Approximate multipliers for energy-efficient computing

Pilipović, Ratko (author), Bulić, Patricio (mentor)

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.

• Language: English
• Keywords: Approximate computing, Arithmetic circuit design, Booth encoding, Logarithmic multipliers, Multipliers, Power-efficient processing, Truncated multipliers
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2021
• PID: 20.500.12556/RUL-130979
• COBISS.SI-ID: 77085699
• Publication date in RUL: 20.09.2021

Secondary language

• Language: Slovenian
• Title: Energijsko učinkovito računanje s približnimi množilniki

Abstract

Množenje je aritmetična operacija, ki je prisotna v različnih aplikacijah, od obdelave multimedijskih vsebin do algoritmov strojnega učenja. Ker so množilniki velika vezja, lahko z uporabo približnih množilnikov zmanjšamo uporabo energije v mnogih pomembnih aplikacijah, ki so odporne na vneseno računsko napako. V disertaciji predlagamo več približnih množilnikov. Pri načrtovanju približnih množilnikov vedno iščemo ravnovesje med natančnostjo in učinkovito izvedbo v strojni opremi, najpogosteje je to energijska učinkovitost. Najprej predstavimo množilnik LOBO, ki združuje Boothovo radix-4 kodiranje in logaritemski približek produkta. Boothovo radix-4 kodiranje zagotavlja majhno napako približka produkta, medtem ko logaritemski približek produkta omogoča učinkovito generiranje delnih produktov. Pri obdelavi in razvrščanju slik dobimo z množilnikom LOBO podobne rezultate kot z nelogaritemskimi množilniki ob manjši porabi energije. Hkrati pa množilnik LOBO nudi še zadovoljivo obnašanje aplikacije v primerjavi z manj natančnimi vendar energijsko učinkovitejšimi logaritemskimi množilniki. Hibridni pristop združevanja nelogaritemskih in logaritemskih približnih množilnikov smo uporabili tudi pri načrtovanju hibridnega radix-4 in logaritmičnega množilnika HRALM, ki je primeren za obdelavo slik. Predlagani množilnik HRALM, generira le dva delna produkta -- višji delni produkt iz treh najpomembnejših bitov dobi z radix-4 kodiranjem, medtem ko z logaritemskim približkom preostalih bitov tvori nižji delni produkt. Rezultati sinteze in ocena napake kažejo, da, podobno kot množilnik LOBO, zaseda vrzel med približnimi nelogaritemskimi in logaritemskimi množilniki. Množilnik HRALM smo preizkusili v več algoritmih za obdelavo slik. Množilnik HRALM zagotavlja podobno vizualno kakovost slike kot približni nelogaritemski množilniki ob manjši porabi energije. Čeprav hibridni množilniki predstavljajo dober kompromis med natančnostjo in energijsko učinkovitostjo, v aplikacijah z visokimi zahtevami po natančnosti niso uporabni. Takšne zahteve najdemo pri obdelavi signalov s pomočjo filtrov IIR. V delu predlagamo množilnik AO-RAD4, približni množilnik z lihim radix-4 kodiranjem, s katerim izboljšamo porabo energije A-utežnega filtra IIR. Množilnik AO-RAD4 doseže manjšo porabo energije z opuščanjem nekaterih delnih produktov. S pozorno nastavitvijo predlaganega množilnika lahko v A-utežnem filtru IIR ob skoraj enakem frekvenčnem odzivu zmanjšamo porabo energije za 70 \%. Uporabnost filtra smo analizirali v merjenju ravni hrupa v okolici. Poskusi so pokazali, da je predlagani približni A-utežni filter mogoče uporabiti za merjenje hrupa brez opaznega poslabšanja karakteristik. Vezje za logaritemsko pretvorbo predstavlja ozko grlo pri načrtovanju logaritemskih množilnikov in določa učinkovitost celotnega množilnika. Za zmanjšanje tega dela vezja predlagamo približni logaritemski množilnik TL z dvostopenjskim krajšanjem operanda. Izvedli smo temeljito oceno strojne zmogljivosti množilnika, napake in njegove uporabnosti pri glajenju slik in razvrščanju slik s konvolucijskimi nevronskimi mrežami. Predlagani množilnik je zelo majhen vendar kljub večji napaki daje še sprejemljive rezultate. Različne karakteristike predlaganih približnih množilnikov, predvsem natančnost in energetska učinkovitost, močno določajo njihove možnosti uporabe. Množilniki AO-RAD4 so zaradi visoke natančnosti primerni za aplikacije, ki zahtevajo precej natančno računanje, na primer pri filtrih. Popolno nasprotje so množilniki TL, ki so s precejšnjo napako in zelo majhno porabo energije primerni za aplikacije, kot so nevronske mreže, ki so zelo odporne napako. Vmes sta tako po napaki kot po energijski učinkovitosti predlagana hibridna množilnika LOBO in HRALM, ki sta se dobro izkazala pri obdelavi slik.

• Keywords: Aproksimativno računanje, načrtovanje aritmetičnih vezij, Boothovo kodiranje, logaritmični množilniki, množilniki, energijsko učinkovito računanje, približni množilniki