Frequency-domain model reduction approach to design IIR digital filters using orthonormal bases

Abstract

A new and numerically efficient technique for the design of reduced-order IIR digital filters is presented. This technique is based on computing the cross-gramian matrix from the state–space representation of the full-order filter in an arbitrary frequency range of operation. The modified cross-gramian is derived in the frequency domain and computed by solving only one Lyapunov equation. This technique does not require a minimal system to start with, and the reduced order can be obtained without calculating the conventional balancing transformation. Instead, orthonormal bases are computed to find the left and right eigenspaces associated with the large eigenvalues of the modified cross-gramian matrix. These eigenspaces are used to convert the high-order to a reduced-order models. The phase linearity is also discussed and it is shown that the proposed method can be used to transform a linear-phase FIR filter to a reduced-order IIR filter and the phase linearity is preserved over the interested frequency band, which is commonly be the passband. A comparison between the conventional time-domain methods such as balanced truncation (BT) and optimal Hankel-norm approximation (OHA); and the proposed frequency-domain model reduction technique is presented through several filters design examples to illustrate the effectiveness of the proposed technique.