Introduction The design of fast-acting 2-D FIR digital lters is a much researched area in Digital Signal Processing. Several authors (see for example

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1 Optimal Parallel 2-D FIR Digital Filter with Separable Terms Vidya Venkatachalam and Jorge L. Aravena Department of lectrical and Computer ngineering Louisiana State University, Baton Rouge, LA 78. Phone: (54) , fax: (54) , April 3, 998 DICS CLASSIFICATION: SP 4.. m-d Filtering All correspondence related to this paper should be addressed to the second author Abstract This paper completely solves the optimal Weighted Least Mean Square (WLMS) design problem using sums of separable terms. For any xed number of separable terms (less than or equal to the rank of the unconstrained solution), the problem is solved as a sequence of separable lter approximations. An ecient computational algorithm based on necessary conditions is presented. The procedure allows a high degree of exibility in the choice of lter orders and the number of separable terms, but it may converge to a local minimum. An improved approximation can be obtained by computing more terms than required and then performing a truncation of the coecient matrix using a singular value analysis. A signicant computational advantage is that the procedure requires neither the solution of the unconstrained WLMS problem nor the singular value analysis of the ideal lter.

2 Introduction The design of fast-acting 2-D FIR digital lters is a much researched area in Digital Signal Processing. Several authors (see for example [], [2], [26], [29]) have proposed new algorithms to achieve good quality designs with reduced computational complexity. A good quality design is obtained by nding the optimal lter coecients that satisfy a given constraint. Reduced computational complexity is obtained by eliminating redundant operations, making acceptable approximations and by putting to use the inherent symmetry properties in the desired lter response. Unfortunately, good quality and reduced complexity are, normally, conicting in nature, and there is a trade-o between them in any standard design technique. With the advances in VLSI technology and the advent of high speed processors which allow a high degree of parallelism, there is new interest ([3]-[8]) in digital lter design algorithms which readily lend themselves to a parallel architecture. Such algorithms provide for a fast implementation without deterioration in lter quality by allowing for several operations to be performed concurrently, thus reducing the trade-o inherent in the standard design techniques. This paper integrates quality of the lter and parallel implementation by approximating a desired lter with sums of simpler and faster lters. The 2-D ltering action is now accomplished by several pairs of 2-D separable lters, all acting concurrently on the image. The transfer function of the k th such separable FIR lter is then given by H k (z ; z 2 ) = N X N X2 a k (n )b k (n 2 )z?n z?n 2 2 () n =?N n 2 =?N2 [3],[4] give the details of one approach to design such lter pairs. This approach entails the use of the Singular Value Decomposition (SVD) of the desired response to nd the optimal separable responses. These responses are then approximated by -D FIR lters, using standard design algorithms (like for example, the Remez algorithm ([3])). In a second stage, the SVD of the lter coecient matrix is used to reduce the number of parallel channels. The main drawback of the SVD design procedure is that it involves an approximation by FIR lters, once the optimal -D frequency responses are determined. This leads to a nal design which is suboptimal. There is, therefore, a need to formulate an algorithm to design truly optimal -D FIR lters and avoid the need for a double SVD. This paper develops a parallel decomposition in terms of separable components by setting the lter design problem as a constrained Weighted Least Mean Square (WLMS) problem in the coecients. Separability of each component is imposed by constraining the coecients of 2-D lters to be rank one matrices. The technique is shown to be equivalent to an SVD with a dierent measure of orthogonality. Notation can become very cumbersome and cloud some developments with unnecessary complexity. For this reason, the next section states the problem in the conventional way and then simplies the notation by introducing an operator based notation which encompasses both 2

3 discrete and continuous cases. This general formulation is solved in section three for the case of one separable lter and extended to a sum of separable lters. Section four presents design results. The nal section contains conclusions. 2 Problem Statement Consider a linear shift invariant 2-D lter with frequency response D(! ;! 2 ). This lter must be approximated by an FIR lter of the form H(! ;! 2 ) = X (k ;k 2 )2Iz The merit index for the design is the cost function = Z Z?? J = Z Z?? W (! ;! 2 )jd(! ;! 2 )? h(k ; k 2 )e?j(k! +k 2! 2 ) W jd? Hj 2 d! d! 2 X (k ;k 2 )2Iz h(k ; k 2 )e?j(k! +k 2! 2 ) j 2 d! d! 2 The index set I z is a subset of the integer numbers and normally is a rectangular region. In this case the collection fh(k ; k 2 )g can be arranged as a matrix. The function W (! ;! 2 ) is a non negative function that can be used to assign more weight to performance in certain regions of the frequency domain. The index is clearly a function of the FIR lter coecients. Its minimization will determine the optimal coecients. This is the standard WLMS problem. With the obvious modications, one can set the problem in the discrete frequency domain, or extend it to m-d lters. For simplicity, the presentation is concentrated on the 2-D case, with appropriate extension to the general m-d case. If one wishes to restrict the minimization to the class of separable lters, then the coecient matrix must be of rank one. Remark 2. It is well known that the general formulation can be simplied, from a computational point of view, if one makes use of symmetry conditions [25], [26]. However, as long as the lter is linear in the coecients and the cost function is quadratic, one can always manipulate the design problem to the general formulation discussed below. 2. The General Formulation We oer here a formulation of the WLMS problem which applies to both discrete and continuous frequency cases and general m-d lters. The main goals are to reduce notational complexity, to highlight the common aspects of the problem, and establish conditions applicable to all cases. The lter to be designed is an element of a 'lter space', F, which is required to be a Hilbert space. For the continuous m-d case this space is L 2 [?; ] m while for the discrete case it will be a uclidian space with dimension depending on the number of frequency points. 3

4 The lter is determined by a set of coecients which are elements of a 'coecient space', C, which will also be a Hilbert space. If the coecients are required to be real, this space is a uclidian space, otherwise it can be taken as a space of complex numbers. If c 2 C is a set of coecients then a feasible lter solution can be represented as H = F(c), where F : C! F is a given map describing the lter in terms of its coecients. For FIR lters, this map is linear. Remark 2.2 In the unconstrained 2-D case, the parameters are normally arranged in a matrix, C. Since N N 2 matrices can also be considered elements of a Hilbert space, N N 2, with inner product < A; B >= trfa Bg; A; B 2 N N 2 ; one can formulate the problem directly in terms of the matrix. For this, one can use the 'stacking' isometry S : N N 2! N N 2. If C 2 N N 2 is an N N 2 matrix, the vector c = S(C) 2 N N 2 is obtained by stacking the columns of C following a left to right order. It follows easily that the 'de-stacking operator' is actually the adjoint S : N N 2! N N 2 and that S S is the identity transformation. For example, if the lter is of the form H(z ; z 2 ) = N X N X2 h(n ; n 2 )z?n z?n 2 2 ; (2) n =?N n 2 =?N2 (where N = 2N +, N 2 = 2N 2 + ), one can dene the matrix of coecients C = [h(k ; k 2 )];?Ni k i Ni; i = ; 2: (3) The denition of the operator FS is then FS(C) = N X N X2 h(n ; n 2 )e j(n! +n 2! 2 ) n =?N n 2 =?N2 Clearly, the same type of representation can be established for the general m? D case. The ideal lter is an element D 2 F, while the WLMS cost function is a weighted distance in F and can be written in the form J(c) = hw(d? F(c)); D? F(c)i F For all cases of practical interest, the map, W, describing the weighting function can be assumed to be self-adjoint and positive semi-denite. The optimal WLMS design consists in determining the coecient ^c 2 C which minimizes the cost function J(c). Using conventional properties one can write J(c) = hw(d); Di F? hf W(D); ci C? hc; F W(D)i C + hc; F WF(c)i C where the notation () denotes the adjoint of the corresponding operator. 4

5 Remark 2.3 Since one must work with dierent spaces, one should use dierent symbols to denote inner products and norms in the various spaces. Thus h; i C denotes inner product in the coecient space. For the sake of simplicity, in the rest of the developments, distinctions are not made when the underlying spaces are clear from context. One can use standard variational techniques and derive necessary and sucient conditions for optimality. Specically one has the standard result Theorem 2.4 The parameter ^c 2 C is an unconstrained solution to the WLMS problem if and only if it satises F WF(^c) = F W(D) Remark 2.5 It is also a standard result that the cost function can be written in terms of an optimal solution as J(c) = hw(d); Di? h^c; F WF^ci + h^c? c; F WF(^c? c)i (4) Hence, minimization of J(c) is equivalent to the minimization of J e (c) = h^c? c; F WF(^c? c)i (5) This expression will be useful in developing a better understanding of the solutions. Clearly, the unconstrained solution to the WLMS problem will be unique if the operator F WF is positive denite. Notice that the map F WF is a linear transformation in the parameter space. For the FIR case, even for the general m? D case, this is a nite dimensional space; the cost function becomes a simple quadratic function in the coecients; and the solution could, in theory, be obtained using matrix inversion techniques. Particularly for m-d lters, this is not a practical approach and researchers have developed many dierent approaches (see for example [25, 26]). 2.. The separable 2? D case The constraint that the lter be a single separable term is easily stated in terms of the matrix of coecients, C. One must have C = ab T ; a 2 N ; b 2 N 2 It is then clear that the vector of coecients obtained by stacking the columns of C is nothing more than the Kronecker product ab 2 N N 2 (i.e., ab = S(ab T )). The vectors a 2 N ; b 2 N 2 are unconstrained. The cost function can be put in the form J(a; b) = hw(d); Di? hf W(D); a bi? ha b; F W(D)i + ha b; F WF(a b)i (6) The optimal design of the one-term separable lter corresponds to the minimization of the cost function in eq.(6) with respect to the unconstrained parameters a; b. 5

6 Remark 2.6 The m-d separable case follows exactly the same model. The lter coecients will also be arranged in a rank one matrix, which now will be of the form C = a a 2 : : :a m. ach vector a k contains the coecients of the k?th -D lter. Their dimensions are determined by the required order in that lter. In order to gain insight into the minimization, one can reformulate the problem in terms of the unconstrained solution. Following remark 2.5, the cost function in q. 6 can be rewritten as J(a; b) = hw(d); Di? h^c; F WF^ci + h^c? a b; F WF(^c? a b)i (7) In a similar manner, the constraint that the lter must consist of k parallel, separable terms, requires a matrix of coecients C k = kx i= a i b T i ; a 2 N ; b 2 N 2 In terms of the vectors obtained by stacking columns, a k? terms approximation is a vector c k = kx i= a i b i ; a i 2 N ; b i 2 N 2 where the vectors a i b i ; i = ; 2; : : : ; k, form a linearly independent set. It should be clear that the number of separable terms must be at most equal to the rank of the unconstrained solution, and must satisfy the constraint k minfn ; N 2 g. The cost in this case has the form J(c k ) = hw(d); Di? h^c; F WF^ci + ^c? * kx a i b i ; F WF(^c? kx i= i= a i b i ) Remark 2.7 Finding the best k terms separable approximation is then equivalent to the approximation of the unconstrained optimal, ^c using a weighted inner product. In particular, if F WF is the identity operator, one must solve the minimization problem J(a ; b ; : : : ; a k ; b k ) =k ^c? kx i= a i b i k 2 Using the 'de-stacking' operators, the cost can be rewritten in terms of matrices as J(a ; b ; : : : ; a k ; b k ) =k ^C? kx i= a i b T i k 2 The solution of this problem is known and can be expressed in terms of the k largest singular values, and corresponding singular vectors, for ^C. However, for an arbitrary matrix F WF, the conventional SVD will not, in general, yield an optimal k terms representation. + (8) 6

7 3 The Optimal Filter with Separable Components This section contains the main theoretical results, establishing the existence of optimal approximations with a specied number of separable terms. The development examines the case of one term and then uses those results to establish the general result. 3. Optimal Separable Filter In the context of the present development, this case is referred to as the one term separable lter. In fact it solves the WLMS problem with the additional constraint that the solution must be a separable lter. For the one term separable lter, the cost function is given by eq. (6). In order to determine an optimal solution, one can use the identity a b = (a I N2 )b Replacing in the expression for J(a; b) (eq. (6)), and using the property, < x; Ay >=< A x; y >, one obtains J(a; b) = hw(d); Di? h(a I N2 )F W(D); bi? hb; (a I N2 )F W(D)i + hb; (a I N2 )F WF(a I N2 )bi (9) For xed a 2 N, the previous equation is a conventional quadratic cost problem in the vector b 2 N 2. The problem will have a unique solution ^b(a) if and only if the matrix (a I N2 )F WF(a I N2 ) is positive denite. The following result shows that this is indeed the case whenever the unconstrained problem has a unique solution. Lemma 3. For any non zero vector a 2 N, the matrix (a I N2 )F WF(aI N2 ) is positive denite if and only if F WF is positive denite. The proof is immediate because hb; (a I N2 )F WF(a I N2 )bi = ha b; F WF(a b)i. The unique solution is ^b(a) = [(a I N2 )F WF(a I N2 ]? (a I N2 )F W(D) () This expression for b can be replaced in the cost function dening J b (a) = J(a; ^b(a)) = hw(d); Di? F W(D); (a I N2 )[(a I N2 )F WF(a I N2 )]? (a I N2 )F W(D) It is immediately apparent that this cost function is independent of the magnitude of the vector a; hence one can restrict its minimization to the unit ball, B a = fa 2 N :k a k= g. Since the unit ball, B a, is compact, the existence of a global minimum can be established by showing that J b (a) is a continuous function on B a. For this purpose, one can use the following steps 7 ()

8 . If the unconstrained WLMS problem has a unique solution, then F WF >. Hence (a) (b) F WF = 2 2 min k p k 2 hp; pi 2 max k p k 2 ; 8p 2 N N 2 (2) 2. Dene X(a) = a I N2, Q(a) = X (a)x(a). Using q( 2) establish (a) min k a kk X(a) k max k a k (b) 2 min k b k2 hb; Q(a)bi 2 max k b k2 ; 8b 2 N 2 ; a 2 B a (c) D?2 max k b k 2 b; Q? (a)b?2 min k b k2 ; 8b 2 N 2 ; a 2 B a (d) If a ; a 2 2 B a, and a = a 2? a then, k a k 2 2 k a k; Q(a 2 )? Q(a ) = X (a )X(a) + X (a)x(a ) + X (a)x(a) and k Q(a 2 )? Q(a ) k 4 2 max k a k 3. Since Q? (a )? Q? (a 2 ) = Q? (a ) (Q(a 2 )? Q(a )) Q? (a 2 ) k Q? (a )? Q? (a 2 ) k 4?4 min 2 max k a k; 8a ; a 2 2 B a 4. Since FW(D) = F WF(^c), the cost function, J b (a), can be written as D J b (a) = hw(d); Di? X (a)(^c); Q? (a)x (a)(^c) Therefore J b (a )? J b (a 2 ) = X (a 2 )(^c); (Q? (a 2 )? Q? (a ))X (a 2 )(^c) + X (a )(^c); Q? (a )X (a)(^c) + X (a)(^c); Q? (a )X (a 2 )(^c) 5. Taking the absolute value, one can see that every inner product in the right hand side can be bounded by k a k. Hence the function is continuous. 8

9 Remark 3.2 Notice that J b (^a) J(a; ^b(a)) J(a; b); 8(a; b) Hence this method indeed computes the globally optimal (one term) separable lter. It is also clear that if for some collection of nonzero vectors fq ; q 2 ; : : : ; q k g, one imposes the additional constraints ha; q i = ; i = ; 2: : : : ; k the resulting domain is the intersection of the unit ball with a collection of subspaces. This is also a compact subset of the unit ball, and the constrained minimization will have a globally optimal solution. This result will be useful in establishing the existence of an optimal decomposition with a given number of terms. Now that the existence of the optimal solution has been established, it is possible to develop necessary conditions which will be useful for the development of computationally ecient algorithms. For this, let ^a; ^b be an optimal solution and a; b any other pair of vectors. Using simple algebraic manipulations, one can write J(a; b)? J(^a; ^b) = D a b? ^a ^b; F WF(a b? ^a ^b) + D^c? ^a ^b; F WF(a b? ^a ^b) + Da b? ^a ^b; F WF(^c? ^a ^b) : By selecting suitable variations one can determine several useful necessary conditions. Taking a b? ^a ^b = ^a ^b one has D D^a 2 ^b; F WF^a ^b + ^c? ^a ^b; F WF^a ^b D + ^a ^b; F WF(^c? ^a ^b) : Using the conventional argument, for small values of, the sign of the right hand side would be determined by the terms linear in. If they are non zero, one could contradict the condition that ^a; ^b are optimal. Hence one must have D ^c? ^a ^b; F WF^a ^b = Taking now a b = ^a b, and using the identity a b = (a I N2 )b, one can write J(a; b)? J(^a; ^b) = D ^a I N2 (b? ^b); F WF(^a I N2 (b? ^b) + D D ^c? ^a ^b; F WF(^a I N2 (b? ^b) + ^a I N2 (b? ^b); F WF(^c? ^a ^b) : The vector b? ^b can be completely arbitrary in N 2. arguments, one now can establish the condition Repeating again the small variation ^a I N2 F WF(^c? ^a ^b) = In a similar way, taking now a b = a ^b and noting that the vectors a b and b a are related by a simple permutation; i.e., a b = P b a; one can write a new necessary condition. These results are summarized in the following theorem 9

10 Theorem 3.3 The optimal pair ^a; ^b satises the necessary conditions. D ^c? ^a ^b; F WF^a ^b = ^a I N2 F WF(^c? ^a ^b) = ^b I N P F WF(^c? ^a ^b) = Remark 3.4 It is easy to see that the rst necessary condition can be derived from any of the other two. For the sake of clarity, it has been kept separate since its shows the orthogonality characterisitic of all LMS solutions. This theorem will be used to establish a numerically simple computational procedure. More details will be presented in section 4, Development of a Computational Algorithm. 3.2 The Optimal Approximation with Several Separable Terms According to remark 2.7, an optimal approximation in terms of k separable terms is equivalent to the determination of a singular value decomposition by using a weighted inner product to determine orthogonality. The following result makes this statement more clear. Theorem 3.5 Assume that the unconstrained optimal ^c can be written in the form with 2 : : : m >. ^c = Xm i= i^a i ^b i Assume further that the terms are F WF conjugate; i.e., D^a i ^bi ; F WF^a j ^bj = ; i 6= j (3) and are normalized so that D^a i ^b i ; F WF^a j ^b j = ; i = j (4) Then, for k m ^c k = kx i= i^a i ^b i is the best k terms approximation, in the sense that any other coecient matrix ~ C of rank less than or equal to k yields a vector ~c = S( ~ C) such that J(~c) J(^ck ).

11 (Note: The normalization condition in q (4) is simply a convenience and can be easily removed.) Proof: If the operator F WF is positive denite, the operation hp ; p 2 i F WF = hp ; F WFp 2 i ; p ; p 2 2 C denes a new inner product in the space C and consequently induces a new denition of orthogonality. Any collection of nonzero vectors p ; p 2 ; : : : ; p k such that hp i ; F WFp j i = ; i 6= j are necessarily linearly independent, since they are orthogonal in the new inner product. In particular if p i = a i b i ; 8i, then the matrix C k = must be exactly of rank k. kx i= If the vectors a i b i satisfy the normalization condition in equation (4), they form an orthonormal basis (in the new inner product) for the subspace V k = spanfa i b i ; i kg Moreover, the vector ^c? i^a i ^b i is clearly orthogonal (in the new inner product) to the vector i^a i ^b i. Hence, the vector ^c k = P k i= i^a i ^b i will be the orthogonal projection of ^c onto this subspace V k. Let now C ~ be a coecient matrix, and ~c = S( C) ~ be the corresponding vector of coecients. Assume that h^c? ~c; F WF(^c? ~c)i < h^c? ^c k ; F WF(^c? ^c k )i Since ^c k is the orthogonal projection of ^c onto V k, the previous inequality implies that the vector d = ~c? ^c cannot belong to subspace V k. In terms of the coecient matrices, one has a i b T i ~C = kx i= ^a i^bt i + D where the matrix D cannot be expressed as a linear combination of the rank one matrices ^a i^bt i. Hence, the rank of the matrix C ~ must be strictly larger than k. Therefore, ^Ck denes the best approximation with rank k. The theorem is established. Since the cost function for the WLMS design (q. 4) J(c) = hw(d? Fc); D? Fci = hw(d); Di? h^c; F WF^ci + h^c? c; F WF(^c? c)i one can easily derive

12 Corollary 3.6 The rst term in the decomposition, ^a ^b, is the optimal separable lter (determined in the previous section). The theorem is constructive and provides sucient conditions for an optimal decomposition of the unconstrained solution of the WLMS problem. The following argument shows that one can always construct an optimal sequence. Hence, it is possible to establish a constructive technique to compute the optimal k terms approximation as a sequence of one term optimizations. Consider an approximation of the form c k+ = P k+ i= a i b i = P k i= a i b i + a k+ b k+ = c k + a k+ b k+ Assume that, ^c k, the optimal approximation with k terms is known (this is the case for k = ). Assume further that the minimization problem min J(^c k + a k+ b k+ ) (5) a k+ ;b k+ with the constraints Da k+ b k+ ; F WF^a i ^b i = ; i k admits a nonzero solution (see remark 3.2 ). It is apparent that if the method is continued until the one term minimization does not permit any improvment, then one has actually constructed a sequence of terms that satis- es the conditions of the theorem (3.5) and has, therefore, computed the optimal solution. Moreover, the optimal solution can be determined sequentially with the one term constrained minimizations. A severe limitation of many lter design tools is their computational complexity. The decomposition into a sequence of smaller problems has denite advantages. However, the one term minimization is a nonlinear programming problem which could still be considered computationally challenging. On the other hand, the results in the previous section show that the unconstrained one term solution always exists. Such a solution must yield a cost which cannot be larger than the constrained case. Hence, it is clear that the optimal solution coincides with an unconstrained one. Moreover, it must coincide with the global optimal. This argument is attractive because it suggests that the optimization with k terms could be solved with a sequence of unconstrained one term minimizations. Its limitation lies in the fact that when using unconstrained minimizations, one does not insure the orthogonality conditions and may end up with suboptimal results. The next section explores this issue and develops an ecient algorithm for the one term minimization. 2

13 4 Development of a Computational Algorithm The previous section establishes sucient conditions for the existence of an optimal approximation with a specied number of separable terms; the solution can be obtained as a sequence of one term optimizations. This is a signicant result; however, from a computational point of view, the one term minimization is still a complicated procedure. This section will develop an ecient algorithm for its solution, based on the necessary conditions in Theorem 3.3. The equations of interest here are ^a I N2 F WF(^c? ^a ^b) = ^b I N P F WF(^c? ^a ^b) = The dependence on the unconstraineed solution ^c is eliminated by the use of the identity (see Theorem 2.4) F WF(^c) = F W(D) Using also the identities a b = (a I N2 )b; a b = P b a, one can write (^a I N2 F WF^a I N2 ) ^b = ^a I N2 F W(D) (6) ^b I N P F WFP ^b IN a = ^b I N P F W(D) (7) The proposed algorithm uses the following steps. Select an arbitrary unit vector a 2 N 2. Given the unitary vector a n 2 N (a) Compute ^b(a n ) as the solution.of q (6), which is a linear equation of size N 2 Remark 4. An equivalent procedure is to minimize the cost function with respect to b for a xed a n. Since this is a quadratic cost function, a conjugate gradient guarantees convergence in at most N 2 steps. (b) Given the vector b n = ^b(a n ), compute ^a(b n ) by solving equation (7) (or by using a minimization procedure). (c) Dene a n+ = ^a(b n) k ^a(b n ) k 3. If else end k a n? a n+ k> tolerance set a n := a n+ and repeat iteration. stop 3

14 It is clear that at every step, one is reducing the cost function. Moreover, the sequence of vectors a n lies on the unit ball in N, which is a compact set, and consequently it must, at least, have a convergent subsequence. In practice, numerous examples with discrete frequency response cases show that the algorithm converges very rapidly to a solution. Moreover, the algorithm appears to be insensitive to the selection of the starting point. However, as is common in nonlinear programming problems, there is no guarantee that it converges to the global optimum. Remark 4.2 The application to the discrete frequency case was analyzed in detail in [3]. It turns out that it is possible to characterize the cases where only real valued separable lters are necessary. For the cases where complex valued vectors are generated, one can constrain the formulation and force only real valued vectors. However, the experimental results showed that constraining the optimization produced slow convergence of the algorithm. It was also established that if a term a b was a solution to the necessary conditions for a half plane symmetric lter, then the conjugate vector a c b c was also a solution. For these lters, in cases where complex valued vectors were generated, the method forms a component lter with real coecients using F(a b + a c b c ). 4. Numerical xamples The operators F; W have been explicitly evaluated in [3] for the case of discrete frequencies! i;k i = 2k i + 2M i ; k i =?M i ; : : : ; M i? ; i = ; 2 The ideal lter D is represented by an (2M + ) (2M 2 + ) matrix. The approximating FIR lter has the form H = ab T T 2 with i (k ; k 2 ) = e? (2k +) 2M i k 2 ; (8) k =?M i ; : : : ; M i? ; k 2 =?Ni ; : : : ; N i ; i = ; 2: Hence F(a b) = 2 a b = ab T T 2 The weighting function is dened by an (2M +)(2M 2 +) matrix, W, with nonnegative entries and the operator W is dened as a Hadamard, or entry-by-entry matrix product, and denoted here by ; i..e, W(D) = W D The numerical examples included here divide the frequency range in 28 points (i.e., M = M 2 = 64) and specify lter matrix coecients of size N = N 2 = 22. Thus the nonseparable case requires 2N N 2 + N + N 2 + = 3 coecients while each separable lter requires only 2N + 2N = 9 coecients. The cases shown below are: 4

15 . A one quadrant fan lter (Figures, 2). This is a good example of an 'almost separable' ideal lter. In this case, the one term approximation yields a very good approximation. In fact, the gure shows one can obtain a very good quality response, comparable to the optimal non-separable response, using only very few (9%) coecients. The maximum error between the exact lter and the one term approximation is less than 2%. In this case the algorithm required 8 iterations which is much smaller than the unconstrained number of parameters. 2. A lter whose support is a rotated ellipse. The ideal response and a computed approximation are shown in Figures 3 and 4 respectively. This lter has axes of :7 and :3 and an external transition band of width of :. It is rotated 3 o counterclockwise about the! axis. The lter is highly non-separable, but [3] established that the solutions to the necessary conditions are always real. One can obtain an approximation with a maximum error of 7% with a relatively small number of terms ( in the case shown in Fig 4). The evolution of the cost function with the number of terms is also examined and shows a steady decrease in the error as the number of terms increases (see Figure 5). This last gure also shows the number of iterations required for convergence for each of the separable lters. Remark 4.3 In order to interpret properly the 7% error, one must consider the fact that the unconstrained solution with the same weighting function also has a very high error. In fact, the corresponding terms approximation derived using SVD analysis of the unconstrained case has a maximal error of 2% (see [3]). 3. A lter with triangular support having axes of :65 and :55, with an internal transition band of width :. This is a half-plane symmetric lter, similar to the one quadrant fan lter, but is also highly non-separable similar to the rotated elliptical. Figure 6 displays the ideal lter and Figure 7 the approximation with 4 separable terms yielding a maximum error of 6:5%. The cost function and the number of iterations as functions of the number of terms are displayed in Figure 8. Remark 4.4 ach of the cases shows that the cost function varies rapidly for the rst few terms and then shows only marginal improvement for each additional term. This property suggests the concept of critical number of terms which appears to be related to the singular value structure of the weighted ideal lter. Remark 4.5 An examination of the data on convergence, shown in Figures 5 and 8, shows that, on the average, the algorithm converges in a number of iterations equal to the order of the separable lter. This speed is comparable to that of the best quadratic algorithms. 5

16 4. The optimality of the approximations is also examined for the last two lters. The procedure is the following: First one computes an approximation with a number of terms greater than the critical number, n c. The resulting matrix of coecients is analyzed for its singular values and a new coecient matrix is determined using the largest n c singular values. The above procedure produces a remarkable reduction in the number of terms required for the lter with triangular support. As can be seen from the Figure 9, one can get a very good approximation with just the rst 4 terms taken from the SVD decomposition of the coecient matrix computed from the 4 terms originally used. An analysis of the singular values of the matrix shows that the remaining singular values are less than % of the highest singular value, and hence do not contribute much to the lter response. For the rotated elliptical lter however, almost no reduction is achievable using this procedure, even though the singular values of its coecient matrix (computed from the original terms), after 5 terms, are less than % of the maximum value. Reduction in the number of terms is not achieved in this case since the originally computed lter has a high error ( 2%). Remark 4.6 The experimental results show that the algorithm does not compute an optimal approximation. In both cases, the approximation obtained by truncating with the SVD analysis presents superior lter characteristics and smaller error (see for example Figures 9 and ). However, the cost function for the one-term obtained using the algorithm is always lower than that for the one-term lter obtained from the SVD-based reduction. This has been observed regardless of the number of terms computed prior to the SVD analysis. The conclusion is that the algorithm based on necessary conditions does converge to an optimal solution, but the algorithm builds up numerical errors as the number of terms is increased. It is also interesting to point out that the separable lters obtained by SVD reduction yield a large cost but look smoother and have better appearance than the optimal one-term solution. This fact is a reection of the acknowledged limitations of the mean square criterion. 5 Conclusions The paper completely characterizes the optimal solution of the WLMS problem using separable terms. The characterization is supplemented with a fast numerical algorithm based on necessary conditions. For any xed number of separable terms (less than or equal to the rank of the unconstrained solution), the problem is solvable as a sequence of separable lter approximations. xtensive numerical results indicate that the algorithm builds up errors as the number of terms increases. However, the technique permits a clear estimation of the number of terms required 6

17 for a good approximation to a given lter. An improved approximation can be obtained by computing a few more terms than required and then performing a truncation of the coecient matrix using a singular value analysis. A signicant computational advantage is that the procedure requires neither the solution of the unconstrained WLMS problem nor the singular value analysis of the ideal lter. Some of the experimental results yield lters with poor characteristics. This is attributable to the known limitations of the LMS criterion. Better designs may be obtained by varying the weighting function, for example using Lawson's type updates ([25, 29]). References [] M.O. Ahmad, and J.D. Wang, "An analytic least mean square solution to the design problem of two-dimensional FIR lters with quadrantally symmetric or antisymmetric frequency response," I Trans. Circuits and Systems, Vol CAS-36, pp , July 989. [2] Wu-Sheng Lu, and Andreas Antoniou, "Two-Dimensional Digital Filters," Marcel Dekker,Inc., New York, 992. [3] A. Antoniou, and W.S. Lu, "Design of two-dimensional Digital Filters by using the Singular-Value Decomposition," I Trans. Circuits Syst., Vol. CAS-34, pp 9-98, Oct [4] W.S. Lu, H.P. Wang, and A. Antoniou, "Design of two-dimensional FIR Digital Filters by using the Singular-Value Decomposition," I Trans. Circuits and Systems, Vol. 37, No., pp 35-46, Jan. 99. [5] W.S. Lu, H.P. Wang, and A. Antoniou, "Design of two-dimensional Digital Filters using Singular-Value Decomposition and Balanced Approximation Method," I Trans. on Signal Processing, Vol. 39, No., pp , Oct. 99. [6] S. Treitel, and J.L. Shanks, "The design of multistage separable planar lters," I Trans. Geosci. lectron., Vol. G-9, pp -27, Jan. 97. [7] P. Karivaratharajan, and M.N.S. Swamy, "Realization of a 2-dimensional FIR digital lter using separable lters," lectron. Lett., Vol. 4, No. 8, pp , April 978. [8] B.R. Suresh, and B.A. Shenoi, "xact realization of 2-dimensional digital lters by separable lters," lectron. Lett., Vol. 2, No., pp , 976. [9] W.S. Lu, and A. Antoniou, "Synthesis of 2-D state-space xed point digital lter structures with minimum roundo noise," I Trans. Circuits Syst., Vol. CAS-33, pp , Oct

18 [] T.S. Huang, J.W. Burnett, and A.G. Deczky, "The importance of phase in image processing lters, I Trans. Acoust., Speech, Signal Process., Vol. ASSP-23, pp , Dec [] T.S. Huang, "Two-dimensional windows," I Trans. Audio lectroacoust., Vol. AU-2, pp 88-89, March 972. [2] A. Antoniou, and W.S. Lu, "Design of 2-D nonrecursive lters using window method," I Proc., Vol. 37, pt. G, pp , Aug. 99. [3] T.C. Speake, and R.M. Mersereau, "A note on the use of windows for two-dimensional FIR lter design," I Trans. Acoust., Speech, Signal Process., Vol. ASSP-29, pp 25-27, Feb. 98. [4] J.F. Kaiser, "Nonrecursive digital lter design using the I? sinh window function," Proc. 974 I Int. Symp. Circuit Theory, pp [5] W. Chen, C.H. Smith, and S.C. Fralick, "A fast computational algorithm for the discrete cosine transform," I Trans. Commun., Vol. COM-25, pp 4-9, Sep [6] J. Makhoul, "A fast cosine transform in one and two dimensions," I Trans. Acoust., Speech, and Signal Process., Vol. ASSP-28, pp 27-34, Feb. 98. [7] A. Gupta, and K.R. Rao, "A fast Recursive Algorithm for the Discrete Sine Transform," I Trans. Acoust., Speech, Signal Process., Vol. ASSP-38, No. 3, pp 553, March 99. [8] A. Gupta, and K.R. Rao, "An cient Algorithm Based on the Discrete Sine Transform," I Trans. on Signal Processing, Vol. 39, No. 2, pp 486, February 99. [9] J.H. McClellan, "The design of two-dimensional lters by transformations," Proc. 7 th Annual Princeton Conf. Information Sciences and Systems, pp , 973. [2] R.M. Mersereau, W.F.G. Mecklenbrauker, and T.F. Quatieri, Jr., "McClellan Transformations for two-dimensional digital ltering: I-Design," I Trans. Circuits Syst., Vol. CAS-23, pp 45-43, July 976. [2] Y. Kamp, and J.P. Thiran, "Chebyshev approximations for two-dimensional nonrecursive digital lters," I Trans. Circuits Syst., Vol. CAS-22, pp 28-28, March 975. [22] T.W. Parks, and J.H. McClellan, "Chebyshev approximation for nonrecursive digital lters with linear phase," I Trans. Circuit Theory, Vol. CT-9, pp 89-94, March 972. [23] J.H. McClellan, and T.W. Parks, "quiripple approximation of fan lters," Geophysics, Vol. 7, pp , 972. [24] L.R. Rabiner, J.H. McClellan, and T.W. Parks, "FIR digital lter design techniques using weighted Chebyshev approximation," Proc. I, Vol. 63, pp 595-6, April

19 [25] G. Gu, J.L. Aravena, "Weighted Least Mean Square Design of Two-dimensional FIR Digital Filters," I Trans. Sig. Proc., Vol SP-42, No, pp , Nov [26] J.L. Aravena and G. Gu," Weighted Least Mean Square Design of 2-D FIR Digital Filters: General Weighting Function," in Proc 27th Asilomar Conference on Signals, Systems and Computers, Asilomar, CA, November 993, pp [27] C. Charalambous, "A unied review of optimization," I Trans. Microwave Theory Tech., Vol. MTT-22, pp 289-3, March 974. [28] C. Charalambous, "Acceleration of the least p th algorithm for minimax optimization with engineering applications," Mathematical Programming, Vol. 7, pp , 979. [29] C. Charalambous, "The performance of an algorithm for minimax design of twodimensional linear phase FIR digital lters," I Trans. Circuits Syst, Vol. CAS-32, pp 6-28, Oct [3] D.J. Shpak and A. Antoniou, "A generalized Remez method for the design of FIR digital lters," I Trans. Circuits Syst., Vol. CAS-37, pp 6-74, Feb. 99. [3] Vidya Venkatachalam, "Parallel and Separable 2-D FIR Digital Filter Design," M.S. Thesis, Louisiana State University, May

20 One quadrant fan filter : Ideal frequency response 2 D rotated elliptical filter : Ideal magnitude frequency response >Magnitude >Magnitude w2<.5 >w.5.5 w2< >w Figure : Ideal magnitude frequency response Figure 3: Ideal magnitude frequency response of 2-D quadrant fan lter of 2-D rotated elliptical lter Magnitude response of the -term separable FIR filter:n=n2=22 Magnitude response of term separable FIR filter:n=n2= >Magnitude -->Magnitude w2< >w w2< Figure 2: Magnitude frequency response of the optimal -term separable FIR quadrant fan l- Figure 4: Magnitude frequency response of the ter optimal -term separable FIR rotated elliptical lter >w Variation of cost function with number of separable terms >Cost function >Number of separable filters 9 Variation of number of iterations with number of separable terms >No of iterations >Number of separable filters 9 Figure 5: Performance variation with number of terms for the -term separable FIR rotated elliptical lter 2

21 Ideal magnitude response of.65/.55 triangular FIR filter >Magnitude.8.6 Magnitude response of 4 conjugate pair svd reduced FIR filter Max error = w2< >Magnitude.5 >w Figure 6: Ideal magnitude frequency response of 2-D triangular lter w2< >w Figure 9: Magnitude frequency response of the 4-term svd reduced FIR triangular lter Magnitude response of 4 conjugate pair FIR separable filter:n=n2=22.2 >Magnitude w2<.5 >w Figure 7: Magnitude frequency response of the optimal 4-term separable FIR triangular lter Magnitude response of 4 separable conjugate pair FIR filter.2 >Magnitude Max error =.229 Variation of cost function with number of conjugate separable filter pairs >Cost function >Number of conjugate filter pairs Variation of number of iterations with number of conjugate filter pairs w2< 8 >No. of iterations.8.5 >w 6 Figure : Magnitude frequency response of the optimal 4-term separable FIR triangular lter >Number of conjugate filter pairs 2 4 Figure 8: Performance variation with number of terms for the 4-term separable FIR triangular lter 2