Multiple-Input Multiple-Output Sampling: Necessary Density Conditions

Abstract

Abstract-We consider the problem of multiple-input multipleoutput (MIMO) sampling of multiband signals. In this problem, a set of input signals is passed through a MIMO channel modeled as a known linear time-invariant system. The inputs are modeled as multiband signals whose spectral supports are sets of finite measure and the channel outputs are sampled on nonuniform sampling sets. The aim is to reconstruct the inputs from the output samples. This sampling scheme is quite general and it encompasses various others including Papoulis' generalized sampling and nonuniform sampling as special cases. We introduce notions of joint upper and lower densities for collections of sampling sets and then derive necessary conditions on these densities for stable sampling and consistent reconstruction of the channel inputs from the sampled outputs. These results generalize classical density results for stable sampling and interpolation due to Landau.

Filter design for mimo sampling and reconstruction

“…For fixed and , define mod and mod for an arbitrary . Then, using the conditions (15), (16), and 25, we see that implying that for all and . Therefore, we obtain for all , where mod .…”

“…Since , (14) requires that have full column rank a.e. As in the continuous-time case [13], it can be easily verified that is continuous if and only if is continuous on , and the following periodicity conditions hold: (15) for all , where mod . As we will see later, in order to achieve continuity of , it is convenient to impose continuity on , and this produces a similar condition on .…”

“…Then, is a Hilbert space equipped with the inner product for all Naturally, is the the norm of . We first review an important notion called stability of MIMO sampling [13], [15].…”

“…Then, perfect reconstruction using an FIR reconstruction filter matrix is possible if and only if , and the minors of have no zero common to all except or . Proof: The coprimeness condition on the minors guarantees the existence of an FIR such that and the periodicity conditions (15) and (16) hold. This is a standard result that can be proved using Bezout's identity [40]- [42].…”

“…The continuous-time inputs can finally be recovered from the discrete-time sequences using a bank of conventional D/A converters. In this paper, we present sufficient conditions for perfect reconstruction in the discrete-time model with uniform subsampling (see [15] and [16] for necessary density conditions on arbitrary nonuniform sampling) and conditions and a solution to the related filter design problem. We will consider only uniform subsampling of the channel outputs.…”