On the connection between multiplier word length limitation and roundoff noise in digital filters

It is shown that for digital filters there exists a direct and an indirect connection between the generation of roundoff noise by a multiplier and the effect that the coefficient word length limitation of this multiplier has upon the response characteristic: rounding can be interpreted as coefficient fluctuation, and any design method requiring fewer digits for the multipliers makes it possible to increase the signal word lengths without an increase in overall complexity. A mathematical formula expressing the noise-to-signal ratio due to roundoff noise generated by a sinusoidal signal, in terms of the attenuation sensitivity, is obtained. It confirms why digital filters with reduced attenuation sensitivity, such as wave digital filters, also produce less roundoff noise.