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# Energy-Delay Complexity of Asynchronous Circuits

Penzes, Paul (2002) Energy-Delay Complexity of Asynchronous Circuits. California Institute of Technology . (Unpublished) http://resolver.caltech.edu/CaltechCSTR:2002.010

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## Abstract

In this thesis, a {\it circuit-level theory of energy-delay complexity} is developed for asynchronous circuits. The energy-delay efficiency of a circuit is characterized using the metric $Et^n$, where $E$ is the energy consumed by the computation, $t$ is the delay of the computation, and $n$ is a positive number that reflects a chosen trade-off between energy and delay. Based on theoretical and experimental evidence, it is argued that for a circuit optimized for minimal $Et^n$, the consumed energy is independent, in first approximation, of the types of gates (NAND, NOR, etc .) used by the circuit and is solely dependent on $n$ and the total amount of wiring capacitance switched during computation. Conversely, the circuit speed is independent, in first approximation, of the wiring capacitance and depends only on $n$ and the types of gates used. The complexity model allows us to compare the energy-delay efficiency of two circuits implementing the same computation. On the other hand, the complexity model itself does not say much about the actual transistor sizes that achieve the optimum. For this reason, the problem of transistor sizing of circuits optimize d for $Et^n$ is investigated, as well. A set of analytical formulas that closely approximate the optimal transistor sizes are explored. An efficient iteration procedure that can further improve the original analytical solution is then studied. Based on these results, a novel transistor-sizing algorithm for energy-delay efficiency is introduced. It is shown that the $Et^n$ metric for the energy-delay efficiency index $n \ge 0$ characterizes {\it any} optimal trade-off between the energy and the delay of a computation. For example, any problem of minimizing the energy of a system for a given target delay can be restated as minimizing $Et^n$ for a certain $n$. The notion of {\it minimum-energy function} is developed and applied to the parallel and sequential composition of circuits in general and, in particular, to circuits optimized through transistor sizing and voltage scaling. Bounds on the energy and delay of the optimized circuits are computed, and necessary and sufficient conditions are given under which these bounds are reached. Necessary and sufficient conditions are also given under which components of a design can be optimized independently so as to yield a global optimum when composed. Through these applications, the utility of the minimum-energy function is demonstrated. The use of this minimum-energy function yields practical insight into ways of improving the overall energy-delay efficiency of circuits.

Item Type: Report or Paper (Technical Report) Computer Science Technical Reports energy, delay, asynchronous circuits CaltechCSTR:2002.010 http://resolver.caltech.edu/CaltechCSTR:2002.010 You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format. 27034 CaltechCSTR Imported from CaltechCSTR 26 Sep 2002 26 Dec 2012 14:13

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