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.