Quantisation

Quantisation Step

$$\Delta =\frac{X_{max}-X_{min}}{L-1}$$

Where $L$ is the number of quantisation levels

Quantisation Error

$$eq(t)=(\frac{2}{\Delta }\ast \frac{1}{T})t$$

Power of Quantisation error:

$$P_q=\frac{1}{\tau }{\int }_0^{\tau }(eq(t){)}^2=\frac{{\Delta }^2}{12}$$

Relative Error of the quantisation noise to a sine wave

$$P_x=\frac{1}{T_P}{\int }_0^{T_p}(A\cos \omega t{)}^{2}dt=\frac{A^2}{2}$$

$T_P$ is one period

$$\frac{P_x}{P_q}=\frac{A^2}{2}\ast \frac{12}{{\Delta }^2}=\frac{6A^2}{{\Delta }^2}$$