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Full Scale gain error is the deviation of the actual measure transfer function from an ideal transfer function of ADC or DAC.
The transfer function of an ADC is a plot of the voltage input to the ADC versus the code's output by the ADC. Such a plot is not continuous but is a plot of 2N codes, where N is the ADC's resolution in bits. If you were to connect the codes by lines (usually at code-transition boundaries), the ideal transfer function would plot a straight line. However, on measurement, we get a line not intersecting the ideal straight line. This is due to the various offsets of ADC.
Offset error, often called 'zero-scale' error, indicates how well the actual transfer function matches the ideal transfer function at a single point. For an ideal data converter, the first transition occurs at 0.5LSB above zero. For an ADC, the zero-scale voltage is applied to the analog input and is increased until the first transition occurs. For a DAC, offset error is the analog output response to an input code of all zeros.
Yes, Gain Error = Full Scale Error & Offset Error = Zero Scale Error
When you actually measure the full scale error and zero scale error, the offset (if any) automatically gets added. However, you can compensate the offset by various techniques, one among them is in the firmware.
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