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31.
We provide a novel approach to the design of fast algorithms for matrix multiplication. The operation of matrix multiplication is reformulated as a convolution, which is implemented using pseudo-number-theoretic transforms. Writing the convolution as multiplication of polynomials evaluated off the unit circle reduces the number of multiplications without producing any error, since the (integer) elements of the product matrix are known to be bounded. The new algorithms are somewhat analogous to the arbitrary precision approximation (APA) algorithms, but have the following advantages: (i) a simple design procedure is specified for them; (ii) they do not suffer from round-off error; and (iii) the reasons for their existence is clear. The new algorithms are also noncommutative; therefore, they may be applied recursively to block matrix multiplication. This work establishes a link between matrix multiplication and fast convolution algorithms and so opens another line of inquiry for the fast matrix multiplication problem. Some numerical examples illustrate the operation of the new proposed algorithms 相似文献
32.
New generalized split Levinson and Schur algorithms for the two-dimensional linear least squares prediction problem on a polar raster are derived. The algorithms compute the prediction filter for estimating a random field at the edge of a disk from noisy observations inside the disk. The covariance function of the random field is assumed to have a Toeplitz-plus-Hankel structure for both its radial part and its transverse (angular) part. This assumption is valid for some types of random fields, such as isotropic random fields. The algorithms generalize the split Levinson and Schur algorithms in two ways: (1) to two dimensions; and (2) to Toeplitz-plus-Hankel covariances 相似文献
33.
Frequency-domain methods are used to study the angles of arrival and departure for multivariable root loci. Explicit equations are obtained. For a special class of poles and zeros, some simpler equations that are generalizations of the single-input-single-output equations are presented. 相似文献
34.
We provide simple and explicit formulae for reconstructing any member of a class of discrete-time signals from the frequencies at which its Fourier phase crosses any specific level of constant phase or a linear-phase line with integer slope, provided that the number of crossings equals the length of the signal support. Unlike previous closed-form solutions, solution of an ill-conditioned system of linear equations is not required. The associated uniqueness results reduce, in special cases, to previous results for reconstruction from Fourier transform real and imaginary part zero crossings 相似文献
35.
Fast algorithms based on the Mersenne and Fermat number-theoretic transforms are used to perform the bilinear transformation of a continuous transfer function to a discrete equivalent. The computations are carried out in finite precision arithmetic, require no multiplications, and can be implemented in parallel using very simple processors. Although the bilinear transform is presently emphasized, similar algorithms are easily derived for any transformation from the s -plane to the z -plane involving the ratio of two polynomials with integer coefficients 相似文献
36.
The discrete phase retrieval problem is to reconstruct a discrete time signal whose support is known and compact from the magnitude of its discrete Fourier transform. We formulate the problem as a linear system of equations; our methods do not require polynomial rooting, tracking zero curves of algebraic functions, or any sort of iteration like previous methods. Our solutions obviate the stagnation problems associated with iterative algorithms, and our solutions are computationally simpler and more stable than alternative noniterative algorithms. Furthermore, our methods can explicitly accommodate noisy Fourier magnitude information through the use of total least squares type techniques. We assume either of the following two types of a priori knowledge of the signal: (1) a band of known values (which may be zeros) or (2) some known values of a subminimum phase signal (whose zeros lie inside a disk of radius greater than unity). We illustrate our methods with nonminimum-phase one-dimensional (1-D) and two-dimensional (2-D) signals 相似文献