A Markov Chain Sequence Generator for Power Macromodeling

Abstract

In this paper, we present a novel sequence generator based on a Markov chain model. Specifically, we formulate the problem of generating a sequence of vectors with given average input probability p, average transition density d, and spatial correlation s as a transition matrix computation problem, in which the matrix elements are subject to constraints derived from the specified statistics. We also give a practical heuristic that computes such a matrix and generates a sequence of l n-bit vectors in O nlDerived from a strongly mixing Markov chain, our generator yields binary vector sequences with accurate statistics, high uniformity, and high randomness. Experimental results show that our sequence generator can cover more than 99% of the parameter space. Sequences of 2,000 48-bit vectors are generated in less than 0.05 seconds, with average deviations of the signal statistics p, d, and s equal to 1.6%, 1.8%, and 2.8%, respectively.Our generator enables the detailed study of power macromodeling. Using our tool and the ISCAS-85 benchmark circuits, we have assessed the sensitivity of power dissipation to the three input statistics p, d, and s. Our investigation reveals that power is most sensitive to transition density, while only occasionally exhibiting high sensitivity to signal probability and spatial correlation. Our experiments also show that input signal imbalance can cause estimation errors as high as 100% in extreme cases, although errors are usually within 25%.

A design automation and power estimation flow for RFID systems

“…A power library of preprofiled hardware functional units is created for each target fabrication process. To accomplish this, three parameters of input signals can be used to accurately estimate power dissipated in digital circuits as suggested by Gupta and Najm [1997], Chen and Roy [1998], and Liu and Papaefthymiou [2004]. They are: average input signal probability, p, average transition density, d, and spatial correlation, s. Transition density represents the frequency of bit changes between two or more values in sequence.…”

A multi-model power estimation engine for accuracy optimization

“…For instance, since the macromodel in [7] assumes that signal statistics are uniformely distributed among inputs and outputs, input signal imbalances may result in significant errors as pointed out in [12]. Furthermore, the macromodel proposed in [1] restricts the training set to a single input stream per characterized point.…”

“…We adopted the procedure described in [12], which allows input stream generation with high-accurate signal probability, transition density and spatial correlation. As suggested in [12], we set each stream to have 2000 vectors. The adopted input-space range discretization was (0.00, 0.05, 0.15, ..., 0.95, 1.0) for both P in and D in .…”

“…This paper claims that one of the main reasons impairing its widespread use is that each macromodeling technique makes assumptions that lead to some sort of intrinsic limitation, thereby affecting its accuracy. In [7], for instance, signal statistics are assumed as uniformly distributed among all inputs and outputs, although input signal imbalances may result in significant errors [12]; the training set is limited to a single input stream per characterized point in [1], even though the choice of a proper set is crucial to obtain high-quality models [16].…”

Methodology for high level estimation of FPGA power consumption

“…Power consumption depends on the input/switching activity and the capacitance. Hence, there have been several approaches, collectively termed Power Macromodeling, that map the power consumption of a circuit to capacitance, input sequence and certain statistics such as, the transition probability of the input sequence [12,13,6,9]. In [17] and [10], the authors used silicon measurements to analyze the power consumption of XC4003 TM and Virtex-II TM FPGAs, respectively.…”