Probability of error of ASK

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Probability of error of ASK using coherent Detection:-

The equation of ASK from it’s basic definition

$S_{ASK}(t)=\left\{\begin{matrix} \sqrt{2P_{s}}\cos 2\pi f_{c}t\ \rightarrow \ symbol\ '1'\\ 0\ \rightarrow \ symbol\ '0' \end{matrix}\right.$ .

The inputs to the optimum filter are $x_{1}(t) \ and \ x_{2}(t)$ when the symbols 1 and 0 are being transmitted at the transmitter.

$\therefore x_{1}(t) = \sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

$x_{2}(t)=0$ .

we know that $P_{e}$ of an optimum filter is $P_{e} = \frac{1}{2}\ erfc(\frac{x_{o1}(t)-x_{o2}(t)}{2\sqrt{2}\sigma })$

now chose the ratio $\rho _{max}^{2} =(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}$

from the Matched filter concept $(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}=\frac{2}{N_{o}}\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df-----EQN(1)$

from Parsevel’s relation $\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df =\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt----EQN(2)$

from equations (1) and (2)

$\rho _{max}^{2}=\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt$

Here x(t) is the signal $x(t) =x_{1}(t)-x_{2}(t)$ .

$x(t) =\sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

over a duration of $[0, T_{b}]$ the symbols are transmitted

$\int_{0}^{T_{b}}\left | x(t) \right |^{2}dt=2P_{s}\int_{0}^{T_{b}}\cos^{2} 2\pi f_{c}t \ dt$ .

$=P_{s}T_{b}$ .

from equation(1) $\rho _{max}^{2} = \frac{2}{N_{o}} P_{s}T_{b}$ .

$\rho _{max} = \sqrt{\frac{2P_{s}T_{b}}{N_{o}}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{2P_{s}T_{b}}{N_{o}}}).\frac{1}{2\sqrt{2}})$

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{P_{s}T_{b}}{4N_{o}}})$ .

we know that carrier signal power is $P_{s} = \frac{A^{2}}{2}$ .

$A=\sqrt{2P_{s}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{A^{2}T_{b}}{8N_{o}}})$ .

$\therefore P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ . since $A^{2}T_{b} = P_{s}T_{b} =E_{b}$ .

probability of error of coherent ASK is $P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ (or) in terms of Q function as $P_{e}= Q(\sqrt{\frac{E_{b}}{2N_{o}}})$ .