# Optimum filter

The function of a receiver in a binary Communication system is to distinguish between two transmitted signals $x_{1}(t)\&space;and&space;\&space;x_{2}(t)$  (or) ($s_{1}(t)\&space;and&space;\&space;s_{2}(t)$) in the presence of noise.

The performance of Receiver is usually measured in terms of the probability of error Pe an the receiver is said to be optimum if it yields the minimum probability of error.

i.e, optimum receiver is the one with minimum probability of error Pe .

optimum receiver takes the form of Matched filter when the noise at the receiver input is white noise.

The block diagram of optimum receiver is as shown in the figure below

the decision boundary is set to $\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

Probability of error of optimum filter:-

The probability of error can be obtained as similar to Integrate and dump receiver. Here we will consider noise as Gaussian Noise.

The output of optimum filter is  $y(t)&space;=&space;x_{o1}(t)+n_{o}(t)$ .

The output of sampler is  $y(T)&space;=&space;\left\{\begin{matrix}&space;x_{o1}(T)+n_{o}(T)&space;\&space;for&space;\&space;binary&space;\&space;i/p&space;\&space;'1'\\&space;x_{o2}(T)+n_{o}(T)&space;\&space;for&space;\&space;binary&space;\&space;i/p&space;\&space;'0'&space;\end{matrix}\right.$

suppose if Binary ‘1’ is transmitted then the input is $x(t)&space;=&space;x_{1}(t)$ , to find the probability of error this transmitted ‘1’ should be received as ‘0’.

this is possible  when the condition  $\left&space;|&space;y(T)&space;\right&space;|&space;<\frac{x_{o1}(T)+x_{o2}(T)}{2}$ is true.

1 will be received as 0    $\Rightarrow&space;x_{o1}(T)+n_{o}(T)&space;<\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$n_{o}(T)&space;<\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

similarly a Binary ‘0’ will be  received as ‘1’ if and only if

$\left&space;|&space;y(T)&space;\right&space;|&space;>\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$\Rightarrow&space;x_{o2}(T)+n_{o}(T)&space;>\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$n_{o}(T)&space;>\frac{x_{o1}(T)-x_{o2}(T)}{2}$ .

the conditions are  summarized in the table

Noe the Probability Distribution Function of Gaussian noise with zero mean and standard deviation $\sigma$  is given by

$f(n_{o}(T))&space;=&space;\frac{1}{\sigma&space;\sqrt{2\pi&space;}}&space;e^{-\frac{n_{o}^{2}(T)}{2}}$ .

Probability of error= probability ‘1’ will be received as ‘0’ =probability ‘0’ will be received as ‘1’.

$\therefore&space;P_{e}&space;=$  area under the curve $n_{o}(T)&space;>\frac{x_{o1}(T)-x_{o2}(T)}{2}$   (or) area under the curve $n_{o}(T)&space;<\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

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