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This paper considers the set of waveforms for which the weighted power-of-p (1 < p < +∞) integral of the Fourier spectrum is bounded. Since different weight functions can be used in different frequency bands, as long as they are positive, the restriction of the waveform by the power-of-p integral is sufficiently general. It is assumed that the waveform belonging to such a set is added to several given linear time-invariant filters. A theory of the optimum approximation of the original waveform is established using the sampling sequences of the corresponding outputs. The measure of the original waveform is established using the sampling sequences of the corresponding outputs. The measure of the approximation error is defined as the upper limit of the absolute value of the difference between the original and the approximating waveforms when the original waveforms are changed one by one over the whole set of waveforms. Considering the sampling points, which substantially affect the approximation at a fixed time t, a theorem is presented which indicates that there exists a waveform which has zero value at all such sampling points and has the absolute value of the approximation error which reaches the above upper limit. It is shown also that a similar result can be established for the multidimensional waveform. In this paper, it is shown that the proposed method of linear approximation is better than any other approximation using the same sample values, including the nonlinear signal processing, as long as the forementioned set of waveforms and the measure for the approximation error are considered. [ABSTRACT FROM AUTHOR]

In this paper we consider a family of output waves obtained by impressing simultaneously an n-dimensional input wave f(X) to the finite number of time-Invariant linear net- works. A problem is considered, where the original wave f(X) is to be approximated using the sampled values of the forementioned output waves. A unified theory is presented with respect to the convergence of the approximate wave to the original wave f(X). The discussion given in this paper may be considered as a basis for the future theory of subband coding and multiplexing of the multidimensional signals, in the sense that we use the sampled values of the multidimensional waves transferred through the linear networks. The approximate wave h(X) for f(X) is a sum of the forementioned sample values of these output waves, being multiplied by certain n-dimensional waves called interpolation functions. The set of sampling points considered in this paper includes most of the practically import- ant arrangements of sapling points, such as hexagonal and octagonal lattices on the two- dimensional plane. The set of f(X) in this paper is defined as the set of n-dimensional waves such that the corresponding Fourier spectrum has the weighted powereofap norm (p > 1) which is less than a given positive value. Under several assumptions, sufficient, necessary, and necessary and sufficient conditions are derived for the wave h(X) always to converge to f(X). These convergence conditions are presented which are imposed on the transfer functions of the time-invariant linear networks through which f(X) propagates as well as the arrangement of the sampling points and the interpolation functions. They are closely related to the rank of the system of linear equations determined by those parameters. [ABSTRACT FROM AUTHOR]

This paper considers approximation formulas of the n-dimensional wave f(X) expressed by the sum of sampled values of f(X) multiplied by certain n-dimensional waves called interpolation functions. A systematic discussion is made to determine whether or not the approximate function converges to the original wave f(X). In this paper it is assumed that the interpolation functions satisfy the causality in regard to a certain axis in the n-dimensional space Rn. Usually, that axis is considered to be the tine axis. These discussions are important from the theoretical viewpoint, since it is closely related to the problem as to whether or not interpolation functions can be realized as impulse responses of physically realizable time-in- variant linear circuits. The set of sampling points is defined as a set of vertices obtained by sampling periodically all the vertices of the n-dimensional parallel polyhedra periodically arranged in the space Rn. This definition of the set of sampling points Includes most of the important arrangements of the sampling points, such as the hexagon and octagon lattices in the two-dimensional plane. In this paper, it is assumed that the interpolation functions have supports, which are parallel translations of each other. The functional forms of the interpolation functions may be different. The set of f(X) considered in this paper is defined as the set of n-dimensional waves whose Fourier spectra have the weighted power-of-p norm (p > 1) smaller than the specified positive value. It is shown first that, under the condition that the squared-sum of the interpolation functions is bounded, the considered approximate expression does not in general converge to the original wave. It is shown that if the I ore-going condition is relaxed, an approximate expression can be found with the impulse response of a causal time-invariant linear network as the interpolation function, which converges to the original wave. [ABSTRACT FROM AUTHOR]

This paper presents a unified theory for the nonuniform sampling theorem for the low-pass band-limited waveform. In most practical cases, the mean interval of the sampling points is in the so-called oversampling state. One can also assume that there exists a certain period in the sampling point sequence. Those properties of the sampling point sequence are assumed in this paper. Yen has derived a sampling theorem for the case where the sampling-point sequence has a period. In his theorem, however, the interpolation function has a spectrum which is a stepwise discontinuous function, not suited to the realization of the interpolation function by a digital or other similar filters. One of the greatest features of the sampling theorem proposed in this paper is that the interpolation function has a continuous spectrum. The problem has been discussed by one of the authors for the case where the sampling frequency and the band-end frequency of the waveform are in an integer ratio. In this paper, the sampling theorem is derived in a unified and integrated form without imposing such a condition. The variance of the approximation error is also derived for the case where a statistically independent additive error is superposed on the sampled value, and it is shown that the proposed sampling theorem is better in that respect. The discussion is extended to the multidimensional band-limited waveform, and a sampling theorem is shown which has an interpolation function with a continuous spectrum. [ABSTRACT FROM AUTHOR]

This paper describes design methods of the digital Nyquist filters based on the squared errors of the impulse response at equally spaced sampling points except one. First of all, for a specified pole of a transfer function, we consider the pole location which causes a problem when we use the residue formula proposed by Kida and Ohsumi to minimize the above mentioned squared errors. Then, assuming about 6 to 8 degrees for the orders for transfer functions, we consider a method which gives a favorable pole arrangement. We make an examination about a simple method of determining a pole arrangement which provides a favorable characteristic such that the squared errors at equally spaced sampling points except one point are about 10-5 and that the stopband attenuation is about 50 dB. Furthermore, in order to realize a large stopband attenuation which is required in the design of digital Nyquist filters, propose a design method of a transfer function based on a quadratic programming specifying the attenuation at a point in the frequency domain specified in advance. We show its effectiveness by examples. Finally, for the design of digital Nyquist filters using an interpolation method based on the sampling theorem of non-equal space derived by Kida and Fukuda, we consider an example which indicates that transfer functions obtained by the above two design methods are appropriate. [ABSTRACT FROM AUTHOR]

An investigation has been conducted about the interpolatory approximation of waveform using a set of sample points in the form of group of K generally non-equidistant sample points which are arranged periodically over a time axis. First, we consider a set I' of waveforms f(t) whose Fourtier spectra F(w) are bandwidth limited in |w| ... w1 and which satisfy ...w1w1 |F(w) |² /W (w) d w ≤ A ( > 0 ) for some positive weight W(w). The interpolation function, which minimizes the supremum of the absolute error between the original waveforms f(t) and corresponding approximation waveforms over P, is derived. Here the approximate waveforms are taken to be the total sum of the values at the sample points mentioned above multiplied by the interpolation functions time-limited to a finite time-band centering at the corresponding sample points. A choice of time limitation of the interpolation function discussed above is fundamental for realization of interpolation functions by a FIR filter. If displacement along the time axis is ignored, the optimal interpolation functions mentions above are divided into K groups and it is shown that they satisfy the orthogonality for a finite sum. Next, using the concept of an inner product expression of a quasi-bilinear form, an extension of the above concept is made and the optimal interpolation function is derived. The extension includes a case in which sample values contain statistical errors and a case in which the original wave is reproduced using sample values of linearity transformed waveforms. [ABSTRACT FROM AUTHOR]

Discusses a method in which as desired waveform is approximated by the impulse response of the transfer function using TBT polynomials as the denominator. Presentation of a selection principle for two parameters for which the squared error in the time domain is minimized; Description of the major problem of waveform approximation in the time domain using impulse response of a lumped parameter network; Examination of the properties of the approximated waveform.

An approximation method is proposed for a class of band-limited waveforms whose Fourier spectra F(ω) are confined in the interval -ω1 ≤ ω ≤ ω1. The side lobes of such waveforms are also small in the range of 0 < ω0 < ¦ω¦ < ω1. First, the approximation waveform is obtained from summation of the product of an interpolating function and the sampling value f(tk) at the sampling point tk (k = 0 ∼ K). One objective of the proposed method is to find an interpolating function that will minimize the envelope of a family of approximation waveforms, which is used as a measure of error associated with the approximation. The optimum interpolating function also satisfies the condition of discrete orthogonality. The proposed approximation technique also provides convergence to the actual waveform with respect to a set of sampling points determined by the conventional sampling theorem. The approximation error is evaluated and weighted in the frequency domain using Fourier series expansion. The lower and upper bounds of approximation error in the frequency domain are derived and their relationships to the original waveform are established. [ABSTRACT FROM AUTHOR]

A study has been conducted on band-limited waves f(t) of bandpass type, the Fourier spectrum F(ω) of which is identically zero outside an occupied band ω... |ω| < ω... + ωb. Based on this study, we have obtained the following results: first, when an interpolation of a wave is attempted using a set of even-numbered samples periodically repeated on both sides of the time axis at a mean distance between samples η/ωb (which in general are arranged ununiformly), the necessary and sufficient condition for said samples to restore a desired band-limited wave having the same occupied bandwidth is demonstrated; second, a concrete formula of interpolation using a set of the prescribed type that satisfies the necessary and sufficient condition above, is introduced. Then a proof that demonstrates the impossibility that a formula which permits a set of samples of an odd number is periodically repeated at a mean interval of η/ωb to restore any desired band-limited wave having the prescribed occupied bandwidth on the basis of interpolation unless ω... or ωb satisfies a special condition. Finally, relations between the possibility of interpolation and combinations of ω... and ωb for sample positions restricted to a certain extent if the mean interval between sampling points is shorter than η/ωb, is discussed. [ABSTRACT FROM AUTHOR]

In the transfer function with zeros restricted on the imaginary axis of S-plane the phase characteristic is essentially determined by that of the denominator polynomial. As long as the denominator polynomial is a Hurwitz polynomial, the tangent of its phase characteristic, i.e., the ratio of the imaginary part (odd part) to the real part (even part), is a reactance function. This paper proposes a waveform approximation method which approximates the phase tangent of the Fourier transform of the object waveform by a reactance function using an optimization technique. The Hurwitz polynomial, which is the sum of the numerator and denominator of the reactance function, is adopted as the denominator polynomial of the approximate transfer function. The numerator polynomial of the approximate transfer function is derived by adjusting the zeros on the imaginary axis by an optimization technique to approximate the amplitude of the Fourier transform of the object waveform. An evaluation measure in the proposed approximation method is presented, describing its problems and physical implications. It is shown that the proposed method can provide a satisfactory approximation for symmetrical waveforms. For asymmetrical waveforms it is shown that the approximation can be improved by adding a delay time. An appropriate weighting function is proposed for the evaluation measure. Lastly, the proposed method is compared with other methods of approximation. [ABSTRACT FROM AUTHOR]