In the present paper, we discuss a method to design a linear phase 1-dimensional Infinite Impulse Response (IIR) filter using orthogonal polynomials. The filter is designed using a set of object functions. These object functions are realized using a set of orthogonal polynomials. The method includes placement of zeros and poles in such a way that the amplitude characteristics are not changed while we change the phase characteristics of the resulting IIR filter. 1. Introduction In the past two to two and half decades, a great deal of work has been carried out in the field of design of linear phase IIR filters. In general, designing exact linear phase IIR filter is not possible, schemes have been proposed to approximate pass band linearity. Conventionally, first the magnitude specifications of an IIR filter are met, and then all pass equalizers are applied to linearize the phase response [1, 2]. Mostly IIR filters are designed with equiripple or maximally flat group delay [3]. But their magnitude characteristics are poor. Optimization techniques are used to simultaneously approximate magnitude and phase response characteristics [4, 5]. To meet with the magnitude and phase characteristics at the same time, generally, linear programming is used [6]. To directly design a linear phase IIR filter, Lu et al. [7] give an iterative procedure, it is based on a weighted least-squares algorithm. Xiao et al. [8] discusse a method to design a linear phase IIR filter with frequency weighted least-square error optimization using Broyden-Fletcher-Goldfarb-Shanno (BFGS) [9] method. The model reduction approach has also been proposed by various authors [10, 11]. A procedure to design linear phase IIR filter from linear phase FIR filter has been discussed by Holford et al. [12] using frequency weighting model reduction for highly selective filters. Holford et al. [12] gives good compromise for order of the filter, pass band maximum ripple, and stop band minimum attenuation. The present paper discusses a technique to design IIR filters with approximately linear phase. An algorithm is presented to design such a filter. The algorithms have been discussed stepwise to make sure that any person with basic programming capabilities can easily design them. We have not used any standard routine of any particular platform; therefore, any freely available programming platform (like C, C++, Scilab, Octave, etc.) can be used to design these filters. The paper is divided into 5 sections. Section 1 is an introduction to the already existing techniques, Section 2 discusses the preliminary
References
[1]
A. Antoniou, Digital Filters: Analysis, Design, and Applications, McGraw-Hill, New York, NY, USA, 2nd edition, 1993.
[2]
C. Charalambous and A. Antoniou, “Equalization of recursive digital filters,” IEE Proceedings Part G, vol. 127, no. 5, pp. 219–225, 1980.
[3]
J. P. Thiran, “Recursive digital filters with maximally flat group delay,” IEEE Trans Circuit Theory, vol. 18, no. 6, pp. 659–664, 1971.
[4]
G. Cortelazzo and M. R. Lightner, “Simultaneous design in both magnitude and group delay of iir and fir filters based on multiple criterion optimization,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 32, no. 5, pp. 949–967, 1984.
[5]
H. Baher, “Digital filters with finite-band approximation to constant amplitude and delay, and arbitrary selectivity,” in Proceeding of the IEEE International Symposium on Circuits ans Systems, pp. 657–659, 1986.
[6]
A. T. Chottera and G. A. Jullien, “A linear programming approach to recursive digital filter design with linear phase,” IEEE Transactions on Circuits and Systems, vol. 29, no. 3, pp. 139–149, 1982.
[7]
W. S. Lu, S. C. Pei, and C. C. Tseng, “A weighted least-squares method for the design of stable 1-d and 2-d iir digital filters,” IEEE Transactions on Signal Processing, vol. 46, no. 1, pp. 1–10, 1998.
[8]
C. Xiao, J. C. Olivier, and P. Agathoklis, “Design of linear phase IIR filters via weighted least-squares approximation,” in Proceedings of the IEEE Interntional Conference on Acoustics, Speech, and Signal Processing (ICASSP '01), vol. 6, pp. 3817–3820, May 2001.
[9]
R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, New York, NY, USA, 2nd edition, 1987.
[10]
B. Beliczynski, I. Kale, and G. D. Cain, “Approximation of FIR by IIR digital filters: an algorithm based on balanced model reduction,” IEEE Transactions on Signal Processing, vol. 40, no. 3, pp. 532–542, 1992.
[11]
M. F. Fahmy, Y. M. Yassin, G. Abdel-Raheem, and N. El-Gayed, “Design of linear-phase IIR filters from FIR specifications,” IEEE Transactions on Signal Processing, vol. 42, no. 2, pp. 437–440, 1994.
[12]
S. Holford and P. Agathoklis, “The use of model reduction techniques for designing iir filters with linear phase in the passband,” IEEE Transactions on Signal Processing, vol. 44, no. 10, pp. 2396–2404, 1996.
[13]
S. Bhooshan and V. Kumar, “A new method to design linear phase FIR filter,” in Proceedings of the IEEE International Conference on Industrial Technology, pp. 1–6, April 2008.
[14]
S. Bhooshan and V. Kumar, “A polynomial approach towards the design of linear phase FIR filters,” in Proceedings of the IEEE International Conference on Industrial Technology, pp. 632–636, December 2006.
[15]
S. Bhooshan and V. Kumar, “Design of chebyshev fir filter based on antenna theory approach,” WSEAS Transactions on Signal Processing, vol. 3, no. 2, pp. 179–184, 2007.
[16]
P. Hardt, “Understanding poles and zeros,” http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf.
[17]
V. Sreeram and P. Agathoklis, “Design of linear-phase IIR filters via impulse-response gramians,” IEEE Transactions on Signal Processing, vol. 40, no. 2, pp. 389–394, 1992.
[18]
L. Li, L. Xie, W. Y. Yan, and Y. C. Soh, “Design of low-order linear-phase IIR filters via orthogonal projection,” IEEE Transactions on Signal Processing, vol. 47, no. 2, pp. 448–457, 1999.
