# Benifit of Delta Sigma DACs

I've researched much about the delta sigma ADC and understand to some extent its workings. The delta sigma modulation in the ADC allows for increasing the ENOB by increasing the SNR through noise shaping the quantization noise out of the band of interest.

However, for a DAC, the quantization noise is already a part of the incoming digital stream. It seems, to me, that the delta sigma modulation scheme would not provide the DAC with any inherent benefits. Yet most of the literature I've looked at have explained the ADC in depth then mention the same can be done with a DAC.

Am I missing something?

A $\Sigma\Delta$-ADC basically works deliberately adding "noise" (not real noise though) to the input signal. By doing so, the ADC can average out signals that are between two quantization levels.

For example, let's say that we have a 0-noise signal at 10.3, then the ADC would measure

10 10 10 10 10 10 10 10 10 10 ...

The ADC would round it down to the nearest quantization level. However, if we were to deliberately add some noise, we might get something like

10 11 10 10 11 10 10 10 11 10 ...

We have on average 3 x 11 and 7 x 10, which yields an average of $10.3$. We managed to capture a signal with a higher resolution, while the core ADC only detects 10's and 11's.

A $\Sigma\Delta$-DAC will do the same thing, though in the other direction. It will deliberately generate a signal with a lot of "noise" - now digitally, such that the average value is what it is supposed to be. For example, if we want to reproduce the 10.3 of earlier, while we only have a DAC that produces only integer values, then we can digitally generate a sequence

11 10 10 10 10 11 11 10 10 10 ...

We finally low-pass filter this signal to get a nice signal. This is also sometimes called dithering. Also note that this is basically a glorified digitally-controlled Pulse-Width Modulator with some better properties.

So the main advantage is that you can turn a low-resolution DAC into a higher resolution one by sacrificing the maximum bandwidth.

• Ah I see, thanks for the numeric explanation! For the filter that follows, I know the ADC implements a digital low pass with a decimation filter to cut off the increasing noise in the unneeded higher frequencies. The DAC doesn’t need this extra step and just does an analog LPF, correct? – Michael E Jul 9 '18 at 7:59
• The "magic" of these DAC's happens in how these seemingly random sequences are generated, which you could arguably also call a "filter". Afterwards, an analog LPF is indeed necessary and sufficient. – Sven B Jul 9 '18 at 8:07