VLSI Architectures of Approximate Arithmetic Units Applied to Parallel Sensors Calibration
Morgana Macedo Azevedo da Rosa, Patrícia da Costa, Eduardo Costa, Rafael Soares, Sérgio Bampi
Abstract
Approximate computing maximizes area and energy savings for a trade-off between quality and efficiency. Approximate arithmetic operators have emerged as an efficient alternative to design low-power VLSI circuits. This paper investigates the design of approximate arithmetic operator units used in the calibration procedure for radio astronomy light sensors — the so-called StEFCal (statistically efficient and fast calibration) method. The StEFCal algorithm comprises arithmetic operations like a divider, square-accumulate (SAC), and multiply-accumulate (MAC) units. The StEFCal circuit of this work explores the following arithmetic operators: i) two approximate squarer units from the literature, i.e., radix-4 (AxRSU) and SquASH, ii) two approximate iterative-based Newton-Raphson (NR) and Goldschmidt (GLD) dividers, iii) one approximate parallel prefix adder (AxPPA), and iv) a new approximate radix-4 multiplier (AxRMU), proposed in this work, explored in the StEFCal multiply-accumulate circuit design. The AxRSU utilizes the parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K1$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K2$ </tex-math></inline-formula> to represent the number of exact encoders for squarer- and conventional-partial products, respectively, subsequently replaced with approximate encoders. The same principle applies to AxRMU, where the parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> indicates the number of exact encoders for conventional-partial products, subsequently exchanged with approximate encoders. We demonstrate the efficiency of StEFCal using the approximate arithmetic operators from the Pareto-optimal front that expresses the area- and power-quality trade-off. The results show that using the AxRSU with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K1=4$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K2=6$ </tex-math></inline-formula> , AxRMU, and AxPPA with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K=16$ </tex-math></inline-formula> and NR with one iteration has an MSE equal to 89.98dB and offers up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$158\times $ </tex-math></inline-formula> energy-savings compared to the exact StEFCal, and up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$25\times $ </tex-math></inline-formula> more energy-savings and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3.33\times $ </tex-math></inline-formula> area-savings compared with our previous work, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$440\times $ </tex-math></inline-formula> energy-savings compared to the accurate state-of-the-art, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$258\times $ </tex-math></inline-formula> compared with the approximate state-of-the-art.