Low roundoff noise augmented IIR filters

The scaling factor in a direct form II infinite impulse response (IIR) digital filters depends on the system poles but not on its zeros. Adding pole-zero cancellation pairs can change the scaling factor and the associated roundoff noise without affecting the desired transfer function. Under l₂ scaling, the upper bound of the noise reduction of direct form II IIR filters augmented with K pole-zero cancellation pairs is found to be about 6 K dB. The above noise reduction is applied to parallel form, where each subfilter is augmented with one pole-zero cancellation pair and then realized in direct form II. An illustrative example shows that the roundoff noise of the resulting augmented realization is lower than that of optimal second-order state-space structure for lowpass elliptic filters with moderate passbands. This augmented realization requires about 3N multipliers for an Nth-order IIR filter. Comparison to the 2N multipliers of parallel form and the 4N multipliers of optimal second-order state-space structure shows that the additional computational complexity of this augmented realization is moderate, and is often well justified by the roundoff noise reduction.