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 which is a noisy input that is
from the figure output of the filter after sampling at seconds is
at output becomes
Now by substituting
by substituting the Equation(2) in Equation(1) over the limits
Now by replacing with the above equation becomes
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.