IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 10, OCTOBER 2001 2301 Optimal Sub-Nyquist Nonuniform Sampling and Reconstruction for Multiband Signals Raman Venkataramani and Yoram Bresler, Fellow, IEEE Abstract—We study the problem of optimal sub-Nyquist sam- pling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau’s lower bound equal to the measure of . For signals with sparse , this rate can be much smaller than the Nyquist rate. Unfortu- nately, the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a recent study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and de- signing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with that tiles under translation. For signals with nontiling , which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with sub-Nyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate. Index Terms—Error bounds, Landau–Nyquist rate, matrix inequalities, multiband, nonuniform periodic sampling, optimal sampling and reconstruction. I. INTRODUCTION T HERE has been a long history of research [1]–[4] devoted to sampling theory, with perhaps the most fundamental and importantpieceofworkinthisareabeingtheclassicalsamplingthe- orem.AlsoknownastheWhittaker–Koteln ´ikov–Shannon(WKS) theorem, it states that a lowpass signal bandlimited to the frequen- cies can be reconstructed perfectly from its samples taken uniformly at no less than the Nyquist rate of [5]. Another importantresultinsamplingtheoryduetoLandauisalowerbound on the sampling density required for any sampling scheme that al- lows perfect reconstruction [6]. For multiband signals, this funda- mentallowerboundisgivenbythetotallength(measure)ofsupport of the Fourier transform of the signal. Landau’s bound applies to an arbitrarily sampling scheme: uniform or not, and the minimum rate isnot necessarilyachievable except asymptotically.Landau’s Manuscript received January 11, 2001; revised June 14, 2001. This work was supported in part by the Joint Services Electronic Program under Grant N00014-96-1-0129, the National Science Foundation under Grant MIP 97-07633, and DARPA under Contract F49620-98-1-0498. The associate editor coordinating the review of this paper and approving it for publication was Dr. Olivier Cappe. The authors are with Coordinated Science Laboratory, Department of Elec- trical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail raman@ifp.uiuc.edu; ybresler@uiuc.edu). Publisher Item Identifier S 1053-587X(01)07767-4. bound is often much lower than the corresponding Nyquist rate. This motivates the study of sub-Nyquist sampling of multiband signals and their perfect reconstruction, cf. [7]–[14] From a practical viewpoint, sub-Nyquist sampling is very important in several Fourier imaging applications such as sensor array imaging, synthetic aperture radar (SAR), and magnetic resonance imaging (MRI), where the physics of the problem provides us samples of the unknown sparse object in its Fourier domain [15]–[18]. Our objective, then, is to reconstruct the object from the Fourier data. It is often expensive or physically impos- sible to collect many samples, and it becomes necessary to sample minimally and exploit the sparsity (i.e., multiband structure) in the object to form its image. These problems are, of course, duals to the problem considered here since the sparsity is in the spatial domain and sparse sampling in the frequency domain. For a given signal , its spectral support is defined as the set of frequencies where the Fourier transform does not vanish, and the spectral span is defined as the smallest interval containing . We consider here only spectral supports that can be expressed as a finite union of finite intervals called bands. The set of multiband signals bandlimited to is de- noted by . Landau’s lower bound for these signals is , where is the Lebesgue measure. However, in general, the Nyquist rate for sampling without aliasing (overlap between translates of by multiples of ) is equal to the width of its spectral span . Hence, for multi- band signals with sparse spectral supports , the Nyquist rate can be much larger than the lower bound . A favorable case is when the widths of the bands and the gaps between them satisfy special relationships so that there is no overlap between uniform translates of by multiples of a quantity . In these cases, when the spectral support is packable, . The most favorable situa- tion of these is when tiles the real line under uniform trans- lations, i.e., is packable without gaps, or “ is an explosion of the interval” [4]. In this (very special) case, , i.e., Landau’s lower bound is achievable by uniform sampling. Instead, the case of interest to us in this paper is the gen- eral case, with . Without loss of gen- erality, we focus on the extreme (worst) case of nonpackable , such that . 1 For such multiband signals, it has been shown that perfect reconstruction is possible from nonuni- formly spaced samples taken at a sub-Nyquist average rate ap- 1 Intermediate cases with are reduced to this case by first sampling the signal at and then considering the problem of further downsampling the discrete-time signal, which now has a nonpackable spectral support. 1053–587X/01$10.00 © 2001 IEEE