Multidimensional moment error bounds for digital communication systems

In many digital communication systems with additive noise and random interference disturbances, the error probability can be expressed as the statistical expectation of the complementary noise distribution function with respect to some function of the random interference variable
Z
. Conditions are presented for constructing a new class of upper and lower error bounds, utilizing an arbitrary number of generalized moments of the random interference. These bounds are based on four forms of the principal representation of Krein in the theory of approximation. General results are given for an
L
-level input system, an arbitrary noise distribution function, and an arbitrary transformation
f(Z)
. A class of simple bounds using polynomial transformations on the centered
Z
for the binary input system with additive Gaussian noise is presented. These bounds depend on the signal-to-noise ratio, the maximum interference distortions, the number of moments, and the polynomial transformation integer. In general the bounds obtained are very tight and need only modest computational efforts. Indeed, this approach yields the tightest possible bounds for a given set of moments of the random interference. Two techniques are discussed for evaluating principal representations as well as relationships to generalized quadrature rules. Numerical results are presented illustrating the tightness of these bounds.