We have designed a PCB containing a data collection circuit for a customer, and their external expert presents this "helpful suggestion":
Question about signal averaging vs. longer integration times: Averaging \$N=20\$ samples provide us with a digital SNR improvement of \$\sqrt{N} = 4.47\$. However, the signal of interest is slowly varying. So, instead we could integrate the signal longer (say, \$N\$ times longer) and reduce the filter bandwidth by \$N\$, resulting in an SNR improvement of \$\sqrt{N}\sqrt{N} = N\$.
Is reducing the sampling rate and corner frequency of the anti-aliasing filter that much more effective than decimation via a digital filter?
Best Answer
No, you can't double-dip like that.
While it is true that a lowpass integrating stage that reduces noise bandwidth by a factor of \$N\$ does improve linear SNR by \$\sqrt{N}\$, you cannot simply cascade two stages to get an SNR improvement of \$N\$. Here's why:
The SNR improvement from averaging \$N\$ consecutive samples is conditional on the characteristics of sampled additive white Gaussian noise, specifically that they are independently distributed. The following relationship holds for the sum of independent Gaussian random variables:
If
$$Y = \sum_i X_i$$
with
$$X_i \sim \mathcal{N}(\mu_i, \sigma_i^2) \quad \forall i$$
then
$$Y \sim \mathcal{N}(\mu_y, \sigma_y^2)$$
where
$$\mu_y = \sum_i \mu_i$$
and
$$\sigma_y^2 = \sum_i \sigma_i^2$$
If the variables are i.i.d, which is to say that \$\mu_i = \mu_0 \; \forall i\$ and \$\sigma_i = \sigma_x \; \forall i\$, then averaging gives
$$\frac{1}{N} \sum_{i=1}^N X_i \sim \mathcal{N}\left(\mu_x,\left(\frac{\sigma_x}{\sqrt{N}}\right)^2\right)$$
As the acquisition system DC offset is properly zeroed, \$\mu_x = 0\$. In this case the noise is reduced by a factor of \$\sqrt{N}\$ by averaging. The same result holds for averaging in the analog domain via integration.
However, this result only applies when the Gaussian random variables are independent (white noise). As soon as a lowpass filter is added in front of the averaging block, the noise contributions become highly correlated, and integration of a signal which is already band-limited results in no additional reduction.