We propose a numerically efficient method for evaluating the random-coding union bound with parameter $s$ on the error probability achievable in the finite-blocklength regime by a pilot-assisted transmission scheme employing Gaussian codebooks and operating over a memoryless block-fading channel. Our method relies on the saddlepoint approximation, which, differently from previous results reported for similar scenarios, is performed with respect to the number of fading blocks (a.k.a. diversity branches) spanned by each codeword, instead of the number of channel uses per block. This different approach avoids a costly numerical averaging of the error probability over the realizations of the fading process and of its pilot-based estimate at the receiver and results in a significant reduction of the number of channel realizations required to estimate the error probability accurately. Our numerical experiments for both single-antenna communication links and massive multiple-input multiple-output (MIMO) networks show that, when two or more diversity branches are available, the error probability can be estimated accurately with the saddlepoint approximation with respect to the number of fading blocks using a numerical method that requires about two orders of magnitude fewer Monte-Carlo samples than with the saddlepoint approximation with respect to the number of channel uses per block.

翻译：我们提出了一种数值高效的方法，用于评估采用高斯码本且在无记忆块衰落信道上运行的导频辅助传输方案在有限码长机制下的错误概率，该错误概率由参数为$s$的随机编码联合界给出。我们的方法基于鞍点近似，与先前针对类似场景报道的结果不同，该方法针对每个码字跨越的衰落块数（又称分集支路）进行鞍点近似，而非每块信道使用次数。这种不同的方法避免了在衰落过程及其基于导频的接收端估计的多种实现上对错误概率进行高成本的数值平均，从而显著减少了准确估计错误概率所需的信道实现数量。我们对单天线通信链路和大规模多输入多输出（MIMO）网络的数值实验表明，当存在两个或更多分集支路时，通过采用针对衰落块数的鞍点近似的数值方法，可以准确估计错误概率，且所需的蒙特卡洛样本量比采用针对每块信道使用次数的鞍点近似的方法少大约两个数量级。