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May 3, 2023 at 3:16 PM

Measurement discretization

We have already mentioned the problem of the resolution limitation due to the discretization of digital instruments, inherent in their reading; now it is worth pointing out that this is only one aspect of the problem of discretization of measurements. The increasingly massive use of digital electronic instrumentation extends the considerations made with regard to visualization, to “measurement” in the broadest sense.

In fact, beyond the problem of visualization, almost all digital electronic instruments perform an analog-to-digital conversion of signals, and consequently create a relative resolution error. It is important to note that this error is present regardless of the presence or characteristics of the display device adopted: a limiting case is an instrument used to acquire measurements to be stored in files, where there is no real display device, but nevertheless there is a resolution error due to the discretization of the measurand.

In almost all cases, the analog-to-digital conversion works on electrical signals that constitute or represent (following the use of a transducer) the measurand. The conversion is performed by an electronic circuit called ADC (analogue-to-digital converter), whose main characteristic is the size (in bits) of the corresponding digital value; the latter is an indication of the resolution of the conversion: a 10-bit ADC converter is able to encode 1024 different values ($$2^{10}$$) within the measuring range, an 8-bit converter is able to encode 256 values ($$2^8$$).

Similarly, resolution errors are also generated in the case of digital-to-analog conversion: in fact, although in theory it is possible the perfect transformation of a digital signal in the analog equivalent, the very fact that we start from a discrete signal prevents the possibility of generating signals of arbitrary value. The electronic circuit called for this conversion is called DAC (digital-to-analog converter). Three general cases can be envisaged:

1. A/D instruments, instruments that provide a conversion of an analog signal representing the measurand, in order to display and store it in digital format (e.g. digital indicators or measurement acquirers);
2. D/A instruments, instruments that generate an analog signal, starting from the relative command in digital format (as in function generators or calibrators used for the calibration of instrumentation);
3. A/D/A instruments, instruments that perform operations (filtering, amplification, conversion, storage) on analog signals, after converting them into digital format, and then make them available again in analog format (e.g. some signal conditioners or in measuring recorders).

In the absence of precise manufacturer’s instructions, the resolution error can be measured in the laboratory by slowly varying the measurand, and detecting the jump in display due to discretization.