Abstract:

This paper introduces two statistical delay variability models for certain hardware adder implementations, namely, the ripple-carry adder (RCA) and the borrow-save adder (BSA). The introduced models take into account correlated variation sources. Initially, we derive a first proposed model, namely, Type-I model, in the form of expressions for the computation of the exact Probability Density Functions (PDFS) of maximum output delays for Gaussian and non-Gaussian variation sources. Furthermore, we present closed formulas for the co-variances between output delays of the aforementioned adder architectures. The introduced derived co-variances are subsequently combined with Clark’s method to derive a second proposed model, Type-II model, which comprises approximations of the maximum delay PDF for an RCA and a BSA. Simulation results and the derived exact Type-I PDFs are found to perfectly agree, while the proposed Clark-based Type-II models present an error for standard deviation of maximum delay that increases as BSA word length increases. Both the introduced models and the simulations prove that BSAs achieve narrower delay distributions than RCAs, i.e., they significantly reduce delay variance. Consequently, BSAs are proven to be suitable for variation-tolerant applications by providing a timing safety margin, when compared to RCA architectures. The underlying analysis indicates that for the case of BSA and either intra-die delay variations only or both intra and inter-die delay variations, the Type-II models introduce non negligible errors, which are as much as 16% of the standard deviation of maximum delay for a 256-digit BSA, as the Type II Gaussian PDF approximations deviate significantly from the exact Type-I PDFs. However, for all RCA and BSA inter-die only variation cases, both types present satisfactory accuracy due to the Gaussian shape of exact PDF.