We have learned that BPSK fails in the Rayleigh fading channel. Can we design a signaling scheme that works in the fading channel?
The reason that BPSK fails is that BPSK encodes information in the phase of the transmit signal, and that the fading channel can change the phase of the signal uniformly randomly. So a natural idea is to encode the information in the magnitude of the transmit signal. One example is the orthogonal signaling, defined as
In this signal scheme, we use two symbols
The received signal is a tuple of two symbols
If
We derive the maximum likelihood detector in this case:
When
and
Therefore, we have
So the maximum likelihood detector has a simple decision rule:
The maximum likelihood detector is again very intuitive: we compare the energy of the two symbols. It is also called the noncoherent detector or the energy detector.
Now we analyze the error probability of such a detector. Again, due to symmetry, we can focus on the case where
Given that
In this signal scheme, the energy per transmit symbol is
In summary, in the fading channel, the orthogonal signal scheme works, but the error rate decays only linearly with the SNR. In contrast, the error rate decays exponentially with the SNR in the AWGN channel.