doi:10.1155/2008/647075 Erratum The PLSI Method of Stabilizing Two-Dimensional Nonsymmetric Half-Plane Recursive Digital Filters

It should be noted that in (23) and all equations in the rest of the paper that contain the term b′ 00, this term is the constant term of the polynomial B′(z1, z2), which is the stable version of the 2D PLSI polynomial B(z1, z2). Since we have assumed that bi j ’s are the coefficients of the 2D PLSI polynomial, the coefficients of the polynomial B′(z1, z2) are b′ i j ’s. Equation (46) in page number 920 is as follows:


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 stable recursive NSHP filters had been a difficult and sometimes more timeconsuming problem. In this paper, we deal with the problem of stabilizing unstable NSHP 2D recursive filters by the planar least squares inverse (PLSI) approach.
The stabilization of one-dimensional (1D) recursive digital filters using least squares inverse (LSI) approach is well known [1]. However, the conjecture, made by Shanks et al. [2], known as Shanks conjecture is yet to be 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 stabilizing can be extended to 2D case, also fails. They have taken the original unstable 2D polynomial to be of degree three in both the variables, and the corresponding PLSI polynomial of degree one was found to be unstable. 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 unstable 2D polynomial. Subsequently, the modified form of the Shanks conjecture [5], which says that the PLSI polynomial should be of the same degree as the original unstable polynomial, has also been proven to be not true [6].
Two of the methods which were extensions from the 1D system theory to 2D case for stabilizing 2D recursive digital filters-namely, the discrete Hilbert transform (DHT) method and the PLSI method-have been left unsolved and not much work was reported on these in 1980s.
Recently, in [7,8] the problem facing the DHT method was resolved. Now it is clear that the DHT method of stabilizing unstable filters works only when the original unstable polynomial is devoid of zeros on the unit circle in the 1D case and on the unit bicircle in the 2D case.
In fact, in [8], a new method of stabilizing multidimensional (N > 2) recursive digital filters has been presented. This new method boils 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 stabilization 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, but a different type of restriction is necessary for the PLSI method, especially for QP filters.
In this paper, we present a method of stabilizing the given 2D NSHP unstable polynomial through the PLSI polynomial approach. It is interesting to note that the PLSI polynomial is always stable provided that the degree of the PLSI polynomial is the same as the given polynomial, whatever may be the relationship among the coefficients in the given polynomial.
The main difference between QP and NSHP filters comes by 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 be superior to QP filters. Based on the region of support R, there are eight classes of NSHP filters. However, all our discussions are based on R + ⊕ filter [10].
In Section 2, we discuss the basic definition of PLSI polynomial for the QP polynomial and NSHP polynomial. We then briefly mention the method of obtaining the PLSI polynomials. In Section 3, we discuss the existence of maximum "b 00 " for the first order and the second order which in turn results in stable polynomial. In Section 4, we present numerical examples for the first order and second order and prove that the PLSI is stable.

OBTAINING PLSI POLYNOMIAL FOR NSHP POLYNOMIALS
In this section, we discuss the basic definition of the PLSI polynomials and the method for obtaining the PLSI in general.
To obtain B(Z 1 , Z 2 ), just like in 1D case, we first form and then we form an error function (see [11]) E from as We then differentiate E with respect to each unknown coefficient b i j and equate each ∂E/∂b i j to zero to get a set of linear algebraic equations of the form In (5), T is a square matrix of order (M + 1) × (M + 1) made up of the 2D autocorrelation functions of A(Z 1 , Z 2 ) as its elements, b is an (M +1)× 1 column matrix of coefficients and a is also an (M + 1) × 1 column matrix like a = a 00 , 0, 0, . . . , 0 t .
Definition 3. A 2D NSHP polynomial It may be noted that the coefficients b mn 's of B(Z 1 , Z 2 ) can be obtained by solving (5) for the vector b, but 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 be used to form (5) by using form-preserving 1D polynomials.
Definition 4. A 1D polynomial A 1 (z)= N k=0 a k Z k is the formpreserving polynomial of a 2D QP polynomial A(Z 1 , Z 2 ): if for every integer set (m, n) in A(Z 1 , Z 2 ) there exists a unique k such that a k = a mn .
It has been proved in [13] that It has also been proved in [13] that if B(Z 1 , Z 2 ) and A(Z 1 , Z 2 ) are two 2D polynomials as defined in Definition 2, then if C(Z) = C(Z L , Z) will be a 1D form-preserving polynomial of C(Z 1 , Z 2 ) if We quoted this concept of form-preserving polynomials because later we are going to use these form preserving 1D polynomials for formulating the matrix T of (5) as well as for testing the stability or instability of the PLSI polynomials.
The above theorem has been proved in [13]. Obtaining the coefficients of 1D LSI polynomial B 1 (Z) corresponding to the 1D polynomial A 1 (Z) is very easy. Since T matrix can be mechanically written down in terms of the coefficients A 1 (Z) [1], this method of deriving the 2D PLSI polynomial B(Z 1 , Z 2 ), corresponding to the given 2D polynomial A(Z 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 (5) for b, the column vector of coefficients of We now elaborate 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 obviously know that B 1 (Z) which is a form-preserving polynomial of B(Z 1 , Z 2 ) will be lacunary with some terms corresponding to certain powers of Z being absent. This is because B 1 (Z) is what we get as B(Z L , Z), with L value being M + N + 1 which is much more than (M + 1). So when we frame the matrix equation mechanically for A 1 (Z) and the corresponding LSI B 1 (Z) of the following type: the column vector b 1 does contain some zeros; and while arriving at (5), we have to delete some rows and corresponding columns of T 1 , some rows of b 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 be deleted. The minimum error (see [1]) is

OBTAINING PLSI POLYNOMIAL FOR FIRST-ORDER NSHP POLYNOMIALS
This theorem can be used to our advantage whenever we want to form the autocorrelation coefficients of a 2D polynomial since obtaining these coefficients from 1D polynomial B 1 (Z) is very simple. What we get from (5) will be the same as what we get from the 1D polynomial B 1 (Z) by deleting proper rows and columns from (12) as mentioned earlier.
be a 2D firstorder NSHP polynomial. This can be written as follows: The autocorrelation coefficients r j 's of B 1 (Z) can be written down as follows: where * indicates the equations that do not contain b 00 .
The autocorrelation equations given in (16) are seven in number. It may be noted that two of these equations do not contain the constant coefficients b 00 of B(Z 1 , Z 2 ). It is easy to verify that B(Z 1 , Z 2 ) has the same autocorrelation coefficients as in (16).
In general, for a polynomial of Nth degree in both variables, 2N 2 number of autocorrelation equations does not contain the constant coefficient b 00 out of the total 4N 2 + 2N + 2 equations.
The following theorem is proved in [14].
We consider the NSHP polynomial (14). In order to determine the stability of this NSHP polynomial, we first map this into first-quadrant filter by finding out the minimum critical angle sector. Once the NSHP is mapped into the first-quadrant filter, then the stability can be determined for the first-quadrant filter as given in Theorem 3.
The corresponding QP polynomial corresponding to According to Theorem 3, the form-preserving 1D polynomial to be tested for stability is The same polynomial G(Z) can be obtained from In (16), since B(Z 1 , Z 2 ) is a PLSI of the constant coefficient 2D NSHP polynomial, b 00 is supposed to have its highest value with the corresponding autocorrelation coefficients being r 0 , r 1 , r 2 , r 3 , r 4 , r 5 , and r 6 . If we want to make sure that it is indeed the highest possible value, we can use the Lagrange multiplier method of optimization that is to be discussed later. This is because, according to (13), the PLSI will be stable only if the error is the minimum, which requires b 00 to be maximum.

EXISTENCE OF MAXIMUM FOR 2D FIRST-AND SECOND-ORDER PLSI POLYNOMIAL OF THE NSHP POLYNOMIAL
We have seen in earlier sections that if B(Z 1 , is a 2D first-order NSHP polynomial, then the form-preserving 1D polynomial B 1 (Z) = B(Z L , Z), when L = 4N + 1, will have the same autocorrelation coefficients as the B(Z 1 , Z 2 ).
In order to prove that the PLSI polynomial B(Z 1 , Z 2 ) is stable, we have to show or prove the existence of a maximum (optimum) value for its constant b 00 . 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 number of unknowns for each case and finally we generalize for Nth order case.
Example 2 (first-order case). Let be the given first-order polynomial. This can be written as (14). The form-preserving 1D polynomial B(Z 1 , Z 2 ) becomes It has seven autocorrelation functions r s 's as given in (16), where r s = N r=0 b r b r+s , s = 0, 1, 2, . . . , N.
Including B 1 (Z), there are totally 2 N number of 1D polynomials (in general) which has the same autocorrelation coefficients r s 's as that of B(Z). Out of these 2 N number of 1D polynomials which are said to form a family, only one polynomial is stable satisfying the condition The stable polynomial is the one which has the maximum value (magnitude) for its constant term. To test the stability, we discuss below 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, For the sake of clarity, we briefly discuss the method as follows. Form the Lagrange function where λ j are the Lagrange multipliers. Then form ∂L b 00 , λ j ∂b 00 = 0, ∂L b 00 , λ j ∂λ j = 0, j = 0, 1, 2, . . . , N.
Equation (28) is called Lagrange equation. Now, ∂L ∂b 00 = 1 + 2b 00 λ 0 + b 01 λ 1 + b λ 4 + b 10 λ 5 + b 11 λ 6 ; (32) There are eight constraint equations including (33) and the Lagrange equation (32). We have 5 b i j 's and 5λ j 's as unknowns with the total of 10. In the above formulas, we have considered the number of λ j 's as only 5 because we do not have to assign λ j for the constraint equation which does not contain b 00 . Thus we have 10 unknowns and 8 equations which can be easily solved, and hence the optimum b 00 exists. So the PLSI is stable.

Example 3 (second-degree case). Let
The form-preserving 1D polynomial of the NSHP PLSI polynomial B(Z 1 , Z 2 ) is B(Z) and is obtained using the transform Form the Lagrange function As seen above, there are 21+1=22 constraint equations including one Lagrange equation. We have 13 b i j 's as unknowns and, in addition, 13 λ j 's making 26 unknowns as a total. In the above, we considered the number of λ j 's as only 13 because we do not have to assign a λ j for the constraint equation which does not contain b 00 . Thus we have 26 unknowns and 22 equations which can be easily solved, and hence the optimum b 00 exists. So, the PLSI is always stable.
Example 4 (Nth-order case). For the 2D NSHP polynomial of Nth order, the total number of constraint equations is 4N 2 + 2N + 2, and out of this 2N 2 number of equations do not contain b 00 .
But the number of the unknowns λ j 's is 2N 2 + 2N + 1 and b i j 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 number of unknowns are more than the number of equations and it can be easily solved, and hence the optimum b 00 exists. So the PLSI polynomial is stable.
In Examples 2, 3, and 4, we have simply stated that the equations can be solved and hence the optimum b 00 exists. Take, for instance, Example 2, the only unique solution for the set of (33), since it contains less number of the unknowns b i j 's than the number of equations, is the one obtainable after solving the set of (5) The vector solution b 1 gives us all the coefficients b i j 's of the PLSI polynomial. But when we couple (32) with (33), if we have more numbers of unknowns than the numbers of equations, then the sets of (32) and (33) together can be solved by a computer-aided nonlinear optimization method by forming and minimizing an artificial objective function In the computer-aided optimization method of solving nonlinear equation, one will be assured of a real solution if the total number of the unknowns, namely, b i j 's and λ j 's, is greater than the number of equations by at least one. This is because 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 b 00 , is chosen to be the same as b 00 , which we already got when we solved (5), we will arrive at the same unique solution as mentioned before. This solution will also satisfy (32). It may be noted that we are not trying to solve (32) and (33) together manually or by using computer. Our interest is in establishing theoretically the fact that an optimum solution for these equations, which will be the same as we had already got by solving (5), does exist. This ensures the stability of the PLSI polynomial.
On the other hand, if the number of the unknowns in (32) and (33) is not greater than the number 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 by solving (5). If this is the case, the PLSI polynomial we have already got will not be stable.
The value of b 00 which we get after using the computeraided nonlinear optimization method is the maximum and it is equal to b 00 provided that the programmer has the freedom, namely, the number of unknowns greater than the number of equations by at least one. We know that b 00 has to be maximum for E min in (13) to be really the least and positive with a 00 being taken as positive.
In the case of 1D, the LSI polynomial is always stable 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 be stable. This is what we have ensured.
Since [14] contains Theorem 3 which enables us to test the stability of a 2D QP polynomial in a simple way, its availability or unavailability now does not make much difference. One can always use the already established methods [10] to test the stability of a 2D QP polynomial by first transforming the NSHP polynomial into a QP polynomial. But in two numerical examples presented in the paper, we used the simple stability test given in [14] successfully.

STABILITY OF 2D NSHP PLSI POLYNOMIALS
In this section, we present examples for 2D NSHP PLSI polynomial of first-and second-order and check their stabilities.
Example 5. Consider the following 2D NSHP polynomial of order 1: is obtained by using the transform The column vector b 1 does contain zeros and while arriving at (5), we have to delete some rows and corresponding columns of T 1 , some rows of b 1 corresponding to zero coefficients in B(Z), and some zeros of a 1 .
After deleting the second and third columns, the corresponding rows of T 1 , and the corresponding rows of b 1 The 2D PLSI is The 2D PLSI polynomial is found to be stable. This is because we have 8 equations and 10 unknown coefficients and hence the optimum exists.
Here T 1 matrix has an order of 21 × 21. The column vector b 1 does contain some zeros and, while arriving at (5), we have to delete some rows and corresponding columns of T 1 , some rows of b 1 corresponding to zero coefficients in B(Z), and some zeros of a 1 .
(50) The PLSI B(Z 1 , Z 2 ) is found to be stable even though the original NSHP polynomial A(Z 1 , Z 2 ) has centrosymmetry among the coefficients in the QP. This is because we have 22 constraint equations and 26 unknown coefficients, and hence the optimum exists.

CONCLUSIONS
In this paper, we dealt with the stabilization of 2D NSHP polynomials by the PLSI approach. The PLSI B(Z 1 , Z 2 ) will be stable provided that the degree of the given polynomial A(Z 1 , Z 2 ) and that of B(Z 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 be stable if the order is greater than two. This is because the number of constraint equations will be more than the number of unknowns in the optimization. Therefore, a restriction is there in the stabilization of QP PLSI polynomial. However, in NSHP, the PLSI will definitely be stable, irrespective of the degree provided that it has the same order as the original polynomial.