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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.

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 [

Xiao et al. [

The model reduction approach has also been proposed by various authors [

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

The proposed linear phase IIR filter design uses linear phase high pass and low pass FIR filters. To design a linear phase low pass IIR filter (

Desired low pass FIR filter characteristics to be used as numerator.

Desired high pass FIR filter characteristics to be used as denominator.

It is clear from (

From (

Detailed procedure to design FIR filters using orthogonal polynomials is discussed in [

Suppose user needs to design a linear phase FIR filter, as shown in Figure

We consider Legendre polynomials in the present discussion, though the procedure is general and can be applied on any orthogonal polynomial. Legendre polynomials are orthogonal between

The object function corresponding to Figure

Object function for LPF of Figure

It is not possible to calculate

We can design the linear phase high pass FIR filter in the same fashion.

Based on the method discussed above, we discuss the procedure to design the IIR filters in detail in the next section.

From here onwards, we represent

As discussed in the last section, low pass and high pass FIR filter characteristics are realized by their corresponding object functions—

Object function for HPF of Figure

The following steps outline the procedure to design the linear phase IIR filters.

Calculate the coefficients

We use finite number of

Polynomials

The discussion in the previous section makes it clear that

Calculate the rational function in

Calculate zeros of the IIR filter by solving

The transfer function of the resulting IIR filter is

To design high pass IIR filter, we have to divide high pass FIR filter characteristics by low pass FIR filter characteristics, that is,

To realize the proposed filter, it is necessary that the filter must be causal in nature. In other words, all the poles of IIR filter must lie within the unit circle. If some of the poles, (

To calculate the frequency response of a system having

Calculation of transfer function from pole-zero plot at frequency

This shift of poles, lying outside the unit circle, changes the phase and magnitude response slightly but on the other hand makes sure that resulting filter is stable in nature.

For clear understanding of the above procedure, we design an IIR filter in the next section.

Suppose we intend to design an IIR filter with following characteristics:

Therefore, as per the discussion of the previous section, first we have to design the low pass and high pass FIR filters. The, assumed, low pass and high pass FIR filter characteristics are as follows.

Low pass FIR (Figure

High pass FIR (Figure

Note that as of now, we show only positive half of the graph the negative half being a mirror image.

Let us design the IIR filter with 20 Legendre polynomial terms used to approximate the object functions (both for low pass and high pass FIR filter characteristics). After following the steps outlined in Section

Magnitude response in dB of low pass IIR filter corresponding to the object function approximated using

Phase response of low pass IIR filter corresponding to the object function approximated using

Let us look at the pole-zero distribution of this filter, which is shown in Figure

Pole-zero distribution of the IIR Filter.

To make sure that the proposed filter is stable, we shift all those poles which lie outside the unit circle to the origin.

Original distribution of poles and zeros is shown in Figure

Pole-zero distribution after shifting poles at origin.

Magnitude and phase response after shifting poles at origin.

Group delay after shifting poles at origin.

We observed that when we move a pole which is lying near the origin (but out of unit circle) to origin, frequency response changes at the point where transition band starts. While when we move the poles lying far away, there is a change near frequency zero of the frequency response. When all the poles of the frequency response are moved to the origin, the overaly effect is very small.

From Figure

Above discussion makes it clear that a postprocessing IIR filter with linear phase can easily be designed by using the orthogonal polynomials. The proposed IIR filter gives good cutoff characteristics. By increasing the number of polynomial terms, we can approximate our object function very closely, which in turn will produce good frequency characteristics both in the pass band and the transition region. The ripples in the pass band become negligible as we increase the number of terms to approximate our object function. Stop band amplitude decreases as we increase the number of terms in our object functions. In all, we may state that the alternate approach discussed in the present paper gives much easier design of IIR filter when compared with the currently available methods [

We are working to develop a mathematical model which can be used to predict the frequency response when poles are shifted from outside the unit circle to inside.

Suppose a polynomial

The mean squared error (MSE) polynomial between the original polynomial and approximated polynomial is given by

Note that

We need to find the coefficients

The authors would like to thank the reviewers for their remarks which helped them in improving the paper.