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

Abstract

Motivated by the fact that uniform quantization is not suitable for signals having non-uniform probability density functions (pdfs), as the Laplacian pdf is, in this paper we have divided the support region of the quantizer into two disjunctive regions and utilized the simplest uniform quantization with equal bit-rates within both regions. In particular, we assumed a narrow central granular region (CGR) covering the peak of the Laplacian pdf and a wider peripheral granular region (PGR) where the pdf is predominantly tailed. We performed optimization of the widths of CGR and PGR via distortion optimization per border–clipping threshold scaling ratio which resulted in an iterative formula enabling the parametrization of our piecewise uniform quantizer (PWUQ). For medium and high bit-rates, we demonstrated the convenience of our PWUQ over the uniform quantizer, paying special attention to the case where 99.99% of the signal amplitudes belong to the support region or clipping region. We believe that the resulting formulas for PWUQ design and performance assessment are greatly beneficial in neural networks where weights and activations are typically modelled by the Laplacian distribution, and where uniform quantization is commonly used to decrease memory footprint.

One-Bit Quantizer Parametrization for Arbitrary Laplacian Sources

“…As the pronounced peak is a specific feature of the Laplacian pdf, the largest number of samples x is concentrated around the mean value μ. Referring to our paper [14] we came to interesting conclusions for medium and high bit-rates when forming two granular regions for the restricted Laplacian pdf -Central Granular Region (CGR) and Peripheral Granular Region (PGR). We have shown that in general standsthe higher the bit rate, the higher the percentage of samples falls in the narrower CGR area.…”

“…Relying on a plethora of previous conclusions about uniform or nonuniform quantization [14], [19]- [21], further enhancements of one-bit quantizer parameterization are intuitively motivated by the better perceiving of the mean and variance for the arbitrary Laplacian pdf, particularly when the pdf of amplitudes being quantized was known in advance. Moreover, as a unique contribution of this paper we emphasize an analysis that outputs exact formulas for the simpler design and performance assessment of the one-bit quantizer.…”

Optimization and Performance Analysis of the 30-Bit Fixed-Point Digital Format

“…we need to know the PDF of the input data to determine the performance of the quantizer. In this paper, we will consider the Laplacian PDF which is widely used for statistical modeling of many types of data [11,12]. The Laplacian PDF is defined by the following expression [11]:…”

“…As the performance of the quantizers depends on the probability density function (PDF) of the input data, the accuracy of the digital representation also depends on the PDF of the input data. This paper considers the Laplacian PDF, which is widely used for statistical modeling of different types of data [11,12].…”