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.

[1]  H. Nicholson,et al.  On the structure of balanced and other principal representations of SISO systems , 1983 .

[2]  Y. Shamash.  Model reduction using the Routh stability criterion and the Padé approximation technique , 1975 .

[3]  R. Aldhaheri.  Model order reduction via real Schur-form decomposition , 1991 .

[4]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[5]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[6]  Balanced realization of non-minimal MIMO discrete-time systems , 1994 .

[7]  Hassan K. Khalil,et al.  Singular perturbation methods in control : analysis and design , 1986 .

[8]  John G. Proakis,et al.  Digital Signal Processing: Principles, Algorithms, and Applications , 1992 .

[9]  Dali Wang,et al.  Model reduction of discrete linear systems via frequency-domain balanced structure , 2000 .

[10]  Bor-Sen Chen,et al.  IIR filter design via optimal Hankel-norm approximation , 1992 .

[11]  Venkatappa Sreeram,et al.  Design of linear-phase IIR filters via impulse-response gramians , 1992, IEEE Trans. Signal Process..

[12]  Rabah W. Aldhaheri.  Design of linear-phase IIR digital filters using singular perturbational model reduction , 2000 .

[13]  H. Nicholson,et al.  On the cross-Gramian for symmetric MIMO systems , 1985 .

[14]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .

[15]  I. Kale,et al.  FIR filter order reduction: balanced model truncation and Hankel-norm optimal approximation , 1994 .

[16]  Wei-Yong Yan,et al.  Design of low-order linear-phase IIR filters via orthogonal projection , 1999, IEEE Trans. Signal Process..

[17]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[18]  G.D. Cain,et al.  Approximation of FIR by IIR digital filters: an algorithm based on balanced model reduction , 1992, IEEE Trans. Signal Process..