What are we doing here?
This second post comes as more in-depth and practical answer to the very simple question:
What is this sensor’s (instrument, setup) resolution? Or, more practically, how can I determine/identify my sensors’ instrumental noise.
After all, forgetting the principles and keeping to the basics, rather than making theoretical estimations, we would like to directly measure the “noise” added by the actual instrument itself (and not by relying on some dummy transducer as we did previously). Indeed, indirect measurements has the following downsides: i/ it is still to prone to extrapolation error and ii/ it is not always possible (typically, for situations with integrated electronics, i.e. when transducer and electronics cannot be separated – a very common situation nowadays). As seen in part 1, direct measurement at least in principle, can be very simply performed by either one of the following two methods:
- corrected difference.
- spectral correlation correlation.
Sadly enough, both methods (‘two channels methods’) suffer from the same limitations. Namely, they work under the hypothesis that the pair of particular sensors used for the characterization consists of two units with perfectly identical self-noise and frequency responses (in other words, the sensors are not required to be perfect i.e. have no deviation but the sensor-to-sensor deviation has to be sufficiently small). More precisely, this implies that the instrumental noise to be measured is larger than the actual signal magnitude multiplied by the module of mismatch between frequency responses. For sensors with high-sensitivity and low noise, this pre-requisite can be very difficult to achieve.
Hence, we will be looking for an approach not requiring such ideal conditions.
Three-sensors method
Using appropriate signal processing, it is possible to isolate the signal and the noise, even in the general case of sensors with different frequency reponse functions. As far as I know, the idea dates back to the old Testament, namely the book by Bendat and Piersol ([1] originally from 1971, see § 6.2.3).
For those that are curious, I will outline the reasoning : let’s assume we have a series of co-located sensors, all measuring the same physical signal (mesurand X) .
The series of output signals ((1,..i ..n) have Fourier transform Yi related to the signal X, individual frequency response Hi and instrumental noise Ni by :
(1) 
Assuming that the mesurand and the instrument noise are un-correlated, the cross PSD between the i-th and j-th sensors reads :
(2) 
This is worth reiterating : using noisy sensors, we can still get noise-freea ( in the sense of unbiased ) estimates of the cross spectral density between sensors. Conversely, only the direct (auto) spectral density will remain contaminated by the instrumental noise (in the sense that averaging will not eliminate its contribution). This is the central idea here : avoiding using direct PSD’s.
Using the previous equation for three sensors (i ,j,k), the frequency response ratio can be found :
(3) 
(i.e. since the cross-spectral density is unbiased, the cross-spectral density ratio is also unbiased).
The last step involves the estimation of the direct to cross-spectral density ratio estimation:
(4) 
In this expression, all quantities are known except for the self-noise Nii, so that we finally arrive at the desired formula :
(5) 