Class Of Bandlimited Signals Research Articles

A sampling theorem for the fractional Fourier transform without band-limiting constraints

The fractional Fourier transform (FRFT), a generalization of the Fourier transform, has proven to be a powerful tool in optics and signal processing. Most existing sampling theories of the FRFT consider the class of band-limited signals. However, in the real world, many analog signals encountered in practical engineering applications are non-bandlimited. The purpose of this paper is to propose a sampling theorem for the FRFT, which can provide a suitable and realistic model of sampling and reconstruction for real applications. First, we construct a class of function spaces and derive basic properties of their basis functions. Then, we establish a sampling theorem without band-limiting constraints for the FRFT in the function spaces. The truncation error of sampling is also analyzed. The validity of the theoretical derivations is demonstrated via simulations.

Sampling and Reconstruction of Signals in Function Spaces Associated With the Linear Canonical Transform

The linear canonical transform (LCT) has been shown to be useful and powerful in signal processing, optics, etc. Many results of this transform are already known, including sampling theory. Most existing sampling theories of the LCT consider the class of bandlimited signals. However, in the real world, many analog signals arising in engineering applications are non-bandlimited. In this correspondence, we propose a sampling and reconstruction strategy for a class of function spaces associated with the LCT, which can provide a suitable and realistic model for real applications. First, we introduce definitions of semi- and fully-discrete convolutions for the LCT. Then, we derive necessary and sufficient conditions pertaining to the LCT, under which integer shifts of a chirp-modulated function generate a Riesz basis for the function spaces. By applying the results, we present a more comprehensive sampling theory for the LCT in the function spaces, and further, a sampling theorem which recovers a signal from its own samples in the function spaces is established. Moreover, some sampling theorems for shift-invariant spaces and some existing sampling theories for bandlimited signals associated with the Fourier transform (FT), the fractional FT, or the LCT are noted as special cases of the derived results. Finally, some potential applications of the derived theory are presented.

Generalizations of Chromatic Derivatives and Series Expansions

Chromatic series expansions of bandlimited functions provide an alternative representation to the Whittaker-Shannon-Kotel'nikov sampling series. Chromatic series share similar properties with Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function. Chromatic derivatives are linear combinations of ordinary derivatives in which the coefficients of the combinations are related to a system of orthonormal polynomials. But unlike Taylor series, chromatic series have more useful applications in signal processing because they represent bandlimited functions more efficiently than Taylor series. The goal of this article is to generalize the notion of chromatic derivatives and then extend chromatic series expansions to a larger class of signals than the class of bandlimited signals. In particular, we extend chromatic series to signals given by integral transforms other than the Fourier transform, such as the Laplace and Hankel transforms.

Signal Sampling and Recovery Under Dependent Errors

The paper examines the impact of the additive correlated noise on the accuracy of the signal reconstruction algorithm originating from the Whittaker-Shannon (WS) sampling interpolation formula. A class of band-limited signals as well as signals which are non-band-limited are taken into consideration. The proposed reconstruction method is a smooth post-filtering correction of the classical WS interpolation series. We assess both the point-wise and global accuracy of the proposed reconstruction algorithm for a broad class of dependent noise processes. This includes short and long-memory stationary errors being independent of the sampling rate. We also examine a class of noise processes for which the correlation function depends on the sampling rate. Whereas the short-memory errors have relatively small influence on the reconstruction accuracy, the long-memory errors can greatly slow down the convergence rate. In the case of the noise model depending on the sampling rate further degradation of the algorithm accuracy is observed. We give quantitative explanations of these phenomena by deriving rates at which the reconstruction error tends to zero. We argue that the obtained rates are close to be optimal. In fact, in a number of special cases they agree with known optimal min-max rates. The problem of the limit distribution of the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> distance of the proposed reconstruction algorithm is also addressed. This result allows us to tackle an important problem of designing non- parametric lack-of-fit tests. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case.

Vector quantization analysis of ΣΔ modulation

When considering a class of finite-dimensional input signals such as bandlimited signals in the discrete Fourier sense within a finite time window [O,T] of observation, we show that a single-loop /spl Sigma//spl Delta/ modulator behaves like the encoder of a vector quantizer. The intrinsic behavior of the modulator can be studied by analyzing the partition it generates in the input space. We show that this partition yields a particular structure that we call the "hyperplane wave structure". As one consequence to this analysis, it can be proved for the considered class of bandlimited signals that the mean squared error (MSE) of reconstruction cannot asymptotically decrease with the oversampling ratio R faster than O(R/sup -4/), regardless of the type of reconstruction used. We then generalize this analysis to n-loop /spl Sigma//spl Delta/ modulators and show that the MSE cannot decrease faster than O(R/sup -(2n+2)/).

Design of subband transmission systems in time and frequency domain

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+akm)(k=1∼N : m=1 ∼M : n=0, ±1, ±2,…; T>0; akm of the output signals gkm(t)(k=1∼N; m=1∼M) are obtained. By multiplying the sampled values by the interpolation function Økm(t)(k=1∼N; m=1 ∼M), which is obtained by the convolution of the time functions Økm(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)=g(t)-y(t) 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 ∼max(t) of the error in a form where the interpolation functions Økm(t) and Øk(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.

A system error bound for the self-truncating reconstructing filter class (Corresp.)

The error signal at the output of a sampling reconstruction filter caused by uncorrelated noise samples is analyzed. This error is combined with a truncation error bound for the class of "self-truncating" reconstruction filters [1] to give a system error bound for a bounded class of band-limited signals.

Compression of Transmission Bandwidth Requirements for a Certain Class of Band-Limited Functions

A study of source-encoding techniques that afford a reduction of data-transmission rates is made with particular emphasis on the compression of transmission bandwidth requirements of band-limited functions. The feasibility of bandwidth compression through analog signal rooting is investigated. It is found that the N th roots of elements of a certain class of entire functions of exponential type possess contour integrals resembling Fourier transforms, the Cauchy principal values of which are compactly supported on an interval one N th the size of that of the original function. Exploring this theoretical result it is found that synthetic roots can be generated, which closely approximate the N th roots of a certain class of band-limited signals and possess spectra that are essentially confined to a bandwidth one N th that of the signal subjected to the rooting operation. A source-encoding algorithm based on this principle is developed that allows the compression of data-transmission requirements for a certain class of band-limited signals. The utility of this algorithm is illustrated by a digital computer simulation of a facsimile telegraphy system. The scanning and recovery of several alphanumeric character images is simulated. Recognizable replicas of the original pattern are retrieved after 6-to-1 bandwidth compression.