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Driven by the need for the compression of weights in neural networks (NNs), which is especially beneficial for edge devices with a constrained resource, and by the need to utilize the simplest possible quantization model, in this paper, we study the performance of three-bit post-training uniform quantization. The goal is to put various choices of the key parameter of the quantizer in question (support region threshold) in one place and provide a detailed overview of this choice’s impact on the performance of post-training quantization for the MNIST dataset. Specifically, we analyze whether it is possible to preserve the accuracy of the two NN models (MLP and CNN) to a great extent with the very simple three-bit uniform quantizer, regardless of the choice of the key parameter. Moreover, our goal is to answer the question of whether it is of the utmost importance in post-training three-bit uniform quantization, as it is in quantization, to determine the optimal support region threshold value of the quantizer to achieve some predefined accuracy of the quantized neural network (QNN). The results show that the choice of the support region threshold value of the three-bit uniform quantizer does not have such a strong impact on the accuracy of the QNNs, which is not the case with two-bit uniform post-training quantization, when applied in MLP for the same classification task. Accordingly, one can anticipate that due to this special property, the post-training quantization model in question can be greatly exploited.

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This paper proposes a low complex forward adaptive loss compression algorithm that works on the frame by frame basis. Particularly, the algorithm we propose performs frame by frame analysis of the input speech signal, estimates and quantizes the gain within the frames in order to enable the quantization by the forward adaptive piecewise linear optimal compandor. In comparison to the solution designed according to the G.711 standard, our algorithm provides not only higher level of the average signal to quantization noise ratio, but also performs a reduction of the PCM bit rate for about 1 bits/sample. Moreover, the algorithm we propose completely satisfies the G.712 standard, since it provides overreaching the curve defined by the G.712 standard in the whole of variance range. Accordingly, we can reasonably believe that our algorithm will find its practical implementation in the high quality coding of signals, represented with less than 8 bits/sample, which as well as speech signals follow Laplacian distribution and have the time varying variances. K e y w o r d s: forward adaptive technique, loss compression algorithm, piecewise linear optimal compandor

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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.

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