Asymptotically optimum quantization with time invariant breakpoints for signal detection

The nonlinear equations whose solution determines the locally optimum detection quantizer design are derived for a general parametric detection problem where the breakpoints are constrained to be time invariant. These quantizers maximize the efficacy of a test based on quantized data. Some specific optimum detection quantizer problems for the case of time-invariant breakpoints have been solved in the past, but only for the special case when the locally optimum nonlinearity factors in a certain way. Examples of observation models that do not satisfy these conditions are given. It is demonstrated that the locally optimum quantizer design for the time-invariant breakpoint constraint is the same as that quantizer design that minimizes the time-average mean-square difference between the quantizer and the locally optimum time-varying nonlinearity. A specific result shows that the optimum quantizer is not symmetric for the quadratic detector for random signals in Gaussian noise