Working wonders with dynamic instruments – Ep 01: Resolution limits

This post comes as an answer to the question I perhaps heard most often when troubleshooting optomechanical devices. When faced with a anything other than an optical sensing device (i.e., an inertial sensor), the optomechanical specialist immediately (and rightfully) wonders:

What is this sensor’s (instrument, setup) resolution?

As it is, this is the perfect example of a question that takes 5 words to ask but requires a relatively lengthy discussion to answer (except for the obvious “it depends” which indeed is only optimal in terms of conciseness). Reading what follows will hopefully provide a satisfying answer for anyone interested in the question.

Inertial Sensors

There are many virtues that make inertia sensors (either velocimeter or accelerometers) attractive. To name only a few, they are compact, relatively economical, have excellent linearity. Importantly, since they are inertia based, they do not require any external reference, making it possible to know exactly which part is moving, because they are delivering a signal corresponding tochanges in position of the attached object (contrary to optical based, or capacitive based sensors etc which deliver a signal correlated to changes in separation length between two objects). Also, they can provide quantities along any direction, and come into a variety of flavor, that is, with or without integrated electronics, the latter being fit for survival in harsh environments, possibly enduring years of operation with high-energy photons exposure.

Now, an immediate question is, how to convert information produced as velocities or acceleration into lengths? There would be integration (single or double) involved, which inevitably poses important limitations:

• Only changes in position are detected, and since the reference (initial value) of position is unknown, so obviously only variation about a time-averaged position can be produced.
• Because of first principles, DC (i.e. constant) components cannot be measured (nor do they need to, unless one wishes to double-check local gravity). The lower limit is dictated by the coil/spring natural frequency (in the case of a velocimeter) or by the crystal leak resistance (in case of a piezoelectric accelerometer).
• Integration involves accumulating signals, but so does errors: hence integrating ‘forever’, i.e. without any high-pass filter, will result in erroneous drifts that will soon render the integrated signals indecipherable. We will come back to this point in a minute.

Resolution and Usable Bandwidth

Now, letting aside for the moment the discussion about the sensor’s usable bandwidth, we need to recognize that inertial sensors have no “threshold” effect, hence their effective resolution will be dictated by their intrinsic noise only. An excellent discussion is made in [1], which I will not repeat here.

For the moment, it suffices to say that noise is by definition a wide-band process, hence before trying to quantify a noise amplitude, one should start by specifying the bandwidth that is of relevance for the problem at hand. Starting with the most simple situation, that is the “white noise of the engineer”, the instrumental noise variance simply scales linearly with the bandwidth:

with:

The approximation assumes small variation of PSD about its mean value (i.e. a “flat” spectrum).

Inertial sensors, have most of their bandwith dominated by thermal noise (i.e. uniform power spectral density), this means a single value is enough to characterize the instrument noise. This, however, is not exactly true at the lower end (where flicker noise dominates, see [2]).

In practice, the RMS amplitude velocity (or acceleration) noise would scale as the bandwidth, and will be independent of the starting or ending frequencies.

Now, this is not the case anymore if a single integration is needed. Namely, for a single integration the variance would read:

Completely opposite to the case without integration, the variance is dominated by the low frequency content. In practical terms, there are orders of magnitude separating lowermost and uppermost frequencies so that only the former is of relevance, and the previous expression can be simplified into

This is even more pronounced when one needs to apply a double integration, in which case the variance due to instrumental noise reads:

where the approximation again only holds if the noise spectrum is flat. By an argument similar to the previous one, this expression can further degenerate into:

Now, clearly, we have a very strong dependency of intrinsic noise on the selected bandwidth, or more accurately on the lower end of the bandwidth. Dividing the lowermost frequency by a factor of 2 will increase the intrinsic noise by at least a factor of 8 (and in practice, a bit more than that as soon as flicker noise comes in).