The minimization of the error associated with a running approximation by a filter bank is one of the most
important problems of the signal processing. In this paper, for a set of vector-signals such that generalized Fourier
transforms have weighted norms smaller than a given positive number, we present the extended optimum running
approximation that minimizes various continuous worst-case measures of approximation error at the same time.
In this discussion, we introduce a new concept of multi-legged-type signal that is a combined-signal of many
one-dimensional band-limited signals. Backbone of this multi-legged-type signal is constituted with a series
of small separable segments of the above one-dimensional signals that are determined by the proposed running
approximation. Based on this concept, we propose an approximation method of the multi-legged-type signals and
we prove that this approximation is the optimum. Then, we define measures of error that become the proposed
measures of error in the position of the backbone made by the corresponding running approximation and become
small about the other errors. Based on these measures of error, we prove that the presented extended optimum
approximation minimizes various continuous worst-case measures of the running approximation error at the same
time. As an application, multiple-input multiple-output/space division multiplexing system is discussed.
In this paper, we define a multi-input multi-output system composed of given analysis-filter matrices, given
sampler matrices and interpolation-filter matrices to be optimized, respectively. It is assumed that input-signal
vectors of this system have a finite number of variables and these input-signal vectors are contained in a certain
given set of input-signal vectors. Firstly, we define new notations which expresses a kind of product between two
matrices or between a vector and a matrix. Using these new notations, we show that the presented approximation
satisfies a certain two conditions and prove that the presented approximation minimizes any upper-limit measure
of error compared to any other linear or nonlinear approximation with same sample values, simultaneously.
The minimization of approximation errors in a FIR multi-dimensional filter bank for band-limited signals is
the important problem of multi-dimensional signal processing. In this paper, we consider the interpolation approximation
that is modeled as a certain multi-dimensional FIR filter bank. Firstly, we introduce the known
optimum approximation of multi-dimensional signals in FIR filter banks using a finite number of sample values.
Secondly, we explain briefly a new concept of multi-legged-type signal that is a combined-signal of many
one-dimensional band-limited signals. Backbone of this multi-legged-type signal is constituted with a set of
small separable segments of the above one-dimensional signals that are determined by the proposed running
approximation. Thirdly, we extend this concept to multi-dimensional hyper signal space. Based on this concept,
we present an approximation method of the extended multi-dimensional multi-legged-type signals and we prove
that this approximation in the hyper domain is the optimum. Finally, we define measures of error that become
the proposed measures of error in the position of the backbone made by the corresponding multi-dimensional
running approximation and become small about the other errors. Based on these measures of error, we prove
that the presented multi-dimensional optimum approximation minimizes various continuous worst-case measures
of the running approximation error at the same time.
We present a running approximation of discrete signals by a FIR filter bank that minimizes various worst-case
measures of error, simultaneously. We assume that the discrete signal is a sampled version of unknown original
band-limited signal that has a main lobe and small side-lobes. To restrict frequency characteristics of signals in
this discussion, we impose two restrictions to a set of signals. One is a restriction to a weighted-energy of the
Fourier transform of the discrete signal treated actually in the approximation and another is a restriction to a
measure like Kullback-Leibler divergence between the initial analog signal and the final discrete approximation
signal. Firstly, we show interpolation functions that have an extended band-width and satisfy condition called
discrete orthogonality. Secondly, we present a set of signals and a running approximation satisfying all of these
conditions of the optimum approximation.
In many physical effects and sensors in biomedical or engineering fields, it is often the case that some small
non-linear characteristics are contained in the system. The previous paper treats approximation of non-linear
filter bank.10 But, a running approximation is not treated. In this paper, we establish a theory of a favorable
interpolation approximation of running filter banks with non-linear analysis filters based on the one-to-one
correspondence between errors in a wide but limited volume and a certain small volume in the variable domain.
Some additional considerations about the optimum interpolation approximation are presented also.
In the iterative CAD design of new materials by digital computers, it is necessary to obtain the differential
coefficients, that is, component-sensitivities caused by the small deviation of inner-components in a given electromagnetic
field expressed by the Maxwell relations. Further, to determine the step-size of the numerical iterative
CAD design that uses the discrete sample values of the wave form at the sample points with the same interval
of the step-size, it is required to estimate the error favorably between the original wave form and its numerical
approximation. In this paper, firstly, we present conservation operators in micro-electromagnetic field and its
macro-expression in the electromagnetic field. Secondly, we present some concrete conservation operators and
make clear that the certain quantities, such as the stored energy of a small inner-component in the closed electromagnetic
field, are closely correlated to the differential coefficients of the electric field and the magnetic field
observed at the outer ports. Secondly, in a single-mode electromagnetic field, we obtain the relation between the
stored energy, and the component-sensitivities caused by the small deviation of the inner-component. Thirdly, we
present a brief survey of the progress in the development of meta material and show the usefulness of combining
the above results with the optimum nonlinear approximation in the iterative design of linear or nonlinear meta
material.
We present the theory of the optimum running approximation of input signals using sample values of the
corresponding output signals of multi-path analysis filters. The presented method uses time-limited running
interpolation functions. As an application, we discuss design of in-silico smart adjusting-systems to support a
doctor about the plan of administering medicine that is useful in personalized medical care. In this paper, firstly,
we define a set of a finite number of signals in the initial set of signals and we present a one to one correspondence
between each signal contained in the set of signals and the corresponding error of approximation of a certain
finite time-interval. Secondly, based on this one-to-one correspondence, we prove that the presented running
approximation minimizes various continuous worst-case measures of error at the same time. Certain reciprocal properties of the approximation are presented. Thirdly, extension to signal-estimation using multiple-input one-output system is presented. Finally, an application to the above in-silico adjusting method is discussed.
In this paper, we present an integrated discussion of the space-limited but approximately band-limited ndimensional
running discrete approximation that minimizes various continuous worst-case measures of error,
simultaneously. Firstly, we introduce the optimum approximation using a fixed finite number of sample values
and a running approximation that scans the sample values along the time-axis. Secondly, we derive another
filter bank having both the set of extended number of transmission paths and a cutoff frequency over the actual
Nyquist frequency. Thirdly, we obtain a continuous space-limited n-dimensional interpolation functions satisfying
condition called extended discrete orthogonality. Finally, we derive a set of signals and discrete FIR filter
bank that satisfy two conditions of the optimum approximation.
We present two topics in this paper. First topic is the optimum running approximation of signals by a FIR filter
bank minimizing various worst-case continuous measures of error, simultaneously. As a direct application, we
obtain a favorable sub-band multi-input multi-output transmission system that is useful to multi-path sensor
network with transmission paths having the minimum transmission power and error of approximation at the same
time at each channel independently. We assume that a Fourier transform F(ω) of a signal f(t) is band-limited
approximately under a Nyquist frequency but its side-lobes over the Nyquist frequency are small. We introduce
a positive low-pass weight function and we define a set of signals Ξ such that a weighted-square-integral of F(ω)
by this weight function is bounded by a given positive constant A.
Firstly, we consider a finite number of signals densely scattered in the initial set of signals and present one-to-one
correspondences between a signal and its running approximation or the error of approximation in a certain
small segment in the time domain. Based on this one-to-one correspondence, we show that any continuous worstcase
measures of error in any time-limited interval can be expressed by the corresponding measures of error in
the small segment in the time axis. Combining this one-to-one correspondence with the optimum approximation
proved by Kida, we present a running approximation minimizing various continuous worst-case measures of error
and continuous upper-limit of many measures of the approximation formula, at the same time.
Firstly, we present the optimum interpolation approximation for multi-dimensional vector signals. The presented
approximation shows high performance such that it minimizes various worst-case measures of error of
approximation simultaneously. Secondly, we consider a set of restricted multi-dimensional vector signals that all
elements of the corresponding generalized spectrum vector are separable-variable functions. For this set of restricted
multi-dimensional vector signals, we present the optimum interpolation approximation. Moreover, based
on this property, putting the variables to be identical with each other in the approximation, we present a certain
optimum interpolation approximation for generalized filter bank with generalized non-linear analysis filters.
This approximation also shows the high performance similar to the above-mentioned approximations. Finally,
as a practical application of the optimum interpolation approximation for multi-dimensional vector signals, we
present a discrete numerical solution of linear partial differential equations with many independent variables.
We present the optimum running-type approximation of FIR filter bank that minimizes various worst-case measures of error, simultaneously, with respect to each of two different sets of signals. The first set is a set of piecewise analytic time-limited but approximately band-limited signals. When a supreme signal that realizes the prescribed worst-case measure of error exists, we prove, firstly, that there exists one-to-one correspondence between error in a wide time interval and error in a small interval. Based on this one-to-one correspondence, we prove that the approximation presented in this paper is the optimum in some sense. Secondly, we consider a set of band-limited signals with a main-lobe and a pair of small side-lobes and obtain similar conclusion.
In this paper, we establish the optimum interpolation approximation for a set of multi-dimensional statistical
orthogonal expansions. Each signal has a bounded linear combination of higher order self-correlations and
mutual-correlations with respect to coefficients of the expansion. For this set of signals, we present the optimum
interpolation approximation that minimizes various worst-case measures of mean-square error among all the linear
and the nonlinear approximations. Finally, as a practical application of the optimum interpolation approximation,
we present a discrete numerical solution of linear partial differential equations with two independent variables.
We begin with a summary of the optimum fixed-type interpolation approximation minimizing the upper bound of various measures of approximation error, simultaneously. The optimum interpolation functions used in this approximation are different from each other and have to cover the entire interval in the time domain to be approximated. Secondly, by applying the above approximation, we present the optimum running-type interpolation approximation for arbitrary long but time-limited signals. The proposed interpolation functions are time-limited and can be realized by FIR filters. Hence, the approximation system can be realized by time-invariant FIR
filter bank. We present one-to-one correspondence between error of approximation in a small interval in the time domain and error of approximation in limited but wide interval in the time domain based on Fredholm integral equation using Pincherle-Goursat kernel. Finally, as a practical application of the optimum fixed-type
interpolation approximation, we present a discrete numerical solution of differential equations.
Extended interpolatory approximation is discussed for some classes of n-dimensional vector signals. Firstly, we present two sufficient conditions of the optimum approximation and prove that the proposed optimum approximation using fixed finite number of sample values satisfies these two conditions. Secondly, we discuss the optimum running approximation of n-dimensional time-limited vector signals based on a certain one-to-one correspondence between a vector signal and the corresponding vector error signal of approximation. The proposed optimum approximation has the minimum measure of error among almost all the linear and the nonlinear approximations using the same measure of error and generalized sample values. Note that the proposed optimum approximation can be realized by flexible FIR filter bank. The term "flexible" means that we can widely choose the number of paths and frequency response of time-invariant FIR analysis filters. Moreover, we can use sample points that are distributed on an arbitrary periodical pattern.
Extended interpolatory approximation is discussed for some classes of n-dimensional statistical signals. Firstly, we present two sufficient conditions of the optimum approximation. Then, as example of this optimum approximation, we consider approximation of n-dimensional statistical signals expressed by linear combination of the finite set of base signals in a n-dimensional space. We assume that these signals have generalized mutual moment smaller than a given positive number. Related topic was discussed in the previous paper. However, discrete running approximation along the time axis that uses shift-invariant interpolation functions with the finite supports is not treated in the previous paper. In the final part of this paper, we discuss best running approximation of n-dimensional signals expressed by linear combination of the finite set of sinusoidal signals in a n-dimensional space. The presented methods have the minimum measure of approximation error among all the linear and the nonlinear approximations using the same measure of error and generalized sample values.
We present the optimum approximation of FIR filter bank that minimizes various measures of error of approximation, simultaneously. The presented approximation is quite flexible in choosing band-width of sub-bands, sample points and analysis filters. A kind of reciprocal relation holds for this approximation. Based on the reciprocal relation, in many examples, we can obtain perfect-reconstruction filter bank, iteratively.
Extended interpolatory approximation is discussed for some classes of signals expressed by the finite sum of sinusoidal signals in the time domain. We assume that these signals have weighted norms smaller than a given positive number. The presented method has the minimum measure of approximation error among all the linear and the nonlinear approximations using the same measure of error and generalized sample values.
We present a necessary and sufficient condition that a given n-dimensional generalized interpolation approximation minimizes various worst-case measures of error of approximation at the same time among all the approximations, including nonlinear approximation, using the same set of sample values. As a typical example of the optimum approximation satisfying the above necessary and sufficient condition, we present n-dimensional generalitd interpolation approximation using the finite number of sample values. Then, we consider n-dimensional generalized discrete interpolation approximation based on n-dimensional FIR filter banks that uses the finite number of sample values in the approximation of each pixel of image but scan the image over the whole pixels. For this scanning-type discrete approximation, we prove that discrete interpolation functions exist that minimize various measures of error of approximation defined at discrete sample points xp=p, simultaneously, where p are the n-dimensional integer vectors. The presented discrete interpolation functions vanish outside the prescribed domain in the integer-vector space. Hence, these interpolation functions are realized by n-dimensional FIR filters. In this discussion, we prove that there exist continuous interpolation functions with extended band-width that interpolate the above discrete interpolation functions and satisfy the condition called discrete orthogonality. This condition is one of the two conditions that constitute the necessary and sufficient condition presented in this paper. Several discrete approximations are presented that satisfy both the conditions constituting the necessary and sufficient condition presented in this paper. The above discrete interpolation functions have much flexibility in their frequency characteristics if appropriate analysis filters are selected.
The approximation in the maximally distinct or under sampled filter banks is considered. In this discussion, the decimated sampling interval T satisfies T≥M, where M is the number of paths of the filter banks. Moreover, we present the optimum transmultiplexer TR. This result is extended to the design of wireless transmultiplexer, used in CDMA systems, for example.
In this paper, we will present a systematic discussion for the optimum interpolatory approximation in a shift-invariant wavelet and/or scaling subspace. Firstly, we will present the optimum interpolation functions which minimize various worst case measure of approximation error among all the linear and the nonlinear approximations using the same sample values of the input signal. Secondly, we will show that the optimum interpolation functions are expressed as the parallel shifts of the finite number of one function. Finally, we will present the optimum interpolation function in wavelet and scaling subspace. These interpolation functions are optimum in the multi-resolution analysis which considers lower resolutions.
In this paper, we will present the optimum interpolation functions minimizing various measures of approximation error simultaneously. For an ordinary interpolatory approximation using sample values of a band-limited signal and a FIR filterbank system having analysis filters Hm((omega) ) (m equals 0,1,...,M - 1), we outline necessary formulation for the time-limited interpolation functions (psi) m(t) realizing the optimum approximation in each limited block separately. Further, under some assumptions, we will present analytic or piece-wise analytic interpolation functions (phi) m(t) minimizing various measures of approximation error defined at discrete time samples tn equals n (n equals 0,+/- 1,+/- 2,...). In this discussion, (phi) m(n) are equal to (psi) m(n) (n equals 0,+/- 1,+/- 2,...). Since (phi) m(t) are time-limited, (phi) m(n) vanish outside of the finite set of n. Hence, one can use FIR filters if one wants to realize discrete synthesis filters which impulse responses are (phi) m(n).
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