Optimal multiresolution quantization for broadcast channels with random index assignment

Abstract

Shannon's classical separation result holds only in the limit of infinite source code dimension and infinite channel code block length. In addition, Shannon theory does not address the design of good source codes when the probability of channel error is nonzero, which is inevitable for finite-length channel codes. Thus, for practical systems, a joint source and channel code design could improve performance for finite dimension source code and finite block length channel code, as well as complexity and delay. Consider a multicast system over a broadcast channel, where different end users typically have different capacities. To support such user or capacity diversity, it is desirable to encode the source to be broadcasted into a scalable bit stream along which multiple resolutions of the source can be reconstructed progressively from left to right. Such source coding technique is called multiresolution source coding. In wireless communications, joint source channel coding (JSCC) has attracted wide attention due to its adaptivity to timevarying channels. However, there are few works on joint source channel coding for network multicast, especially for the optimal source coding over broadcast channels.In this work, we aim at designing and analyzing the optimal multiresolution vector quantization (MRVQ) in conjunction with the subsequent broadcast channel over which the coded scalable bit stream would be transmitted. By adopting random index assignment (RIA) to link MRVQ for the source with superposition coding for the broadcast channel, we establish a closed-form formula of end-to-end distortion for a tandem system of MRVQ and a broadcast channel. From this formula we analyze the intrinsic structure of end-to-end distortion (EED) in a communication system and derive two necessary conditions for optimal multiresolution vector quantization over broadcast channels with random index assignment. According to the two necessary conditions, we propose a greedy iterative algorithm for jointly designed MRVQ with channel conditions, which depends on the channel only through several types of average channel error probabilities rather than the complete knowledge of the channel. Experiments show that MRVQ designed by the proposed algorithm significantly outperforms conventional MRVQ designed without channel information.By building an closed-form formula for the weighted EED with RIA, it also makes the computational complexity incurred during the performance analysis feasible. In comparison with MRVQ design for a fixed index assignment, the computation complexity for quantization design is significantly reduced by using random index assignment. In addition, simulations indicate that our proposed algorithm shows better robustness against channel mismatch than MRVQ design with a fixed index assignment, simply due to the nature of using only the average channel information. Therefore, we conclude that our proposed algorithm is more appropriate in both wireless communications and applications where the complete knowledge of the...

Designing Optimal Multiresolution Quantizers with Error Detecting Codes

“…Remark 2: (8) and (9) serve as a generalization of the scenario without error detection considered in [16]. The substitution of p bd = p d1 = p d2 = p ud = 0 indeed simplifies (8) and (9) to the form in [16] without presence of error detection to imply that all errors are undetectable.…”

“…However, their analytical results were for vector quantizers that were designed independent of channel statistics in the high-rate asymptotic case. On the other hand, efforts by Yu et al and Teng et al in applying RIA led to the derivation of closed-form nonasymptotic EED formulae for both the point-to-point [15] and broadcast [16] channels. With the closed-form EED formula, theoretical analysis of optimal noisy channel quantizers became tractable, and algorithm design required only the average channel error probability, as opposed to the entire matrix of transitional probabilities necessary for the fixed case.…”

“…Under RIA, formulations in [15] and [16] further attributed a large 1536-1276/13$31.00 c 2013 IEEE portion of the distortion contribution from channel errors to a structural parameter named the scatter factor of the noisy channel quantizer. It was demonstrated in [15] and [16] that under RIA, the design of the optimal noisy channel quantizer became a tradeoff between balancing distortion contributions from the scatter factor and the quantization itself.…”

“…Under RIA, formulations in [15] and [16] further attributed a large 1536-1276/13$31.00 c 2013 IEEE portion of the distortion contribution from channel errors to a structural parameter named the scatter factor of the noisy channel quantizer. It was demonstrated in [15] and [16] that under RIA, the design of the optimal noisy channel quantizer became a tradeoff between balancing distortion contributions from the scatter factor and the quantization itself. The result of considering such a tradeoff led to significantly reduced EED in comparison to a system employing quantizers designed via Lloyd-Max. Although an optimal noisy channel quantizer partially mitigates distortion contribution from the scatter factor at the expense of larger quantization distortions, it cannot entirely eliminate the effect of the scatter factor on the EED.…”

“…1. Like [15] and [16], RIA is adopted to link MRVQ at the source with superposition coding (SPC) at the CRC-coded broadcast channel. Our contributions can be summarized as follows.…”