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FAST FOURIER TRANSFORMS: A TUTORIAL REVIEW AND A STATE OF THE ART
, 1990
"... The publication of the CooleyTukey fast Fourier transform (FIT) algorithm in 1965 has opened a new area in digital signal processing by reducing the order of complexity of some crucial computational tasks like Fourier transform and convolution from N 2 to N log2 N, where N is the problem size. The ..."
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Cited by 129 (2 self)
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The publication of the CooleyTukey fast Fourier transform (FIT) algorithm in 1965 has opened a new area in digital signal processing by reducing the order of complexity of some crucial computational tasks like Fourier transform and convolution from N 2 to N log2 N, where N is the problem size. The development of the major algorithms (CooleyTukey and splitradix FFT, prime factor algorithm and Winograd fast Fourier transform) is reviewed. Then, an attempt is made to indicate the state of the art on the subject, showing the standing of research, open problems and implementations.
Matched spectralnulls codes for partial response channels
 IEEE Trans. Inform. Theory
, 1991
"... AbstractA new family of codes is described that improve the reliability of digital communication over noisy, partialresponse channels. The codes are intended for use on channels where the input alphabet size is limited. These channels arise in the context of digital data recording and certain data ..."
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Cited by 21 (7 self)
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AbstractA new family of codes is described that improve the reliability of digital communication over noisy, partialresponse channels. The codes are intended for use on channels where the input alphabet size is limited. These channels arise in the context of digital data recording and certain data transmission applications. The codescalled matchedspectralnull codessatisfy the property that the frequencies at which the code power spectral density vanishes correspond precisely to the frequencies at which the channel transfer function is zero. It is shown that matchedspectralnul1 sequences provide a distance gain on the order of 3 dB and higher for a broad class of partialresponse channels, including many of those of primary interest in practical applications. The embodiment of the matchedspectralnull coded partialresponse system incorporates a slidingblock code and a Viterbi detector based upon a reducedcomplexity trellis structure, both derived from canonical diagrams that characterize spectralnull sequences. The detectors are shown to achieve the same asymptotic average performance as maximumlikelihood sequencedetectors, and the slidingblock codes exclude quasicatastrophic trellis sequences in order to reduce the required path memory length and improve “worstcase ” detector performance. Several examples are described in detail. Index TermsSpectralnull codes, partialresponse channels.
A New Algorithm for Multiplication in Finite Fields
 I E E E Trans. Computers
, 1989
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Design Of RNS Frequency Sampling Filter Banks
 ICASSP, Session DSP1P
, 1997
"... Frequency sampling filters (FSF) are of interest to the designers of multirate filter banks due to their intrinsic loworder, complexity, and linear phase behavior. Fast FSFs residing in smaller packages will be required to support future highbandwidth, mobile image and signal processing application ..."
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Cited by 5 (4 self)
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Frequency sampling filters (FSF) are of interest to the designers of multirate filter banks due to their intrinsic loworder, complexity, and linear phase behavior. Fast FSFs residing in smaller packages will be required to support future highbandwidth, mobile image and signal processing applications. Since FSF designs rely on the exact annihilation of selected poleszeros, a new facilitating technology is required which is fast, compact, and numerically exact. Exact FSF polezero annihilation is guaranteed by implementing polynomial filters over an integer ring in the residue arithmetic system (RNS). The design methodology is evaluated as an ASIC. Based on an FPGA technology, at least an 86% complexity reduction can be achieved with even greater advantages gained as a custom VLSI. An RNSbased FSF implementation of an eight channel cochlea filter bank is presented which demonstrates both the performance and packaging advantages of the new FSF paradigm. . 1. INTRODUCTION A classical f...
Multifrequency Phase Unwrapping from Noisy Data: Adaptive Local Maximum Likelihood Approach
"... Abstract. The paper introduces a new approach to absolute phase estimation from frequency diverse wrapped observations. We adopt a discontinuity preserving nonparametric regression technique, where the phase is reconstructed based on a local maximum likelihood criterion. It is shown that this criter ..."
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Abstract. The paper introduces a new approach to absolute phase estimation from frequency diverse wrapped observations. We adopt a discontinuity preserving nonparametric regression technique, where the phase is reconstructed based on a local maximum likelihood criterion. It is shown that this criterion, applied to the multifrequency data, besides filtering the noise, yields a 2πQperiodic solution, where Q>1 is an integer. The filtering algorithm is based on local polynomial (LPA) approximation for the design of nonlinear filters (estimators) and the adaptation of these filters to the unknown spatially smoothness of the absolute phase. Depending on the value of Q and of the original phase range, we may obtain complete or partial phase unwrapping. In the latter case, we apply the recently introduced robust (in the sense of discontinuity preserving) PUMA unwrapping algorithm [1]. Simulations give evidence that the proposed method yields stateoftheart performance, enabling phase unwrapping in extraordinary difficult situations when all other algorithms fail.
Some Primality Testing Algorithms
 Notices of the AMS
, 1993
"... We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now i ..."
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We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now incorporated in computer algebra systems (CAS) as standard. In this review I give some details of the implementations of these algorithms and a number of examples where the algorithms prove inadequate. The algebra systems reviewed are Mathematica, Maple V, Axiom and Pari/GP. The versions we were able to use were Mathematica 2.1 for Sparc, copyright dates 19881992; Maple V Release 2, copyright dates 19811993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing area of research. For further reading and comprehensive bibliographies, the interested re...
UnusualLength NumberTheoretic Transforms Using Recursive Extensions of Rader's Algorithm
, 1995
"... A novel decomposition of NTT blocklengths is proposed using repeated applications of Rader's algorithm to reduce the problem to that of realising a single smalllength NTT. An e#cient implementation of this smalllength NTT is achieved by an initial basis conversion of the data, so that the new ..."
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Cited by 2 (1 self)
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A novel decomposition of NTT blocklengths is proposed using repeated applications of Rader's algorithm to reduce the problem to that of realising a single smalllength NTT. An e#cient implementation of this smalllength NTT is achieved by an initial basis conversion of the data, so that the new basis corresponds to the kernel of the smalllength NTT. Multiplication by powers of the kernel become rotations and all arithmetic is e#ciently performed within the new basis. More generally, this extension of Rader's algorithm is suitable for NTT or DFT applications where an e#cient implementation of a particular smalllength NTT/DFT module exists. 1 Introduction The NumberTheoretic Transform (NTT) has been suggested as an alternative to the DFT for computing cyclic convolution [14, 9] and is suitable for inclusion within signal processing, errorcorrection and residue number systems. E#cient architectures are possible for Fermat and Mersenne Transforms [8, 9, 1], where multiplication within ...
Fast BlumBlumShub Sequence Generation Using Montgomery Multiplication
 In IEEE Proceedings of Computers and Digital Techniques
, 2000
"... VLSI modules are proposed for fast, efficient generation of highthroughput BlumBlumShub (BBS) and BBSlike sequences using Montgomery Multiplication, where postprocessing associated with Montgomery’s algorithm can be eliminated. 2 1 ..."
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VLSI modules are proposed for fast, efficient generation of highthroughput BlumBlumShub (BBS) and BBSlike sequences using Montgomery Multiplication, where postprocessing associated with Montgomery’s algorithm can be eliminated. 2 1
Fast and Precise Computations of Discrete Fourier Transforms using Cyclotomic Integers
 In Proc. ACM Symp. Theory of Comp
, 1997
"... Many applications of fast fourier transforms (FFT's), such as computertomography, geophysical signal processing, high resolution imaging radars, and prediction filters, require high precision output. The usual method of fixed point computation of FFT's of vectors of length 2 ` leads to ..."
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Many applications of fast fourier transforms (FFT's), such as computertomography, geophysical signal processing, high resolution imaging radars, and prediction filters, require high precision output. The usual method of fixed point computation of FFT's of vectors of length 2 ` leads to an average loss of `=2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real time applications. Several researchers have noted that calculation of FFT's with algebraic integers avoids computational noise entirely, see, e.g., [3]. We will show that complex numbers can be approximated accurately by cyclotomic integers, and combine this idea with Chinese remaindering strategies in the cyclotomic integers to, roughly, give a O(b 1+ffl L log(L)) algorithm to compute bbit precision FFT's of length L. The first part of the paper will describe the FFT strategy, assuming good approximation...