<p>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.</p>

<p>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.</p>