Error probability:
- We know that BPSK signal is represented as follows:
Binary 1:x1(t)=√2PscosωC t$
Binary 0: x2(t)=−√2PscosωCt
Therefore x2(t)=−x1(t)
- By using the matched filter for detection of BPSK signal.The expression for error probability of an optimum filter is,

- The expression for the signal to noise ration of matched filter is given by,

- Using the Rayleigh’s energy theorem,

- The limits of integration of the last term in equation (3) are 0 to T because x (t) is present only over one bit interval T. Substituting equation (3) into equation (2) we get,

and for BPSK x2(t)=−x1(t)

- Substituting this value of x (t) into equation (4) we get,

- Substitute this into equation (5) to get

- The value of second term in the RHS in equation (6) is zero


Bandwidth of BPSK:
- From the frequency spectrum of BPSK signal ,it is clear that the bandwidth of a BPSK signal is given by,
BW= Highest frequency – Lowest frequency in main lobe =(fc+fb)−(fc−fb)
∴ BW=2 fb
Where fb=1/Tb
- Thus the minimum bandwidth of BPSK signal is equal to twice the highest frequency contained in the baseband signal.