Linear Coding of LTI Sources Over Vector Gaussian Channels: A Majorization Approach
Information Theory
Summary
The authors study how to design systems that send information about a changing vector (a source) over multiple noisy communication channels while keeping power limits in mind. They look at two scenarios: one where each channel has its own power limit, and another where there is a total power limit shared among all channels. They find criteria for when the system can reliably estimate the source state without errors growing too large, using math related to the source's unstable parts and the channels' quality. The authors also develop methods to optimally allocate power across channels, including a water-filling approach for equal noise cases, and show that the challenge depends not just on the system’s complexity but also on how balanced the unstable parts are. Their results guide how to build encoder-decoder pairs that work well under these constraints.
Authors
Shihao Jin, Junhui Li, Shinji Hara, Wei Chen
Abstract
We study the design of linear time-invariant (LTI) encoder-decoder pairs for transmitting the state of a discrete-time LTI vector source over power-constrained parallel Gaussian channels with feedback. Two types of power constraints are considered. Under individual subchannel power constraints, a necessary and sufficient condition for designing an encoder-decoder pair that achieves bounded estimation error covariance (EEC) is established via two coupled majorization inequalities involving the subchannel signal-to-noise ratios and the antistable poles of the source. Under total channel power constraint, we derive the minimum total power required for a feasible encoder-decoder design by exploiting partial-order progamming under majorization order. An analytical optimal power allocation is obtained for the case of equal noise variances, which admits a water-filling interpretation; for general noise case, a sequential water-filling algorithm is developed. Our results reveal that the difficulty of transmitting a discrete-time LTI source via LTI coding is governed not only by its topological entropy, but also by the evenness of the log-magnitudes of its antistable poles. The design methods for feasible encoder-decoder pairs are also provided.