Reduction of l2-sensitivity for three-dimensional separable-denominator digital filters

The problem of reducing the deviation from a desired transfer function caused by the coefficient quantization errors is investigated for a three-dimensional (3-D) separable in denominator digital filter. To begin with, a 3-D transfer function with separable denominator is represented with the cascade connection of three one-dimensional (1-D) transfer functions by applying a minimal decomposition technique, and the multi-input multi-output (MIMO) 1-D transfer function located in the middle of the cascade connection is realized by a minimal state-space model. Next, the l
₂
-sensitivity of the state-space model is analyzed, and the minimization problem of the l
₂
-sensitivity subject to l
₂
-scaling constraints is formulated. This problem is then converted into an unconstrained optimization problem by using linear-algebraic techniques, and an efficient quasi-Newton algorithm is applied to solve it. A numerical example is presented to illustrate the validity and effectiveness of the proposed technique.