Generalized Sampling Theorem Research Articles

Sampling expansions and generalized translation invariance

The generalized sampling theorem of Kramer is derived and interpreted in the context of the theory of linear systems satisfying a generalized form of translation invariance. The results are extended to the form of expansions developed by Papoulis and by Campbell.

Restoring causal signals by analytic continuation: A generalized sampling theorem for causal signals

A band-limited part x 0 (t) of a non-band-limited causal signal x(t) is used to completely restore x(t) by using analytic continuation. As a consequence, a generalized sampling theorem for causal signals is derived; the total causal signal x(t) can be completely restored by sampling the band-limited part x 0 (t) with finite sampling frequency.

Generalization of sampling theorem by blurring operation

AbstractThe sampling theorem established by Someya and Shannon is presntly finding wide applications in various fields. However, it often happens in actual application that the assumption of band‐limitation which is a prerequisite of the theorem is difficult to verify. From such a viewpoint, this paper generalizes the sampling theorem to remedy the problem. the blurring operation originally has the function of positively suppressing the higher‐order harmonic components. It is a regular linear operation for which the inverse operation exists. Utilizing this property and combining the operation with the traditional sampling theorem, a new sampling theorem is derived, by which only those components of the object function in the limited band can be extracted. the generalized sampling function derived first in this paper is a natural extension of the traditional sampling function. With its analytical structure as well as the clearly stated method of computation, the generalized sampling theorem derived in this paper is applicable widely to actual problems.

On the Generalized Sampling Theorem, Generalized Frequency, and Generalized Band‐Limited Spectrum Function

AbstractA generalized sampling theorem is presented, by which a continuous function can be reconstructed from its sampled values and sampled higher order derivatives. Several remarks are made on the generalized sampling formula presented here. The generalized frequency and the generalized band‐limited spectrum function are discussed in relation to the generalized sampling theorem. The condition for establishing the generalized sampling theorem is stated in terms of the generalized frequency. The bound for truncation error of the generalized sampling expansion is briefly mentioned.

A generalized sampling theorem with application to computer-generated transparencies*

A set of generalized Dirac functions (Δ functions) is introduced which resemble the Dirac function (δ function) in that their masses equal unity. Different members out of this set are determined by different values of a certain parameter. The sampling theorem—which tells us that a band-limited function can be synthesized by a proper low-pass filtering of a sequence of equidistant δ functions having variable masses, these masses being equal to the corresponding sample values of the band-limited function—no longer holds, if the practically unrealizable δ functions are replaced by realizable Δ functions. It must be replaced by a generalized sampling theorem, which tells us what relationship exists between the parameters of the Δ functions and the sample values of the function to be generated at the ouput of the low-pass filter. Once a set of Δ functions has been chosen, this relationship can be determined explicitly. An application to the synthesis of coherent optical fields by means of computer-generated transparencies is given.

A generalized sampling theorem with application to computer-generated transparencies

A set of generalized Dirac functions (Δ functions) is introduced which resemble the Dirac function (δ function) in that their masses equal unity. Different members out of this set are determined by different values of a certain parameter. The sampling theorem—which tells us that a band-limited function can be synthesized by a proper low-pass filtering of a sequence of equidistant δ functions having variable masses, these masses being equal to the corresponding sample values of the band-limited function—no longer holds, if the practically unrealizable δ functions are replaced by realized Δ functions. It must be replaced by a generalized sampling theorem, which tells us what relationship exists between the parameters of the Δ functions and the sample values of the function to be generated at the ouput of the low-pass filter. Once a set of Δ functions has been chosen, this relationship can be determined explicitly. An application to the synthesis of coherent optical fields by means of computer-generated transparencies is given.

一般化された標本化定理の公式の諸例について (第2報)

一般化された標本化定理の公式の諸例について (第2報)

一般化された標本化定理の公式の諸例について(第1報)

一般化された標本化定理の公式の諸例について(第1報)

一般化された標本化定理

一般化された標本化定理

Application of the sampling theorem to boundary value problems

The generalized sampling theorem is used to facilitate the solution of a conjugated boundary value problem of the Graetz type. The analysis is applied to determine the effects of axial conduction on the temperature field in a fluid in laminar flow in a tube. This represents the first application of the sampling theorem outside of the area of communications theory.

A Generalized Sampling Theorem for Non-Stationary Random Processes

A Generalized Sampling Theorem for Non-Stationary Random Processes

On the equivalence of Kramer's and Shannon's sampling theorems (Corresp.)

The question of Campbell concerning the possibility of applying Shannon's sampling theorem to the same functions that can be sampled by Kramer's generalized sampling theorem is investigated and illustrated for more functions which are solutions of second-order differential equations with singular coefficients. A few definitions are introduced and some theorems are proved to show, in a more precise way, that under certain conditions the two sampling theorems are equivalent.

Some applications for Kramer's generalized sampling theorem

Kramer's generalization of Shannon's sampling theorem takes us from a signal represented by a finite Fourier transform to a signal represented by another and more general finite integral transform. In this paper we will attempt to show that the already obtained results for Kramer's theorem are of use in the field of finite integral transforms. Also by introducing such transforms one can treat some communications problems. An example is the case of representing a signal which is the output of time variant filter.

Bounds for Truncation Error of the Sampling Expansion

Bounds for Truncation Error of the Sampling Expansion

Error-Free Recovery of Signals from Irregularly Spaced Samples

Error free recovery of signals from irregularly spaced samples in terms of completeness of sets of nonharmonic exponentials

A Generalized Sampling Theorem

A Generalized Sampling Theorem

On Nonuniform Sampling of Bandwidth-Limited Signals

The purpose of this investigation is to examine four special nonuniform sampling processes in detail, and to deduce some interesting properties of bandwidth-limited signals. The main results are contained in four generalized sampling theorems. These theorems not only contain the nature of determination (unique-specification, over-specification, and underspecification) of signals but also include explicit reconstruction formulas. From the reconstruction formulas, the complexity and accuracy of the nonuniform sampling processes discussed can be estimated. In addition, these theorems lead to observations regarding the allowable shapes, the prediction, and the energy of bandwidth-limited signals in general. A minimum-energy signal is introduced which has certain advantages as compared to the ordinary time-limited signals when a finite number of sample values are given. Finally, a statement due to Cauchy on the sampling of bandwidth-limited signals is generalized to include a wider class of nonuniform sample point distributions and modified to give more exact information regarding the nature of determination of signals.