Here are some results for 3,4, and 5 bit quantisers over a range of errors:
Curiously, the natural binary results are a little better (about 1dB less Eb/No for the same SNR). Another surprise is that at low Eb/No (high BERs) the SNRs are about the same for each quantiser. For example around 9dB SNR at Eb/No = -2dB, for 5,4 and 3 bits.
Here is a plot of 2 to 7 bit natural binary quantisers over a wide Eb/No range. Up to about Eb/No of 4dB (a BER of 1%), the 3-7 bit quantisers all work about the same! At lower BER (higher Eb/No), the quantisation noise starts to dominate and the higher resolutions quantisers work better. Each extra bit adds about 6dB of improved SNR.
Channel errors dominate the SNR at BER greater than 1% (Eb/No=4dB). In some sense the extra quantiser bits are “wasted”. This may not be true in terms of subjective decoded speech quality. The occasional large error tends to drag the SNR measure down, as large errors dominate the noise power. Subjectively, this might be a click, followed by several seconds of relatively clean speech. So more (subjective) testing is required to determine if natural or Gray coding is best for Codec 2 parameters. The SNR results suggest there is not much advantage either way.
Here is a plot of the error from the natural and Gray coded quantisers at Eb/No=-2dB. Occasionally, the Gray coded error is very large (around 1.0), compared to the natural coded error which has a maximum of around 0.5.
This example of a 3 bit quantiser helps us understand why. The natural binary and Gray coding is listed below the quantiser values:
|Natural Binary Code||000||001||010||011||100||101||110||111|
Although Gray codes are robust to some bit errors (for example 000 and 001), they also have some large jumps, for example the 000 and 100 codes are only 1 bit error apart but jump the entire quantiser range. Natural binary has an exponentially declining error step for each bit.