Definition (Closed-loop prediction gain)

In the difference quantization scheme we define the closed-loop performance gain as $$G_{\text{clp}}=\frac{E[X_{n}^{2}]}{E[e_{n}^{2}]}$$Here we can see the larger $G_{\text{clp}}$, the more $e_{n}$ is compact and easier to quantize.

Definition (Coding gain)

In the difference quantization scheme we define the coding gain of our closed-loop quantizer as $$G_{Q}=\frac{E[e_{n}^{2}]}{E[(e_{n}-\hat{e}_{n})^{2}]}$$

Definition (System SNR)

For a closed-loop predictive quantizer we define the system SNR in terms of the closed-loop prediction gain, $G_{\text{clp}}$ and coding gain $G_{Q}$ (not in dB) $$\text{SNR}_{\text{sys}}=G_{\text{clp}}G_{Q}=\frac{E[X_{n}^{2}]}{E[e_{n}^{2}]}\frac{E[e_{n}^{2}]}{E[(e_{n}-\hat{e}_{n})^{2}]}$$or simply in dB as: $$\text{SNR}_{\text{sys}}=10\log_{10}G_{\text{clp}}+\text{SNR}_{Q} \ \ \ [\text{dB}]$$