the Am29500 family for signal processing applications is discussed. The architectural requirements of FFT processors are then outlined. A parallel processing architecture using pipelining is developed and the microprogramming of the system is described. Timing and implementation details, together with some practical test results, are given. The paper concentrates mainly on radix-2 decimation-in-time (DIT) FFT computations, but the architecture described can be applied to variable-radix processors running DIT or DIF (decimation-in-frequency) algorithms.

FFT performance in the presence of noise.

Contamination of signals by noise degrades the performance of fast Fourier transform (FFT) analysis of biological systems. Thus, we have developed simulation techniques to investigate the effects of noise on FFT computations. Continuous and discrete representations of forcing-noise and response-noise signals are derived. The FFT is used to estimate magnitude and phase of noisy digital signals constructed using the discrete representations. The estimates are then compared to the estimates obtained from the noise-free digital signals. The following factors are shown to have an important influence on estimation accuracy: the inclusion of noninteger as well as integer harmonic noise, the signal series length, the relative noise-to-forcing and response signal magnitude ratios, and the degree of noise signal cross correlation between the forcing and response signals. We demonstrate that the FFT estimation accuracy of magnitude and phase is similar for integer and noninteger noise harmonics, it varies directly with signal series length, and inversely with the noise-to-forcing and response signal magnitude ratios.

Fast convolution methods for hexagonally sampled two‐dimensional signals

AbstractFor fast convolution operations on one‐dimensional signal sequences there is the fast cyclic convolution (FCC) method based on sectioning. Also, for a two‐dimensional signal sequence in the convolution operation of a rectangularly sampled signal sequence and a convolving sequence arranged in rectangular blocks, an FCC method based on rectangular sectioning has been reported. On the other hand, two‐dimensional signals band‐limited over a circular region are sampled hexagonally. This arrangement of the convolving sequence in hexagonal blocks to provide it with 12‐fold symmetry is most effective. This paper describes an FCC method based on sectioning for such types of hexagonally sampled signal sequence and convolving sequence arranged in hexagonal blocks, Presupposing the use of a fast Fourier transformation (FFT) of base number 2 in the discrete Fourier transformation in FCC, two types of FCC (hexagonal sectioning and parallelogrammatic sectioning) are proposed and overlap‐and‐save algorithms for these two are given. Also, it is shown that there exists an optimal sectioning block size for which the number of numerical operations becomes minimum in both methods. Next, for both types, the number of numerical operations are calculated and compared. Due to the limited memory capacity useable in FFT calculation, if convolution is not possible with optimal sectioning, the hexagonal sectioning method is advantageous. However, when the memory size is sufficiently large, it is difficult to say which of the two is better.

Multivariate signal analysis for process noise source identification

In the present work sponsored by Swedish National Board for Energy Source Development a method for identification of noise sources has been used — the multivariate auto regressive process (AR). Analyses were performed using simulated noise and operating nuclear power plant process noise. The simulated noise signals are generated mathematically in the same way as the system structure identified by the AR-method. This fact makes the interpretation of the result easy. The analysis of the simulated noise shows convincingly the possibility of the method to identify a simulated stochastic system and to clarify the influence of the different noise sources. Auto power spectrum of nuclear power plant signals calculated by the AR-method are in good agreement with FFT (Fast Fourier Transform) calculations. The contributions to the noise from the different sources calculated by the AR-program are in agreement with expected characteristics from physical considerations. With the aid of power plant signals examples are given how the result of the analyses are influenced by the number of measurement signals. Analyses of 3 and 1 signals are given for comparison. Given examples also emphasize the importance of chosing a suitable selection of measured signals. Experience from the test runs shows that non stationary noise signals or unsuitably chosen order may cause the computation not to converge. This can in general be overcome by choosing a different order of measured signals and/or increase the number of time series data used. The analyses of simulated noise were performed using 500 time series data while for the power plant noise 2 000 – 3 000 data were required to reach convergence in the calculations. The difference in requirements of time series data shows that the power plant signals do not have the same ideal statistics as simulated noise.