ESTIMATION OF MEAN SQUARED ERRORS OF NON-BINARY A/D ENCODERS THROUGH FREDHOLM DETERMINANTS OF PIECEWISE-LINEAR TRANSFORMATIONS II: GENERAL CASE

Abstract We conduct a theoretical analysis of the performance of $\beta $ -encoders. The $\beta $ -encoders are A/D (analogue-to-digital) encoders, the design of which is based on the expansion of real numbers with noninteger radix. For the practical use of such encoders, it is important to have theoretical upper bounds of their errors. We investigate the generating function of the Perron–Frobenius operator of the corresponding one-dimensional map and deduce the invariant measure of it. Using this, we derive an approximate value of the upper bound of the mean squared error of the quantization process of such encoders. We also discuss the results from a numerical viewpoint.