Showing papers on "Prime-factor FFT algorithm published in 2006"

Fast three-step phase-shifting algorithm

Abstract: We propose a new three-step phase-shifting algorithm, which is much faster than the traditional three-step algorithm. We achieve the speed advantage by using a simple intensity ratio function to replace the arctangent function in the traditional algorithm. The phase error caused by this new algorithm is compensated for by use of a look-up-table (LUT). Our experimental results show that both the new algorithm and the traditional algorithm generate similar results, but the new algorithm is 3.4 times faster. By implementing this new algorithm in a high-resolution, real-time 3D shape measurement system, we were able to achieve a measurement speed of 40 frames per second (fps) at a resolution of 532 × 500 pixels, all with an ordinary personal computer.

An efficient locally pipelined FFT processor

Abstract: The fast Fourier transform (FFT) is a very important algorithm in digital signal processing. The locally pipelined (LPPL) architecture is an efficient structure for FFT processor designing in a real-time embedded system. Two basic building blocks, to the LPPL FFT processor, the butterfly in pipeline, and address generating, are discussed in this brief. Based on the "deep" feedback to butterfly-2, a novel approach for pipelined architecture, the radix-2 single-path deep delay feedback architecture is proposed. For length-N discrete Fourier transform computation, the dominant hardware requirements are minimal for complex multipliers log/sub 4/N-1 and adders 2log/sub 4/N. As an integral need of the LPPL FFT processor design, address generating and coefficient store-load structures are also presented.

FFT Algorithm with High Accuracy for Harmonic Analysis in Power System

Abstract: The Fast Fourier Transform(FFT) cannot be directly used to the harmonic analysis of electric power system because of its higher error,especially the phase error when used with a sample sequence which is not synchronized with the signal.To reduce the influence of an unsynchronized sample sequence on FFT and to improve the precision of harmonics in electric power system,this paper improves the algorithm by using windows and interpolation methods.The paper firstly addressed the leakage and picket fence effects of FFT briefly and then analyzes the interpolation algorithm on Rife Vincent(I) window in detail.Besides,the computing formula for the harmonic parameter estimation are given as well.The simulating result demonstrates that the improved algorithm holds a very high precision wien used for the unsynchronized sample sequence.

Efficient computation of quadratic-phase integrals in optics

Abstract: We present a fast NlogN time algorithm for computing quadratic-phase integrals. This three-parameter class of integrals models propagation in free space in the Fresnel approximation, passage through thin lenses, and propagation in quadratic graded-index media as well as any combination of any number of these and is therefore of importance in optics. By carefully managing the sampling rate, one need not choose N much larger than the space-bandwidth product of the signals, despite the highly oscillatory integral kernel. The only deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus the algorithm computes quadratic-phase integrals with a performance similar to that of the fast-Fourier-transform algorithm in computing the Fourier transform, in terms of both speed and accuracy.

The Fast Fourier and Hilbert-Huang Transforms: A Comparison

Abstract: The conversion of time domain data via the fast Fourier (FFT) and Hilbert-Huang (HHT) transforms is compared. The FFT treats amplitude vs. time information globally as it transforms the data to an amplitude vs. frequency description. The HHT is not constrained by the assumptions of stationarity and linearity, required for the FFT, and generates both amplitude and frequency information as a function of time. The behavior and flexibility of these two transforms are examined for a number of different time domain signal types.

Memory-efficient and high-speed split-radix FFT/IFFT processor based on pipelined CORDIC rotations

Abstract: The author presents a CORDIC-based split-radix fast Fourier transform (FFT)/inverse FFT (IFFT) processor dedicated to the computation of 2048/4096/8192-point discrete Fourier transforms (DFTs). The arithmetic unit of a butterfly processor and a twiddle factor generator are based on a CORDIC algorithm. An efficient implementation of the CORDIC-based split-radix FFT algorithm is demonstrated. The chip of 2048/4096/8192-point FFT/IFFT core processor is fabricated in a 0.18 µm CMOS technology. The core size is 4860×7883 µm2 and contains about 200 822 gates for logic and memory, and the power dissipation is 350 mW with a clock rate of 150 MHz at 1.8 V. All control signals are generated internally on-chip. The processor performs 8192-point FFT/IFFT every 138 µs and 2048-point FFT/IFFT every 34.5 µs, respectively, which exceeds orthogonal frequency division multiplexer symbol rates. The modified-pipelining CORDIC arithmetic unit is employed for complex multiplication. A CORDIC twiddle factor generator is proposed and implemented for reducing the size of ROM required for storing the twiddle factors. Compared with conventional FFT implementations, the power consumption is reduced by 25%.

Fast fourier transform twiddle multiplication

Abstract: An FFT engine implementing a cycle count method of applying twiddle multiplications in multi-stages. When implementing a multistage FFT, the intermediate values need to be multiplied by various twiddle factors. The FFT engine utilizes a minimal number of multipliers to perform the twiddle multiplications in an efficient pipeline. Optimizing a number of complex multipliers based on an FFT radix and a number of values in each row of memory allows the FFT function to be performed using a reasonable amount of area and in a minimal number of cycles. Strategic ordering and grouping of the values allows the FFT operation to be performed in a fewer number of cycles.

The "Invaders' Algorithm: Range of Values Modulation for Accelerated Correlation

Abstract: In this paper, we present an algorithm that allows the simultaneous calculation of several cross correlations. The algorithm works by shifting the range of values of different images/signals to occupy different orders of magnitude and then combining them to form a single composite image/signal. Because additional signals are placed in the space usually occupied by a single signal, we call this the "invaders algorithm," to imply that extra signals invade the space that normally belongs to a single signal. After correlation is performed, the individual results are recovered by performing the inverse operation. The limitations of the algorithm are imposed by the finite length of the mantissa of the hardware used, the precision of the algorithm that performs the cross correlation (e.g., the precision of the fast Fourier transform (FFT)) and by the actual values of the images/signals that are to be combined. The algorithm does not require any special hardware or special FFT algorithm. For typical 250 times 256 images, an acceleration by a factor of at least two in the calculation of their cross correlations is guaranteed using an ordinary PC or a laptop. As for smaller sized templates, tenfold accelerations may be achieved

Achieving efficient polynomial multiplication in fermat fields using the fast Fourier transform

Abstract: We introduce an efficient way of performing polynomial multiplication in a class of finite fields GF(pm) in the frequency domain. The Fast Fourier Transform (FFT) based frequency domain multiplication technique, originally proposed for integer multiplication, provides an extremely efficient method for multiplication with the best known asymptotic complexity, i.e. O(n log n log log n). Unfortunately, the original FFT method bears significant overhead due to the conversions between the time and the frequency domains, which makes it impractical to perform multiplication of relatively short (160 - 1024 bits) integer operands as used in many applications. In this work, we introduce an efficient way of performing polynomial multiplication in finite fields using the FFT. We show that, with careful selection of parameters, all the multiplications required for the FFT computations can be avoided and polynomial multiplication in finite fields can be achieved with only O(m) multiplications in addition to O(m log m) simple shift, addition and subtraction operations. We show that, especially in constrained devices where multiplication is expensive, polynomial multiplication in the suggested finite fields using the FFT outperforms both the schoolbook and Karatsuba methods for practically small finite fields, e.g., relevant to elliptic curve cryptography.

The Fast Fourier and Hilbert-Huang Transforms: A Comparison

Abstract: The conversion of time domain data via the fast Fourier (FFT) and Hilbert-Huang (HHT) transforms is compared. The FFT treats amplitude vs. time information globally as it transforms the data to an amplitude vs. frequency description. The HHT is not constrained by the assumptions of stationarity and linearity, required for the FFT, and generates both amplitude and frequency information as a function of time. The behavior and flexibility of these two transforms are examined for a number of different time domain signal types.

A Split Vector-Radix Algorithm for the 3-D Discrete Hartley Transform

Abstract: In this paper, we propose a three-dimensional (3-D) split vector-radix fast Hartley transform (FHT) algorithm. The main idea behind the proposed algorithm is that the radix-2/4 approach is introduced in the decomposition of the 3-D discrete Hartley transform by using an appropriate index mapping and the Kronecker product. This provides an algorithm based on a mixture of radix-(2times2times2) and radix-(4times4times4) index maps and has a butterfly that is characterized by simple closed-form expressions. This algorithm offers substantial reductions in the numbers of multiplications, additions, data transfers, and twiddle factor evaluations or accesses to the look-up table, without a significant increase in the structural complexity compared to that of the existing 3-D vector radix FHT algorithm

An incremental algorithm for signal reconstruction from short-time fourier transform magnitude.

Abstract: We present an algorithm for reconstructing a time-domain signal from the magnitude of a short-time Fourier transform (STFT). In contrast to existing algorithms based on alternating projections, we offer a novel approach involving numerical root-finding combined with explicit smoothness assumptions. Our technique produces high-quality reconstructions that have lower signal-to-noise ratios when compared to other existing algorithms. If there is little redundancy in the given STFT, in particular, the algorithm can produce signals which also sound significantly better perceptually, as compared to existing work.

Arabic Character Recognition using Modified Fourier Spectrum (MFS)

Abstract: Arabic character recognition algorithm using Modified Fourier Spectrum (MFS) is presented. The MFS descriptors are estimated by applying the Fast Fourier Transform (FFT) to the Arabic character primary part contour. Ten descriptors are estimated from the Fourier spectrum of the character primary part contour by subtracting the imaginary part from the real part (and not from the amplitude of the Fourier spectrum as is usually the case). These descriptors are then used in the training and testing of Arabic characters. The computation of the MFS descriptors requires less computation time than the computation of the Fourier descriptors. Experimental results have shown that the MFS features are suitable for Arabic character recognition. Average recognition rate of 95.9% was achieved for the model classes. The analysis of the errors indicates that this recognition rate can be improved by using the "hole" feature of a character and use cleaning corrupted data.

A Reconfigurable Butterfly Architecture for Fourier and Fermat Transforms

Abstract: Reconfiguration is an essential part of Soft- Ware Radio (SWR) technology. Thanks to this technique, systems are designed for change in operating mode with the aim to carry out several types of computations. In this SWR context, the Fast Fourier Transform (FFT) operator was defined as a common operator for many classical telecommunications operations [1]. In this paper we propose a new architecture for this operator that makes it a device intended to perform two different transforms. The first one is the Fast Fourier Transform (FFT) used for the classical operations in the complex field. The second one is the Fermat Number Transform (FNT) in the Galois Field (GF) for channel coding and decoding.

An FFT-based spectral analysis method for linear discrete dynamic systems with non-proportional damping

Abstract: This paper proposes a fast Fourier transforms (FFT)-based spectral analysis method for the dynamic analysis of linear discrete dynamic systems which have non-proportional viscous damping and are subjected to non-zero initial conditions. To evaluate the proposed FFT-based spectral analysis method, the forced vibration of a three degree-of-freedom (DOF) system is considered as an illustrative problem. The accuracy of the proposed FFT-based spectral analysis method is evaluated by comparing the forced vibration responses obtained by the present FFT-based spectral analysis method with those obtained by using the well-known Runge-Kutta method and modal analysis method.

A Low Multiplier and Multiplication Costs 256-point FFT Implementation with Simplified Radix-24 SDF Architecture

Abstract: In this paper, we propose a low multiplier and multiplication complexities 256-point fast Fourier transform (FFT) architecture, especially for WiMAX 802.16a systems. Based on the radix-16 FFT algorithm, the proposed FFT architecture utilizes cascaded simplified radix-24 single-path delay feedback (SDF) structures. The control circuit of the proposed simplified radix-24 SDF FFT architecture is simple. The hardware requirement of the proposed FFT architecture only needs 1 complex multiplier and 56 complex adders for supporting 256-point computations. The computation complexity of multiplications and the hardware complexity of the proposed FFT architecture need less complexity than both complexities of the previous FFT structures in 256-point FFT applications. In hardware verifications, the output throughput rate of our FFT design processes up to 35.5M samples/sec with Xilinx Virtex2 1500 FPGA, and it processes up to 51.5M samples/sec with UIMC 0.18?m standard cell technology. The throughput rate of this implementation is suitable for WiMLAX 802.16a application, whose maximum sample rate is 32MHz.

Fast Fourier Transform Algorithm for Low-Power and Area-Efficient Implementation

Abstract: This paper proposes the new radix-2 4 FFT algorithm and an efficient pipeline FFT architecture based on the algorithm for wideband OFDM systems. The proposed pipeline architecture has the same number of multipliers as that of the radix-2 2 algorithm. However, the multiplication complexity is reduced more than 30% by using the newly proposed CSD constant multipliers instead of the programmable multipliers. From the synthesis simulations of a standard 0.35μm CMOS SAMSUNG process, the proposed CSD constant complex multiplier achieved a reduction of more than 60% of the power consumption/area when compared with the conventional programmable complex multiplier.

Split-radix FFT/IFFT processor

Abstract: This invention presents a CORDIC-based split-radix FFT/IFFT (Fast Fourier Transform/Inverse Fast Fourier Transform) processor dedicated to the computation of 2048/4096/8192-point DFT (Discrete Fourier Transform). The arithmetic unit of butterfly processor and twiddle factor generator are based on CORDIC (Coordinate Rotation Digital Computer) algorithm. An efficient implementation of CORDIC-based split-radix FFT algorithm is demonstrated. All control signals are generated internally on-chip. The modified-pipelining CORDIC arithmetic unit is employed for the complex multiplication. A CORDIC twiddle factor generator is proposed and implemented for saving the size of ROM (Read Only Memory) required for storing the twiddle factors. Compared with conventional FFT implementations, the power consumption is reduced by 25%.

On Errors and Bias of Fourier Transform Methods in Quadratic Term Structure Models

Abstract: We analyze and compare the performance of the Fourier transform method in affine and quadratic term structure models. We explain why the method of the reduction to FFT in dimension one is efficient for ATSMs of type $A_0(n)$ but may lead to sizable errors for QTSMs unless computational errors are taken into account properly. We suggest a certain improvement and generalization which make FFT more accurate and, for the same precision, faster than Leippold and Wu (2002) method. We deduce simple general recommendations for the choice of parameters of computational schemes for QTSMs, which ensure a given precision, and an approximate formula for the bias which FFT produces.

The Cylindrical Taylor-Interpolation FFT Algorithm

Abstract: The cylindrical Taylor-interpolation FFT (TI-FFT) algorithm for the computation of the near-field and far-field in the "quasi-cylindrical" geometry has been developed. The cylindrical modal expansion of the vector potential is shown to be in the form that can make use of the cylindrical TI-FFT. The near-field on an arbitrary cylindrical surface is obtained from the vector potential and through the inverse FFT (IFFT). The far-field is obtained through the near-field far-field (NF-FF) transform. The cylindrical TI-FFT is valuable for back-scattering problems and "quasi-cylindrical" geometry. Like the planar TI-FFT algorithm, the cylindrical TI-FFT algorithm also has advantages of a N2 log2 N2 computational complexity, a low sampling rate, no basis functions and free of singularity problems.

An efficient fast full search block matching algorithm using FFT algorithms

Abstract: Summary Motion estimation is the most computationally expensive operation in the coding and transmitting of video streams, and the search for efficient motion estimation (in terms of computational complexity and compression efficiency) algorithm has been a challenging problem for years. The challenge is to decrease the computational complexity of the full search as much as possible without losing too much performance and quality at the output. In this paper, we propose a fast algorithm which achieves exactly the same optimal result as the direct full search algorithm. The key idea is to express a robust matching criteria sum square difference (SSD) in terms of cross correlation operations. Speed is obtained from computing the cross correlations in the frequency domain via the Fast Fourier Transform (FFT).

Fast evaluation of trigonometric polynomials from hyperbolic crosses

Abstract: 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.

Performance Models and Search Methods for Optimal FFT Implementations

Abstract: This thesis considers systematic methodologies for finding optimized implementations for the fast Fourier transform (FFT). By employing rewrite rules (e.g., the CooleyTukey formula), we obtain a divide and conquer procedure (decomposition) that breaks down the initial transform into combinations of different smaller size sub-transforms, which are graphically represented as breakdown trees. Recursive application of the rewrite rules generates a set of algorithms and alternative codes for the FFT computation. The set of "all" possible implementations (within the given set of the rules) results in pairing the possible breakdown trees with the code implementation alternatives. To evaluate the quality of these implementations, we develop analytical and experimental performance models. Based on these models, we derive methods dynamic programming, soft decision dynamic programming and exhaustive search to find the implementation with minimal runtime. Our test results demonstrate that good algorithms and codes, accurate performance evaluation models, and effective search methods, combined together provide a system framework (library) to derive automatically fast FFT implementations.

Faster than the FFT: The chirp-z RAG-n Discrete Fast Fourier Transform

Abstract: DFT and FFTs are important but resource intensive building blocks and have found many application in communication systems ranging from fast convolution to coding of OFDM signals. It has recently be shown that the n-Dimensional Reduced Adder Graph (RAG-n) technique is beneficially in many applications such as FIR or IIR filters, where multiplier can be grouped in multiplier blocks. This paper explores how the RAG-n technique can be applied to DFT algorithms. A RAG-n fast discrete Fourier transform will be shown to be of low latency and complexity and posses a VLSI attractive regular data flow when implemented with the Bluestein chirp-z algorithm. VHDL code synthesis results for Xilinx Virtex II FPGAs are provided and demonstrate the superior properties when compared with Xilinx FFT IP cores. Index Terms – Fast Fourier Transform, OFDM, FPGA, n-Dimensional Reduced Adder Graph

A Novel VLSI Architecture for Image Compression

Abstract: In analog signal processing of multimedia applications, audio need not be compressed but video must be compressed in order to conserve the bandwidth. For this purpose fast cosine transform (FCT) is preferred, which is developed using fast Fourier transform (FFT). But it has several limitations of delays, area and power. In this paper the similarities among fast Fourier transform (FFT), fast cosine transform (FCT) and fast Hartley transform (FHT) have been studied. The converged very large scale integration (VLSI) architecture for these hybrid transforms for video compression are designed, simulated and synthesized. Implementing FCT, FFT, and FST from fast Hartley transform is developed and hybrid transforms architecture is proposed. Further more, delays and area overheads have been calculated. For all these transforms the layouts have been drawn using magma tools with 0.13 mum technology. Compression ratio for image with each transform and Hybrid transform is also implemented

The fast Fourier transform for experimentalists. Part VI. Chirp of a bat

Abstract: Two assumptions underlie the Fourier transform process: stationarity and linearity. When signals deviate from these conditions, the transform outcomes are suspect. A chirp, which by definition has a frequency that varies with time, doesn't satisfy these requirements, and its fast Fourier transform (FFT) doesn't adequately express the changing nature of the signal's frequency content. In this analysis of a bat chirp, I first examine how the FFT handles a chirp and then how we can use a sequence of windows that individually span only a portion of the total time-domain signal to generate a frequency versus time description of the signal. The trade-off in this kind of windowing is between dynamic response and resolution: we obtain improved dynamics if we use shorter windows, whereas we get better resolution with longer windows. This article and this series concludes with a brief look at the Hilbert-Huang transform, which isn't constrained by the same assumptions as the FFT. This transform process consists of two independent sets of operations. The first, called empirical mode decomposition, generates a set of intrinsic mode functions (IMFs), from the data. The second step extracts phase information from each IMF and its Hilbert transform. The derivative of the phase with respect to time yields the instantaneous frequency. The net effect of these operations is to transform the time-domain data to frequency versus time data instead of the amplitude versus frequency the FFT obtains

R2²SDF FFT Implementation with Coefficient Memory Reduction Scheme

Abstract: Fast Fourier transform (FFT) is a key building block for orthogonal frequency division multiplexing (OFDM) systems. Due to the development of wireless portable devices, it is important to minimize the size and power of a FFT processor. One of the methods to satisfy such demands is reducing the size of twiddle coefficient memory. This paper presents an effective coefficient memory reduction scheme for a R22SDF FFT implementation. When applying a conventional method to an N- point R22SDF FFT, the number of twiddle coefficients is 3N/4. However, the proposed scheme requires only (N/8+1) coefficients and its additional hardware architecture is very simple. The effectiveness of the proposed method is verified by implementation results on a FPGA.