Editing Bit error rate (section)

==Mathematical draft==
The BER is the likelihood of a bit misinterpretation due to electrical noise <math>w(t)</math>. Considering a bipolar NRZ transmission, we have

<math>x_1(t) = A + w(t)</math> for a "1" and <math>x_0(t) = -A + w(t)</math> for a "0". Each of <math>x_1(t)</math> and <math>x_0(t)</math> has a period of <math>T</math>.

Knowing that the noise has a bilateral spectral density  <math>\frac{N_0}{2} </math>,

<math>x_1(t)</math> is <math>\mathcal{N}\left(A,\frac{N_0}{2T}\right)</math>

and <math>x_0(t)</math> is <math>\mathcal{N}\left(-A,\frac{N_0}{2T}\right)</math>.

Returning to BER, we have the likelihood of a bit misinterpretation <math>p_e = p(0|1) p_1 + p(1|0) p_0</math>.

<math> p(1|0) = 0.5\, \operatorname{erfc}\left(\frac{A+\lambda}{\sqrt{N_o/T}}\right)</math> and  <math> p(0|1) = 0.5\, \operatorname{erfc}\left(\frac{A-\lambda}{\sqrt{N_o/T}}\right)</math>

where <math> \lambda</math> is the threshold of decision, set to 0 when <math>p_1 = p_0 = 0.5</math>.

We can use the average energy of the signal <math>E = A^2 T</math> to find the final expression :

<math>p_e = 0.5\, \operatorname{erfc}\left(\sqrt{\frac{E}{N_o}}\right).</math>
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