Winograd Fourier Transform Algorithm Research Articles

Design of 3780 Points FFT Processor for DTMB Receiver

In this paper, a design of FFT processor for Digital terrestrial multimedia/television broadcasting (DTMB) receiver is presented. This processor is based on mixed-radix algorithms, prime factor algorithms and Winograd Fourier Transform algorithm (WFTA). Due to the adopted in-place algorithm in this design, the area consumption of the processor is reduced, and the simulation shows that SQNR of the processor is 60.5dB on the condition that the input and output data are both 13 bits.

A novel 3780-point FFT processor scheme for the time domain synchronous OFDM system

The 3780-point FFT is a main component of the time domain synchronous OFDM (TDS-OFDM) system and the key technology in the Chinese Digital Multimedia/TV Broadcasting-Terrestrial (DMB-T) national standard. Since 3780 is not a power of 2, the classical radix-2 or radix-4 FFT algorithm cannot be applied directly. Hence, the Winograd Fourier transform algorithm (WFTA) and the Good-Thomas prime factor algorithm (PFA) are used to implement the 3780-point FFT processor. However, the structure based on WFTA and PFA has a large computational complexity and requires many DSPs in hardware implementation. In this paper, a novel 3780-point FFT processor scheme is proposed, in which a 60×63 iterative WFTA architecture with different mapping methods is imported to replace the PFA architecture, and an optimized CoOrdinate Rotation DIgital Computer (CORDIC) module is used for the twiddle factor multiplications. Compared to the traditional scheme, our proposed 3780-point FFT processor scheme reduces the number of multiplications by 45% at the cost of 1% increase in the number of additions. All DSPs are replaced by the optimized CORDIC module and ROM. Simulation results show that the proposed 3780-point FFT processing scheme satisfies the requirement of the DMB-T standard, and is an efficient architecture for the TDS-OFDM system.

Hardware Efficient Fast Computation of the Discrete Fourier Transform

In this paper, a new systolic array for prime N-length DFT is first proposed, and then combined with Winograd Fourier Transform algorithm (WFTA) to control the increase of the hardware cost when the transform length is large. The proposed new DFT design is both fast and hardware efficient. Compared with the recently reported DFT design with computational complexity of O(log N), the proposed design saves the average number of required multiplications by 30 to 60% and reduces the average computation time by more than 2 times, when the transform length changes from 16 to 2048.

Evaluation of generation system reliability indices by fast transform techniques

Two computationally efficient Fourier transform methods namely, the mixed radix Fast Fourier transform algorithm (FFTA) and the Winograd Fourier transform algorithm (WFTA) for evaluating the reliability of a power system are presented in this paper. Both the methods are used to evaluate the reliability of the IEEE reliability test system and they are compared with the direct discrete Fourier transform method and the basic recursive approach in terms of accuracy and computer time. A sensitivity analysis has been performed by variation of a few system variables to compare the two methods.

VLSI architecture for the Winograd Fourier transform algorithm

A simple systotic architecture for the computation of the DFT using the Winograd Fourier Transform algorithm is presented. The architecture is shown to be problem-size independent and to satisfy the limited bandwidth constraint. By satisfying the above constraints, it is then shown to be naturally scalable within the limits allowed by the Winograd algorithm. The proposed architecture is compared with existing VLSI solutions using Winograd's technique. Lastly, the feasibility of deriving and using Winograd type algorithms for larger primes and prime-powers is studied.

Programming the WFTA for two-dimensional data

The author introduces a two-dimensional Winograd Fourier transform algorithm (WFTA) technique and compares it with more traditional fast Fourier transform (FFT) implementations. Techniques for programming the WFTA in two dimensions are introduced. For completeness, enumerations of the Winograd small-n transposes, required when applying these techniques, are included. It is found that the WFTA implementation does save CPU time when implemented on a general-purpose computer for the following reasons: (1) the number of multiplications is far fewer than for common algorithms; (2) integer arithmetic can be used, at least for several stages of input additions, with no loss of accuracy; and (3) the fact that many operations are written out explicitly allows a programmer to save on the computation of indexes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Prime-factor algorithm and winograd fourier transform algorithm for real symmetric and antisymmetric sequences

In the paper, algorithms for computing discrete Fourier transform (DFT) of 1-D real symmetric and antisymmetric sequences, using the prime-factor algorithm (PFA) and the Winograd Fourier transform algorithm (WFTA), are presented. These algorithms are obtained from different factorisations of the Fourier matrix, and it is shown that symmetry conditions exist at each stage which are used to construct efficient algorithms for computing DFTs.

Efficient realisation of discrete Fourier transforms using the recursive discrete Hartley transform

In the paper, we present the results of our study using a recursive discrete Hartley transform technique to compute discrete Fourier transforms. We also introduce an improved in-place and in-order prime-factor mapping to effectively realise composite-length DFTs. In using these new techniques, the speed of computation is comparable to that of the Winograd Fourier transform algorithm (WFTA), whereas the program size of the present approach is much smaller than that of the WFTA. This approach is most suitable for the cases where there are restrictions on program lengths.

A high-quality digital filterbank for speech recognition which runs in real time on a standard microprocessor

Asynchronous digital filterbank or LPC output is widely used for high-quality research and commercial speech recognition applications. Special-purpose hardware is usually applied. In this paper, a decimation/DFT filterbank system is described which may be implemented in real time on a standard microprocessor. Key features of the approach are a dithering technique, which allows output sampling at a rate lower than would be theoretically predicted as adequate, and the Winograd Fourier transform algorithm (WFTA) implementation of the DFT. The contributions of this paper are the dithering idea, the demonstration that it is feasible to implement a high-quality digital filter-bank on a standard microprocessor, and the discussion of starting-phase error. The nonlinear effects of dithering are discussed in detail, and it is shown that the dithering technique significantly reduces starting phase error, which can be important in very close discrete utterance recognition (DUR) situations.

Fast mersenne number transforms for the computation of discrete fourier transforms

In this paper we present results on the computation of Discrete Fourier Transforms (DFT) using Mersenne Number Transforms (MNT). It is shown that in the case of Mersenne-composite Number Transforms, the number of multiplications per point for real input data is never more than one, even for sequence lengths exceeding one thousand points. The computation time per point for a length (2 P + 1)-point DFT is simply equal to the time for one MNT multiplication and 3 MNT additions if a high-speed, parallel hardware module is used to implement the MNT unit. This new approach allows a large choice of wordlengths and in addition the control of data flow is extremely simple. We also present the results obtained by using Winograd's Fourier Transform Algorithm and the nested MNT to compute efficiently the DFT's of long sequences. We also show that the number of additions can be reduced significantly if Pseudo Mersenne-Number Transforms are used for the computation of DFTs.

An approach to DFT calculations using standard microprocessors

The use of the DFT as an everyday tool is now commonplace, principally due to advances in hardware technology. Special-purpose VLSI chips for signal processing are available. In this paper, we describe an approach which marries the Winograd Fourier Transform Algorithm (WFTA) with a state-of-the-art 16-bit general-purpose microprocessor for the purpose of DFT calculation. The heart of the approach is the real-input 240-point WFTA, which has been carefully optimized for time and space. In particular, an implementation for the 10-MHz M68000 executes in 10.8 ms. A simple hardware module is described which implements the optimized software. The use of the module for the inverse transform and for the complex case is also discussed. Advantages to the approach taken in this paper are the low cost/performance ratio and the general-purpose nature of the system that allows many non-signal-processing functions to be performed by the microprocessor.

The design of optimal DFT algorithms using dynamic programming

A broad class of efficient discrete Fourier transform algorithms is developed by partitioning short DFT algorithms into factors. The factored short DFT's are combined into longer DFT's using multi-dimensional index maps. By exploiting a property which allows some of the factors to commute, a large set of possible DFT algorithms is generated which contains both the prime factor algorithm (PFA) and the Winograd Fourier transform algorithm (WFTA) as special cases. The problem of finding an algorithm from this class which is optimal with respect to the specific add, multiply, and data transfer characteristics of a particular implementation is posed, and a highly effective dynamic programming algorithm is presented as a solution.

Fast discrete cosine transform algorithm for systolic arrays

A fast algorithm for an N-point discrete cosine transform (DCT) is derived from a 4N-point Winograd Fourier transform algorithm (WFTA). This algorithm, which has the same form as Winograd's Fourier transform and convolution algorithms, is suitable for a high-speed implementation using one-bit systolic arrays.

Selection criteria for efficient implementation of FFT algorithms

The number of real operations and memory is presented for three efficient Fortran algorithms which compute the mixed radix discrete Fourier transform (DFT). It is shown that Singleton's mixed radix algorithm (MFFT) is the most flexible and uses the least memory, while the Winograd Fourier transform algorithm (WFTA) and Kolba-Parks prime factor algorithm (PFA) are the most efficient.

A new technique for WFTA input/output reordering

The Winograd Fourier transform algorithm (WFTA) is receiving intensive study. The advantage of this class of transform is its potential high throughput due to reduced multiplication count. However, both input and output reorderings have to be performed when the algorithm is implemented. In this work, a technique for the WFTA input/output reorderings is developed. This technique is flexible to the choice of the base numbers of WFTA and capable of operating at high speed in digital hardware. This technique requires no extra memory for reordering when implementing the WFTA in the residue number system if the moduli set of RNS is carefully chosen to contain the base numbers of WFTA.

Fixed-point error analysis of winograd Fourier transform algorithms

The quantization error introduced by the Winograd Fourier transform algorithm (WFTA) when implemented in fixed-point arithmetic is studied and compared with that of the fast Fourier transform (FFT). The effect of ordering the computational modules and the relative contributions of data quantization error and coefficient quantization error are determined. In addition, the quantization error introduced by the Good-Winogzad (GW) algorithm, which uses Good's prime-factor decomposition for the discrete Fourier transform (DFT) together with Winograd's short length DFT algorithms, is studied. Error introduced by the WFTA is, in all cases, worse than that of the FFT. In general, the WFTA requires one or two more bits for data representation to give an error similar to that of the FFT. Error introduced by the GW algorithm is approximately the same as that of the FFT.

Further corrections to "An introduction to programming the winograd Fourier transform algorithm (WFTA)"

Further corrections to "An introduction to programming the winograd Fourier transform algorithm (WFTA)"

Comments on "An introduction to programming the winograd Fourier transform algorithm (WFTA)"

This correspondence points out some inconsistencies between definitions and algorithms presented in the paper by H. F. Silverman.

Corrections and an addendum to "An introduction to programming the winograd Fourier transform algorithm (WFTA)"

Corrections and an addendum to "An introduction to programming the winograd Fourier transform algorithm (WFTA)"

A comparative study of time efficient FFT and WFTA programs for general purpose computers

Time efficient autogen software implementations of the fast Fourier transform (FFT) and the Winograd Fourier transform algorithm (WFTA) are examined and compared in detail. Both high-level language (optimized Fortran IV) and assembler implementations are considered on two general purpose computers, the DEC PDP-11/55 and the IBM- 370/168, both having floating-point multiply/add time ratios of about 1.17. It is shown that although the WFTA reduces the number of multiplications relative to the FFT, a substantial increase in data transfer, both memory/register and register/register, together with a smaller increase in additions and data reordering overhead, combine to give WFTA execution times about 20-40 percent longer than those for the FFT. These results are explained by examining the internal computational kernel structure for both algorithms and relating the arithmetic operation sequencing to the computer instructions necessary to implement the software. It is concluded that for floating-point software implementations on the class of general purpose computers considered, the WFTA offers neither time nor space advantages over the radix-4 FFT.