Can gain calibration be used in PC in the outputs of 2 ADCs?

Question

Some amplifiers don't have any analog differential amplifies (for example the g.USBamp). Instead it deducts 2 separate ADCs (not time multiplexed) in different channels producing both differential amplification and common mode rejection. This is true in the g.USBamp ampflier which has a +- 250mV ADC where no programmable gain amplifiers are used. The method used is instead like this. For separate ADC channels say $A$ and $B$. If you subtract the signal of channel $A$ from the signal of channel $B$, you will automatically cancel any signal $C$ that is common to both: $A = A_{0} + C$, $B = B_{0} + C$, $B - A = B_{0} + C - A_{0} - C = B_{0} - A_{0}$.

The subtraction is usually done in DSP/firmware but for BCI2000 (Brain Computer Interface software). It doesn't use the internal DSP but subtract the 2 ADC/channels at the output using PC software. I'd like to ask the following:

1. Can you use gain calibration in the output using PC software? Assuming no gain calibration is used in the DSP or it still doesn't have the CMRR you need or are satisfied with.

2. If the gain calibration and ADC deduction are in the DSP. But the BCI2000 software doesn't make use of ADC subtraction right in the DSP, but on PC software specifically the Spatial Signal Processing module. Can gain mismatch occur in the physical output paths of ADCs? I know it occurs in the inputs. For example. If you want to have a common mode rejection of, say, 50 dB, the paths need to be gain matched to 0.03 dB. But gain calibration in the DSP (if available) can fix it and increase the CMRR to say 100 dB. But if the ADC/channel subtraction was not used in DSP but in the outputs. Can there be further gain mismatch in the output paths before the signal ends up in the PC?

Answer 1

Let's assume that you have a differential signal $x[n]$ with common mode noise $c[n]$. We can write the two ADC signals as

$$a[n] = x[n] + c[n]\\ b[n] = g (-x[n] + c[n]) $$

where $g$ is the gain mismatch between the two paths (whatever they may include).

Provided that signal is indeed out of phase, sufficiently large, uncorrelated with the noise, and that gain is the only issue here, we can estimate the gain mismatch by minimizing the power of the sum $<(a+b)^2>$ and we get the calibration factor as

$$ \frac{1}{g} \approx -\frac{<a\cdot b>}{<b^2>} $$

Can you use gain calibration in the output using PC software?

Sure. It's just an algorithm and a simple one at that. You can implement it on a the DSP, on a PC (or Mac), in the cloud or even on your phone.

... Can gain mismatch occur in the physical output paths of ADCs? ... Can there be further gain mismatch in the output paths before the signal ends up in the PC?

That would be unusual. Once it's a through the ADC it's juts a set of fixed point numbers and not a physical signal any more. These numbers can be transmitted, moved or copied from one place to another without any loss of information or additional noise. It's possible (but not likely) that the DSP or the drivers do some pre-processing that would prevent proper gain calibration. The only way to know for sure is to read the documentation and data sheets of your specific devices and software modules.