The receiver interested in the transmitted bit stream must perform two tasks when received waveform begins.
Let's assume a perfect channel for the moment: The received signal equals the transmitted one. If bit 0 were sent using the baseband BPSK signal set, the integrator outputs would be
Can you develop a receiver for BPSK signal sets that requires only one multiplier-integrator combination?
In BPSK, the signals are negatives of each other: . Consequently, the output of each multiplier-integrator combination is the negative of the other. Choosing the largest therefore amounts to choosing which one is positive. We only need to calculate one of these. If it is positive, we are done. If it is negative, we choose the other signal.
What is the corresponding result when the amplitude-modulated BPSK signal set is used?
The matched filter outputs are because the sinusoid has less power than a pulse having the same amplitude.
Clearly, this receiver would always choose the bit correctly. Channel attenuation would not affect this correctness; it would only make the values smaller, but all that matters is which is largest.