The Heisenberg Uncertainty Principle provides a well-known constraint on time-frequency analysis. A long history of coefficient and signal modeling errors exists for IIR (Infinite-duration Impulse Response) digital filters. Here, we use the Uncertainty Principle to show that the implemented filter’s wordlength is directly related to the output quantization noise power produced in scaled IIR filter implementations, and that this error power is directly related to a sensitivity measure developed by the first author [3]. The Uncertainty Principle ties the wordlengths of the IIR digital filter implementation to the normalizing frequency, fs. As the sampling frequency, fs, increases, the quality of the digital filter must correspondingly increase, because the passband bandwidth decreases. We know that this requires an increase in the order of the required filter. But, it also necessitates a more precise amplitude representation. We develop this relationship both theoretically and implementationally in this paper. We give a basic result that can be used to predict, and thus to compensate for, this problem. This prediction extends to any implementation structure. We find that for fixed point implementations of even low sample rate audio systems, the impact can be significant.