Sampling Theorem For Bandlimited Signals Research Articles

Hermite interpolation theorems for band-limited functions of the linear canonical transforms with equidistant samples

We establish convergence analysis for Hermite-type interpolations for L^{2} ( mathbb {R})-entire functions of exponential type whose linear canonical transforms (LCT) are compactly supported. The results bridges the theoretical gap in implementing the derivative sampling theorems for band-limited signals in the LCT domain. Both complex analysis and real analysis techniques are established to derive the convergence analysis. The truncation error is also investigated and rigorous estimates for it are given. Nevertheless, the convergence rate is O(1/sqrt{N}), which is slow. Consequently the work on regularization techniques is required.

Approximate Hermite Interpolations for Compactly Supported Linear Canonical Transforms

There has been several Lagrange and Hermite type interpolations of entire functions whose linear canonical transforms have compact supports inℝ. There interpolation representations are called sampling theorems for band-limited signals in signal analysis. The truncation, amplitude, and jitter errors associated with the Lagrange type interpolations are investigated rigorously. Nevertheless, the amplitude and jitter errors arising from perturbing samples and nodes are not studied before. The aim of this work is to establish rigorous analysis of their types of perturbation errors, which is important from both practical and theoretical points of view. We derive precise estimates for both types of errors and expose various numerical examples.

$$\sin (\omega x)$$ sin ( ω x ) Can Approximate Almost Every Finite Set of Samples

Consider a set of points $$(x_1,y_1),\ldots ,(x_n,y_n)$$ with distinct $$0 \le x_i \le 1$$ and with $$-1 < y_i < 1$$ . The question of whether the function $$y = \sin (\omega x)$$ can approximate these points arbitrarily closely for a suitable choice of $$\omega $$ is considered. It is shown that such approximation is possible if and only if the set $$\{x_1,\ldots ,x_n\}$$ is linearly independent over the rationals. Furthermore, a constructive sufficient condition for such approximation is provided. The results provide a sort of counterpoint to the classical sampling theorem for bandlimited signals. They also provide a stronger statement than the well-known result that the collection of functions $$\{\sin (\omega x) : \omega < \infty \}$$ has infinite pseudo-dimension.

Multichannel sampling and reconstruction of bandlimited signals in the linear canonical transform domain

Multichannel sampling for bandlimited signals is fundamental in the theory of multichannel parallel analogue-to-digital (A/D) environment and multiplexing wireless communication environment. The analysis and application of multichannel sampling in the traditional Fourier domain have been extensively studied, but so far none of the research papers covering the reconstruction of multichannel sampling in the linear canonical transform (LCT) domain have been published. This study is to explore the multichannel sampling and reconstruction of bandlimited signals in the LCT domain. First, the multichannel sampling theorem for bandlimited signals with LCT is proposed, which is the generalisation of classical generalised Papoulis sampling expansion. Considering the signal reconstruction purpose, the authors present two schemes for the multichannel sampling with LCT. The first scheme is based on the conventional Fourier series and inverse LCT, whereas the other uses the generalised convolution systems. Second, by designing different LCT filters, the authors obtain the reconstruction method for the uniform sampling theorem, sampling from the signal and its derivative by using the derived multichannel sampling theorem and properties of the LCT. Last, the simulations are carried out to verify the correctness of the results. Moreover, the potential applications of the multichannel sampling are also presented.

Sampling of Bandlimited Signals in Fractional Fourier Transform Domain

Fractional Fourier transformed bandlimited signals are shown to form a reproducing kernel Hilbert space. Basic properties of the kernel function are applied to the study of a sampling problem in the fractional Fourier transform (FRFT) domain. An orthogonal sampling basis for the class of bandlimited signals in the FRFT domain is then given. A nonuniform sampling theorem for bandlimited signals in the FRFT domain is also presented. Numerical experiments are given to demonstrate the effectiveness of the proposed nonuniform sampling theorem.

On Bandlimited Signals Associated With Linear Canonical Transform

We first show that the bandlimited signals associated with linear canonical transform (LCT) form a reproducing kernel Hilbert space. An orthogonal basis for the class of bandlimited signals associated with LCT is then proposed by use of the reproducing kernel, with respect to which the coordinates of signal are actually values taken by the signal at certain instants of time. Finally, a nonuniform sampling theorem for bandlimited signals associated with LCT is presented.

Multi-channel sampling theorems for band-limited signals with fractional Fourier transform

Multi-channel sampling for band-limited signals is fundamental in the theory of multi-channel parallel A/D environment and multiplexing wireless communication environment. As the fractional Fourier transform has been found wide applications in signal processing fields, it is necessary to consider the multi-channel sampling theorem based on the fractional Fourier transform. In this paper, the multi-channel sampling theorem for the fractional band-limited signal is firstly proposed, which is the generalization of the well-known sampling theorem for the fractional Fourier transform. Since the periodic nonuniformly sampled signal in the fractional Fourier domain has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived multi-channel sampling theorem and the specific space-shifting and phase-shifting properties of the fractional Fourier transform. Moreover, by designing different fractional Fourier filters, we can obtain reconstruction methods for other sampling strategies.

Signal reconstruction from the undersampled signal samples

It is well-known from the celebrated Shannon sampling theorem for bandlimited signals that if the sampling rate is below the Nyquist rate, aliasing takes place and the original signal cannot be reconstructed back by simply passing the signal samples through an ideal lowpass filter. However, researchers such as Stern and Gori have shown the existence of some classes of signals for which the signals are sampled below the Nyquist rate but perfect signal reconstruction is still possible from the given signal samples. Here, we present a generalized lowpass sampling theorem and show that Stern’s and Gori’s lowpass sampling theorems are special cases of it. A sampling theorem for the bandpass signals in the linear canonical transform domains is also presented and its special cases are discussed. Using a modification of the conventional natural sampling waveform with a specific width of the pulses, it is shown that the sampling rate in our generalized lowpass sampling theorem and hence in the Stern’s and the classical Shannon sampling theorems can be further reduced by a factor of two, while for the bandpass signals, the reduction in the sampling rate by some factor is possible only under some restricted conditions.

SAMPLING AND OVERSAMPLING IN SHIFT-INVARIANT AND MULTIRESOLUTION SPACES I: VALIDATION OF SAMPLING SCHEMES

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.

Interpolation of Non-Uniformly Sampled Sequences and Reconstruction of Sampled Sequences with Any Uniform Interval Based on Non-Uniform Sampling Theorem

In this paper we psesent a new non-uniform sampling theorem for band-limited signals, together with an associated interpolation formula of non-uniformly sampled sequences, and propose a practical algorithm for reconstructing sampled sequences of any uniform interval from a non-uniform one by using the frequency sampling technique and FFT algorithm. After a brief review of the ordinary uniform sampling theorem, the non-uniform sampling theorem is given, together with an aliasing free condition being made clear. Based on the fundamentals, an interpolation formula of non-uniformly sampled sequences is derived and a practical implementation algorithm to reconstruct uniformly sampled sequences of any desired interval from a non-uniformly sampled version is proposed by way of the frequency sampling technique and FFT algorithm. The principle as well as the effectiveness of the proposed algorithm are ascertained through numerical experiments with sinusoidal and FM signals of finite bandwidths.

On bandlimited signals with fractional Fourier transform

We study bandlimited signals with fractional Fourier transform (FRFT). We show that if a nonzero signal f is bandlimited with FRFT F/sub /spl alpha// for a certain real /spl alpha/, then it is not bandlimited with FRFT F/sub /spl beta// for any /spl beta/ with /spl beta//spl ne//spl plusmn//spl alpha/+n/spl pi/ for any integer n. This is a generalization of the fact that a nonzero signal can not be both timelimited and bandlimited. We also provide sampling theorems for bandlimited signals with FRFT that are similar to the Shannon sampling theorem.

Time-limited sampling theorems for band-limited signals (Corresp.)

A procedure is given for the recovery of a band-limited signal from any set of samples by which it is uniquely determined. Containing, as a special case, Shannon's sampling theorem, it applies even to situations, such as time-limited sampling, for which no sampling basis exists, and extends to those eases the essential features of the Shannon representation needed for approximation from finite data. A second reconstruction, based on sequences of time-limited equispaced sample values, is also presented, and the extension to stochastic processes is discussed.

Sampling, data transmission, and the Nyquist rate

The sampling theorem for bandlimited signals of finite energy can be interpreted in two ways, associated with the names of Nyquist and Shannon. 1) Every signal of finite energy and bandwidth W Hz may be completely recovered, in a simple way, from a knowledge of its samples taken at the rate of 2W per second (Nyquist rate). Moreover, the recovery is stable, in the sense that a small error in reading sample values produces only a correspondingly small error in the recovered signal. 2) Every square-summable sequence of numbers may be transmitted at the rate of 2W per second over an ideal channel of bandwidth W Hz, by being represented as the samples of an easily constructed band-limited signal of finite energy. The practical importance of these results, together with the restrictions implicit in the sampling theorem, make it natural to ask whether the above rates cannot be improved, by passing to differently chosen sampling instants, or to bandpass or multiband (rather than bandlimited) signals, or to more elaborate computations. In this paper we draw a distinction between reconstructing a signal from its samples, and doing so in a stable way, and we argue that only stable sampling is meaningful in practice. We then prove that: 1) stable sampling cannot be performed at a rate lower than the Nyquist, 2) data cannot be transmitted as samples at a rate higher than the Nyquist, regardless of the location of sampling instants, the nature of the set of frequencies which the signals occupy, or the method of construction. These conclusions apply not merely to finite-energy, but also to bounded, signals.

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