Accurate MMSE Expressions for Short-Packet Pilot-Based Channel Estimation

Motivated by increasing the packet efficiency of pilot-based wireless communication, we derive accurate expressions of the channel-estimation minimal mean square error (MMSE) for finite blocklength communication over continuous fading. We used the time vectorial representation and quantified the dependence of the MMSE expression on various system parameters of interest (e.g., packet size, fading rate, signal-to-noise ratio). It is shown that using the conventional infinite-blocklength expression substantially undervalues the channel estimation MMSE values – emphasizing the need of the new accurate expression. We provided both a closed-form expression of the MMSE, for Gauss-Markov fading model, and an associated approximation using Kac-Murdock-Szego (KMS) matrix inversion.