SAMPAT05, Samsun, Turkey, July 10 – 15, 2005 Optimal Sampling and Reconstruction in Multiple-Input-Multiple-Output (MIMO) Systems 1 Yoram Bresler Coordinated Science Laboratory and Dept. of Electrical and Computer Eng. Univ. of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A. ybresler@uiuc.edu Raman Venkataramani 2 Seagate Technology, Pittsburgh, PA 15222 U.S.A. ramanv@ieee.org Abstract — We consider a sampling scheme where a set of multiband input signals are passed through a MIMO liner time-invariant system and the outputs are sampled. MIMO sampling is a very general scheme that encom- passes various other schemes, including Papoulis’ gener- alized sampling and nonuniform sampling as special cases. We present necessary density conditions for stable MIMO sampling. These results generalize Landau’s classical den- sity results for stable sampling and interpolation. Under the additional assumption of periodic (but possibly non- uniform) sampling sets, we present necessary and sufficient conditions on the system and sampling rates for stable MIMO sampling. I. Introduction Given a multiple-input multiple-output (MIMO) chan- nel with observable outputs, the problem of multichan- nel deconvolution or multichannel separation of a con- volutive mixture is to invert or equalize the channel to recover the channel inputs. Applications include mul- tiuser or multiaccess wireless communications and space- time coding with antenna arrays, or telephone digital sub- scriber loops, multisensor biomedical signals, multi-track magnetic recording, multiple speaker (or other acoustic source) separation with microphone arrays, geophysical data processing, and multichannel image restoration. To enable digital processing for the inversion of the channel, its continuous-time outputs are sampled prior to processing, and the goal is to reconstruct the continuous- time channel inputs. We assume that the channel char- acteristics are known (e.g., can be estimated accurately using known test input signals.) We call this problem MIMO sampling. Our problem is formulated as follows. Let x r (t), r = 1,...,R, be a collection of complex-valued signals whose spectral supports are sets F r ⊆ R of finite measure. We refer to such signals as multiband signals because in prac- tice F is a finite union of intervals. These R signals are in- put to a MIMO channel consisting of linear time-invariant filters to produce P outputs y p (t)= R  r=1 g pr (t) ∗ x r (t), p =1,...,P (1) where ∗ denotes convolution, and {g pr } are square- integrable impulse responses. Each output y p (t) is sub- sequently sampled on a discrete set Λ p = {λ np : n ∈ Z} 1 This work was supported in part by a grant from DARPA under contract F49620-98-1-0498 administered by AFSOR, and by NSF Infrastructure Grant CDA-24396. 2 This work was performed while R. Venkataramani was at the University of Illinois. and these samples are then used to reconstruct the in- puts. This sampling scheme is very general and subsumes various other sampling schemes as special cases. For in- stance Papoulis’ generalized sampling [1] is essentially a single-input multiple-output (SIMO) sampling scheme, i.e., MIMO sampling with R = 1. An extension of Pa- poulis’ sampling expansion to vector valued inputs [2] is also a special case with all inputs having identical low- pass spectra, i.e., F r =[−B,B]. In this paper, we present an overview of our recent re- sults on MIMO sampling [3–6]. In particular, we present necessary conditions [4] on {Λ p } and the channel for stable reconstruction of the inputs x r (t) from the MIMO out- put samples {y p (λ np )}. Similar results are also available [4](but not reviewed here) for the dual problem of consis- tent reconstruction : necessary conditions on {Λ p }, to en- sure that ∃x r (t) such that y p (λ np )= c np for any sequence {c np : n ∈ Z,p =1,...,P }∈ l 2 . Under the additional assumption of periodic (but possibly non-uniform) sam- pling sets, we present conditions that are both necessary and sufficient [5]. Finally, we review results on optimum reconstruction under these conditions [6]. Landau [7] proved the following fundamental result for sampling of multiband signals. Let x ∈ L 2 (R d ) be a con- tinuous function whose Fourier transform is supported on a measurable set F⊂ R d . Then, for stable reconstruction of any such x from its samples, it is necessary that the density of Λ be no less than the measure of F . We refer to this problem as classical sampling. Gr¨ochenig and Razafinjatovo [8] provided a simpler proof of Landau’s classical result for the case that F has zero boundary measure. We extended the idea of [8] (removing the restriction of zero boundary measure) to derive necessary density results for MIMO sampling of multiband signals [4]. Our results for single variate functions (d = 1) easily extend to multivariate functions. For stable sampling, we prove that a family of 2 P − 1 bounds hold—a lower bound on the joint lower density of each nonempty set of P output sampling sets. These bounds generalize Landau’s necessary density results for classical sampling. Since the MIMO sampling scheme is extremely general, and encompasses various sampling schemes such as Papoulis’ generalized sampling, and mul- ticoset or periodic nonuniform sampling as special cases, we automatically have necessary conditions for all these sampling schemes as well.