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Infinite Impulse Response (IIR) l_p design

Module by: Ricardo Vargas. E-mail the author

In contrast to FIR filters, an Infinite Impulse Response (IIR) filter is defined by two ordered vectors aRNaRN and bRM+1bRM+1 (where 0<M,N<0<M,N<), with frequency response given by

H ( ω ) = B ( ω ) A ( ω ) = n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n H ( ω ) = B ( ω ) A ( ω ) = n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n
(1)

Hence the general lplp approximation problem is

min a n , b n n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n - D ( ω ) p min a n , b n n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n - D ( ω ) p
(2)

which can be posed as a weighted least squares problem of the form

min a n , b n w ( ω ) · n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n - D ( ω ) 2 2 min a n , b n w ( ω ) · n = 0 M b n e - j ω n 1 + n = 1 N a n e - j ω n - D ( ω ) 2 2
(3)

It is possible to design similar problems to the ones outlined in (Reference) for FIR filters. However, it is worth keeping in mind the additonal complications that IIR design involves, including the nonlinear least squares problem presented in Section 1 below.

Least squares IIR literature review

The weighted nonlinear formulation presented in Equation 3 suggests the possibility of taking advantage of the flexibilities in design from the FIR problems. However this point comes at the expense of having to solve at each iteration a weighted nonlinear l2l2 problem. Solving least squares approximations with rational functions is a nontrivial problem that has been studied extensively in diverse areas including statistics, applied mathematics and electrical engineering. One of the contributions of this document is a presentation in (Reference) on the subject of l2l2 IIR filter design that captures and organizes previous relevant work. It also sets the framework for the proposed methods used in this document.

In the context of IIR digital filters there are three main groups of approaches to Equation 3. (Reference) presents relevant work in the form of traditional optimization techniques. These are methods derived mainly from the applied mathematics community and are in general efficient and well understood. However the generality of such methods occasionally comes at the expense of being inefficient for some particular problems. Among the methods found in literature, the Davidon-Flecther-Powell (DFP) algorithm [3], the damped Gauss-Newton method [2], [17], the Levenberg-Marquardt algorithm [14], [16], and the method of Kumaresan [6], [4] form the basis of a number of methods to solve Equation 2.

A different approach to Equation 2 from traditional optimization methods consists in linearizing


Equation 3 by transforming the problem into a simpler, linear form. While in principle this proposition seems inadequate (as the original problem is being transformed), (Reference) presents some logical attemps at linearizing Equation 3 and how they connect with the original problem. The concept of equation error (a weighted form of the solution error that one is actually interested in solving) has been introduced and employed by a number of authors. In the context of filter design, E. Levy [8] presented an equation error linearization formulation in 1959 applied to analog filters. An alternative equation error approach presented by C. S. Burrus [11] in 1987 is based on the methods by Prony [1] and Pade [10]. The method by Burrus can be applied to frequency domain digital filter design, and is used in selected stages in some of the algorithms presented in this work.

An extension of the equation error methods is the group of iterative prefiltering algorithms presented in (Reference). These methods build on equation error methods by weighting (or prefiltering) their equation error formulation iteratively, with the intention to converge to the minimum of the solution error. Sanathanan and Koerner [12] presented in 1963 an algorithm (SK) that builds on an extension of Levy's method by iterating on Levy's formulation. Sid-Ahmed, Chottera and Jullien [9] presented in 1978 a similar algorithm to the SK method but applied to the digital filter problem.

A popular and well understood method is the one by Steiglitz and McBride [7], [13] introduced in 1966. The SMB method is time-domain based, and has been extended to a number of applications, including the frequency domain filter design problem [15]. Steiglitz and McBride used a two-phase method based on linearization. Initially (in Mode-1) their algorithm is essentially that of Sanathanan and Koerner but in time. This approach often diverges when close to the solution; therefore their method can optionally switch to Mode-2, where a more traditional derivative-based approach is used.

A more recent linearization algorithm was presented by L. Jackson [5] in 2008. His approach is an iterative prefiltering method based directly in frequency domain, and uses diagonalization of certain matrices for efficiency.

While intuitive and relatively efficient, most linearization methods share a common problem: they often diverge close to the solution (this effect has been noted by a number of authors; a thorough review is presented in [15]). (Reference) presents the quasilinearization method derived by A. Soewito [15] in 1990. This algorithm is robust, efficient and well-tailored for the least squares IIR problem, and is the method of choice for this work.

References

1. de Prony, Baron (Gaspard Clair Francois Marie Riche). (1795). Essai Experimental et Analytique: Sur les lois de la Dilatabilite des fluides elastiques et sur celles de la Force expansive de la vapeur de l'eau et de la vapeur de l'alkool, a differentes temperatures. Journal de l'Ecole Polytechnique (Paris), 1(2), 24-76.
2. Dennis, J. E. and Schnabel, R. B. (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Philadelphia, PA: SIAM.
3. Fletcher, R. and Powell, M. J. D. (1963). A Rapidly Convergent Descent Method for Minimization. Computer Journal, 6(2), 163-168.
4. Hildebrand, F. B. (1974). Introduction to Numerical Analysis. McGraw-Hill.
5. Jackson, Leland B. (2008). Frequency-Domain Steiglitz-McBride Method for Least-Squares IIR Filter Design, ARMA Modeling, and Periodogram Smoothing. IEEE Signal Processing Letters, 15, 49-52.
6. Kumaresan, R. and Burrus, C. S. (1991). Fitting a Pole-Zero Filter Model to Arbitrary Frequency Response Samples. Proc. ASILOMAR, 1649-1652.
7. L. E.McBride, H. W. Schaefgen and Steiglitz, K. (1966, December). Time-Domain Approximation by Iterative Methods. IEEE Transactions on Circuit Theory, CT-13(4), 381-87.
8. Levy, E. C. (1959, May). Complex-Curve Fitting. IRE Transactions on Automatic Control, AC-4(1), 37-43.
9. M. A. Sid-Ahmed, A. Chottera and Jullien, G. A. (1978, October). Computational Techniques for Least-Square Design of Recursive Digital Filters. IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-26(5), 477-480.
10. Pade, Henri Eugene. (1892). Sur la Representation Approchee d'une Fonction par des Fractions Rationnelles. Annales Scientifiques de L'Ecole Normale Superieure (Paris), 9(3), 1-98.
11. Parks, T. W. and Burrus, C. S. (1987). Digital Filter Design. John Wiley and Sons.
12. Sanathanan, C. K. and Koerner, J. (1963, January). Transfer Function Synthesis as a Ratio of Two Complex Polynomials. IEEE Transactions on Automatic Control, AC-8, 56-58.
13. Steiglitz, K. and McBride, L. E. (1965, October). A Technique for the Identification of Linear Systems. IEEE Transactions on Automatic Control, AC-10, 461-64.
14. Spanos, J. T. and Mingori, D. L. (1993, January). Newton Algorithm for Fitting Transfer Functions to Frequency Measurements. Journal of Guidance, Control and Dynamics, 16(1), 34-39.
15. Soewito, Atmadji W. (1990, December). Least Square Digital Filter Design in the Frequency Domain. PhD. Thesis. Rice University.
16. Sorensen, D. C. (1982). Newton's Method with a Dodel Trust Region Modification. SIAM Journal of Numerical Analysis, 16, 409-426.
17. T. P. Krauss, L. Shure et al. (1994). 2. Signal Processing Toolbox User's Guide. (pp. 143-145). The MathWorks.

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