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The mapping of into the modulation symbols is done by the following formula:
where is the phase rotation,
is the decimal input symbol, and
is the modulation order.
The input should be in range between 0 and M-1.
If the input is binary, the conversion from binary to decimal should be done before.
The Gray mapping provides a more robust modulation scheme. For example:
- Let us 00 was transmitted. According to QPSK signal constellation, it is roughly 0,7+0,7i.
- However, let's say because of strong noise 0,4-0,1i was detected.
- It will be interpreted as 11 in the binary mapping and 01 in the Gray mapping
Yes, the situation will be symmetric if the symbol is detected as -0,4-0,1i, however, the probability of this case is smaller.
MathWorks company provides several algorithms to demodulate BPSK, QPSK, 8-PSK and other M-PSK modulations.
To reduce the number of implemented schemes the following way is used in our project:
- calculate LLRs (soft decision)
- map LLR to bits according to the sign of LLR (inverse of NRZ)
So, this works. We guess the complexity issues are not the critical part due to hard output demodulators are not so popular. This phenomenon depends on channel decoders properties: e.g., Convolutional codes, Turbo convolutional codes and LDPC codes work better with LLR.