Analysis of quantization noise feedback in causal transform coding

The performances of the LDU (Lower-Diagonal-Upper (LDU) factorization) transform were recently shown to be equivalent to those of the Karhunen-Lo`eve Transform (KLT, which is optimal for Gaussian sources) in the limit of high rates [1, 2, 3]. In this paper, we further investigate the performances of the LDU for actual transform coding (TC) schemes. The results of [2] showed that the LDU should be implemented in closed loop around the quantizers, though this leads to a noise feedback effect, similar to that occuring in DPCM systems. We develop in this paper novel analyses of these effects on the distortion-rate functions and coding gains. The proposed analyses compare the results of [2] obtained for an hypothetical TC system for which the bit allocation is optimal and the rate is high, to those obtained for practical TC systems whose bit allocation is nearly optimal. By means of a theorem and numerical results, evidence is given that ordering the subsignals in the source vector by order of decreasing variance minimizes the quantization noise feedback. For the investigated practical systems, we show that deviations from the high rate assumptions arise below _ _ b/s. The effects of the noise feedback become non negligible below _ _ b/s. The LDU competes with the KLT above _ ___ b/s.