The relationship between bit depth and SNR in ADC

An 8-bit audio signal has a theoretical maximum SNR of 48.16 dB (20log(2^8)).

My question then would be, does this work in reverse, i.e. would an analogue audio (or video) signal with an SNR of <=48 dB only be worth sampling at 8 bits? Or is this purely just a measure of SNR in the derived digital signal after quantisation?

• Pretty much. Sampling a 48dB SNR with 8 bits (48dB) quantisation will give only 45dB SNR (as quantisation noise is not correlated with the signal noise, noise powers add, increasing noise by 3dB) and that 3dB loss may matter. But there would be no gain beyond 9 or 10 bits.
– user16324
Sep 27, 2020 at 23:11
• @BrianDrummond: That sounds like the answer to me! Sep 27, 2020 at 23:59

An 8-bit audio signal has a theoretical maximum SNR of 48.16 dB

On the face of it, an 8-bit digitization process would increase "noise" by 3 dB (as per Brian Drummond's comment). However, when you take into account the natural "crest factor" for audio, the "noise" problem is worse: -

Image from here.

would an analogue audio (or video) signal with an SNR of <=48 dB only be worth sampling at 8 bits?

The picture above is telling you that the main parts of the audio signal will be about 15 dB lower in RMS value compared to the peak value so, if you are listening to a piece of analogue music, the quieter areas will appear to have more noise. But, that regular noise might not be as annoying if you inserted an 8-bit ADC/DAC process in the signal chain. The added quantization noise will be more annoying and you may lose all audio information at very low signal levels.

So, if you listened to the audio from the 8-bit DAC output, it could be significantly worse (as a listening experience) than what you would expect if you analysed the degradation of signals based on peak/maximum levels.

The quieter areas would "appear" heavily distorted and noisy. The ADC/DAC system required to barely degrade the original signal would need to be significantly greater in bit depth than 8-bits, possibly more like 12 bits.

However, if the peak signal was quite loud and a volume control was used to lower the input signal by 20 dB (that's an approximate quartering of perceived loudness) then you would probably want to use a 16 bit ADC/DAC system.

• +1. I didn't feel like tackling peak-mean ratio in a comment. Or that, if correctly dithered, audio information well below the quantisation limit CAN be preserved (albeit noisy) as shown by Decca (Tony Griffiths?) in the early 80s. Nowadays you'd go straight to 16 bit because there's really no cost penalty.
– user16324
Sep 28, 2020 at 11:02
• Interesting, makes a lot of sense. In that case, what would the crest factor be for a typical analogue video signal? My guess would be 3 dB based on the idea that the APL might usually be about 50%. Sep 28, 2020 at 17:41
• You’ll need to look into that but, be aware that the worst manifestation of the problem is with lower level signals. Sep 28, 2020 at 17:43