Error analysis of recursive digital filters implemented with logarithmic number systems

This paper presents an analysis of roundoff error accumulation in the direct realization of a logarithmic-arithmetic digital filter and formulas for computing the ratio of the rms value of the roundoff error and that of the output signal of the digital filter. It is shown that for the case of stochastic input, the theoretical error to signal ratio is less using logarithmic number systems as compared to using floating-point number systems. The theoretical results of the roundoff error analysis were verified by simulation experiments. The experimental results demonstrate that a logarithmic number system gives filtering performance superior to that of a floating-point system of equivalent word length and range, and that the limitation in implementing digital filters with logarithmic number systems is a function of the range of the numbers which may be represented.