Approximate computing has been extensively employed in arithmetic circuits such as a ripple carry adder (RCA); in an approximate adder, errors are introduced by design. VLSI circuits are also prone to errors caused by external and physical phenomena (such as cosmic rays or a stuck-at); hereafter, these errors are referred to as functional. This paper investigates the combined effects of single functional error (SFE) in an approximate cell and an RCA. The exact and approximate cell designs are considered using a state transition diagram-based analysis to identify relationships between the types of functional error in a cell against the expected behavior for all possible cases. A probabilistic analysis for an exact RCA is proposed; it shows excellent agreement with the simulation results for different metrics such as the error rate (ER). Next, an error analysis is performed on the RCA by considering the number of approximate cells as well as the location of the erroneous cell; the simulation results and modeling analysis of the exact RCA show and prove that the ER and MED (Mean Error Distance) of odd-numbered cases is higher than for evennumbered cases. Moreover, the MED for an approximate RCA in the presence of an SFE is higher than for the exact RCA. Simulation and analytical results show that an approximate RCA in the presence of an SFE can have a serious degradation in accuracy performance if the approximate cell type is not properly selected. In addition, the binary tree-based mathematical analysis and pseudocode are provided to support the exhaustive simulation results for the ER of the RCA using different approximate cells.

Approximate cells can be used to design Ripple Carry Adders (RCAs) for realizing approximate addition in energy-efficient CMOS digital circuits. As inputs of approximate cells could be non-commutative in nature, approximate adders may show different output values under a commutative operation, and this may have a significant effect on the generated sum. This paper presents a detailed analysis of the commutative addition in RCAs made of different approximate cells. Initially, the impact of a non-commutative addition (NCA) to RCAs by approximate cells is assessed by exhaustive simulation at adder level. The results show that at most 17% of additions executed using AFA3 suffer from the non-commutative property, while the values for other adder cells can reach 75%~99%. Then, an extensive analysis using images from a publicly available library is performed by comparing three-image additions with two-image additions. As a further evaluation, the adders are assessed in an image denoising application. As expected, the effect of NCA is especially pronounced for some non-commutative adders, such as AA2 and AMA4. NCA is also cumulative with the number of approximate additions, thereby causing a significant variation in the output image quality. In terms of metrics, the largest average difference in mean error distance (DMED) for three-image addition is 5.3 times higher than for two-image addition. Rankings of the non-commutative approximate adders show that AMA3 and AFA1 based adders are the best schemes with respect to commutative addition; they both also show good performance in image denoising.