A continuously operating coherent Binary Phase shift Keying (BPSK) system is mad

Question

A continuously operating coherent Binary Phase shift Keying (BPSK) system is made by an engineer.
He claims that it is having an average error probability of 10^-5. The system produced 1000 errors in a day when tested with a data rate of 1000 bits/s and the single-sided noise power spectral density is No=10^-10 W/Hz. He also claim that the system is capable of maintaining the error rate even if the received power is as low as 10^-6 W.
Do you agree with his claims? Give justifications to your answer.

Explanation / Answer

For determining error-rates mathematically, some definitions will be needed:

{displaystyle E_{b}} E_{b} = Energy per bit
{displaystyle E_{s}} E_{s} = Energy per symbol = {displaystyle nE_{b}} nE_{b} with n bits per symbol
{displaystyle T_{b}} T_{b} = Bit duration
{displaystyle T_{s}} T_{s} = Symbol duration
{displaystyle N_{0}/2} N_{0}/2 = Noise power spectral density (W/Hz)
{displaystyle P_{b}} P_{b} = Probability of bit-error
{displaystyle P_{s}} P_{s} = Probability of symbol-error
{displaystyle Q(x)} Q(x) will give the probability that a single sample taken from a random process with zero-mean and unit-variance Gaussian probability density function will be greater or equal to {displaystyle x} x. It is a scaled form of the complementary Gaussian error function:

{displaystyle Q(x)={rac {1}{sqrt {2pi }}}int _{x}^{infty }e^{-t^{2}/2},dt={rac {1}{2}}operatorname {erfc} left({rac {x}{sqrt {2}}} ight), xgeq 0} {displaystyle Q(x)={rac {1}{sqrt {2pi }}}int _{x}^{infty }e^{-t^{2}/2},dt={rac {1}{2}}operatorname {erfc} left({rac {x}{sqrt {2}}} ight), xgeq 0}.

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