Combining the rms-signal and rms-noise terms, the ideal ADC SNR in decibels is:
Again, where did the bits go? The ADC vendors enthusiastically explain the missing-bits phenomenon, because they bench-test their devices to see how good the SNR is. Fundamentally, they find that the device noise from resistors and transistors creeps into the results. Vendors test their ADC SNR by inputting their data into the following formula:
These theoretical and tested SNR formulas are complete, but they provide only part of what you need to know about how many bits your converter is truly giving you. THD (total harmonic distortion), another ADC specification you need to watch, is the ratio of the rms sum of the powers of the harmonic components, or spurs, to the input-signal power:
or
where HDx is the magnitude of distortion at the Xth harmonic, PS is the signal power of the first harmonic, and PO is the power of harmonics two through eight. Significant ADC INL (integral-nonlinearity) errors typically appear in the THD results.
Finally, SINAD (signal-to-noise and distortion) is the ratio of the fundamental input signal’s rms amplitude to the rms sum of all other speCTRal components below half of the sampling Frequency, excluding dc. The theoretical minimum for SINAD is equal to the ideal SNR, or 6.02N+1.76 dB, with SAR and pipeline converters. For delta-sigma converters, the ideal SINAD equals 6.02N+1.76 dB+10 log10(fS/(2BW)), where fS is the converter sampling frequency and BW is the maximum bandwidth of interest. The not-so-ideal value of SINAD is 
or
.
where PS is the fundamental signal power, PN is the power of all the noise spectral components, and PD is the power of all the distortion spectral components.
So, the next time you’re looking for lost bits, remember that it is the combination of SNR, THD, and SINAD that gives you the complete picture of the real bits in your ADC—regardless of whether it’s SAR, pipeline, or delta-sigma technology and regardless of the number of bits that the first page of the data sheet mentions.
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