we know that one form of Optimum filter is Matched filter, we will now derive another form of Optimum filter that is different from Matched filter Let the input to the Optimum filter is $v(t)$ which is a noisy input that is $v(t)=x(t)+n(t)$

from the figure output of the filter after sampling at $t=T_{b}$ seconds is $v_{o}(T_{b})$

$v_{o}(t)=v(t)*h(t)$

$v_{o}(t)&space;=&space;\int_{-\infty&space;}^{T_{b}}v(\tau&space;)&space;h(t-\tau&space;)d\tau$

at $t=&space;T_{b}$ output becomes  $v_{o}(T_{b})&space;=&space;\int_{-\infty&space;}^{T_{b}}v(\tau&space;)&space;h(T_{b}-\tau&space;)d\tau&space;-----Equation(1)$

Now by substituting $h(\tau&space;)=x_{2}(T_{b}-\tau&space;)-x_{1}(T_{b}-\tau&space;)$

$h(T_{b}-\tau&space;)=x_{2}(T_{b}-T_{b}+\tau&space;)-x_{1}(T_{b}-T_{b}+\tau&space;)$

$h(T_{b}-\tau&space;)=x_{2}(\tau&space;)-x_{1}(\tau&space;)------Equation(2)$

by substituting the  Equation(2)  in Equation(1) over the limits $[0,T_{b}]$

$v_{o}(T_{b})&space;=&space;\int_{0&space;}^{T_{b}}v(\tau&space;)&space;(x_{2}(\tau&space;)-x_{1}(\tau&space;))d\tau$

$v_{o}(T_{b})&space;=&space;\int_{0&space;}^{T_{b}}v(\tau&space;)&space;x_{2}(\tau&space;)&space;d\tau-\int_{0}^{T_{b}}v(\tau&space;)x_{1}(\tau&space;)d\tau$

Now by replacing $\tau$ with $t$ the above equation becomes

$v_{o}(T_{b})&space;=&space;\int_{0&space;}^{T_{b}}v(t&space;)&space;x_{2}(t&space;)&space;dt-\int_{0}^{T_{b}}v(t&space;)x_{1}(t&space;)dt-----Equation(3)$

The equation (3) suggests that the Optimum Receiver can be implemented as shown in the figure, this form of the Receiver is called  as correlation Receiver. This receiver requires the integration operation be ideal with zero initial conditions. Correlation Receiver performs coherent-detection.

in general Correlation Receiver can be approximated with Integrate and dump filter.

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