# Why is the Mean Square Error (MSE) used in the Peak Signal-to-Noise Ratio (PSNR) calculation rather than the Root Mean Square Error (RMSE)?

Peak signal-to-noise ratio (PSNR) is calculated with

$$\text{PSNR} = 10 \log \frac{\;\text{MAX}^2}{\text{MSE}}\;,$$

with MSE the mean square error and MAX the maximum signal value.

Why is the MSE used in calculating the PSNR, rather than RMSE (root mean square error)?

$$\text{PSNR} = 10 \log \frac{\text{MAX}^2}{\text{MSE}} = 20 \log \frac{\text{MAX}}{\text{MSE}^{1/2}} = 20 \log \frac{\text{MAX}}{\text{RMSE}}$$