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PARAMETRIC IIR FILTER DESIGN METHOD BASED ON ARBITRARY MAGNITUDE SPECIFICATION

IP.com Disclosure Number: IPCOM000007863D
Original Publication Date: 1996-Nov-01
Included in the Prior Art Database: 2002-Apr-30
Document File: 6 page(s) / 323K

Publishing Venue

Motorola

Related People

Dan Hoary: AUTHOR [+2]

Abstract

The purpose of the algorithm described in this paper is to design an IIR filter network which matches an arbitrary tiequency response, based on a user spec- ification, with minimal number of IIR sections. Since the goal is to use a minimum number of second order IIR sections, while maintaining a minimum error between the actual response and specification, the result approximates a minimum cost implemen- tation. Use of the IIR parametric filter [l] avoids stability and coefftcient scaling problems. Step 1 is an iterative error minimization process for obtaining an initial magnitude lit which agrees with the user specification. Reduction of the IIR filter network horn N fixed bands to M parametric bands, where M c N, makes up Step 2. Finally, Step 3 utilizes the Steepest Descent Method to minimize the error between actual response and the user specification. This procedure optimizes the complex frequency response (magnitude and phase) by finding the approximate location of the minimum on a multi- dimensional error surface. Minimiition is performed using the parametric IIR form (with respect to each section's center frequency, gain level and Q-factor). An example application of this would be to reduce the number of bands in a N section, lixed band equal- izer to an M section, parametric band equalizer, where again M < N.

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MOlVROLA Technical Developments

PARAMETRIC IIR FILTER DESIGN METHOD BASED ON ARBITRARY MAGNITUDE SPECIFICATION

by Dan Hoary and John Lane

tives of the error with respect to thefl, Q and gain parameters of the parametric form of the IIR filter and hence avoids stability and scaling problems. This leads to a direct implementation of the filter net- work with inherent stability (i.e., no possibility of poles outside of the unit circle).

INTRODUCTION

  The purpose of the algorithm described in this paper is to design an IIR filter network which matches an arbitrary tiequency response, based on a user spec- ification, with minimal number of IIR sections. Since the goal is to use a minimum number of second order IIR sections, while maintaining a minimum error between the actual response and specification, the result approximates a minimum cost implemen- tation. Use of the IIR parametric filter [l] avoids stability and coefftcient scaling problems. Step 1 is an iterative error minimization process for obtaining an initial magnitude lit which agrees with the user specification. Reduction of the IIR filter network horn N fixed bands to M parametric bands, where M c N, makes up Step 2. Finally, Step 3 utilizes the Steepest Descent Method to minimize the error between actual response and the user specification. This procedure optimizes the complex frequency response (magnitude and phase) by finding the approximate location of the minimum on a multi- dimensional error surface. Minimiition is performed using the parametric IIR form (with respect to each section's center frequency, gain level and Q-factor). An example application of this would be to reduce the number of bands in a N section, lixed band equal- izer to an M section, parametric band equalizer, where again M < N.

ALGORITHM DESCRIPTION

The present method is divided into three steps, as shown in Figure 1.

STEP1

  Starting from the user specified frequency and gain levels, each hequency-gain level pair is assigned to a single second order IIR bandpass section. An iterative algorithm is used to optimize gain levels (while center hequencies are kept fixed), to mini- mize the magnitude response error. The following recursion formula is used:

Gk(n) = Ck (n - 1) + fl e, (n) ek (n) = Gk - 20 bh&f@k~

where the G',(n) are the new gain levels calculated by Step 1 for the kth filter section, based on the set of user input specification gains, G,(n). !H(&j is the total transfer function magnitude response of the fil- ter network at hequency f, corresponding to the cen- ter frequency of the kth bandpass filter section. The above formulas are calculated recursively until the error ek(n) is reduced to some minimum threshold value, or for a prespecihed number ofiterations.

BACKGROUND

  A method of filter design using cascaded IIR filter sections was developed in 1970 by Kenneth Steiglitz [2]. The Steiglitz method uses a Steepest Descent search [3] on the magnitude ofthe filter trans- fer function. Als...