Error spectrum shaping quantizers with non-ideal reconstruction filters and saturating quantizers

Abstract

Error spectrum shaping (ESS) quantizers reduce the quantization error of oversampled signals by using quantizer error feedback to position the quantizer error power outside the frequency band of interest. Previous analyses of ESS quantizers do not consider either the limitations of realizable reconstruction filters or the effects of quantizer saturation. Consequently, predictions of ESS distortion were often'optimistic. This paper extends the original analysis to include the use of non-ideal reconstruction filters. Simulations show that this extended analysis predicts accurately the distortion of non-saturating ESS quantizers. An attempt to account analytically for saturation effects led to a difficult non-linear design problem. A constraint on the energy of the feedback filter impulse response was effective in suppressing saturation and led to a straightforward linear optimization problem.

On the DPCM compression of Gaussian autoregressive sequences

“…However, it can be seen from Figure 6 that the standard system puts more of the quantization noise to these bands. As noted by [12], putting more of the quantization noise to suppressed bands, while reducing the quantization distortion, has the unfortunate effect of increasing |C(w)| 2 σ 2 q (and hence, the rate) and a compromise is necessary. Note that the formulation presented in this paper results in the rate-distortion optimal compromise.…”

“…• Fixed-rate quantization vs. entropy coded uniform quantization: DPCM variants incorporating fixed-rate quantizers (i.e., quantizers that utilize a fixed number of levels without entropy coding) have been analyzed [6,9,12,13]. In principle, one can formulate the ratedistortion optimization for these systems as a distortion minimization subject to a given target rate constraint (as determined by the number of quantizer levels).…”

“…The quantized signal is then filtered by an ideal postfilter at the decoder in order to eliminate the out-of-band quantization error. As pointed out by [12], the procedure has practical shortcomings because ideal postfilters at the decoder are not feasible and quantization saturation effects at the encoder have to be accounted for, i.e., assuming a practical quantizer with a fixed number of levels the magnitude of the feedback term should not become arbitrarily large.…”

“…It should also be noted that[12] restricts the search for the shaping filter to FIR filters as opposed to the more general formulation presented in this paper.…”

“…If one allows a fixed-rate uniform quantizer (N levels, ∆ stepsize) and assumes that a granular region the size of a given multiple (say 2k) of the input standard deviation is sufficient to ignore overload distortions, then the rate constraint can be established by 2kσp/∆ = N which will lead to a rate-distortion optimization formulation similar to this paper 11. As mentioned in Section 2.2,[12] tries to curb saturation effects by heuristically constraining the squared magnitude of the shaping filter. Note that this procedure will still not lead to the identified trade-off as the constraint is not established on the prefilter.…”