This paper investigates the problem of zero-delay joint source-channel coding of a vector Gauss-Markov source over a multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channel with feedback. In contrast to the classical problem of causal estimation using noisy observations, we examine a system where the source can be encoded before transmission. An encoder, equipped with feedback of past channel outputs, observes the source state and encodes the information in a causal manner as inputs to the channel while adhering to a power constraint. The objective of the code is to estimate the source state with minimum mean square error at the infinite horizon. This work shows a fundamental theorem for two scenarios: for the transmission of an unstable vector Gauss-Markov source over either a multiple-input single-output (MISO) or a single-input multiple-output (SIMO) AWGN channel, finite estimation error is achievable if and only if the sum of logs of the unstable eigenvalues of the state gain matrix is less than the Shannon channel capacity. We prove these results by showing an optimal linear innovations encoder that can be applied to sources and channels of any dimension and analyzing it together with the corresponding Kalman filter decoder.

翻译：本文研究了向量高斯-马尔可夫源在多输入多输出加性高斯白噪声反馈信道上的零延迟联合信源信道编码问题。与利用噪声观测进行因果估计的经典问题不同，我们考察了信源在传输前可进行编码的系统。编码器利用历史信道输出的反馈，以因果方式观测信源状态并编码为信道输入，同时满足功率约束。编码目标是在无限时域内以最小均方误差估计信源状态。本文针对两种场景证明了基本定理：对于不稳定向量高斯-马尔可夫源在多输入单输出或单输入多输出加性高斯白噪声信道上的传输，当且仅当状态增益矩阵不稳定特征值的对数之和小于香农信道容量时，可达到有限估计误差。我们通过证明一种可适用于任意维度信源与信道的最优线性新息编码器，并结合相应卡尔曼滤波器解码器进行分析，推导出这些结果。