Optimum block quantisation in signal processing

Block quantisation is a process whereby groups of continuous variables are quantised in a given number of discrete levels for digital processing. In problems involving block quantisation certain constraints concerned with integer bit assignment that guarantee a realistic solution are difficult to incorporate into the development of algorithms for optimal solutions. In this kind of situation the existing methods produce solutions by an appropriate algorithm which treats the number of bits per sample as a continuous variable, and a correcting process necessarily follows to assign integer bits per sample, thereby presenting the solution in a realistic form. Deviation from optimality, however, is often a common side effect of the entire procedure. In the paper we present a recursive algorithm for optimum block quantisation, based on dynamic programming, by means of which integer bit assignment constraints are easily met. Further to a computational solution, a dynamic programming analytical solution is also derived. Compared with existing analytical solutions derived according to entirely different considerations, it is shown that the new approach, without the integer bit assignment constraint, leads to precisely the same conditions of optimality and results in similar closed-form solutions for the block quantisation problem.