Sensitivity Analysis of Digital Filter Structures

A reasonable coefficient sensitivity measure for state space, recursive, finite wordlength, digital filters is the sum of the $L_2 $ norm of all first-order partial derivatives of the system function with respect to the system parameters. This measure is actually a linear lower bound approximation to the output quantization noise power. An important feature of this measure is that it can be broken down into evaluations of ARMA auto- and cross-covariance sequences, both of which can be computed efficiently and in closed form. This efficient closed form computation is a big improvement over the computational methods used by previous researchers. Their limited methods produced only approximations to the sensitivity measure and wasted computer time (i.e., these methods are open form solutions). The direct form II sensitivity, which is shown to be approx-imately inversely proportional to the sum of products of system pole and zero distances, can, as a result, usually be reduced by the judicious placement of added pole/zero cancellation pairs. These cancellation pairs provide extra degrees of freedom which are used to minimize the sensitivity measure while not affecting the system function. This new filter still has the convenient direct form II structure.