2136 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 5, MAY 2007 On the Approximation of Inner Products From Sampled Data Hagai Kirshner and Moshe Porat, Senior Member, IEEE Abstract—Most signal processing applications are based on dis- crete-time signals although the origin of many sources of informa- tion is analog. In this paper, we consider the task of signal repre- sentation by a set of functions. Focusing on the representation co- efficients of the original continuous-time signal, the question con- sidered herein is to what extent the sampling process keeps alge- braic relations, such as inner product, intact. By interpreting the sampling process as a bounded operator, a vector-like interpreta- tion for this approximation problem has been derived, giving rise to an optimal discrete approximation scheme different from the Rie- mann-type sum often used. The objective of this optimal scheme is in the min-max sense and no bandlimitedness constraints are im- posed. Tight upper bounds on this optimal and the Riemann-type sum approximation schemes are then derived. We further consider the case of a finite number of samples and formulate a closed- form solution for such a case. The results of this work provide a tool for finding the optimal scheme for approximating an inner product, and to determine the maximum potential representation error induced by the sampling process. The maximum representa- tion error can also be determined for the Riemann-type sum ap- proximation scheme. Examples of practical applications are given and discussed. Index Terms—Approximation, inner-product, sampling, signal representation. I. INTRODUCTION S IGNAL processing applications are concerned mainly with discrete-time signals although the origin of many sources of information is analog. Examples for such signals are speech, optics, biomedical signals, and images. In this regard, one may consider analog signal representation schemes such as wavelets and Gabor [1]–[9], and E-splines [10], [11]. Extracting repre- sentation coefficients for such schemes involves inner product calculations within the analog domain; whereas utilizing the available discrete-time data provides an approximation only. Those coefficients play a key role in signal processing appli- cations in characterizing and processing analog signals, in com- paring analog signals, in initializing the wavelet transform, and Manuscript received December 31, 2005; revised July 19, 2006. The asso- ciate editor coordinating the review of this manuscript and approving it for pub- lication was Dr. Antonia Papandreou-Suppappola. This work was supported in part by the Harmonic Analysis and Statistics for Signal and Image Processing (HASSIP) Research Program HPRN-CT-2002-00285 of the European Commis- sion, by the H. and R. Sohnis Cardiology Research Fund, and by the Ollendorff Minerva Centre. Minerva is funded through the BMBF. The authors are with the Department of Electrical Engineering, the Tech- nion—Israel Institute of Technology, Haifa 32000, Israel (e-mail: kirshner@tx. technion.ac.il; mp@ee.technion.ac.il). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2007.892706 in interpolation methods to name a few. Further motivation is also found in [12]. It is stated there that although many results and problems in Fourier and Gabor analysis are formulated in the continuous-time domain, a more suitable setting for practical computations is the discrete-time finite case. Nevertheless, such problems arise not only in time-frequency and signal analysis applications but also in numerically solving differential equa- tions [13]–[15]. The question raised then is how to optimally approximate an inner product within the analog domain while having the sam- pled version of a signal as the only available data? We will be considering the ideal uniform sampling scheme although other schemes such as nonuniform sampling, generalized sampling, and sampling in arbitrary spaces [10], [16]–[32] may be applied as well. Indeed, there are cases for which one can reconstruct the original continuous-time signal albeit the sampling process. One such case is the well-known bandlimited one. Another case arises in shift-invariant spaces for which the generating function complies with a certain condition [9]. However, no approximation scheme other than the Riemann-type sum (and related interpolation methods) is known to exist in the general case. That is, applying (1) where is the original signal, is a known analysis function, and is the sampling interval. With regard to (1), Fornasier [14] derives a convergence rate for approximating an inner product by means of a Riemann- type sum. However, within the context of representation coef- ficients, the function is analytically known and it seems pos- sible to further investigate this approximation scheme. We adopt the basic linear form of (1) and investigate whether the ideal sampling process keeps algebraic relations shared in the analog domain intact. We consider the operation widely used in signal representation, the inner product, and propose a new discrete ap- proximation method for its calculation, allowing for an optimal approach to this operation. The outline of this paper is as follows. Section II describes the problem at hand. In Section III, intertwining inner product relations are derived for several Hilbert spaces. Those relations are then utilized in Section IV for deriving an optimal min-max approximation scheme for the inner product, as well as for analyzing the Riemann-type sum approximation scheme. In Section V, a closed-form solution for the min-max approxima- tion scheme is given for the case where only a finite number of samples is available. Some examples are given in Section VI. 1053-587X/$25.00 © 2007 IEEE