Coded Kalman Filtering over MIMO Gaussian Channels with Feedback
arxiv(2024)
Abstract
We consider the problem of remotely stabilizing a linear dynamical system. In
this setting, a sensor co-located with the system communicates the system's
state to a controller over a noisy communication channel with feedback. The
objective of the controller (decoder) is to use the channel outputs to estimate
the vector state with finite zero-delay mean squared error (MSE) at the
infinite horizon. It has been shown in [1] that for a vector Gauss-Markov
source and either a single-input multiple-output (SIMO) or a multiple-input
single-output (MISO) channel, linear codes require the minimum capacity to
achieve finite MSE. This paper considers the more general problem of linear
zero-delay joint-source channel coding (JSCC) of a vector-valued source over a
multiple-input multiple-output (MIMO) Gaussian channel with feedback. We study
sufficient and necessary conditions for linear codes to achieve finite MSE. For
sufficiency, we introduce a coding scheme where each unstable source mode is
allocated to a single channel for estimation. Our proof for the necessity of
this scheme relies on a matrix-algebraic conjecture that we prove to be true if
either the source or channel is scalar. We show that linear codes achieve
finite MSE for a scalar source over a MIMO channel if and only if the best
scalar sub-channel can achieve finite MSE. Finally, we provide a new
counter-example demonstrating that linear codes are generally sub-optimal for
coding over MIMO channels.
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