The joint detection uses Kalman filtering (KF) to estimate the prior probability of control outputs to assist channel decoding. In this paper, we regard the joint detection as maximum a posteriori (MAP) decoding and derive the lower and upper bounds based on the pairwise error probability considering system interference, quantization interval, and weight distribution. We first derive the limiting bounds as the signal-to-noise ratio (SNR) goes to infinity and the system interference goes to zero. Then, we construct an infinite-state Markov chain to describe the consecutive packet losses of the control systems to derive the MAP bounds. Finally, the MAP bounds are approximated as the bounds of the transition probability from the state with no packet loss to the state with consecutive single packet loss. The simulation results show that the MAP performance of $\left(64,16\right)$ polar code and 16-bit CRC coincides with the limiting upper bound as the SNR increases and has $3.0$dB performance gain compared with the normal approximation of the finite block rate at block error rate $10^{-3}$.

翻译：联合检测利用卡尔曼滤波估计控制输出的先验概率以辅助信道解码。本文将联合检测视为最大后验概率解码，并基于考虑系统干扰、量化区间与权重分布的成对错误概率推导其上下界。首先，推导了信噪比趋于无穷且系统干扰趋于零时的极限界。随后，构建无限状态马尔可夫链描述控制系统的连续丢包行为，以推导最大后验概率界。最终，将最大后验概率界近似为从无丢包状态到连续单次丢包状态转移概率的界。仿真结果表明：$\left(64,16\right)$极化码与16位循环冗余校验的最大后验概率性能随信噪比提升趋近极限上界，在误块率$10^{-3}$处较有限块长速率正态近似获得$3.0$dB性能增益。