Probabilistic Error Analysis of Approximate Recursive Multipliers

Abstract

Approximate multipliers are gaining importance in energy-efficient computing and require careful error analysis. In this paper, we present the error probability analysis for recursive approximate multipliers with approximate partial products. Since these multipliers are constructed from smaller approximate multiplier building blocks, we propose to derive the error probability in an arbitrary bit-width multiplier from the probabilistic model of the basic building block and the probability distributions of inputs. The analysis is based on common features of recursive multipliers identified by carefully studying the behavioral model of state-of-the-art designs. By building further upon the analysis, Probability Mass Function (PMF) of error is computed by individually considering all possible error cases and their inter-dependencies. We further discuss the generalizations for approximate adder trees, signed multipliers, squarers and constant multipliers. The proposed analysis is validated by applying it to several state-of-the-art approximate multipliers and comparing with corresponding simulation results. The results show that the proposed analysis serves as an effective tool for predicting, evaluating and comparing the accuracy of various multipliers. Results show that for the majority of the recursive multipliers, we get accurate error performance evaluation. We also predict the multipliers’ performance in an image processing application to demonstrate its practical significance.