# Probability of error of ASK

Probability of error of ASK using coherent Detection:-

The equation of ASK from it’s basic definition

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

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

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

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

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

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

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

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

from equations (1) and (2)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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