A full scale sinusoidal signal is applied to a $10$ -bit $\mathrm{ADC}$. The fundamental signal component in the $\mathrm{ADC}$ output has a normalized power of $1 \mathrm{~W}$, and the total noise and distortion normalized power is $10 \mu \mathrm{W}$. The effective number of bits (rounded off to the nearest integer) of the $\mathrm{ADC}$ is $\_\_\_\_\_\_\_$.
- $7$
- $8$
- $9$
- $10$