Optimal feedback quantizers for -dimensional systems with discrete-valued input

This paper addresses an optimal design problem of feedback quantizers for a class of n -dimensional ( n -D) systems whose input signals are restricted to discrete-valued ones. First, for an arbitrary given quantizer, we analyze the maximum discrepancy between outputs of both the discrete-valued input n -D system connected with the quantizer and its corresponding continuous-valued input one. Based on this result, an optimal feedback quantizer which minimizes the discrepancy is derived in a closed form. A numerical example is given to demonstrate its effectiveness even in the case of coarse quantization. Second, we apply the optimal feedback quantizer to generate binary halftone images to verify its applicability and potential to real problems.