Document Type

Article

Department

Mathematics (HMC)

Publication Date

1971

Abstract

The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for $N = 2$, and the optimal signal sets are determined for $M = 2,3,4,6$ and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ${\bf s}_j $, $j = 1,2, \cdots ,M$, on the unit sphere $S_1 $ in $C^N $. If $W_{jk} $ is the half space determined by ${\bf s}_j $ and ${\bf s}_k $ and containing ${\bf s}_j $, i.e., $W_{jk} = \{ {{\bf r} \in C^N :| {\langle {{\bf r},{\bf s}_j } \rangle } |\geqq | {\langle {{\bf r},{\bf s}_k } \rangle } |} \}$, then $\{ {\mathfrak{R}_j = \cap _{k \ne j} W_{jk} :j = 1,2, \cdots ,M} \}$, the maximum likelihood decision regions, partition $S_1 $. For additive complex Gaussian noise ${\bf n}$ and a received signal ${\bf r} = {\bf s}_j e^{i\theta } + {\bf n}$, where $\theta $ is uniformly distributed over $[ {0,2\pi } ]$, the probability of correct decoding for the signal-to-noise ratio $A^2 $ is \[ P_C = \frac{1} {{\pi ^N }}\int_0^\infty {r^{2N - 1} } e^{ - \left( {r^2 + A^2 } \right)} U( r )dr, \]where\[U( r ) = \frac{1} {M}\sum\limits_{j = 1}^M {\int_{Rj} {I_0 } } \left( {2Ar\left| {\left\langle {{\bf s},{\bf s}_j } \right\rangle } \right|} \right)d\sigma ( {\bf s} ) \]$R_j = \mathfrac{R}_j \cap S_1 $. For $N = 2$, it is proved that \[ U( r )\leqq \int_{C_\alpha } {I_0 } \left( {2Ar\left| {\left\langle {{\bf s},{\bf s}_j } \right\rangle } \right|} \right)d\sigma ( s ) - \frac{{2K}} {M} \cdot h\left( {\frac{1} {{2K}}\left[ {M\sigma \left( {C_\alpha } \right) - \sigma \left( {S_1 } \right)} \right]} \right), \] where\[ C_\alpha = \left\{ {{\bf s} \in S_1 :\left| {\left\langle {{\bf s},{\bf s}_j } \right\rangle } \right|\geqq \alpha } \right\}, \]$2K$ is the total number of half spaces that actually determine the decision regions, and h is the strictly increasing, strictly convex function of $\sigma ( {C_\alpha \cap W} )$ (where W is a half space not containing ${\bf s}_j $), given by\[ h = \int_{C_\alpha ^ * } {I_0 } \left( {2Ar\left| {\left\langle {{\bf s},{\bf s}_j } \right\rangle } \right|} \right)d\sigma ( {\bf s} ), \]$C_\alpha ^ * = C_\alpha \cap W$. Conditions for equality are established and these give rise to the globally optimal signal sets for $M = 2,3,4,6$ and 12.

Rights Information

© 1971 Society for Industrial and Applied Mathematics

Download

Share

COinS