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Frequency, Resolution, and Power Consumption

To measure its own clock frequency, a digital system would typically feed its clock to a counter and periodically compare its output with that of another counter clocked by an accurate reference. The digital nature of the counting process, however, limits the resolution of the measurement due to temporal quantization. Thus, to obtain a high resolution frequency measurement, one must either use a high nominal frequency or sample the counter over long intervals.

More formally, the resolution error $\epsilon$ of a frequency measurement is bounded by

\begin{displaymath}\epsilon \geq \frac{1}{F_0\cdot \tau},\end{displaymath}

where, $F_0$ is the nominal frequency of the clock and $\tau$ is the interval over which the counter is sampled. In other words, if a system wants to know the frequency error with a resolution of $\epsilon_0$, then it needs to wait for at least

\begin{displaymath}\tau \geq \frac{1}{F_0 \cdot \epsilon_0}\end{displaymath}

seconds. Numerically, a nominal clock frequency of 1MHz estimated to a resolution of 0.04ppm will require at least 25 seconds of contiguous counting. Power consumption is also related to nominal frequency through the linear relationship:
P = P_0 + (C \cdot V^2\cdot F_0),
\end{displaymath} (1)

where $P_0$ is the power lost due to leakage, $C$ is the effective load capacitance and $V$ is the driving voltage. Further, as energy consumed is $P\cdot\tau$, there exists a strict tradeoff between the energy consumption and the achievable timing resolution. Or in other words, if you lower the nominal frequency, you lower the power consumption, but at the same time decrease the achievable time resolution.

Thomas Schmid 2008-11-14