Showing papers on "Fast Fourier transform published in 1978"

z-transform DFT filters and FFT's

Abstract: The paper shows how discrete Fourier transformation can be implemented as a filter bank in a way which reduces the number of filter coefficients. A particular implementation of such a filter bank is directly related to the normal complex FFT algorithm. The principle developed further leads to types of DFT filter banks which utilize a minimum of complex coefficients. These implementations lead to new forms of FFT's, among which is a \cos/\sin FFT for a real signal which only employs real coefficients. The new FFT algorithms use only half as many real multiplications as does the classical FFT.

Source coding of the discrete Fourier transform

Abstract: Distortion-rate theory is used to derive absolute performance bounds and encoding guidelines for direct fixed-rate minimum mean-square error data compression of the discrete Fourier transform (DFT) of a stationary real or circularly complex sequence. Both real-part-imaginary-part and magnitude-phase-angle encoding are treated. General source coding theorems are proved in order to justify using the optimal test channel transition probability distribution for allocating the information rate among the DFT coefficients and for calculating arbitrary performance measures on actual optimal codes. This technique has yielded a theoretical measure of the relative importance of phase angle over the magnitude in magnitude-phase-angle data compression. The result is that the phase angle must be encoded with 0.954 nats, or 1.37 bits, more rate than the magnitude for rates exceeding 3.0 nats per complex element. This result and the optimal error bounds are compared to empirical results for efficient quantization schemes.

On Computing the Discrete Cosine Transform

Abstract: Haralick has shown that the discrete cosine transform of N points can be computed more rapidly by taking two N-point fast Fourier transforms (FFT's) than by taking one 2N-point FFT as Ahmed had proposed. In this correspondence, we show that if Haralick had made use of the fact that the FFT's of real sequences can be computed more rapidly than general FFT's, the result would have been reversed. A modified algorithm is also presented.

Computation of convolutions and discrete Fourier transforms by polynomial transforms

Abstract: Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete two-dimensional convolutions with a minimum number of operations. Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms.

Space-time trade-offs on the FFT algorithm

Abstract: The performance of the fast Fourier transfmm algorithm is examined under limitations on computational space and time. It is shown that if the algorithm with n inputs, n as a power of two, is implemented with S temporary locations where S=o(n/ \log n) , then the computation time T grows faster than n \log n . Furthermore, T can grow as fast as n^{2} if S=S_{min} + O(1) where S_{min}=l+\log_{2}n , the minimum necessary. These results are obtained by deriving tight bounds on T versus S and n .

An algorithm for the numerical evaluation of the Hankel transform

Abstract: A procedure is proposed for the numerical evaluation of the Hankel (Fourier-Bessel) transform of any integer order using the FFT algorithm The basis for the procedure is the "projection-slice" theorem associated with the two-dimensional Fourirer transform

A High-Speed FFT Processor

Abstract: A high-speed, low-power, fast Fourier transform (FFT) processor is described in this paper. The FFT processor is designed around parallel arithmetic functions (16 by 16 multiplier and 16-bit adders) and can operate up to a 17.0-MHz clock rate. It performs a 128-point FFT in 250 μs at 16-MHz clock rate and therefore can be used for applications such as frequency-division multiplexing/timedivision multiplexing (FDM/TDM) transmultiplexer. The processor was designed and tested according to the design specifications. Its standalone feature permits its use in a variety of systems employing spectral analysis. The high-speed requirements are met by a real-time address generation scheme. The design can be used for a higher order FFT by providing extra memory space.

A new algorithm for the radix-3 FFT

Abstract: A radix-3 FFT which has no multiplications in the three-point DFT's is introduced. It uses arithmetic with numbers of the form a + bμ, where μ is a complex cube root of unity. The application to fast convolution of real sequences is discussed.

An Improved Method for Digital SSB-FDM Modulation and Demodulation

Abstract: A new method for digital implementation of SSB-FDM modulation and demodulation is presented, which utilizes the FFT algorithm to reduce the multiplication rate. The hardware realization of the presented method is shown to be the simplest; it consists of only an FFT processor and a set of complex bandpass filters which operate at the same rate as the input baseband sequences; no signal conversions are required prior to FFT processing. Simple and practical approaches are presented for design and implementation of the EFT processor and the set of complex bandpass filters. Design examples, which are made for implementation of a TDM-FDM translator at the supergroup level, show that the proposed method can attain the minimum multiplication rate among methods previously reported.

The fast decoding of Reed-Solomon codes using Fermat theoretic transforms and continued fractions

Abstract: It is shown that Reed-Solomon (RS) codes can be decoded by using a fast Fourier transform (FFT) algorithm over finite fields GF(F_{n}) , where F_{n} is a Fermat prime, and continued fractions. This new transform decoding method is simpler than the standard method for RS codes. The computing time of this new decoding algorithm in software can be faster than the standard decoding method for RS codes.

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

Abstract: 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.

FDM/TDM Transmultiplexer

Abstract: A FDM/TDM transmultiplexer uses sampling rate multiplication to increase the sampling rate for time division multiplexed (TDM) to frequency division multiplexed (FDM) conversion and decrease the sampling rate for FDM to TDM conversion. The rate multiplication filters are realized digitally in order to exploit the computational advantage of Fast Fourier Transform (FFT) algorithm, and channel filtering is implemented by a single time-shared sixth-order elliptic digital recursive filter. A novel FFT processor and recursive filter are disclosed which may be used in the system.

Automatic network measurements in the time domain

Abstract: The time-domain automatic network analyser (TDANA) is described. It utilizes time-domain measurements and the fast Fourier transform to obtain frequency-domain scattering parameters. A TDANA consists of a pulse generator, sampling oscilloscope, and a minicomputer. The present state of the art for TDANA's includes frequency coverage from dc to 18 GHz in a single instrument and complex scattering parameter measurement uncertainties of the order of 1 percent.

FFT Parallel processor having mutually connected, multiple identical cards

Abstract: A modular fast Fourier transform processor which employs identical processor cards, each of which has a butterfly, memory units which send and receive data to and from the butterfly and a pair of input ports and a pair of output ports. The cards are interconnected through these ports to execute a fast Fourier transform decomposition in accord with Singleton's algorithm. The speed and capacity of the processor may be increased in a gradual manner by employing more of the processor cards.

Cordic FFT processor

Abstract: An apparatus for performing a Fourier transform using Cordic techniques. Digital words are pipelined through serial add/subtract stages to provide vector rotations without trignometric lookup tables or multiply operations. The throughput of an FFT butterfly calculation is increased over prior art digital processors. A plurality of apparatus may be pipelined in a system to further increase the throughput rate. Also, the apparatus may be programmed to perform vector rotations through a plurality of angles thus providing the capability to compute FFT's of varying numbers of points.

Digital TDM-FDM Translator with Multistage Structure

Abstract: In this paper, a new digital signal processing algorithm for the digital TDM-FDM translator is proposed. The digital TDM-FDM translator, which performs a direct translation between two multiplex formats in the telephone network (time-division-multiplexing (TDM) and frequency-division-multiplexing (FDM)) by using digital techniques, has advantages in accuracy and stability of characteristics over equivalent analog equipments. However from the economical point of view, it largely depends on the cost reduction of semiconductor devices and LSI technologies. The proposed algorthm can be realized using only two digital filters and does not require product modulators or Fast Fourier Transform (FFT) processors. The required number of multiplications, which is closely related to the quantity of hardware, is considerably reduced by the multistage structure of this algorithm. The reduction in the kind of required digital hardware and the required number of multiplications makes it possible to efficiently utilize the new hardware realization techniques of digital filters or multipliers using read-only memories and simple logic devices. Since it is foreseen that cost reduction of memory devices will be more rapid than that of logic devices, the proposed algorithm is expected to be advantageous with regard to cost over existing algorithms where complex multiplier logic is required. The estimation of the computation rate is carried out with reference to a practical case. The computer simulation results are also shown.

Fixed-point error analysis of winograd Fourier transform algorithms

Abstract: 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.

Calculation of narrow-band spectra by direct decimation

Abstract: The calculation of the spectrum of a narrow-band signal which is embedded in a broad-band sequence usually requites substantial computation and storage if executed by performing an FFT or DFT's directly on the broad-band sequence. In this paper a new scheme using direct decimation is proposed which computes the narrow-band spectrum with good resolution while requiring only modest computation and storage. The performance of the proposed scheme is analyzed. Examples are presented which demonstrate the efficiency of this scheme when compared with the FFT, DFT, zoom transform, and complex modulation scheme.

Convolution using a conjugate symmetry property for the generalized discrete Fourier transform

Abstract: Often, signals which lie in a ring S are convolved using a generalized discrete Fourier transform (DFT) over an extension ring R in order to allow longer sequence lengths. In this paper, a conjugate symmetry property which generalizes the well known property of the complex DFT for real data is presented for this situation. This property is used to obtain a technique for computing the DFT of μ sequences with values in a ring S using a single DFT in an extension ring R of degree μ over S. From this result, a method to compute the convolution of length μn S-sequences using a length n DFT in R is derived. Example of the application to the complex DFT and to a number theoretic transform are presented to illustrate the theory.

A generalized scheme for an all-digital time-division multiplex to frequency-division multiplex translator

Abstract: A generalized scheme for an all-digital time-division multiplex (TDM) to frequency-division multiplex (FDM) translator is discussed. It is shown that existing systems for this translation, e.g., those in which a fast Fourier transform (FFT) processor is used, can be derived from this generalized scheme as special cases. Several new Implementations can also be derived from this scheme when different processors are used. A specific example using a Hadamard processor is discussed.

The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

Abstract: It is shown that \sqrt\[8]{2} is an element of order 2^{n+4} in GF(F_{n}) , where F_{n}=2^{2^{n}}+1 is a Fermat prime for n=3,4 . Hence it can be used to define a fast Fourier transform (FFT) of as many as 2^{n+4} symbols in GF(F_{n}) . Since \sqrt[8]{2} is a root of unity of order 2^{n+4} in GF(F_{n}) , this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the case n=2 , where \sqrt{2} is in GF(F_{2})=GF(17) , is given.

Full‐photon‐counting Rayleigh spectrometer: a correlation and/or fast Fourier transform instrument

Abstract: A fast, flexible, and relatively inexpensive minicomputer based Rayleigh spectrometer has been constructed. This spectrometer can utilize photon pulses in a most efficient way by performing autocorrelation and/or FFT with any desired number of channels. Its capability has been demonstrated by measuring large latex particles, small sodium dodecyl sulfate micelles, and a polydisperse latex‐γ‐globulin system. Detailed timing diagrams and circuit schematics are also included.

Enhancement of Noisy Images Using an Interpolative Model in Two Dimensions

Abstract: The use of two-dimensional interpolative models for recursive enhancement of noisy images is considered. The two-dimensional data is decorrelated either row-wise or column-wise to obtain an equivalent one-dimensional interpolative model. An adaptive identification scheme to identify the parameters of the model is presented which are then used in a simple estimation scheme for recovering the image data. A modification of the FFT algorithm as well as the DCT algorithm are used in decorrelating the data. The performance of these transforms for the present application is compared in terms of the number of computations required and the mean-square error between the enhanced image and the original noise-free image. Several examples are presented to illustrate the applicability of the algorithm.

Correcting errors in the optical path difference in Fourier spectroscopy: a new accurate method

Abstract: A new computational method for calculating and correcting the errors of the optical path difference in Fourier spectrometers is presented. This method only requires an one-sided interferogram and a single well-separated line in the spectrum. The method also cancels out the linear phase error. The practical theory of the method is included, and an example of the progress of the method is illustrated by simulations. The method is also verified by several simulations in order to estimate its usefulness and accuracy. An example of the use of this method in practice is also given.

An algorithm for generating fluctuations having any arbitrary power spectrum

Abstract: A calculational scheme is given for generating fluctuations which have any specified power spectrum. The fast computer-based algorithm makes use of a random number generator and fast Fourier transform (FFT) routine.

Optimal sectioning procedure for the implementation of 2-D digital filters

Abstract: The sectioning approach to the implementation of two-dimensional convolution operations is reviewed. An optimal sectioning algorithm is presented and its computational requirements (CPU and I/O) completely characterized. Procedures are then given to enable the optimal choice of sectioning parameters for an arbitrary computer system. This procedure is Illustrated for a large scientific compUter (CDC 7600), and the sectioning implementation compared with the traditional FFT approach. Computer programs are included for both the sectioning algorithm and the determination of optimal sectioning parameters.

Relative evaluation of various number theoretic transforms for digital filtering applications

Abstract: The main existence conditions for pseudo Mersenne and pseudo Fermat number transforms defined in a ring submultiple of a pseudo Mersenne or pseudo Fermat number are defined. The computational complexity of various multiplication-free number theoretic transforms (NTT's) used for implementing digital filters is evaluated. It is shown that Fermat number transforms (FNT's) with root \sqrt{2} and some complex pseudo Mersenne and pseudo Fermat number transforms with root 1 + j yield optimum processing efficiency and allow significant computational savings over direct filter evaluation.