New solutions to the signal design problem for coherent channels

A fundamental problem of statistical communication theory is that of selecting the optimal set of
M
equipowered finite duration waveforms to minimize the error rate for a coherent channel perturbed by white Gaussian noise, the signals being restricted to
D
degrees of freedom, where
D < M
. It is demonstrated that the concept of placing lower-dimensional regular simplicities mutually orthogonal to one another (the technique of known solutions) can be extended. In fact, such signal structures are solutions to the signal design problem for the cases where the dimensionality is restricted to be at most
M-K
, where
K
is any fixed integer between
2
and
M/2
. Of the several solutions for each
K
, the most preferred of these is determined at large signal-to-noise ratio, and in certain cases numerical computation has shown this preference to agree with that at small signal-to-noise ratio. Finally, the effect on probability of error due to successive reductions in the allowed dimensionality of the signal set is analyzed for the optimal solutions known to date.