Quantization of Multiple Sources Using Nonnegative Integer Bit Allocation

Abstract

Asymptotically optimal real-valued bit allocation among a set of quantizers for a finite collection of sources was derived in 1963 by Huang and Schultheiss, and an algorithm for obtaining an optimal nonnegative integer-valued bit allocation was given by Fox in 1966. We prove that, for a given bit budget, the set of optimal nonnegative integer-valued bit allocations is equal to the set of nonnegative integer-valued bit allocation vectors which minimize the Euclidean distance to the optimal real-valued bit-allocation vector of Huang and Schultheiss. We also give an algorithm for finding optimal nonnegative integer-valued bit allocations. The algorithm has lower computational complexity than Fox's algorithm, as the bit budget grows. Finally, 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 (MSE) using optimal integer-valued bit allocation from the MSE using optimal real-valued bit allocation. It is shown that, for asymptotically large transmission rates, optimal integer-valued bit allocations do not necessarily achieve the same performance as that predicted by Huang-Schultheiss for optimal real-valued bit allocations.

Limited Feedback of Precoder and Bit Loading for MIMO Systems: A Joint Design

“…Thus . APPENDIX D PROOF OF (13) Using (12), the th column of the quantized precoder can be obtained from quantized as (28) Using (12) and (28), we have and (29) where and . Let us use Gram-Schmidt process to obtain from and from .…”

“…The solution of rate allocation given by (18) is not an integer in general. We can quantize it to an integer using the method in [28].…”

“…The above two steps are iterated until the average mean square error is below a given threshold or when there is little improvement. The codewords generated in this way are real-valued; like rate allocation obtained in Section III they can be quantized to integer codewords using the method in [28].…”

Bit Allocation and Statistical Precoding for Correlated MIMO Channels With Limited Feedback

“…The iterations can be ended when there is little improvement in the average BER. The resulting BA vectors b i can be quantized to have integer entries subject to the sum rate constraint using the method in [34]. Note that the derivation of BER bounds in the previous section needs the assumption M r > M. When we use the BA system in practice, M need not be smaller than M r ; it can be equal to M r .…”

Optimized rate allocation for state estimation over noisy channels

“…The problem we are addressing here is related to classical rate allocation problems in communications [6], [7]. To quantify the relation between rate and performance, we resort to high-rate quantization theory [6], [8], [9].…”