Simulating quantization of ADC using device SNR

Question

Suppose I have ADC microphone with a given SNR in the sense that is related to its quantization. From this tutorial we have the following connection how bit-depth determines SNR, Eq. (9): $$ \mathrm{SNR} = 6.02 \times N + 1.76 \ \mathrm{dB} $$ where $N$ is the bit-depth. I would like to invert this relation in Python. Generate a float32 signal and quantize it as if it was recorded by an ADC microphone with the given SNR.

My original Python code was:

bits = int((snr_mic - 1.76)/6.02)
quant = 2**bits
out_audio = np.floor(audio * quant).astype(int)/quant

Later, I modified the code to this:

bits = np.floor((snr_mic - 1.76)/6.02) + 1
quant = 2**bits
out_audio = (audio * quant).astype(int)/quant

Since it enforces a signal simulated to be recorded with 0 dB SNR mic to be all 0's. However, even signals simulated to be recorded with 5 dB of SNR mic was all 0's. So I probably have bug in my code, probably related to the fine difference between floor and int for negative numbers.

What is the correct way to do it and have nonzero quantized signals for a given nonzero SNR mic?

Answer 1

5 dB is not a sensible value for a formula whose minimum result would be 7.78 dB for $N=1$. So, your test case is "broken", and correctly, you get a 0-bit quantization (i.e., no values at all).

Even for higher SNRs: This sounds pretty reasonable!

Your audio will rarely come close to maximum amplitude. Think about this: if you only had, say, two bits, then that's 1 bit for the sign (think your audio as shifted to be centered around 0 if it helps), and 1 bit leading to two amplitude steps. If your audio recording always keeps more than 6 dB headroom (to avoid clipping!), then that quantization leads to all-zero.

Audio ADCs are pretty good, and I'd have no problem calling anything with a QSNR < 32 dB "atrocious".

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Note that your considerations also has another shortcoming:

The formula you cite only applies to signals that are uniformly distributed over exactly the full ADC range. You'll be very hard tasked finding an audio signal that even remotely looks like that! For clipping / linearity / crest factor reasons, you need to have headroom for the vast majority of audio samples. Then, speech, music and many noise sources do not resemble uniformly distributed values:

^{Unnormalized 100-bin histogram over the values in Devin Townsend's Kingdom, a song that definitely tries to also achieve high amplitudes.}

^{Unnormalized 100-bin histogram over the values in Nightwish's Angels Fall First, a song that contains clean vocals and acoustic guitar playing.}

As you can see, assuming all values are equally probable is very far from the truth; accordingly, the actual quantization noise (the difference between quantized and original signal) does not follow the oft-cited formula. Whether or not that's a problem to you, I can't tell: it depends on what you want to show. However, "I have an audio SNR of x dB, thus I must have an ADC with (x-1.76)/6.02" is a fallacy.