Measurement, AD conversion, optimal filtering: abstract view from stochastic processes

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 14:58

Summary:

This video describes, in an abstract way, the process of ‘measurement’ when the electronics of a ‘data acquisition card’ or ‘probe’ are connected to the ‘continuous-time world’.

Basically, it will have to absorb energy from the measured variable and, well, if the signal has a certain power spectral density and the thermal/electromagnetic noise has another, then there are optimal continuous-time filters (Kalman/Wiener) that could be implemented... But, of course, that would require a different analog circuit for each process... which was fine with 1960’s technology but, as of today, it is much cheaper to acquire with an off-the-shelf analog to digital conversion circuit, and then perform digital signal processing later on (and the optimal filter will obviously be the Kalman filter in discrete-time, well, with suitable sampled-data considerations to be addressed in forthcoming videos).

From an abstract point of view, if the ‘acquisition time’ of the electronics is ‘fast’ (and indeed it is, at least for an industrial engineering application: you can nowadays sample at mega- or giga-Hertz frequencies), the physical signals could be assumed to be ‘quasi-constant’ so that the optimal filter would simply ‘average’ over those milli- or micro-seconds of acquisition, so that ‘ideally’ the acquisition electronics should perform the finite-time averaging $ŷ:={\int }_0^{T_s}y(\tau )d\tau$; actually, to have an average, approximately not depending on sampling period, we may approximate the measurement process to $ŷ:=\frac{1}{T_s}{\int }_0^{T_s}y(\tau )d\tau$. What might be actually done ‘in practice’ (electronics detail) is not of interest at this point (we are in an abstract setting).

After that acquisition, since the signals will not actually be constant, a digital filter should be incorporated to try to approximate what could be done with the said Wiener/Kalman optimal filters in continuous time.

A concrete numerical example of these ideas is developed in the videos [medex1] and [medex2].

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.