Given an arbitrary receiver, the evaluation of P(e) can be carried out in a very general way starting from the knowledge of the decision regions in the space RN and of f(r|si) (i=1,2,..,M).
The evaluation requires in general the solution of integrals over N dimensional domains and, is therefore difficult.
The evaluation can be simplified if several assumptions, which are reasonable in some cases of interest, can be done.
If the signals are equiprobable: (fig.1)
If the P(c|si) =P(c) (are constant with respect to i) namely, if the receiver exhibits congruent decision regions: (fig.2)
It can be shown in the binary case that P(e) depends only on the geometric representation of the transmitted signals
Example: binary baseband signalling and binary bandpass signalling
Signals with the same geometric representation are equivalent in terms of performances
We can utilize the shape of the signals to satisfy other requirements (for example the simplicity to synthesize the signals or reduce ISI)
It can be shown that the property holds also for M > 2
Two sets of signals whose geometric representation overlapping (with rigid movement) will exhibt the same optimum performances
→ Are equivalent namely MV receivers have the same performances.
Two equivalent sets of signals do not have the same RX MV but they exhibit the same performances
For monodimensional signalling (PAM, ASK) and bidimensional (MPSK, QAM) ones the PM can be kept constant when M grows by increasing E bav /N0 → Such signalling schemes are not power efficient.
13. Cyclic Codes