Efficient Computation of Time-Index Powered Weighted Sums Using Cascaded Accumulators

2025-09-18

AI Summary

Efficient computation of the time-indexed weighted sum $sum_{n=0}^{N-1} n^K v[n]$ is challenging: direct evaluation requires $O(KN)$ general-purpose multiplications, while lookup-table or block-storage approaches incur prohibitive memory overhead. Method: This paper proposes a recursive algorithm based on a cascade of accumulators, eliminating the need for lookup tables or full data buffering. The algorithm leverages the mathematical structure of accumulator chains to implicitly encode polynomial index weights—$n^K$—within state updates, enabling sample-by-sample streaming computation. Contribution/Results: The method reduces computational complexity to $O(N)$ using only $K+1$ constant multiplications, achieves low latency and minimal hardware resource usage, and supports real-time processing. Experimental results demonstrate substantial performance gains over existing approaches for large $N$, balancing high computational efficiency with hardware amenability.

Technology Category

Machine Learning: Hardware-aware MLSearch and Optimization: Distributed SearchCognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applicationsSearch and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

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 $K{ imes}N$ 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.

Problem

• Efficiently computing time-index powered weighted sums
• Reducing multiplicative cost for large data blocks
• Enabling real-time implementation without storage requirements

Innovation

• Cascaded accumulators eliminate storage needs
• Reduces multiplicative cost to K+1 operations
• Enables efficient real-time sample processing