Optimum Filter in Digital Communication Systems

In this article, we will learn about Optimum filter or Optimum Receiver, I have already explained about the Integrator and Dump Filter. So, its advance form is Optimum filter.

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Structure of Optimum Filter

So, in the Optimum Filter the input is \(x(t)+n(t)\) we are seeing the performance in the presence of noise. So, we are taking two inputs \(x_1(t) ,x_2(t)\) and \(y(t)\) is the output of the Optimum filter. 

\(y(t)=x_{o1}(t)+n_o(t); x_1(t)\)

\(y(t)=x_{o2}(t)+n_o(t); x_2(t)\)

Working of Optimum Filter

The optimum filter is used to minimize the probability of error, and these can be achieved by amplifying the SNR (Signal to noise Ratio). The Received signal be binary waveform. Assume that the polar NRZ Signal is used to represent binary 1's and 0's i.e.,

for binary '1'

            

            \(x_1(t)=+A\) for one-bit period T

and for binary '0'

            \(x_2(t)=-A\) for one-bit period T

Hence, the input signal \(x(t)\) will be either \(x_1(t)\) or \(x_2(t)\) depending upon the polarity of the NRZ signal. Figure above Shows the block diagram of a receiver for such a binary coded signal.

The noise \(n(t)\) is added to the signal \(x(t)\) in the channel during the transmission. Thus, Input to the optimum filter/optimum receiver is 

\(x(t)+n(t)\)

In the Absence of the noise \(n(t)\), the output of the receiver will be :

\(y(t) = x_{o1}(t)\) if \(x(t)=x_1(t)\)

\(y(t) = x_{o2}(t)\) if \(x(t)=x_2(t)\)

Hence, in the absence of noise, decisions are taken clearly. If noise is present then, we select closer to \(x_{o1}(t)\) than \(x_{o2}(t)\) for the input \(x_1(t)\) and we select \(x_2(t)\) if \(y(t)\) is closer to \(x_{o2}(t)\) than \(x_{o1}(t)\).

Therefore, the decision boundary will be in the middle of \(x_{o1}(t)\) and \(x_{o2}(t)\). It expressed as, 

Decision Boundary = \(\frac{x_{o1}(t)+x_{o2}(t)}{2}\)