How Well Can We Know A MIMO Channel ? 1 Introduction

Wireless channels are notorious for their spatiotemporal variation – so much so, that the usual recourse is to assume stochastic models [1–3] and devise signaling methods which work well under uncertainty [4–6]. This state of affairs is especially frustrating for multiple input multiple output (MIMO) channels because of the large potential gains possible were only the channel well known [4, 7–11]. Furthermore, MIMO channels can be exploited for better inter-user isolation in multi-user systems [7, 12, 13] and even, potentially, perfect wireless secrecy [6] – a seeming oxymoron. Precise channel knowledge is essential for such applications. It is these tantalizing benefits which prompt us to ask the basic question: how well can we know a MIMO channel? The underlying physics of wireless channel composition is what makes these sorts of questions so interesting – stochastic variation of wireless channels is caused by the stochastic movement of scatterers and transceivers. Scatterers could be vehicles, buildings, lampposts, trees, animals, people, sometimes even the atmosphere – anything that interacts with radio waves and moves. And it is exactly this dependence of the channel on the motion of macroscopic objects which suggests that knowing the wireless channel and tracking its changes might not be the seeming fool’s errand which has historically caused wireless theorists to collectively throw up our hands and invoke probability theory. Specifically, the motion of the macroscopic objects which cause scattering is constrained by the energy available to move them. In particular, we can show that unpredictable motion is limited owing to the energy necessary to change the momentum of an object. Thus, a skyscraper may sway, a vehicle may speed and a person may shift position, but all will do so in a relatively predictable manner – which implies that the information rate necessary to specify their positions could be relatively small. The information and information rates which are necessary to specify relevant constituent parts of the channel constitute exactly an upper bound on the amount of information that must be extracted from measurements to specify the channel. Thus, by first deriving bounds on the information necessary to specify and track relevant channel constituents and then comparing them to the amount of information that can be extracted from channel interrogation, we can determine precisely whether knowing a fading MIMO channel well is possible or impossible. We suspect that because constituents of most scattering environments are relatively massive objects and energy to move them stochastically is limited, the information rates necessary to track changes and disseminate them appropriately could be manageable. Establishing exactly when the channel can be known with an eye toward the benefits which can accrue from such knowledge is our aim. To do so, we must first quantify the effects of inaccurate channel knowledge and establish just how well the channel must be known in order to realize the full benefits of MIMO systems. We will explore these issues somewhat anecdotally in this proposal to obtain a feel for the necessary precision, but our ultimate goal is to determine how we may specify the channel state in bits – essentially a quantization problem. We must then consider the modeling of scattering channels at a physical level as an assortment of spatially distributed scattering objects (perhaps grouped into “scattering centers” [14]) whose ensemble properties and evolution we wish to track. We will argue that energy considerations constrain the stochastic motion of scattering center constituents as well as transceivers