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

Deijany Rodriguez Linares, Oksana Moryakova, Håkan Johansson

TL;DR

The paper addresses efficient real-time computation of time-indexed weighted sums $S = \sum_{n=0}^{N-1} n^{K} v[n]$ by introducing a cascade of $K+1$ accumulators. It proves that $S$ can be expressed as $S = \sum_{k=1}^{K+1} c_k A_k[N-1]$ with $A_k[N-1]$ the accumulators' outputs and closed-form coefficients $c_k = \sum_{j=0}^{k-1} (-1)^j \binom{k-1}{j} (N + j)^K$, enabling only $K+1$ multiplications after accumulation. This yields a real-time, storage-light solution that avoids forming $n^K$ or buffering the input, reducing multiplicative complexity from $M\times N$ to a constant $K+1$ multiplications. Hardware analysis shows substantial resource savings and feasibility for streaming, low-power DSP systems, particularly when $N$ is large or memory is constrained.

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 $K{\times}N$ 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.

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

TL;DR

The paper addresses efficient real-time computation of time-indexed weighted sums by introducing a cascade of accumulators. It proves that can be expressed as with the accumulators' outputs and closed-form coefficients , enabling only multiplications after accumulation. This yields a real-time, storage-light solution that avoids forming or buffering the input, reducing multiplicative complexity from to a constant multiplications. Hardware analysis shows substantial resource savings and feasibility for streaming, low-power DSP systems, particularly when is large or memory is constrained.

Abstract

This letter presents a novel approach for \mbox{efficiently} computing time-index powered weighted sums of the form using cascaded accumulators. Traditional direct computation requires general multiplications, which become prohibitive for large , 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 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.

Paper Structure

This paper contains 7 sections, 1 theorem, 19 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Cascaded Accumulator's Output
  3. Proposed Computation of Time-Index Powered Weighted Sums via Cascaded Accumulators
  4. Derivation of $\{c_k\}$ in Closed Form via Stirling Numbers
  5. Complexity Analysis
  6. Hardware Cost
  7. Conclusion

Key Result

Theorem 1

Let $K \in \mathbb{Z}_{\ge 0}$ and $N \ge 1$. Then the representation in eq:c_kA_k is equivalent to and the coefficients $\{c_k\}$ exist and are unique provided that $N \ge K{+}1$.

Figures (3)

  • Figure 1: $K{+}1$ cascaded accumulators.
  • Figure 2: Structure for computing time-index weighted sums via $K{+}1$ cascaded accumulators ($\mathrm{ACC}$).
  • Figure 3: Arithmetic complexity. Proposed (P) against baseline (B) [optimal addition-chain exponentiation, see Footnote \ref{['foo:chain']}]. Vertical axis is logarithmic.

Theorems & Definitions (4)

  • Theorem 1: Equivalence and Uniqueness
  • proof
  • Remark 1
  • Remark 2