# Correlated Quantization Noise

Normally, when we are dealing with quantization noise, we assume quantization noise as uncorrelated, which is white.

In this case, we assume SNR = 6.02N + 1.76 where N = resolution

However, if quantization noise is correlated to input signal (ex : direct digital synthesis),it's likely that q noise is not white, rather spurious.

1. What is SNR in case of coherent q noise?

2. Can we precict exact power of each spur?

3. How to calculate THD from SNR?

• I disagree with the premise that "direct digital synthesis" necessarily has "quantization noise is correlated to input signal" any more than any audio DSP correlates quantization error to the signal. It all depends on the ratio of the amplitude of the signal (either processed or synthesized) to the quantizer step size. If that ratio is large enough, the quantization error can be about anything. I guess, with synthesis, if the waveform is static in amplitude and with a period of exactly an integer number of samples, then the quantization error will also be periodic with the same frequency. Dec 18, 2023 at 2:50
• @robertbristow-johnson Thank you for your comment. So if the quantization error will be periodic with the same frequency, how do we define SQNR in this case? Dec 18, 2023 at 4:27
• Same as otherwise. Dec 18, 2023 at 5:00

In this case, we assume SNR = 6.02N + 1.76 where N = resolution

That's a bit of over simplification at best. This is only true if the input signal is a full scale sine wave at a frequency that's not an integer divider of the sample rate.

It assumes that the quantization error is uniformly distributed on $$[-\Delta/2,+\Delta/2]$$ where $$\Delta$$ is the quantization interval, in which case the Quantization noise power will always be

$$\sigma_q^2 = \frac{\Delta^2}{12} \tag{1}$$

The resulting SNR will depend mainly on the power of the signal.

There are cases when the assumption of a uniform distribution is wrong, particularly if the quantization interval is large and the signal itself has very "sparse" or "spiky" or "not normal distributed"

In this case, the only way to really predict it is to quantize the signal and explicitly calculate the quantization power. However for a rounding quantizer we can bound it as

$$0 \le \sigma_q^2 \le \frac{\Delta^2}{4} \tag{2}$$

The spectrum is primarily a function of whether the signal is periodic or not. If the signal is periodic, so will be the quantization noise and hence the noise will have a harmonic spectrum. The height of the individual peaks will depend on the exact frequency, amplitude and phase of the sine wave and the easiest way to determine this is, again, just to quantize the signal of interest and do a spectral analysis.

For sine wave you will only get odd harmonics. The overall quantization noise power is still bound by equation (2).