Sampling rate effect on quantizer SNR

I am currently reading the 'Coding' chapter on Rabiner & Schafer's Book on speech processing.

In one of its exercises, the reader is given a simple A/D converter using 16-bit uniform quantization. The converter uses a sampling rate of $Fs = 8000 Hz$. Then it asks what is the effect on $SNR_Q$ (quantization signal-to-noise ration) if we double the sampling rate.

From what I understand, doubling the sampling rate has obviously no effect on the quantizer's range $X_{max}$ as well as the quantization step $\Delta$. Furthermore, the theoretical model for $SNR_Q$ on uniform quantizers is built on top of the hypothesis that the quantization error $e(n)$ follows a uniform probability density, so:

$$ SNR_Q = \frac{\mathcal{E}(\sum_n x^2(n))}{\mathcal{E}(\sum_n e^2(n))} $$

should remain constant, since both its numerator and denominator are calculated off expected values.

Is my thinking correct or does the change of the sampling rate break any of my assumptions?

Answer

You are correct. When using uniform sampling, the sampling rate is unrelated to the quantization noise. Only the quantization step and the signal's dynamic range are relevant for calculating the quantization SNR.