Quantization of multiple sources using integer bit allocation

Asymptotically optimal bit allocation among a set of quantizers for a finite collection of sources was determined by Huang and Schultheiss (1963). Their solution, however, gives a real-valued bit allocation, whereas in practice, integer-valued bit allocations are needed. We compare the performance of the Huang-Schultheiss solution to that of an optimal integer-valued bit allocation. Specifically, we derive upper and lower bounds on the deviation of the mean squared error using optimal integer-valued bit allocation from the mean squared error using optimal real-valued bit allocation. One consequence shown is that optimal integer-valued bit allocations do not necessarily achieve the same performance as that predicted by Huang-Schultheiss, for asymptotically large transmission rates. We also prove that integer bit allocation vectors that minimize the Euclidean distance to the optimal real-valued bit allocation vector are optimal integer bit allocations.