Optimal added noise for minimizing distortion in quantizer-array linear estimation

For linearly estimating the input signal, the optimization of suprathreshold stochastic resonance in an array of N parallel M-ary quantizers is theoretically investigated. We first prove that the mean square error (MSE) of the designed quantizer-array is a convex functional with respect to the cumulative probability function (CDF) of the added noise. Then, for an arbitrary input signal and the quantization interval being N times the decoding step size, we theoretically demonstrate that minimum MSE distortion can be obtained for optimal added uniform noise. Furthermore, for a uniform input signal, the optimality condition also holds if the system parameters and the boundary of the signal satisfy an inequality constraint condition. Moreover, under this condition, the optimal parameters of the quantizer-array can be also determined exactly. By applying both the optimal added noise and optimal parameters to the quantizer-array, the MSE can be further improved for larger array size N or quantization order M. Finally, the optimal added noise is also discussed for minimizing distortions of the quantizer-array linear estimation in the presence of the background noise.