I wanted to go back to the previous post where we investigated the trade-offs between speed, resolution and power consumption in a digital-to-analog converter (DAC). That gave us a bound in that triangle.
What if we use another starting point in our argument rather than noise?
Same type of DAC
Consider the DAC in the figure. Same type of DAC as last time. We have a current through a resistor forming the voltage. What is the minimum possible quanta through the resistor? Leading question … Assume the least significant bit is determined by one single electron during the sampling period.
The average current for a single electron is given by
where is the sample frequency and aC is the elementary charge of the electron. (Let us ignore quantum effects and those things. Let us be traditional and assume we know where the electron is, before we open the box with the cat, …)
So, is this a small or a large value?
Well, assume a resistive load of . The (average) voltage for that particular charge, corresponding to one least significant bit, would be
Assume 100 Ohms in the load, and a sample frequency of 600 MHz. We get
which is 10 nV. This would correspond to something like:
- In a 16-bit converter, this would mean that the peak voltage is some 655 uV.
- In a 20-bit converter, sampled at 1.2 GHz, the peak voltage is some 21 mV.
The voltages can simply not be less than that for the given sample frequencies and resolutions. Otherwise we have to split the electron (buying Swedes a cake).
I think it is actuall rather interesting: the faster you sample, the less electrons will be there for each least significant bit (LSB).
With a full-scale sinusoid signal in place, we can find the average power as
which is then the absolute minimum possible power that must be consumed to obtain a certain resolution.
Are we having fun yet?
Just for the fun of it, let us rewrite the formula a bit
where we have assumed that we also have to guarantee the bandwidth, not just the sample frequency and voltage levels.
In the equation, we can identify the 26-mV term (q/kT), a noise power kT/C, and some constants. Possibly, it could be related to the previous post. Philosophically (?) one could also think what the thermal noise looks like when we push single electrons back and forth.
How fast do we need to sample over a 100-Ohm load to get a 1-V drop with a single electron? (Once again, from a mathematical point of view, and possibly not the correct physical description of the scenario).
Ok, so 60 PHz is rather fast… if we would correlate with light, it would end up in the UV domain, 100 times less than visible light. And now we kind of entering the photoelectric effects, sort of …