3.4 The $ET^{n}$Metric

Power consumption, delay, throughput and energy consumption are metrics commonly used to compare systems. Considering each of these metrics in isolation does not permit a fair comparison of systems because of the ability of CMOS circuits to trade performance for energy. When multiple criteria need to be optimized simultaneously, it is common to optimize their weighted product. In the case of energy and time, this product may be represented as the metric $M$ for a circuit configuration $C$ such that:

$$M(C)=ET^{n}$$

Here $n$ is a weight that represents the relative importance of the two criteria. The $ET^{n}$ metric was first proposed by Martin, Nystroem and Penzes [64]. Since energy and time can be traded off for each other, consider the infinitesimally small quantity of energy $\Delta E$ that needs to be expended to reduce the time for a computation by an infinitesimally small amount $\Delta T$. Using Newton's binomial expansion and ignoring products and higher powers of $\Delta E$ and $\Delta T$ we get:

$$M(C')=(E+\Delta E)(T-\Delta T)^{n}=ET^{n}-nE\Delta T+T\Delta E$$

If this new operating point is equivalent to the old operating point under the metric $M$:

$$ET^{n}-nE\Delta T+T\Delta E=ET^{n}$$

Rearranging this equation yields:

$$\frac{\Delta E}{E}=\frac{n\Delta T}{T}$$ (3.4)

Intuitively, this means that a small reduction in time is considered $n$ times more valuable than a corresponding reduction in energy. For example, if $n=1$, a 1% reduction in time is considered worth paying a 1% increase in energy. If $n=2$, then it is acceptable to pay for a 1% increase in performance with a 2% increase in energy consumption. In general, when $n=1$, energy and delay are equally important, when $n>1$ performance is valued more than energy and when $0<n<1$ energy savings are considered more important than performance. The case of $n=0$ optimizes just for energy and $n=-1$ optimizes for power. Other negative values of $n$ are not useful for optimization since $E/T^{n}$ changes in opposite directions for improvements in energy and delay.