The PLSI Method of Stabilizing Two-Dimensional Nonsymmetric Half-Plane Recursive Digital Filters

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1 EURASIP Journal on Applied Signal Processing 23:9, c 23 Hindawi Pulishing Corporation The PLSI Method of Stailizing Two-Dimensional Nonsymmetric Half-Plane Recursive Digital Filters N. Gangatharan Faculty of Engineering, Multimedia University, Jalan Multimedia, 631 Cyerjaya, Selangor, Malaysia n.gangatharan@mmu.edu.my P. S. Reddy Faculty of Engineering, Multimedia University, Jalan Multimedia, 631 Cyerjaya, Selangor, Malaysia suarami.reddy@mmu.edu.my Received 19 April 22 and in revised form 19 Feruary 23 Two-dimensional 2D recursive digital filters find applications in image processing as in medical X-ray processing. Nonsymmetric half-plane NSHP filters have definitely positive magnitude characteristics as opposed to quarter-plane QP filters. In this paper, we provide methods for stailizing the given 2D NSHP polynomial y the planar least squares inverse PLSI method. We have proved in this paper that if the given 2D unstale NSHP polynomial and its PLSI are of the same degree, the PLSI polynomial is always stale, irrespective of whether the coefficients of the given polynomial have relationship among its coefficients or not. Examples are given for 2D first-order and second-order cases to prove our results. The generalization is done for the Nth order polynomial. Keywords and phrases: PLSI, NSHP, staility, Lagrange, autocorrelation. 1. INTRODUCTION The two-dimensional 2D filters find numerous applications like in image processing, seismic record processing, medical X-ray processing, and so forth. The nonsymmetric half-plane NSHP 2D recursive filters have assured positive magnitude characteristics and so they are preferred to quarter-plane QP filters. But the design of stale recursive NSHP filters had een a difficult and sometimes more timeconsuming prolem. In this paper, we deal with the prolem of stailizing unstale NSHP 2D recursive filters y the planar least squares inverse PLSI approach. The stailization of one-dimensional 1D recursive digital filters using least squares inverse LSI approach is well known [1]. However, the conjecture, made y Shanks et al. [2], known as Shanks conjecture is yet to e proved or totally disproved. Genin and Kamp [3] were the first to give a counterexample showing that the Shanks conjecture, which says that 1D technique of stailizing can e extended to 2D case, also fails. They have taken the original unstale 2D polynomial to e of degree three in oth the variales, and the corresponding PLSI polynomial of degree one was found to e unstale. Later they have produced three more counterexamples of that kind [4], where the chosen PLSI polynomial is of lower degree than the degree of the original unstale 2D polynomial. Susequently, the modified form of the Shanks conjecture [5], which says that the PLSI polynomial should e of the same degree as the original unstale polynomial, has also een proven to e not true [6]. Two of the methods which were extensions from the 1D system theory to 2D case for stailizing 2D recursive digital filters namely, the discrete Hilert transform DHT method and the PLSI method have een left unsolved and not much work was reported on these in 198s. Recently, in [7, 8] the prolem facing the DHT method was resolved. Now it is clear that the DHT method of stailizing unstale filters works only when the original unstale polynomial is devoid of zeros on the unit circle in the 1D case and on the unit icircle in the 2D case. In fact, in [8], a new method of stailizing multidimensional N > 2 recursive digital filters has een presented. This new method oils down to the same DHT method when applied to 1D and 2D filters. More recently, in [9] a complete solution to the PLSI method of stailization was reported. It was proved that a restriction of the kind imposed on the DHT method on the original 2D polynomial is not needed for the PLSI method to work, ut a different type of restriction is necessary for the PLSI method, especially for QP filters. In this paper, we present a method of stailizing the given

2 The PLSI Method of Stailizing NSHP Recursive Digital Filters 915 2D NSHP unstale polynomial through the PLSI polynomial approach. It is interesting to note that the PLSI polynomial is always stale provided that the degree of the PLSI polynomial is the same as the given polynomial, whatever may e the relationship among the coefficients in the given polynomial. Definition 1.A2DNSHPpolynomialofdegreeN is given y AZ 1,Z 2 N N R+ a mn Z m 1 Z n 2,whereR + {m, n } {m>, n<}. The main difference etween QP and NSHP filters comes y the way in which the output masks are defined. The output mask of NSHP is more general than that of QP filters. Hence NSHP filters will e superior to QP filters. Based on the region of support R, there are eight classes of NSHP filters. However, all our discussions are ased on R + filter [1]. In Section 2, we discuss the asic definition of PLSI polynomial for the QP polynomial and NSHP polynomial. We then riefly mention the method of otaining the PLSI polynomials. In Section 3, we discuss the existence of maximum for the first order and the second order which in turn results in stale polynomial. In Section 4,we present numerical examples for the first order and second order and prove that the PLSI is stale. 2. OBTAINING PLSI POLYNOMIAL FOR NSHP POLYNOMIALS In this section, we discuss the asic definition of the PLSI polynomials and the method for otaining the PLSI in general. Definition 2. In the case of QP filters, if AZ 1,Z 2 Ni Nj a ij Z1Z i j 2 is a given 2D polynomial of degree N, then the polynomial BZ 1,Z 2 M Mj i ij Z1Z i j 2 of the degree M forms the PLSI of AZ 1,Z 2 if B Z 1,Z 2 1 A Z 1,Z 2 for Z 1 1, Z To otain BZ 1,Z 2, just like in 1D case, we first form C Z 1,Z 2 A Z1,Z 2 B Z1,Z and then we form an error function see [11] E from CZ 1,Z 2, where C M+N Z 1,Z 2 i M+N j c ij Z i 1Z j algeraic equations of the form T a. 5 In 5, T is a square matrix of order M +1 M +1 made up of the 2D autocorrelation functions of AZ 1,Z 2 as its elements, is an M +1 1columnmatrixofcoefficients of BZ 1,Z 2 like {,,..., mm } t, 6 and a is also an M +1 1columnmatrixlike Definition 3. A 2D NSHP polynomial a { a,,,..., } t. 7 B N N Z 1,Z 2 mn Z1 m Z2 n 8 R+ of degree N is the PLSI of AZ 1,Z 2 N N R+ a mn Z m 1 Z n 2, also of degree N, wherer + {m, n } {m>, n<} if 1holds. It may e noted that the coefficients mn s of BZ 1,Z 2 can e otained y solving 5 for the vector, ut the formulation of 5 as explained in Section 1 is rather very tedious, especially for larger values of M and N. Here, we indicate the method see [12] that can e used to form 5 y using form-preserving 1D polynomials. Definition 4. A1DpolynomialA 1 z N k a k Z k is the formpreserving polynomial of a 2D QP polynomial AZ 1,Z 2 : N 2 A N 1 Z 1,Z 2 a mn Z1 m Z2 n 9 m n if for every integer set m, ninaz 1,Z 2 there exists a unique k such that a k a mn. Ithaseenprovedin[13] that A 1 Z AZ L 1,Zforms the form-preserving polynomial of AZ 1,Z 2 ifl N It has also een proved in [13] that if BZ 1,Z 2 andaz 1,Z 2 are two 2D polynomials as defined in Definition 2, then if C Z 1,Z 2 A Z1,Z 2 B Z1,Z 2, 1 as E 2 1 c + cij. 2 4 i j CZ CZ L,Z will e a 1D form-preserving polynomial of CZ 1,Z 2 if L N + M We then differentiate E with respect to each unknown coefficient ij and equate each E/ ij to zero to get a set of linear We quoted this concept of form-preserving polynomials ecause later we are going to use these form preserving 1D

3 916 EURASIP Journal on Applied Signal Processing polynomials for formulating the matrix T of 5 as well as for testing the staility or instaility of the PLSI polynomials. Theorem 1. If BZ 1,Z 2 is the PLSI polynomial of AZ 1,Z 2, then B 1 Z is the LSI of A 1 Z if L M+N +1,whereB 1 Z BZ L,Z and A 1 Z AZ L,Z. The aove theorem has een proved in [13]. Otaining the coefficients of 1D LSI polynomial B 1 Z corresponding to the 1D polynomial A 1 Z is very easy. Since T matrix canemechanicallywrittendownintermsofthecoefficients A 1 Z [1], this method of deriving the 2D PLSI polynomial BZ 1,Z 2, corresponding to the given 2D polynomial AZ 1,Z 2, is highly recommended. Once we form the T matrix of 5, of course after deleting certain rows and columns of a corresponding coefficient matrix pertaining to A 1 Z, we can easily solve 5for, the column vector of coefficients of BZ 1,Z 2. We now elaorate on the deletion of certain rows and corresponding columns from the coefficient matrix, mechanically written from the coefficients of the polynomial A 1 Z. We oviously know that B 1 Z which is a form-preserving polynomial of BZ 1,Z 2 will e lacunary with some terms corresponding to certain powers of Z eing asent. This is ecause B 1 ZiswhatwegetasBZ L,Z, with L value eing M + N +1whichismuchmorethanM +1.Sowhenwe frame the matrix equation mechanically for A 1 Z and the corresponding LSI B 1 Z of the following type: T 1 1 a 1, 12 the column vector 1 does contain some zeros; and while arriving at 5, we have to delete some rows and corresponding columns of T 1,somerowsof 1 corresponding to zero coefficients in B 1 Z, and some rows of a 1. Also if N>M, the last N M rows and corresponding columns of T 1 are to e deleted. The minimum error see [1] is E min 1 a OBTAINING PLSI POLYNOMIAL FOR FIRST-ORDER NSHP POLYNOMIALS Theorem 2. If BZ 1,Z 2 N N R+ mn Z1 m Z2 n is a 2D NSHP polynomial of degree N, then its form-preserving 1D polynomial B 1 Z BZ L,Z, when L 4N +1,willhavethesame autocorrelation coefficients as the BZ 1,Z 2. This theorem can e used to our advantage whenever we want to form the autocorrelation coefficients of a 2D polynomial since otaining these coefficients from 1D polynomial B 1 Z is very simple. What we get from 5 will e the same as what we get from the 1D polynomial B 1 Z y deleting proper rows and columns from 12as mentioned earlier. Example 1. Let BZ 1,Z R+ mn Z m 1 Z n 2 e a 2D firstorder NSHP polynomial. This can e written as follows: B Z 1,Z 2 + Z Z Z 1 Z 2 + Z 1 Z Then, B 1 Z BZ 4N+1,Z BZ 5,Zwilleequalto B 1 Z + Z +++Z Z Z The autocorrelation coefficients r j s of B 1 Z canewritten down as follows: r, r 1, 11 r 2, 1 r 3, + 1 r 4, r 5, 11 r 6, where indicates the equations that do not contain. 16 The autocorrelation equations given in 16 are seven in numer. It may e noted that two of these equations do not contain the constant coefficients of BZ 1,Z 2. It is easy to verify that BZ 1,Z 2 has the same autocorrelation coefficients as in 16. In general, for a polynomial of Nth degree in oth variales, 2N 2 numer of autocorrelation equations does not contain the constant coefficient out of the total 4N 2 + 2N +2equations. The following theorem is proved in [14]. Theorem 3. A 2D first-quadrant polynomial BZ 1,Z 2 of degree N is stale if and only if BZ 2N+1,Z is stale. It may e noted that if, in BZ 1,Z 2, the degree of Z 1 is M and of Z 2 is N, then BZ 1,Z 2 is stale if and only if BZ L,Z, where L M + N + 1 is stale. We consider the NSHP polynomial 14. In order to determine the staility of this NSHP polynomial, we first map this into first-quadrant filter y finding out the minimum critical angle sector. Once the NSHP is mapped into the first-quadrant filter, then the staility can e determined for the first-quadrant filter as given in Theorem 3. The corresponding QP polynomial corresponding to BZ 1,Z 2 [1]is G Z 1,Z 2 + Z Z 1 Z Z 1 Z Z According to Theorem 3, the form-preserving 1D polynomial to e tested for staility is GZ + Z + 1 Z Z 6 + Z 4. 18

4 The PLSI Method of Stailizing NSHP Recursive Digital Filters 917 The same polynomial GZ can e otained from BZ 1,Z 2 as BZ 4N+1,Z, where N 1. If the degree of Z 1 is different from Z 2, then the transformation is BZ 2M+2N+1,Z. Thus we have the following theorem. Theorem 4. An NSHP polynomial BZ 1,Z 2 of degree N is stale if and only if its form-preserving polynomial BZ 4N+1,Z is stale. In 16, since BZ 1,Z 2 is a PLSI of the constant coefficient 2D NSHP polynomial, is supposed to have its highest value with the corresponding autocorrelation coefficients eing r, r 1, r 2, r 3, r 4, r 5,andr 6. If we want to make sure that it is indeed the highest possile value, we can use the Lagrange multiplier method of optimization that is to e discussed later. This is ecause, according to 13, the PLSI will e stale only if the error is the minimum, which requires to e maximum. 4. EXISTENCE OF MAXIMUM FOR 2D FIRST- AND SECOND-ORDER PLSI POLYNOMIAL OF THE NSHP POLYNOMIAL We have seen in earlier sections that if BZ 1,Z 2 1 1R+ ij Z1Z i j 2 is a 2D first-order NSHP polynomial, then the form-preserving 1D polynomial B 1 Z BZ L,Z, when L 4N +1, will have the same autocorrelation coefficients as the BZ 1,Z 2. In order to prove that the PLSI polynomial BZ 1,Z 2 is stale, we have to show or prove the existence of a maximum optimum value for its constant. So, we discuss in this section Lagrange multiplier method of optimization and the existence of solution for the equations. First, we arrive at a figure for the numer of unknowns for each case and finally we generalize for Nth order case. Example 2 first-order case. Let BZ 1,Z ij Z1Z i j 2 19 R+ e the given first-order polynomial. This can e written as 14. The form-preserving 1D polynomial BZ 1,Z 2 ecomes B 1 Z B Z 4N+1,Z B Z 5,Z since N 1, 2 B 1 Z + Z +++Z Z Z It has seven autocorrelation functions r s sasgivenin16, where r s N r r r+s, s, 1, 2,...,N. Including B 1 Z, there are totally 2 N numer of 1D polynomials in general which has the same autocorrelation coefficients r s s as that of BZ. Out of these 2 N numer of 1D polynomials which are said to form a family, only one polynomial is stale satisfying the condition BZ, Z The stale polynomial is the one which has the maximum value magnitude for its constant term. To test the staility, we discuss elow the Lagrange multiplier method. In this method, one has to maximize a function f as satisfying the constraints g i given as where that is, f 23 g i r r r+s r s, s, 1, 2,...,N, 24 N r s r r+s, s, 1, 2,...,N, 25 r g i, i, 1, 2,...,N. 26 For the sake of clarity, we riefly discuss the method as follows. Form the Lagrange function L N,λ j f + λ j g j, 27 j where λ j are the Lagrange multipliers. Then form,λ j, 28,λ j, j, 1, 2,...,N. 29 j Equation 28 is called Lagrange equation. Now, L f + λ g + λ 1 g 1 + λ 2 g λ 6 g 6 ; 3 [ L 2 + λ ] r [ + λ r [ 1] + λ2 11 r ] 2 [ ] [ ] + λ 3 1 r 3 + λ4 + 1 r λ + λ 1 + λ 4 + 1λ λ 6 ; 32

5 918 EURASIP Journal on Applied Signal Processing Lλ r, Lλ r 1, 1 Lλ 2 11 r 2, 2 Lλ 3 1 r 3, 3 Lλ r 4, 4 Lλ r 5, 5 Lλ 6 11 r There are eight constraint equations including 33 and the Lagrange equation 32. We have 5 ij s and 5λ j s as unknowns with the total of 1. In the aove formulas, we have considered the numer of λ j s as only 5 ecause we do not have to assign λ j for the constraint equation which does not contain. Thus we have 1 unknowns and 8 equations which can e easily solved, and hence the optimum exists. So the PLSI is stale. Example 3 second-degree case. Let B Z 1,Z 2 + Z Z Z Z 1 Z Z 1 Z Z Z 2 1Z Z 2 1Z Z 1 Z Z 2 1Z Z 1 Z Z 2 1Z The form-preserving 1D polynomial of the NSHP PLSI polynomial BZ 1,Z 2 isbz and is otained using the transform BZ B Z 4N+1,Z, N 2, BZ + Z + 2 Z Z Z Z Z Z Z Z Z Z Z Form the Lagrange function L N,λ j f + λ j g j +, 36 j where λ j s are Lagrange multipliers. To find the optimum and hence stale PLSI, otain the constraint equations and unknowns. The constraint equations are,,,..., As seen aove, there are constraint equations including one Lagrange equation. We have 13 ij s as unknowns and, in addition, 13 λ j s making 26 unknowns as a total. In the aove, we considered the numer of λ j s as only 13 ecause we do not have to assign a λ j for the constraint equation which does not contain. Thuswehave26unknowns and 22 equations which can e easily solved, and hence the optimum exists. So, the PLSI is always stale. Example 4Nth-order case. For the 2D NSHP polynomial of Nth order, the total numer of constraint equations is 4N 2 +2N + 2, and out of this 2N 2 numer of equations do not contain. But the numer of the unknowns λ j s is 2N 2 +2N +1 and ij is 2N 2 +2N + 1 and hence the total is 4N 2 +4N +2. The highest order of the form-preserving 1D polynomial is 4N 2 +2N for the Nth-order NSHP polynomial. Since 4N 2 +4N +2> 4N 2 +2N + 2, the numer of unknowns are more than the numer of equations and it can e easily solved, and hence the optimum exists. So the PLSI polynomial is stale. In Examples 2, 3, and4, we have simply stated that the equations can e solved and hence the optimum exists. Take, for instance, Example 2, the only unique solution for the set of 33, since it contains less numer of the unknowns ij s than the numer of equations, is the one otainale after solving the set of 5 The vector solution 1 gives us all the coefficients ij s of the PLSI polynomial. But when we couple 32 with 33, if we have more numers of unknowns than the numers of equations, then the sets of 32 and33 together can e solved y a computer-aided nonlinear optimization method y forming and minimizing an artificial ojective function F i λ i. 38 In the computer-aided optimization method of solving nonlinear equation, one will e assured of a real solution if the total numer of the unknowns, namely, ij s and λ j s, is greater than the numer of equations y at least one. This is ecause the programmer has the freedom to choose the value of at least one coefficient as he likes. And if the value of this one coefficient unknown, say that of, is chosen to e the same as, which we already got when we solved 5, we will arrive at the same unique solution as mentioned efore. This solution will also satisfy 32. It may e noted that we

6 The PLSI Method of Stailizing NSHP Recursive Digital Filters 919 are not trying to solve 32 and33 together manually or y using computer. Our interest is in estalishing theoretically the fact that an optimum solution for these equations, which will e the same as we had already got y solving 5, does exist. This ensures the staility of the PLSI polynomial. On the other hand, if the numer of the unknowns in 32 and 33 is not greater than the numer of equations, the nonlinear computer-aided optimization, since the programmer has no degree of freedom, sometimes may not give us any real solution at all or it may give some other real solution other than what we had got y solving 5. If this is the case, the PLSI polynomial we have already got will not e stale. The value of which we get after using the computeraided nonlinear optimization method is the maximum and it is equal to provided that the programmer has the freedom, namely, the numer of unknowns greater than the numer of equations y at least one. We know that has to e maximum for E min in 13 to e really the least and positive with a eing taken as positive. In the case of 1D, the LSI polynomial is always stale when its constant term is the maximum. Similarly, if we can ensure that, also in the case of 2D polynomial, the PLSI polynomial has its maximum constant term, the PLSI will e stale. This is what we have ensured. Since [14] contains Theorem 3 which enales us to test the staility of a 2D QP polynomial in a simple way, its availaility or unavailaility now does not make much difference. One can always use the already estalished methods [1] to test the staility of a 2D QP polynomial y first transforming the NSHP polynomial into a QP polynomial. But in two numerical examples presented in the paper, we used the simple staility test given in [14] successfully. 5. STABILITY OF 2D NSHP PLSI POLYNOMIALS Inthissection,wepresentexamplesfor2DNSHPPLSIpolynomial of first-and second-order and check their stailities. Example 5. Consider the following 2D NSHP polynomial of order 1: A Z 1,Z 2.9Z1 Z Z 2 +.6Z 1 +.8Z 1 Z Let the PLSI polynomial e BZ 1,Z 2 + Z Z Z 1 Z 2 + Z 1 Z2 1. The form-preserving 1D polynomial AZofAZ 1,Z 2 is otained y using the transform Thus, Z 2 Z, Z 1 Z 4N+1 Z 5 since N 1. 4 AZ.8Z 6 +.6Z 5 +.9Z Z The polynomial AZ is unstale as some of the roots are inside the unit circle. The LSI of AZ is BZ and to find out BZ, we compute autocorrelation coefficients of AZ as follows: Now we form 12 as follows: γ 2.26, γ 1 1.2, γ 2.72, γ 3.54, γ 4.63, γ 5.66, γ The column vector 1 does contain zeros and while arriving at 5, we have to delete some rows and corresponding columns of T 1,somerowsof 1 corresponding to zero coefficients in BZ, and some zeros of a 1. After deleting the second and third columns, the corresponding rows of T 1, and the corresponding rows of 1 and a 1,weget , , , BZ.3492Z Z Z Z

7 92 EURASIP Journal on Applied Signal Processing The2DPLSIis B Z 1,Z Z1 Z Z Z Z 1 Z The 2D PLSI polynomial is found to e stale. This is ecause we have 8 equations and 1 unknown coefficients and hence the optimum exists. Example 6. Consider the following 2D NSHP polynomial of order 2: A Z 1,Z Z2 +.3Z Z Z 1 Z 2 +.9Z 1 Z Z Z 2 1Z 2 +.6Z 2 1Z Z 2 1Z Z 1 Z Z 2 1Z Z 1 Z Z 2 1Z As can e seen, we have assumed centrosymmetry among the coefficients in the QP. Let the PLSI polynomial e BZ 1,Z 2, where 34 holds. The form-preserving 1D polynomial AZofAZ 1,Z 2 isotained y using the transform Z 2 Z and Z 1 Z 4N+1 Z 9 since N 2: AZ.6+.9Z +.3Z Z 7 +.5Z 8 +.9Z Z 1 +.9Z Z Z Z Z Z 2. The polynomial AZ is unstale. Now, the LSI of AZ isbz andtofindoutbz, we compute the autocorrelation function of AZ as given in Example 5 for first order. The autocorrelation functions are γ 8.77, γ , γ , γ , γ , γ , γ 6 3., γ , γ 8 4.4, γ , γ , γ , γ 12.93, γ 13.42, γ 14.3, γ , γ , γ , γ , γ , γ There are 21 autocorrelation coefficients. Now, we form 12. Here T 1 matrix has an order of The column vector 1 does contain some zeros and, while arriving at 5, we have to delete some rows and corresponding columns of T 1,somerowsof 1 corresponding to zero coefficients in BZ, and some zeros of a 1. After deleting 8 columns containing s, the corresponding rows of T 1, and the corresponding rows of 1 and a 1,we get ,

8 The PLSI Method of Stailizing NSHP Recursive Digital Filters 921 Now, the PLSI BZis BZ Z Z Z Z 9.39Z 1.588Z Z Z Z Z Z 2. 5 The PLSI BZ 1,Z 2 is found to e stale even though the original NSHP polynomial AZ 1,Z 2 has centrosymmetry among the coefficients in the QP. This is ecause we have 22 constraint equations and 26 unknown coefficients, and hence the optimum exists. 6. CONCLUSIONS In this paper, we dealt with the stailization of 2D NSHP polynomials y the PLSI approach. The PLSI BZ 1,Z 2 will e stale provided that the degree of the given polynomial AZ 1,Z 2 and that of BZ 1,Z 2 are the same. In the case of QP filters, if there is symmetry among the coefficients, either in the original polynomial or the corresponding PLSI, then the PLSI need not e stale if the order is greater than two. This is ecause the numer of constraint equations will e more than the numer of unknowns in the optimization. Therefore, a restriction is there in the stailization of QP PLSI polynomial. However, in NSHP, the PLSI will definitely e stale, irrespective of the degree provided that it has the same order as the original polynomial. REFERENCES [1] E. A. Roinson, Statistical Communication and Detection, Griffin, London, UK, [2] J. L. Shanks, S. Treitel, and J. Justice, Staility and synthesis of two-dimensional recursive filters, IEEE Transactions on Audio and Electro acoustics, vol. 2, no. 2, pp , [3] Y. Genin and Y. Kamp, Counter example in the least-square inverse stailization of 2-D recursive filters, Electronics Letters, vol. 11, pp , July [4] Y. Genin and Y. Kamp, Comments on On the staility of the least mean-square inverse process in two-dimensional digital filters, IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 25, no. 1, pp , [5] E. I. Jury, An overview of Shanks conjecture and comments on its validity, in Proc. 1th Asilomor Conf. Circuits Systems and Computers, Pacific Grove, Calif, USA, Novemer [6] A. H. Kayran and R. King, Comments on least-squares inverse polynomials and a counter example for a Jury s conjecture, Electronics Letters, vol. 16, no. 21, pp , 198. [7] N. Damera-Venkata, M. Venkataraman, M. S. Hrishikesh, and P. S. Reddy, Stailization of 2-D recursive digital filters y the DHT method, IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol. 46, no. 1, pp , [8] N. Damera-Venkata, M. Venkataraman, M. S. Hrishikesh, and P. S. Reddy, A new transform for the stailization and staility testing of multidimensional recursive digital filters, IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, vol. 47, no. 9, pp , 2. [9] E. M. A. Gnanamuthu, N. Gangatharan, and P. S. Reddy, The PLSI method of stailizing 2-D recursive digital filters A complete solution, Sumitted to IEEE journal of circuits and systems. [1] T. S. Huang, Ed., Two-Dimensional Digital Signal Processing I. Linear Filters, vol. 42 of Topics in Applied Physics, Springer- Verlag, Berlin, Germany, [11] E.I.Jury,R.V.Kolavenu,andB.D.O.Anderson, Stailization of certain two dimensional recursive digital filters, Proc. IEEE, vol. 65, pp , June [12] S. S. Rao, Optimization Theory and Applications, Wiley Eastern Limited, New Delhi, India, [13] P. S. Reddy, D. Reddy, and M. Swamy, Proof of a modified form of Shank s conjecture on the staility of 2-D planar least square inverse polynomials and its implications, IEEE Trans. Circuits and Systems, vol. 31, no. 12, pp , [14] P. Rangarajan, P. Muthukumaraswamy, N Gangatharan, and P. S. Reddy, A simple staility test for 2-D recursive digital filters, sumitted to International Journal of Circuit Theory and Applications. N. Gangatharan was orn in Kanyakumari, India, He received the B.E. degree in electronics and communication engineering in 1988 and the M.E. degree in microwave and optical engineering in 199, oth from Madurai Kamaraj University, India. He is a second ranker in the M.E. degree examinations of Madurai Kamaraj University in 199. He received his second M.E. degree in computer science and engineering from Manonmaniam Sundaranar University, Tirunelveli, India in 1997 and his MBA degree in finance and marketing from Madurai Kamaraj University, India in He was on the faculty of Electronics and Communication Engineering at the Indian Engineering College, Vadakkangulam, India, as a Professor during In 21, he joined as a Lecturer the Faculty of Engineering, Multimedia University, Cyerjaya Campus, Malaysia. Since 21, he has een working toward the Ph.D. degree at the same university. His current research interests concern digital signal processing, particularly in the stailization of multidimensional recursive digital filters. His other research areas are in multimedia signal processing and QoS for networked multimedia systems. P. S. Reddy was orn in India. He received his B.E. degree Honors in electrical engineering with a gold medal from Andhra University, India, He is a first ranker in the M.S. degree examinations in engineering at University of Madras, India, in He received his Ph.D. degree in electrical engineering from Indian Institute of Technology IIT, Madras, India in He was a Senior Fellow of the Indian government during , during which he was trained for a teaching profession at Guindy Engineering College, India. During , he was on the faculty of IIT, Madras, India, where he was a Professor. He was a Postdoctoral Fellow at the University of Aachen, Germany, during , and at Concordia University, Montreal, Canada, during He was a Visiting Scientist at Concordia University during During , he has een a Professor and Chairman of the Department of Electrical and Computer Engineering ECE at various engineering colleges in India. He worked as a Professor at Multimedia University, Malaysia, during He is currently a Postgraduate Professor in the Department of ECE at SRM Engineering College, Chennai, India. He has pulished aout 45 research papers in international journals. His research interests are in the areas of digital signal processing and communication engineering.