J. H. Cozzens and L. A. Finkelstein, "Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers," IEEE Transactions on Information Theory, vol. IT-31, no. 5, pp. 580-588, Sept. 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....see, e.g. 12] Then R[x] x 2 1) represents C. 12 DANIEL J. BERNSTEIN Dubois and Venetsanopoulos in [32] suggested an alternative for the radix 3 FFT, namely R[x] x 2 x 1) compare this to Sch onhage s trick. Martens in [69] suggested R[x] x 2 x 1) Cozzens and Finkelstein in [30] pointed out the following alternative: Z[x] x 4 1) is dense in C. One can multiply in Z= 2 65536 1) as follows. Map to the isomorphic ring (Z[x] x 4096 1) 2 32 x) lift to Z[x] x 4096 1) map to C[x] x 4096 1) apply the FFT. The product can be recovered from approximate FFT ....

John H. Cozzens, Larry A. Finkelstein, Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers, IEEE Transactions on Information Theory 31 (1985), 580-588. MR 86k:94005.

....yields approximations within a distance of at most 3:42=M . Several examples are included which show that the algorithm is very fast in practice. For instance, 50000 complex approximations take less than 0.7 seconds on a SPARC 5. ii 1 Introduction In a pioneering paper Cozzens and Finkelstein [1] suggest to use cyclotomic integers in the ring Z[i] n ff 0 ff 1 i : ff 2 n Gamma1 Gamma1 i 2 n Gamma1 Gamma1 fi fi fi ff i 2 Z o , i = e 2i=2 n , to approximate the input as well as program constants for residue number system processing of functions such as the Discrete ....

....for the case i = e 2i=8 . To respect dynamic range requirements, the algorithm has to have an additional input M 2 N and should output approximations in the set Z[i] M which is defined as the set of linear combinations of powers of i with integer coefficients bounded in absolute value by M . In [1] Cozzens and Finkelstein present an algorithm whose basic ingredient is exhaustive search, and hence, is impractical for larger values of M . Games [2, 3] develops a greedy algorithm, a rough sketch of which is as follows: in a first step a small element of Z[i] M is found. Then the algorithm ....

J. H. Cozzens and L. A. Finkelstein. Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers. IEEE Transactions on Information Theory, 31(5):580--588, 1985.

....August 1996 Abstract We present a new method of approximating complex numbers by cyclotomic integers in Z[e 2i=2 n ] whose coefficients with respect to the basis given by powers of e 2i=2 n are bounded in absolute value by a given integer M . It has been suggested by Cozzens and Finkelstein [5] that such approximations reduce the dynamic range requirements of the discrete Fourier transform. For fixed n our algorithm gives approximations with an error of O(1=M 2 n Gamma2 Gamma1 ) This proves a heuristic formula of Cozzens and Finkelstein. We will also prove a matching lower bound ....

....is possible. If the approximation errors are not too large, this procedure yields outputs of guaranteed good precision. An important class of computations for which this approach has been developed in detail is the fast Fourier transform [4] FFT for short. The basic idea is as follows (see [5, 6, 7] and the references therein) the input vector as well as the roots of unity involved are approximated by Gaussian integers, i.e. elements of the ring Z[i] This step is accomplished by scaling the complex numbers by a large number, and then rounding to the nearest Gaussian integer. Since the ....

[Article contains additional citation context not shown here]

J. H. Cozzens and L. A. Finkelstein. Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers. IEEE Transactions on Information Theory, 31(5):580--588, 1985.

....on the direct implementation of algorithm [3,4] In these implementations for the calculation of the cas function, different types of approximations have been introduced. Processing with algebraic integers, in which the signal sample is represented by a set of small integers, was introduced in [5]. Algebraic integers are roots of monic polynomials that have integer coefficients with leading coefficient equal to unity. The motivation for introducing this new mapping of real numbers is to drastically reduce the dynamic range of each of the independent computations. In this letter, we ....

Cozzens, J.H., and Finkelstein, L.A.: `Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers', IEEE Trans. Info. Theory, 1985, 31, (5), pp. 580-588

....and expensive than fixed point processors. Any improvement of fixedpoint computation so as to be more robust and accurate could lead to significantly less expensive chips in high precision FFT processors. These issues have led to suggestions for alternate FFT computations by many scientists [1, 7, 8, 18, 19, 20, 21]. Much of the research has concentrated on Residue Number System (RNS) processors. The idea is as follows: the complex numbers constituting the input and the twiddle factors (i.e. the roots of unity involved) are approximated by Gaussian integers, that is, complex numbers of the form Z iZ, where ....

....problems. The bottleneck of the standard RNS approach is that it is not possible to approximate complex numbers by Gaussian integers with arbitrary precision, since the Gaussian integers form a discrete subset of the set C of complex numbers. In a pioneering paper Cozzens and Finkelstein [7] suggested that the fourth root of unity i could be replaced by a 2 n th root i, n 3. Hence, Gaussian integers are replaced by integral linear combinations of i. These constitute a ring which we call Z[i] The fundamental difference between the set of Gaussian integers and Z[i] is that the ....

J.H. Cozzens and L.A. Finkelstein. Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers. IEEE Trans. Inform. Theory, 31(5):580--588, 1985.

....22, 1996 Abstract. We present a new method of approximating complex numbers by cyclotomic integers in Z[e 2 i=2 n ] whose coefficients with respect to the basis given by powers of e 2 i=2 n are bounded in absolute value by a given integer M . It has been suggested by Cozzens and Finkelstein [5] that such approximations reduce the dynamic range requirements of the discrete Fourier transform. For fixed n our algorithm gives approximations with an error of O(1=M 2 n Gamma2 Gamma1 ) This proves a heuristic formula of Cozzens and Finkelstein. We will also prove a matching lower bound ....

....is possible. If the approximation errors are not too large, this procedure yields outputs of guaranteed good precision. An important class of computations for which this approach has been developed in detail is the fast Fourier transform [4] FFT for short. The basic idea is as follows (see [5, 6, 7] and the references therein) the input vector as well as the roots of unity involved are approximated by Gaussian integers, i.e. elements of the ring Approximation of Complex Numbers 2 Z[i] This step is accomplished by scaling the complex numbers by a large number, and then rounding to the ....

[Article contains additional citation context not shown here]

J. H. Cozzens and L. A. Finkelstein. Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers. IEEE Transactions on Information Theory, 31(5):580--588, 1985.

....ALGEBRAIC INTEGERS U. Meyer Baese, J. Mellott, and F. Taylor High Speed Digital Architecture Laboratory University of Florida Gainesville 32611 6130, U.S. e mail: fuwe,jon,fjtg alpha.ee. ufl.edu ABSTRACT Algebraic integers have been proven beneficial to DFT and nonrecursive FIR filter designs [2, 4] since algebraic integers can be dense in C , resulting in short word width, high speed designs. This paper uses another property of algebraic integers: algebraic integers can produceexact pole zero cancellation pairs that are used in recursive FIR, frequency sampling filter designs. 1. ....

....addition operation. However, the multiplication of polynomials given above is recognized as cyclic convolution. An interesting property of the multiplication given above is that if B(x) x l then the product is simply a cyclic rotation of the coefficients of A(x) Cozzens and Finkenstein [2] have shown that the quotient ring of algebraic integers produce for N 8 a dense set in the complex plane. The benefit for a DFT implementation is that greater U. Meyer Baese was supported by a European Space Agency fellowship. 8 16 24 32 40 48 56 64 72 80 88 10 20 30 40 50 60 70 80 90 100 110 ....

[Article contains additional citation context not shown here]

J. Cozzens and L. Finkelstein,"Computing the Discrete Fourier Transform Using Residue Number Systems in a Ring of Algebraic Integers". IEEE Transactions on Information Theory, 31(5):580-588, Sept. 1985.

....are included which show that the algorithm is very fast in practice. For instance, 50000 complex approximations take less than 0.7 seconds on a SPARC 5. Key words. Fast Fourier transforms, cyclotomic fields, continued fractions. 1. Introduction In a pioneering paper Cozzens and Finkelstein [1] suggest to use cyclotomic integers in the ring Z[i] n ff 0 ff 1 i : ff 2 n Gamma1 Gamma1 i 2 n Gamma1 Gamma1 fi fi fi ff i 2 Z o , i = e 2 i=2 n , to approximate the input as well as program constants for residue number system processing of functions such as the Discrete ....

....the case i = e 2 i=8 . To respect dynamic range requirements, the algorithm has to have an additional input M 2 N and should output approximations in the set Z[i] M which is defined as the set of linear combinations of powers of i with integer coefficients bounded in absolute value by M . In [1] Cozzens and Finkelstein present an algorithm whose basic ingredient is exhaustive search, and hence, is impractical for larger values of M . Games [2, 3] develops a greedy algorithm, a rough sketch of which is as follows: in a first step a small element of Z[i] M is found. Then the algorithm ....

J. H. Cozzens and L. A. Finkelstein. Computing the discrete fourier transform using residue number systems in a ring of algebraic integers. IEEE Transactions on Information Theory, 31(5):580--588, 1985.

No context found.

J. H. Cozzens and L. A. Finkelstein, "Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers," IEEE Transactions on Information Theory, vol. IT-31, no. 5, pp. 580-588, Sept. 1985.

No context found.

Cozzens, J.H., L.A. Finkelstein. "Computing the Discrete Fourier transform Using Residue Number Systems in a Ring of Algebraic Integers." IEEE Trans. Information Theory. vol. IT-31 pp. 580-587, 1985.

No context found.

J.H.Cozzens and L.A.Finkelstein, Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers, IEEE Trans. on Information Theory, vol. 31, 1985, pp. 580-588.

No context found.

J. Cozzens and L. Finkelstein, "Computing the discrete fourier transform using residue number systems in a ring of algebraic integers," IEEE Transactions on Information Theory 31, pp. 580--8, Sept. 1985.

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