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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]

This paper presents a comprehensive discussion of the approximation of the n-dimensional wave f(X) using the sampled values of the output wave obtained by exciting a series of time-invariant linear circuits by the wave f(X). It is assumed that the approximate wave h(X) of f(X) is given by the sum of sample values of the output wave multiplied by certain n-dimensional waves. For simplicity, n-dimensional waves to be multiplied with the sample values are called the interpolation functions. The set of sample points treated in this paper is defined as a subset obtained by sampling periodically the vertices of the n-dimensional parallelepipeds placed periodically in the space Rn. Such a set of rumpling points includes the most of the typical arrangements of the sampling points, such as the hexagonal and the octagonal lattices on the two-dimensional space. It is assumed that the sample values contain statistically independent errors such as the observation error and/or the quantization error. Moreover, it is assumed that the interpolation functions have the supports which are parallel-translations of each other. First, it is assumed that the functional forms of these interpolation functions may be different. Further, a set of n-dimensional waves is considered where the corresponding spectrum has the weighted p-norms smaller than the prescribed positive constant. The standard deviations of the difference between f(X) and their approximations are considered. As the measure of the approximation error, the upper limit of the standard deviation obtained by varying the original waves over the given set of waves is adopted. In the following sections it is shown that the interpolation functions minimizing the forementioned measure of error can be expressed as the parallel-translations of a finite number of functions. [ABSTRACT FROM AUTHOR]

Consider a system in which N time-invariant linear circuits Hk(ω)(k=1∼N) are connected in parallel and to each of which M parallel time-invariant linear circuits Hkm(ω)(k=1 ∼N; m=1∼M) are connected in cascade. Then consider that a certain class of band-limited signals f(t) is impressed on such a system, and the equal-interval sampled values gkm(nT + αkm)(k=1∼N; m=1∼M; n=0, ±1, ±2, …; T>0; αkm of the output signals gkm(t)(k=1∼N; m=1∼M) are obtained. By multiplying the sampled values by the interpolation function ψkm(t)(k= 1∼N : m=1∼M), which is obtained by the convolution of the time functions φkm(t)(k=1∼N : m=1∼M), and by summing up, an approximation formula y(t) is determined. A unified investigation is made on the approximation problem using y(t) for the response g(t) when the original signal f(t) passes through a given filter H(ω). First, the error e(t)=&verbar;g(t)-y(t)&verbar; is considered for given Hk(ω)(k=1∼N), and Hkm(ω)(k=1∼N : m=1∼M) over the set of f(t) belonging to the considered set of waveforms, and a formula is derived which represents the upper bound ell;max(t) of the error in a form where the interpolation functions φkm(t) and φk(t) are separated. Then, based on the derived formula, an effective design method for the interpolation filter is established, based on the coefficient equation and linear programming, assuming that all of the interpolation filters are constructed as FIR type. Finally, the optimization of the system with a hierarchical structure is discussed. [ABSTRACT FROM AUTHOR]

Assume that signal f(t) is impressed on N parallel time-invariant liner networks Hm(ω) (m = 1∼N), and consider the uniformly sampled values gm(nT) (m=1∼N; n=0, ± 1, ±2; T > 0) of the output signal gm(t) (m=1∼N). This paper discusses the following problem from a unified viewpoint. The response g(t), when the signal f(t) is passed through the given filter H(ω), is to be approximated by and expression y (t) which is the sum of the forementioned sample value multiplied by time functions ø mn(t) (m-1 ∼ N; n=0, ± 1, ± 2,). It is assumed that the signal f(t) belongs to the set 1 of the signals for which the weighted square integral of the Fourier spectrum F(ω) is not greater than a positive number A. In the foregoing, ømn(t) is called the interpolation function. First, it is shown that given Hm(ω)(m=1∼N), the time-limited interpolation function, which minimizes the upper limit emax(t) of the error e(t) =g(t) - y(t) over all f(t) belonging to T, is obtained by shifting the impulse responses øm(t) (m=1∼N) of certain N linear time-invariant interpolation filters øm (ω) (m=1∼N) and øm(ω) (m=1∼N) are optimized so that the measure emax(t) of the error is minimized, the upper and the lower bounds of the optimal emax(t) are shown. Considering an application where N is large, the optimization is discussed in the sense that the equivalent multiplicity is reduced at the sacrifice of the approximation error. [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]

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]