This letter presents a novel approach for \mbox{efficiently} computing time-index powered weighted sums of the form $\sum_{n=0}^{N-1} n^{K} v[n]$ using cascaded accumulators. Traditional direct computation requires $K{\times}N$ general multiplications, which become prohibitive for large $N$, while alternative strategies based on lookup tables or signal reversal require storing entire data blocks. By exploiting accumulator properties, the proposed method eliminates the need for such storage and reduces the multiplicative cost to only $K{+}1$ constant multiplications, enabling efficient real-time implementation. The approach is particularly useful when such sums need to be efficiently computed in sample-by-sample processing systems.

翻译：本文提出一种新颖方法，利用级联累加器高效计算形如$\sum_{n=0}^{N-1} n^{K} v[n]$的时间指数加权和。传统直接计算方法需要$K{\times}N$次通用乘法运算，当$N$较大时计算量将变得不可接受；而基于查找表或信号反转的替代策略需要存储整个数据块。通过利用累加器的特性，所提方法消除了此类存储需求，并将乘法运算成本降至仅需$K{+}1$次常数乘法，从而实现了高效的实时计算。该方法在需要逐样本处理的系统中进行此类求和计算时具有显著优势。