A statistical framework to minimise and predict the range values of quantisation errors in fixed-point FIR filters architectures

Filters datapath signals and coefficients are quantised when implemented in hardware to limit and reduce excessive hardware requirements. In this paper, we formulate quantisation errors (noises) in fixed-point arithmetic using a novel analytical model. The latter extends a conventional signal quantisation statistical model by assuming a Gaussian distribution noise. The paper gives the mathematical expressions to compute the statistical parameters and range values of the quantisation errors at any point in a multistage FIR filters structure depending on the wordlengths fractional precisions. Three case studies are included to vindicate the model's validity and accuracy in predicting the quantisation error parameters in the absence of filter coefficients quantisation. To counter the effects of the latter, we present a novel approach, called errors cancellation. The approach tends to represent the filter coefficients using different wordlengths to minimise the dynamic of the error filter's output. This allows limiting the quantisation effects to the signals quantisation only, which is statistically accurately modelled. The validity and the efficiency of the approach along with our analytical model are shown using two further case studies. Through our errors cancellation approach and analytical model, a hardware designer can now minimise the effects of the filter coefficients quantisation and predict subsequently the range values of the computation errors depending on the fractional precision used. He can also preset the latter to achieve the sought computations accuracy.