Abstract: We consider the computational cut-off rate for the
complex Rayleigh flat fading spatio-temporal channel under a peak
power constraint, where the propagation matrix is unknown at both
transmitter and receiver. Determination of the cut-off rate requires
maximization of an average error exponent over all possible
probability distributions on the signal matrices. This error
exponent is monotone decreasing in a measure of dissimilarity between
pairs of signal matrices. For low SNR the dissimilarity function
reduces to a trace norm of the difference between the outerproduct of
each signal matrix. Under the practical constraint of finite
dimensional signal constellations, different characterizations of the
optimal constellation are obtained. For arbitrary finite dimension,
the optimal constellation must admit an equalizer distribution, i.e.,
a positive set of signal probabilities which equalizes the
conditional decoding error probabilities. The cut-off rate
optimization reduces to maximization of a quadratic form over the
feasible set formed from such constellations. When the number of
signal matrices in the constellation is less than the ratio of the
number of time samples to the number of transmit antennas, the
optimal cut-off rate attaining constellation is a set of equiprobable
mutually-orthogonal unitary matrices. When the SNR is below a
specified threshold the matrices in the constellation are rank one
and the cut-off rate is achieved by applying all transmit power to a
single antenna and using orthogonal signaling. Finally, we derive
recursive necessary and sufficient conditions for a constellation to
lie in the feasible set. This motivates a greedy procedure for
constructing a sequence of feasible constellations with monotone
increasing cut-off rate objective function.
Status:
Technical Memorandum, Bell Laboratories, Lucent Technologies,
April 2000. Submitted to IEEE Trans. Info. Th.
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