Approximate multipliers for energy-efficient computing

Multiplication represents a ubiquitous arithmetic operation found in various applications. Many error-tolerant applications can significantly benefit in terms of power and area consumption by replacing the exact multiplier with an approximate one. Approximate multipliers offer means for achieving energy-efficient computing in applications that exhibit an inherent tolerance to inaccuracy. The study presented in this thesis proposes several approximate multipliers that exhibit a different trade-off between accuracy and energy consumption. We first present the logarithmic-booth (LOBO) multiplier, which combines the radix-4 Booth encoding and logarithmic product approximation. Radix-4 Booth encoding ensures low approximation error, while logarithmic product approximation enables efficient generation of high-radix partial products. The synthesis results from simulation using TSMC 180 nm cell library reveal that LOBO exhibits lower energy consumption than non-logarithmic approximate multipliers. At the same time, LOBO has the same level of applicability in image processing and classification applications as non-logarithmic multipliers. Compared with approximate logarithmic multipliers, LOBO consumes more energy, while it outperforms them in image processing and image classification. Driven by the achievements of the LOBO multiplier, we propose hybrid radix-4 and the logarithmic multiplier (HRALM). The radix-4 encoding generates higher partial product from the three most significant bits, while logarithmic product approximation produces lower partial product from remaining multiplicand bits. The synthesis results and error assessment show that the proposed multiplier, like LOBO, occupies the gap between approximate non-logarithmic multipliers and logarithmic multipliers. In several image processing algorithms, the proposed multiplier outperforms approximate logarithmic multipliers. Compared to approximate non-logarithmic multipliers, the proposed multipliers delivers similar perceived quality while it consumes less energy. Although the previous multipliers offer a compromise between accurate and efficient design, the non-logarithmic multipliers deliver better results in applications that require high accuracy. A similar accuracy demands hold for sensor data processing with digital IIR filters. We propose a novel non-logarithmic approximate odd radix-4 (AO-RAD4) multiplier, which aims to improve the energy consumption of the A-weighting digital IIR filter - an essential element in noise level measurement. The AO-RAD4 multiplier employs partial product perforation to consume less energy. By carefully placing the proposed multiplier in the A-weighting filter, we can decrease energy consumption by 70\% and achieve a nearly identical frequency response as the exact A-weighting filter. The experiments for sound-level measurement showed that the resulting A-weighting filter could be used for noise measurement without any notable performance degradation. During the LOBO development, we identified that the circuitry for logarithmic conversion represents the main bottleneck in the design of logarithmic multipliers. We present a two-stage operand trimming approximate logarithmic multiplier with an improved design of the logarithmic conversion circuitry. The multiplier trims the least significant parts of input operands in the first stage and the mantissas of the obtained operands' approximations in the second stage. We conducted a thorough evaluation of the multiplier's hardware performance, the error performance and applicability in the image blurring and image classification with convolutional neural networks. The proposed multiplier outperforms state-of-the-art approximate multipliers in terms of area and energy consumption. At the same time, it demonstrates acceptable behaviour in image smoothing and image classification with convolutional neural networks. The different trade-off between the accuracy and energy-efficient design of the proposed multipliers determines their application. Due to their high accuracy, the AO-RAD4 multipliers are suitable for application with high accuracy demands. In contrast to AO-RAD4, the TL multipliers are applicable in the highly error-tolerant application, as they offer small energy consumption with an increased approximation error. Finally, the hybrid multipliers, LOBO and HRALM, deliver a good trade-off between accuracy and energy-efficient design and are suitable for less error-tolerant applications, e.g. image processing.