Iterative algorithm for designing asymptotically optimal uniform scalar quantisation of the one‐sided Rayleigh density

Abstract

Design of optimal and asymptotically optimal quantisation subject to the mean squared error (MSE) criterion is a complex issue, even in the case of uniform scalar quantisation (USQ). The reason is that the MSE distortion dependence on the key designing parameter of USQ for source densities with infinite supports are complex and limit analytical optimisation of USQs. This issue of USQ design has been addressed for some source densities derived from the generalised gamma density. However, to the best of our knowledge, USQ for the one-sided Rayleigh density has not been studied in detail. This has prompted our research so that this study provides a detailed analysis of USQ for the one-sided Rayleigh density and proposes an iterative algorithm for its asymptotically optimal design. To estimate signal to quantisation noise ratio, we derive an asymptotic formula having reasonable accuracy for rates higher than 3 bits/sample. Our analysis can be useful in digitalto-analogue and analogue-to-digital conversion in diversity systems, orthogonal frequency division multiplexing systems and medical image processing. 1 INTRODUCTION Uniform scalar quantisation (USQ) is the earliest, the simplest and the most researched type of quantisation commonly studied for the source densities derived from the generalised gamma density (Gaussian, Laplacian, the two-sided Rayleigh density) [1-6]. Detailed analyses were conducted in the field of USQ for some of these densities. However, to the best of our knowledge, USQ for the one-sided Rayleigh density has not been thoroughly studied although this density is widely used to model: fading in diversity systems [7, 8], the amplitude of orthogonal frequency division multiplexing (OFDM) signals [9-13] and the noise variance in magnetic resonance imaging [14-16]. This implies that studying USQ for the one-sided Rayleigh density with the goal to maximise its performance would be important for digital-to-analogue and analogue-to-digital conversion (DAC/ADC) in diversity systems, OFDM systems and medical image processing. Also, as shown in [9-13], due to Rayleigh distribution of OFDM signals, the peak power can be much larger than the average power resulting in the high value of peak-to-average power ratio (PAPR) that can adversely affect the OFDM system. The value of PAPR can be reduced by using This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Whether the Support Region of Three-Bit Uniform Quantizer Has a Strong Impact on Post-Training Quantization for MNIST Dataset?

“…The uniform fixed-rate scalar quantizer, or briefly, uniform quantizer (UQ), is the simplest quantizer model, which, for a given bit rate, R is characterized with only one parameter—the support region threshold. However, its design, or determining the support region threshold to provide the smallest possible mean-squared error (MSE) distortion, is not simple for source densities with infinite support [ 22 , 25 , 35 , 36 ]. The reason is that expressions for the MSE distortion of UQ for source densities with infinite support are not simple, so the distortion minimization per support region threshold does not generally provide the derivation of a closed-form expression for the optimal support region threshold [ 22 , 35 , 36 ].…”

“…However, its design, or determining the support region threshold to provide the smallest possible mean-squared error (MSE) distortion, is not simple for source densities with infinite support [ 22 , 25 , 35 , 36 ]. The reason is that expressions for the MSE distortion of UQ for source densities with infinite support are not simple, so the distortion minimization per support region threshold does not generally provide the derivation of a closed-form expression for the optimal support region threshold [ 22 , 35 , 36 ]. Note that the above-mentioned problem is more prominent as R increases since the expression for the UQ performance assessment obtains more terms, and, accordingly, determining the optimal support region threshold becomes more complex.…”

“…Note that the above-mentioned problem is more prominent as R increases since the expression for the UQ performance assessment obtains more terms, and, accordingly, determining the optimal support region threshold becomes more complex. As highlighted in [ 34 , 35 ], this issue has greater importance in uniform quantization than in a nonuniform one. However, the question that arises in this paper is whether it is of the utmost importance to determine the optimal support region threshold value to achieve some predefined performance of post-training three-bit uniform quantization.…”

“…For symmetrical data distributions, symmetrical quantizers with an even number of quantization steps are preferable [ 14 , 24 , 25 , 26 , 27 , 28 , 29 , 32 , 34 , 35 , 38 , 40 ]. Since it can be expected that most of the real data is asymmetrical, one could not conjecture that the preferable quantizer is also asymmetrical.…”

Iterative Algorithm for Parameterization of Two-Region Piecewise Uniform Quantizer for the Laplacian Source

“…It is well known that a nonuniform quantizer model, well accommodated to the signal's amplitude dynamic and a nonuniform pdf, has lower quantization error compared to the uniform quantizer (UQ) model with an equal number of quantization levels or equal bit-rates [2,11,13,18,[20][21][22][23][24][25][26][27]. However, due to the fact that UQ is the simplest quantizer model, it has been intensively studied, for instance in [23,24,[28][29][30][31][32]. Moreover, the high complexity of nonuniform quantizers can outweigh the potential performance advantages over uniform quantizers [21].…”