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An increasingly important figure-of-merit of a VLSI system is "power awareness," which is its ability to scale power consumption in response to changing operating conditions. These changes might be brought about by the time-varying nature of inputs, desired output quality, or just environmental conditions. Regardless of whether they were engineered for being power aware, systems display variations in power consumption as conditions change. This implies, by the definition above, that all systems are naturally power aware to some extent. However, one would expect that some systems are "more" power aware than others. Equivalently, we should be able to re-engineer systems to increase their power awareness. In this paper, we attempt to quantitatively define power awareness and how such awareness can be enhanced using a systematic technique. We illustrate this technique by applying it to VLSI systems at several levels of the system hierarchy - multipliers, register files, digital filters, dynamic voltage-scaled processors, and data-gathering wireless networks. It is seen that, as a result, the power awareness of these preceding systems can be significantly enhanced leading to increases in battery lifetimes in the range of 60-200%.