__Concept:__

**Optimum threshold voltage**

For an Additive White Gaussian Noise function, the optimum threshold value is given by:

\({v_{th}} = \frac{{{a_1} + {a_2}}}{2} + \frac{{\sigma _n^2}}{{{a_1} - {a_2}}}\ln \left[ {\frac{{P\left( 0 \right)}}{{P\left( 1 \right)}}} \right]\)

P(0): Probability of the error when ‘0’ transmitted

P(1): Probability of the error when ‘1’ transmitted

__Calculation:__

Given that P(1) = 0.4 then P(0) = 0.6

The variance of the noise is 0.4V^{2}

Given that binary ‘1’ is represented for +2V and binary ‘0’ is represented for -2V

For transmission of binary ‘1’

a_{1} = E[X + N]

a_{1} = E[X] + E[N]

Given that mean of noise is zero i.e, E[N] = 0

a_{1} = E[2] + E[N] = 2 + 0

**a**_{1} = 2

For transmission of binary ‘0’

a_{2} = E[X + N]

a_{2 }= E[- 2] + E[N]

**a**_{2} = - 2

Optimum threshold voltage is

\({v_{th}} = \frac{{2 - 2}}{2} + \frac{{0.4}}{{2 - \left( { - 2} \right)}}\ln \left[ {\frac{{0.6}}{{0.4}}} \right]\)

\({v_{th}} = \frac{{0.4}}{{2 - \left( { - 2} \right)}}\ln \left[ {\frac{{0.6}}{{0.4}}} \right]\)

**V**_{th }= 0.04 V