With this post, I just want to compile a future lecture on digital-to-analog converters and the fact that we have a trade-off between speed, resolution and power consumption. What does this trade-off look like in it’s simplest form?
You have a digital-to-analog converter (DAC) with bits of resolution. The sample (or perhaps more correctly, updating) frequency is . The (maximum) output voltage swing is 0 to V. It is a current-steering dumping its output current over a resistance, typically some 50 Ohm or a derivative from that.
Say that we want to produce a Nyquist converter, and guarantee same bandwidth as the Nyquist range. This implies that the (-3-dB) bandwidth is half the sample frequency.
Now, given the load, the total thermal noise power (over 1 Ohm) from the resistive load is typically given by the
noise, with being the total load capacitance. ( at room temperature.) The total noise power dissipated in the resistive load is
(Yes, the resistance is there again now, don’t always use that kT-over-C-noise thing – depends what you want to compare with…) where, due to the above stated requirements,
which implies the “simpler” expression, and arguably more straight-forward, of the total noise power.
(Notice the single-sided integration of the noise density.)
Signal vs Noise
The maximum signal power that can be delivered to the output is obtained with a full-scale sinusoid. The power becomes
Further on, a break-point in terms of noise is for that case when the thermal noise power equals the quantization noise power.
We will combine some equations and we land at
where we can see that disappeared again, phew!
Notice now that we have a relationship between Power, Speed, and Resolution. One of those FOMs, we’ve seen quite often before: the faster we operate, the less resolution given a certain signal power. By boosting the signal power (i.e., power consumption) we can move that boundary.
and if we like, we can express this in a logarithmic scale.
Or expressed in dBm (0dBm = 1 mW)
[Caveat: possibly a factor two missing…]
Trying to illustrate this would require some more clever graphs. Instead I’ve chosen to use a spread sheet [click the picture for better resolution] and I have highlighted (red-ish) the cells that require more than 0 dBm of output power to meet the frequency-resolution requirement. For example, a 10-MHz DAC with a 16-bit resolution would be on the safe side of 0 dBm. 17-bits would not. Other color schemes could be added to compare with e.g. WLAN, GSM, GPS, etc., to get a feeling for the levels involved.
Simple maths, but large consequences. So, check your specification…