Optimal binary index assignments for a class of equiprobable scalar and vector quantizers

Abstract

The problem of scalar and vector quantization in conjunction with a noisy binary symmetric channel is considered. The issue is the assignment of the shortest possible distinct binary sequences to quantization levels or vectors so as to minimize the mean squared error caused by channel errors. By formulating the assignment as a matrix (or vector in the scalar case) and showing that the mean squared error due to channel errors is determined by the projections of its columns onto the eigenspaces of the multidimensional channel transition matrix, a class of source/quantizer pairs is identi ed for which the optimal index assignment has a simple and natural form. Among other things, this provides a simpler and more accessible proof of the result of Crimmins et al. that the natural binary code is an optimal index assignment for the uniform scalar quantizer and uniform source. It also provides a potentially useful approach to further developments in sourcechannel coding. Index terms: source-channel coding, natural binary code, vector quantization.

Optimal multiresolution quantization for broadcast channels with random index assignment

“…They also asserted the optimality of the NBC for the binary symmetric channel. Crimmins et al [25] proved the optimality of the NBC as asserted in [24], and McLaughlin, Neuhoff, and Ashley [28] generalized this result to uniform vector quantizers. Various other analytical results on index assignments without channel-optimized encoders or decoders have been given in [26], [27].…”

Channel-optimized scalar quantizers with erasure correcting codes

“…In [16], it is proved that NBC is optimal for uniform scalar quantizers and uniform source. In [17], McLaughlin et al extended it to uniform vector quantizers. Farber and Zeger [8] also proved the optimality of NBC for uniform source and quantizers with uniform encoders and channel-optimized decoders.…”

“…The usual goal to design an index *Correspondence: ye.li@siat.ac.cn 1 Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China Full list of author information is available at the end of the article assignment for noisy channels is to minimize the endto-end MSD over all possible index assignments. Some famous index assignments, such as natural binary code (NBC), Gray code and randomly chosen index assignments, are studied on a binary symmetric channel (BSC) in [16][17][18][19][20][21]. In [16], it is proved that NBC is optimal for uniform scalar quantizers and uniform source.…”

Quantizers with uniform encoders and channel optimized decoders

“…that satisfy the centroid condition), for the Natural Binary Code (Theorem 7.2), the Folded Binary Code (Theorem 7.4), the Gray Code (Theorem 7.6), and for a randomly chosen index assignment (Theorem 7.8). Finally we extend the (uniform scalar quantizer) proof in [2] by showing that the Natural Binary Code is an optimal index assignment (Theorem 7.10).…”

“…Crimmins et al [1] studied the uniform scalar quantizer for the uniform source and proved the Yamaguchi-Huang assertion, that the Natural Binary Code is the best possible index assignment in the mean squared sense for the binary symmetric channel. McLaughlin, Neuhoff, and Ashley [2] generalized this result for certain uniform vector quantizers and uniform vector sources. Other than these papers, there are no others presently known in the literature giving index assignment optimality results.…”