General theory for integer-type algorithm for higher order differential equations

Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four arithmetical operations on integers. This algorithm is based on a band-diagonal matrix representation of the differential operator P(x,d/dx), though it is quite different from the usual Galerkin methods. This representation is made for the respective CONSs of the input Hilbert space H and the output Hilbert space H' of P(x,d/dx). This band-diagonal matrix enables the construction of a recursive algorithm for solving the ODE. However, a solution of the simultaneous linear equations represented by this matrix does not necessarily correspond to the true solution of ODE. We show that when this solution is an l^2 sequence, it corresponds to the true solution of ODE. We invent a method based on an integer-type algorithm for extracting only l^2 components. Further, the concrete choice of Hilbert spaces H and H' is also given for our algorithm when p_m is a polynomial or a rational function with rational coefficients. We check how our algorithm works based on several numerical demonstrations related to special functions, where the results show that the accuracy of our method is extremely high.Comment: Errors concerning numbering of figures are fixe

Wavelets and their use

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

Robust high-capacity audio watermarking based on FFT amplitude modification

This paper proposes a novel robust audio watermarking algorithm to embed data and extract it in a bit-exact manner based on changing the magnitudes of the FFT spectrum. The key point is selecting a frequency band for embedding based on the comparison between the original and the MP3 compressed/decompressed signal and on a suitable scaling factor. The experimental results show that the method has a very high capacity (about 5 kbps), without significant perceptual distortion (ODG about -0.25) and provides robustness against common audio signal processing such as added noise, filtering and MPEG compression (MP3). Furthermore, the proposed method has a larger capacity (number of embedded bits to number of host bits rate) than recent image data hiding methods