This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shift-invariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,

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We consider wavelet estimation of the time--dependent (evolutionary) power spectrum of a locally stationary time series. Hereby, wavelets are used to provide an adaptive local smoothing of a short--time periodogram in the time--frequency plane. For this, in contrast to classical nonparametric (linear) approaches we use non--linear thresholding of the empirical wavelet coe#cients. We show how these techniques allow for both adaptively reconstructing the local structure in the time--frequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which results into a near--optimal L 2 --minimax rate for the resulting spectral estimator. Our approach is based on a 2--d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a time--varying spectrum motivated from mobile radio propagation. 1 Introduction Estimating power spectra which (slowly) change over ...

Abstract. Two digital filters H(z) and F(z) are said to be biorthogonal partners of each other if their cascade H(z)F(z) satisfies the Nyquist or zero-crossing property. Biorthogonal partners arise in many different contexts such as filter bank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. In this paper we first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs, and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the above mentioned applications. 1 1.

In this paper, we introduce vector-valued multiresolution analysis and vector-valued wavelets for vector-valued signal spaces. We construct vector-valued wavelets by using paraunitary vector filter bank theory. In particular, we construct vector-valued Meyer wavelets that are bandlimited. We classify and construct vector-valued wavelets with sampling property. As an application of vector-valued wavelets, multiwavelets can be constructed from vector-valued wavelets. We show that certain linear combinations of known scalar-valued wavelets may yield multiwavelets. We then present discrete vector wavelet transforms for discrete-time vector-valued (or blocked) signals, which can be thought of as a family of unitary vector transforms. In applications of vector wavelet transforms in two dimensional transform theory, the nonseparability can be easily handled. 1 Introduction Wavelet theory has been studied extensively in both theory and applications in the last ten years. The main advantage of...

Abstract. 1 The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well known form is Shannon’s uniform sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed.

. Under the appropriate definition of sampling density D # , a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D # # 1. This result is similar to Landau's result for the PaleyWiener space B 1/2 . If the shift invariant space consists of polynomial splines, then we show that D # < 1 is su#cient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B 1/2 . In the sampling problem one seeks to recover a function f from its samples {f(x j ) : j # ZZ}. Clearly, this problem is ill-posed, and becomes meaningful only after some a priori conditions on f . Until recently, the standard assumption is that f is band-limited, i.e., supp f # [-#, #] [7, 19, 29]. Although this is a reasonable assumption, in many applications di#erent hypothesis on the functions are desirable, e.g., for taking into account real acquisition and reconstruction devices, for obtaini...

This paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal (interpolating). They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator.

We consider the problem of lconstructing a multidimensional vector function fln: "* from a finite set of linear measures. These can be irregularly sampled responses of several linear filters. Traditional approaches reconstruct in an a priori given space, e.g., the space of bandlimited functions. Instead, we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion J. First, we plsent the solution of the generalized interpolation problem. Latel; we also consider the approximation plblem, and we show that both lead to the same class of solutions.

In this paper we present several techniques to calculate the wavelet coefficients of a function from its samples. Interpolation, quadrature formulae and filtering methods are discussed and compared. 1 Introduction 1.1 Multiresolution analysis We will first briefly review wavelets and multiresolution analysis. For more detailed treatments, one can consult [9, 15, 24, 26, 28]. A multiresolution analysis of L 2 (IR) is defined as a set of closed subspaces V j with j 2 ZZ that exhibit the following properties: 1. V j ae V j+1 , 2. v(x) 2 V j , v(2x) 2 V j+1 , 3. v(x) 2 V 0 , v(x + 1) 2 V 0 , 4. +1 [ j=\Gamma1 V j is dense in L 2 (IR), 5. +1 " j=\Gamma1 V j = f0g, To appear in the proceedings of the Joint Statistical Meetings, San Francisco, August 1993. y Research Assistant of the National Fund of Scientific Research Belgium and partially supported by ONR Grant N00014-90-J-1343. 6. A scaling function '(x) 2 V 0 exists such that the set f'(x \Gamma l) j l 2 ZZg is a Riesz b...