Consider the common source stage as shown below:
We model the thermal and flicker noise of M1 by two current sources –
We also represent thermal noise of $R_D$ by current source
The output noise voltage per unit Bandwidth is given by –
The noise mechanisms are added as “power” quantities because they are uncorrelated. The value given in equation (1) represents the noise power in 1 Hz at a frequency f. the total output noise can be obtained by integration over bandwidth of interest.
To calculate the input referred noise voltage, we have,
The first term in equation (2) is the thermal noise of resistor equal to (2/3$g_m$) placed in series with gate. Similarly third term corresponds to the noise of a resistor equal to ($g_m^2$ $R_D$) $^-$$^1$. Therefore the total input referred thermal noise of the circuit in unit bandwidth is equal to 4KT$R_T$.