i) For MOS devices operating in saturation region the channel noise can be modeled by a current source connected between the drain and source terminal and expressed as
$\bar{T_n^2}=4KT\gamma g_m$

where, K= boltzmann constant=$1.38*10^{-23}$ J/K

T=Temperature.

$\gamma$=2/3 for long channel device in saturation

$\hspace{0.3cm}$=2 for sub-micron devices.

$g_m$=transconductance.

ii) The dependance of thermal noise upon Temperature(T) suggests that low temperature operation decrease the noise in analog circuit.

iii) **Flicker Noise:**

As charge carrier moves at the interface the random charge trapping by the energy states introduces a noise in the drain current called flicker noise. Flicker noise is modelled by a current source across the drain and source and expressed as

$\bar{V_n^2}=\frac{K}{C_{ox}WL}\, \frac{1}{f}$

where, K= process dependant constant.

W,L= width and length of MOS device.

iv) Flicker noise reduces with increasing frequency and at a point it starts falling much below the thermal noise. The frequency at which flicker noise is equal to thermal noise is called corner frequency($f_c$) of flicker noise.

$4KT(2/3 gm)= \frac{K}{C_{OX}WL}\, \frac{1}{f_c}g^2_m$

i.e $f_c=\frac{K}{C_{OX}WL}gm \frac{3}{8KT}$

This result implies that $f_c$ generally depends on devices dimensions and bias current.

**Equation for output noise voltage of CS stage**

We model the thermal and flicker noise of $M_1$ by 2 current ->

$\hspace{2cm}\bar{I_{n,th}^2}=4KT(\frac{2}{3})g_m$

$\hspace{2cm}\bar{I_{n,1/f}^2}=\frac{Kg_m^2}{C_{OX}WLf}$

We also represent thermal noise of $R_D$ by current source.

$\hspace{2cm}\bar{I_{n,R_D}^2}=\frac{4KT}{R_D}$

The output noise voltage per unit BW=

$\bar{V_{n,out}^2}=\Big( 4KT\frac{2}{3}g_m+\frac{Kg_m^2}{C_{OX}WLf}+\frac{4KT}{R_D} \Big)R_D^2 \hspace{2cm}$.......(1)

We have,

$\bar{V_{n,in}^2}=\frac{\bar{V_{n,out}^2}}{A_v^2}$

$=\Big( 4KT\frac{2}{3}g_m+\frac{Kg_m^2}{C_{OX}WLf}+\frac{4KT}{R_D} \Big)R_D^2 \,\, \frac{1}{g_m^2\,R_D^2}\,\hspace{2cm}$

Input reffered noise:

$\therefore\,\,\,V_{n,in}^2=\Big( 4KT\frac{2}{3g_m}+\frac{K}{C_{OX}WLf}+\frac{4KT}{g_m^2\,R_D} \Big)\hspace{2cm}$.......(2)