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 0.4V2
Given that binary ‘1’ is represented for +2V and binary ‘0’ is represented for -2V
For transmission of binary ‘1’
a1 = E[X + N]
a1 = E[X] + E[N]
Given that mean of noise is zero i.e, E[N] = 0
a1 = E[2] + E[N] = 2 + 0
a1 = 2
For transmission of binary ‘0’
a2 = E[X + N]
a2 = E[- 2] + E[N]
a2 = - 2
Optimum threshold voltage is
\({v_{th}} = \frac{{2 - 2}}{2} + \frac{{0.4}}{{2 - \left( { - 2} \right)}}\ln \left[ {\frac{{0.6}}{{0.4}}} \right]\)
\({v_{th}} = \frac{{0.4}}{{2 - \left( { - 2} \right)}}\ln \left[ {\frac{{0.6}}{{0.4}}} \right]\)
Vth = 0.04 V