Approximate multipliers attract a large interest in the scientific literature that proposes several circuits built with approximate 4-2 compressors. Due to the large number of proposed solutions, the designer who wishes to use an approximate 4-2 compressor is faced with the problem of selecting the right topology. In this paper, we present a comprehensive survey and comparison of approximate 4-2 compressors previously proposed in literature. We present also a novel approximate compressor, so that a total of twelve different approximate 4-2 compressors are analyzed. The investigated circuits are employed to design $8\times 8$ and $16\times 16$ multipliers, implemented in 28nm CMOS technology. For each operand size we analyze two multiplier configurations, with different levels of approximations, both signed and unsigned. Our study highlights that there is no unique winning approximate compressor topology since the best solution depends on the required precision, on the signedness of the multiplier and on the considered error metric.

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Comparison and Extension of Approximate 4-2 Compressors for Low-Power Approximate Multipliers

Current research in the area of Neural Networks (NN) has resulted in performance advancements for a variety of complex problems. Especially, embedded system applications rely more and more on the utilization of convolutional NNs to provide services such as image/audio classification and object detection. The core arithmetic computation performed during NN inference is the multiply-accumulate (MAC) operation. In order to meet tighter and tighter throughput constraints, NN accelerators integrate thousands of MAC units resulting in a significant increase in power consumption. Approximate computing is established as a design alternative to improve the efficiency of computing systems by trading computational accuracy for high energy savings. In this work, we bring approximate computing principles and NN inference together by designing NN specific approximate multipliers that feature multiple accuracy levels at run-time. We propose a time-efficient automated framework for mapping the NN weights to the accuracy levels of the approximate reconfigurable accelerator. The proposed weight-oriented approximation mapping is able to satisfy tight accuracy loss thresholds, while significantly reducing energy consumption without any need for intensive NN retraining. Our approach is evaluated against several NNs demonstrating that it delivers high energy savings (17.8% on average) with a minimal loss in inference accuracy (0.5%).

Weight-Oriented Approximation for Energy-Efficient Neural Network Inference Accelerators

This paper proposes a new design methodology to explore the state-of-the-art approximate adders for accelerator architectures conceived in the realm of multiplier-less multiple constant multiplication optimization problem. The proposed methodology is composed of: 1) a search heuristic to seek faster and feasible approximate configurations for the architectures under evaluation; 2) low-power techniques regarding hybrid approximate adders design for accelerators based on trees of shift-and-add operations; 3) high-performance evaluation by exploring parallel prefix adders and low power analysis through the use of the adder optimized by a commercial synthesis tool in the precise part of the approximate adders; and 4) energy efficiency analysis by considering both the approximate techniques and voltage over scaling estimation. Furthermore, improvements are proposed for the state-of-the-art approximate adders under evaluation in this paper. Two case studies are considered to assess the proposed methodology: 1) Gaussian image filter and 2) Sobel operator. The precise and approximate image filters were described in very high-speed integrated circuits hardware description language regarding the proposed methodology. Results are shown after synthesis to a 45-nm standard cell-based technology, where energy reductions ranging from 7.7% up to 73.2% were experienced for multiple levels of quality considering the applications under analysis.

Design Methodology to Explore Hybrid Approximate Adders for Energy-Efficient Image and Video Processing Accelerators

A cross-layer design space exploration (DSE) method based on a proposed co-simulation technique is presented herein. The proposed method is demonstrated evaluating the impacts on both coding efficiency and power dissipation of applying distinct approximate logic operators in a s $\mu {\mathrm{ m}}$ of absolute differences (SAD) kernel that accelerates an H.265/HEVC (high-efficiency video coding) encoder. The proposed method simulates the gate-level circuit dynamically inside the application, with realistic results of the impact of the adder-tree approximate logic implementation on both quality and encoder bit-rate results. A comprehensive DSE is shown herein, with 13 types of 6 classes of approximate adders in the SAD accelerator hardware blocks. Over 3,000 logic variants of approximations at gate-level were developed. Actual video sequences as inputs to the x265 software encoder are co-simulated, to dynamically capture the video motion-estimation (ME) behavior in the presence of logic approximations. While the prior art that only estimates the impact of the approximate logic on power, area, and quality on static designs with statistical assumptions, which are agnostic to the actual algorithm data-dependent behavior in the application, our method explores accurately the trade-off between power dissipation and coding efficiency dynamically over the entire HEVC encoding. Our approach shows that the lower-part-or and error-tolerant adder I approximate adders, as well as truncation-to-zero deliver better compression-power trade-offs, with substantial differences from the static analysis.

A Cross-Layer Gate-Level-to-Application Co-Simulation for Design Space Exploration of Approximate Circuits in HEVC Video Encoders

The approximate computing paradigm emerged as a key alternative for trading off accuracy and energy efficiency. Error-tolerant applications, such as multimedia and signal processing, can process the information with lower-than-standard accuracy at the circuit level while still fulfilling a good and acceptable service quality at the application level. The automatic detection of R-peaks in an electrocardiogram (ECG) signal is the essential step preceding ECG processing and analysis. The Haar discrete wavelet transform (HDWT) is a low-complexity pre-processing filter suitable to detect ECG R-peaks in embedded systems like wearable devices, which are incredibly energy-constrained. This work presents an approximate HDWT hardware architecture for ECG processing at very high energy efficiency. Our best-proposal employing pruning within the approximate HDWT hardware architecture requires just seven additions. The use of a truncation technique to improve energy efficiency is also investigated herein by observing the evolution of the signal-to-noise ratio and the ultimate impact in the ECG peak-detection application. This research finds that our HDWT approximate hardware architecture proposal accepts higher truncation levels than the original HDWT. In summary: Our results show about 9 times energy reduction when combining our HDWT matrix approximation proposal with the pruning and the highest acceptable level of truncation while still maintaining the R-peak detection performance accuracy of 99.68% on average.

Approximate Pruned and Truncated Haar Discrete Wavelet Transform VLSI Hardware for Energy-Efficient ECG Signal Processing

This work explores the 8–2 adder compressor in the addition tree of the Sum of Squared Differences (SSD) architecture. SSD is a distortion metric of the motion estimation in the High-Efficiency Video Coding (HEVC) standard. Distortion metrics are the most time-consuming operations of the encoder. SSD hardware architecture consists of a sum tree that accumulates the calculation of the partial values. The addition tree in the SSD opens an opportunity of exploring efficient addition schemes such as the combinations of efficient adder compressors. The SSD architectures herein presented are compared regarding power dissipation using real video sequences. Our work overcomes the state-of-the-art related work which they investigated different SSD hardware architectures concluding that employing the synthesis tool arithmetic operators are the best choice to reduce the power dissipation. According to our results, we reveal that the SSD employing the 8–2 hierarchical adder compressor combined with a Brent-Kung adder in the final sum saves on average 29.4 % of total power dissipation, when comparing with SSD implemented using the arithmetic operators automatically selected by the synthesis tool.

Exploring Efficient Adder Compressors for Power-Efficient Sum of Squared Differences Design

Multipliers are present in a large variety of applications. However, it is usually responsible for most of the power dissipation. On the other hand, the squared multiplier is a particular case of the general-purpose multiplier, in which both operands are the same, proportioning many architecture optimizations. This paper introduces the radix-2m squared array multiplier architecture. Our architecture proposal for the squared multiplier is the first to reduces the partial products by splitting the operands into mbit groups. Our squared multiplier architecture explores different adder schemes in the multipliers adder tree. As a case study, we demonstrate our squared multiplier proposal for m=2 (radix-4). We investigated the Wallace and Dadda addition trees employing as final carry propagating adder (CPA) the Ripple Carry adder (RCA) and with the adder automatically selected by the synthesis tool. Our best radix-4 squared multiplier proposal employs the Dadda technique and the RCA to implement the adder tree, showing significant energy savings of 20.5%, 56.5%, and 47.4%, for 8, 16, and 32 bits, respectively, compared to the squared multiplier automatically selected by the synthesis tool. Furthermore, our best radix-4 squared multiplier proposal outperforms the Vedic squared multiplier with energy savings in about 21.5%, 71.0%, and 9.0%, respectively, for 8, 16, and 32 bits.

The Radix-2 m Squared Multiplier

This paper proposes an energy-efficient Haar transform architecture using efficient adders circuits. Nine levels of decomposition in a fixed-point format are used in the architectures. The Processing Module (PM) of Haar explores efficient Parallel Prefix Adders (PPA) and adder compressors. Since power-of-two operations approximate one of the inputs of the PM module, efficient adder compressors implement them. It allowed a very high precision adjustment of the transform coefficient. The architectures were described in VHDL and synthesized using the ST 65nm CMOS cell library. The results show that the Haar architecture with 4-2 and 5-2 compressors in the PM module, both with the recombination line with a Sklansky (SK) adder, is more efficient. It has a reduction of 33.6% in the cell area, with a 35.7% reduction in total power in comparison with the literature. Moreover, the Haar architecture with 4-2 and 5-2 adder compressors is more efficient than the one using the tool-selected implementation of the behavioral "+" (sum) in VHDL, with 23% of power reduction.

Morgana M. A. da Rosa

Papers

Design Methodology to Explore Hybrid Approximate Adders for Energy-Efficient Image and Video Processing Accelerators

Approximate Pruned and Truncated Haar Discrete Wavelet Transform VLSI Hardware for Energy-Efficient ECG Signal Processing

Exploring Efficient Adder Compressors for Power-Efficient Sum of Squared Differences Design

The Radix-2 m Squared Multiplier

Energy-Efficient Haar Transform Architectures Using Efficient Addition Schemes