Minimization of Frequency-Weighted l 2 -Sensitivity Subject to l 2 -Scaling Constraints for Two-Dimensional State-Space Digital Filters

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1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ 007 Minimization of Frequency-Weighted l -Sensitivity Subject to l -Scaling Constraints for Two-Dimensional State-Space Digital Filters Taao Hinamoto, Toru Oumi, Osemehian I. Omoifo and Wu-Sheng Lu Abstract This paper investigates the problem of frequencyweighted l -sensitivity minimization subject to l -scaling constraints for two-dimensional (-D) state-space digital filters described by the Roesser model. It is shown that the Fornasini- Marchesini second model can be imbedded in the Roesser model. Two iterative methods are developed to solve the constrained optimization problem encountered. The first iterative method introduces a Lagrange function and optimizes it using some matrix-theoretic techniques and an efficient bisection method. The second iterative method converts the problem into an unconstrained optimization formulation by using linear-algebraic techniques and solves it by applying an efficient quasi-newton algorithm. The optimal filter structure with minimum frequencyweighted l -sensitivity and no overflow is then synthesized by an appropriate coordinate transformation. Case studies are presented to demonstrate the validity and effectiveness of the proposed techniques. Index Terms -D digital filters, Roesser s model, Fornasini- Marchesini s second model, frequency-weighted l -sensitivity minimization, l -scaling constraints, no overflow, Lagrange function, bisection method, quasi-newton method I. INTRODUCTION It is well nown that there exist infinitely many minimal state-space realizations for a given transfer function, and some inherent properties such as controllability, observability, stability, etc. are invariant within these realizations. However, performance measures such as coefficient sensitivity, output roundoff noise, overflow oscillations, etc. may be significantly varying among the realizations. Consider a transfer function with coefficients of infinite accuracy, which meets certain design specifications including stability. When the transfer function is implemented by a state-space model with a finite binary representation, truncation or rounding of the state-space model is required to satisfy the finite word length (FWL) constraints. As a result, the characteristics of the stable filter might be so altered that the filter may become unstable. This motivates the study of the coefficient sensitivity minimization problem. To date, several techniques have been reported for synthesizing the state-space descriptions with minimum coefficient sensitivity. The techniques can be divided into two Manuscript received mm dd, 007; revised mm dd, 007. T. Hinamoto, T. Oumi and O. I. Omoifo are with the Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima , Japan. ( {hinamoto,oumi,osei}@hiroshima-u.ac.jp, Phone: , Fax: ) W.-S. Lu is with the Depertment of Electrical and Computer Engineering, University of Victoria, Victoria, B.C, Canada, V8W 3P6. ( wslu@ece.uvic.ca, Phone: , Fax: ) main classes: those for l /l -mixed sensitivity minimization -5 and those for l -sensitivity minimization 6-. In 6-0, it has been argued that the sensitivity measure based on a sole l -norm is more natural and reasonable relative to the l /l -mixed sensitivity minimization. For -D statespace digital filters, the l /l -mixed sensitivity minimization problem -7 and l -sensitivity minimization problem 0,7-0 have also been investigated. It has been realized that solutions for frequency-weighted sensitivity minimization would be of practical use as these solutions allow to emphasize or de-emphasize the filter s sensitivity in certain frequency regions of interest. Synthesis procedures of the optimal FWL -D filter structures that minimize the frequency-weighted sensitivity measure have been considered 5-8. However, the minimization methods proposed in the above wor do not impose constraints on the scaling of the design variables. As a result, elimination of overflow cannot be ensured. More recently, the minimization problem of l -sensitivity subject to l -scaling constraints has been explored for -D and a class of -D state-space digital filters -3. It is well nown that the use of scaling constraints can be beneficial for suppressing overflow 4,5. However, frequency-weighted sensitivity measure has not yet been considered in -3. In this paper, we investigate the problem of minimizing a frequency-weighted l -sensitivity measure subject to l -scaling constraints for -D state-space digital filters described by the Roesser local state-space (LSS) model 6. We then proceed by introducing an expression for evaluating the frequencyweighted l -sensitivity, and formulating the minimization problem for the frequency-weighted l -sensitivity measure subject to l -scaling constraints. Next, two iterative methods are developed for solving the constrained optimization problem. The first iterative method introduces a Lagrange function, and maes use of some matrix-theoretic techniques and an efficient bisection method. The second iterative method relies on a technique that converts the constrained optimization problem into an unconstrained optimization formulation and utilizes an efficient quasi-newton method with closed-form formula for gradient evaluation. Finally, case studies are presented to demonstrate the validity and effectiveness of the proposed techniques. One of the contributions made in this paper is to show that either the Fornasini-Marchesini (FM) second LSS model 7 or its transposed-structure model,8 can be imbedded in the Roesser model as a special case. This justifies the use of the Roesser model in our studies. Another

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ 007 contribution is that a bisection method is applied to obtain the Lagrange multipliers, which maes it possible to attain considerably faster convergence than the algorithm reported in. Moreover, unlie and, the present paper investigates a frequency-weighted l -sensitivity measure under l -scaling constraints. Although this extension is technically manageable, to the best of our nowledge, this is the first time a frequency-weighted l -sensitivity measure under l -scaling constraints is addressed in the l /l framewor for state-space digital filters. Throughout the paper, I n stands for the identity matrix of dimension n n, is used to denote the direct sum of matrices, the transpose (conjugate transpose) of a matrix A is indicated by A T (A ), and the trace and ith diagonal element of a square matrix A are denoted by tra and (A) ii, respectively. II. PROBLEM FORMULATION. System Models Consider a stable, separately locally controllable and separately locally observable LSS model for -D recursive digital filters x h (i +,j) A A x h (i, j) b + u(i, j) x v (i, j +) A 3 A 4 x v (i, j) b y(i, j) x h (i, j) c c + du(i, j) x v (i, j) () which was originally proposed by Roesser 6,9, x h (i, j) is an m horizontal state vector, x v (i, j) is an n vertical state vector, u(i, j) is a scalar input, y(i, j) is a scalar output, and A, A, A 3, A 4, b, b, c, c,andd are m m, m n, n m, n n, m, n, m, n, and real constant matrices, respectively. A bloc diagram of the LSS model in (39) is shown in Fig.. The Fig.. The bloc diagram of the Roesser LSS model. transfer function of the LSS model in (39) is given by H(z,z )c(z A) b + d () A, b, andc are (m + n) (m + n), (m + n), and (m + n) real constant matrices defined by A A b A, b, c c c, A 3 A 4 b respectively, and Z z I m z I n. For the sae of simplicity, the LSS model in (39) is denoted hereafter by (A, b, c,d) m,n. Alternatively, an LSS model for a class of -D recursive digital filters can be described by,8 x(i +,j+) A A x(i, j +) y(i, j) c c x(i +,j) (3) b + u(i, j) d x(i, j) is an N local state vector, u(i, j) is a scalar input, y(i, j) is a scalar output, and A, A, b, c, c,andd are N N, N N, N, N, N, and real constant matrices, respectively. The transfer function of the LSS model in (3) is given by D(z,z )(z c + z c ) If we define (I n z A z A ) b + d. (4) x(i, j +)x h (i, j), x(i +,j)x v (i, j) (5) then the LSS model in (3) can then be imbedded in that of (39) as a special case as follows: x h (i +,j) A A x h (i, j) b + u(i, j) x v (i, j +) A A x v (i, j) b y(i, j) x h (i, j) c c + du(i, j) x v (i, j) (6) m n N. It is noted that D(z,z ) T can be viewed as a transfer function of the FM second LSS model 7, which reveals that the LSS model of D(z,z ) T can be realized by a transposed structure of that in (6). Therefore, the LSS model in (39) is more general than either the LSS model in (3) or the FM second LSS model 7 (and vice versa with the same dimension, but increased number of coefficients 7). In addition, we note that the technique reported in has merely treated the l -sensitivity minimization problem for the LSS model in (3) subject to l -scaling constraints. Recall that the total numbers of the coefficients in (39) and (3) are (m + n) +(m + n)+ and N +3N +, respectively. This means that the LSS model in (39) has less number of the coefficients than that of (3) when their local state vectors possess the same dimension, i.e., m + n N. Under these circumstances, it is worthwhile to consider the more general problem of minimizing the frequency-weighted l -sensitivity for the LSS model in (39) subject to l -scaling constraints, because its solutions will mae it possible to emphasize or deemphasize the filter s sensitivity in certain frequency regions of interest, and because the LSS model in (39) owns less number of the coefficients in case m + n N.. A Frequency-Weighted l -Sensitivity Measure Suppose that the LSS model in (39) is implemented by FWL fixed-point arithmetic with a B-bit fractional representation, and is realized with coefficient matrices

3 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ Ã A +ΔA, b b +Δb (7) c c +Δc, d d +Δd ΔA, Δb, Δc, andδd stand for the quantization errors of the coefficient matrices. The transfer function of the FWL realization is then expressed as H(z,z ) c(z Ã) b + d. (8) Let {p i,i,,,m} be the set of the ideal parameters of a realization and let { p i,i,,,m} be its FWL version p i p i +Δp i with Δp i indicating the corresponding parameter perturbation. If all Δp i are sufficiently small in magnitude, then the first-order approximation of the Taylor series expansion yields ΔH(z,z ) H(z,z ) H(z,z ) M H(z,z ) Δp i. p i i Obviously, smaller H(z,z )/ p i for i,,,m yield smaller transfer function error ΔH(z,z ). For a fixed-point implementation of B bits, the parameter perturbations can be considered to be independent random-variables uniformly distributed within the range B, B. Under these circumstances, a measure of the transfer function error can statistically be defined as σ ΔH (πj) z z (9) E ΔH(z,z ) dz dz z z (0) E( ) denotes the ensemble-average operation. Since Δp i s are uniformly-distributed independent random variables, it follows that E ΔH(z,z ) M i H(z,z ) p i σ E(Δp i ) B. Thus, if the measure S o (πj) z M z i H(z,z ) p i σ () dz dz () z z is minimized then the minimum variance σδh can be attained because of the relation σδh S oσ. The measure in () is referred to as an l -sensitivity measure. The frequency-weighted l -sensitivity of the LSS model in (39) is defined as follows. Definition :LetX be an m n real matrix and let f(x) be a scalar complex function of X, that is differentiable with respect to all the entries of X. The sensitivity function of f(x) with respect to X is then defined as 5 S X f(x) X, (S X ) ij f(x) (3) x ij x ij denotes the (i, j)th entry of matrix X. Definition : In order to tae into account the sensitivity of the transfer function in a specified frequency band, or even at some discrete frequency points, the weighted sensitivity functions are defined as 5,7 δh(z,z ) W A (z,z ) H(z,z ) δa A δh(z,z ) W B (z,z ) H(z,z ) δb b δh(z,z ) δc T W C (z,z ) H(z,z ) c T (4) W A (z,z ), W B (z,z ), and W C (z,z ) are scalar, stable, causal functions of the complex variables z and z. Notice that δ in (4) is not meant to be a derivative operator, but rather a notation for defining the weighted parameter sensitivity. Definition 3 :LetX(z,z ) be an m n complex matrix valued function of the complex variables z and z.thel norm of X(z,z ) is then defined by X(z,z ) ( tr (πj) Γ X(z,z )X (z,z ) dz dz Γ z z ) (5) j and Γ i {z i : z i } for i,. From () and Definitions -3, the overall frequencyweighted l -sensitivity measure for the LSS model in (39) can be defined as S δh(z,z ) δa + δh(z,z ) δb + δh(z,z ) δc T W A (z,z )F (z,z )G(z,z ) T + W B (z,z )G T (z,z ) + W C(z,z )F (z,z ) (6) F (z,z )(Z A) b, G(z,z )c (Z A). It follows that the frequency-weighted l -sensitivity measure in (6) can be written as S trm A +trw B +trk C (7) M A, W B,andK C are obtained by the following general expression: X (πj) Y (z,z )Y (z,z ) dz dz Γ z z Γ with Y (z,z )W A (z,z )F (z,z )G(z,z ) T for X M A, Y (z,z )W B (z,z )G T (z,z ) for X W B,and Y (z,z )W C (z,z )F (z,z ) for X K C..3 Problem Formulation Define a state-space coordinate transformation by 6,9 x h (i, j) T 0 x h (i, j) (8) x v (i, j) 0 T 4 x v (i, j)

4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ T and T 4 are m m and n n nonsingular matrices, respectively. New realizations can then be characterized as (A, b, c,d) m,n with A T AT, b T b, c ct (9) T T T 4. For a new realization, the frequencyweighted l -sensitivity measure in (7) is changed to S(P )trm A (P )P +trw B P +trk C P trn A (P )P +trw B P +trk C P (0) with P TT T P P 4 and M A (P ) (πj) Y (z,z )P Y (z,z ) dz dz Γ z z N A (P ) (πj) Γ Γ Y (z,z )PY(z,z ) dz dz Γ z z Y (z,z )W A (z,z )F (z,z )G(z,z ) T. If l -scaling constraints are imposed on the horizontal and vertical state vectors x h (i, j) and x v (i, j), we require that 30 (K ) ξξ (T K T T ) ξξ for ξ,,,m (K 4 ) ζζ (T 4 K 4T T 4 ) ζζ for ζ,,,n () K (πj) F (z,z )F (z,z ) dz dz Γ Γ z z K K K 3 K 4 is the local controllability Gramian for the LSS model in (39) with an m m submatrix K and an n n submatrix K 4 along its diagonal 9. Thus, the l -scaling constrained frequency-weighted l - sensitivity minimization problem can be formulated as follows: Given matrices A, b, and c, obtain a bloc-diagonal nonsingular matrix T T T 4 which minimizes S(P ) in (0) subject to l -scaling constraints in (). III. PROBLEM SOLUTION 3. A Constrained Optimization Method Solving the optimization problem formulated above consists of several steps. First, we relax the problem of minimizing S(P ) in (0) subject to l -scaling constraints in () into the problem minimize S(P ) in (0) subject to trk P m and trk 4P 4 n. () If trk P m(trk 4P 4 n) is satisfied, then an m m (n n) orthogonal matrix U (U 4 ) matrix can always be constructed so that T P / U (T 4 P / 4 U 4 ) satisfies l -scaling constraints in (). This justifies the relaxation made in (). In order to solve problem () for P P P 4,wedefine the Lagrange function of the problem as J(P,λ,λ 4 ) trm (P )P +trm 4 (P )P 4 +trw B P + trw 4B P 4 +trk C P +trk 4CP 4 (3) + λ (trk P m)+λ 4 (trk 4 P 4 n) λ and λ 4 are the Lagrange multipliers, and M (P ) M (P ) M A (P ) M 3 (P ) M 4 (P ) W B W B KC K C W B, K C W 3B W 4B K 3C K 4C with an m m submatrix and an n n submatrix along its diagonal for each matrix. Using the formula for evaluating matrix gradient 3, p.75 tr(mx) M T X (4) tr(mx ) (X MX ) T X we compute J(P,λ,λ 4 ) P J(P,λ,λ 4 ) P 4 M (P ) P N (P )P + W B P K C P λ P K P M 4 (P ) P 4 N 4 (P )P 4 + W 4B P 4 K 4C P 4 λ 4 P 4 K 4 P 4 (5) N (P ) N (P ) N A (P ) N 3 (P ) N 4 (P ) with an m m submatrix N (P ) and an n n submatrix N 4 (P ) along its diagonal. Setting J(P,λ,λ 4 )/ P 0 and J(P,λ,λ 4 )/ P 4 0, it follows that P F (P )P G (P,λ ) (6) P 4 F 4 (P )P 4 G 4 (P,λ 4 ) F (P )M (P )+W B, F 4 (P )M 4 (P )+W 4B G (P,λ )N (P )+K C + λ K G 4 (P,λ 4 )N 4 (P )+K 4C + λ 4 K 4. The equations in (6) are highly nonlinear with respect to P and P 4. Namely, P i F i (P )P i for i, 4 has a rational type R/D of nonlinearity, the degree of the nonlinearity of R is m + n + and the degree of the nonlinearity of D is m+n, while G i (P,λ i ) for i, 4 depends on P linearly. An effective approach for solving these equations is to relax them into the following recursive second-order matrix equations: P (+) F (P () )P (+) G (P (),λ (+) ) (7) P (+) 4 F 4 (P () )P (+) 4 G 4 (P (),λ (+) 4 )

5 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ with initial condition P (0) P (0) P (0) 4 I m+n. Noting that PWP M has the unique solution 5 P W W MW W (8) W > 0 and M 0 are symmetric, the unique solutions P (+) and P (+) 4 of (7) are found to be P (+) F (P () ) F (P () ) G (P (),λ (+) )F (P () ) F (P () ) P (+) 4 F 4 (P () ) F 4 (P () ) G 4 (P (),λ (+) 4 )F 4 (P () ) F 4 (P () ). (9) Here, the Lagrange multipliers λ (+) and λ (+) 4 can be efficiently obtained using a bisection method 3 so that f (λ (+) () )m tr K G () (λ (+) ) 0 f 4 (λ (+) () 4 ) n tr K 4 G () 4 (λ (+) 4 ) 0 are satisfied K () F (P () ) K F (P () ) K () 4 F 4 (P () ) K 4 F 4 (P () ) (30) G () (λ (+) )F (P () ) G (P (),λ (+) )F (P () ) G () 4 (λ(+) 4 )F 4 (P () ) G 4 (P (),λ (+) 4 )F 4 (P () ). The iteration process continues until J(P (),λ (+),λ (+) 4 ) J(P ( ),λ (),λ() 4 ) <ε (3) for a prescribed tolerance ε>0. If the iteration is terminated at step, thenp () is claimed to be a solution point. It is noted that a straightforward extension of the iterative algorithm reported in for updating λ and λ 4 does not wor well in this case, we do not require trkp m+n, but both trk P m and trk 4P 4 n. In addition, the bisection method offers an exponential convergence rate of (/) L L is the number of iterations used. As such, accurate solutions of the Lagrange multipliers λ (+) and λ (+) 4 in (30) can be identified with just a few iterations. In our simulation studies, the bisection method was found considerably faster than the iterative algorithm proposed in. All the Gramians can be evaluated by truncating the corresponding infinite summations, see Appendix I for details. 3. An Unconstrained Optimization Method By defining ˆT ˆT ˆT 4 (T T 4 ) T (K K 4 ), (3) it follows that T i K i T i T T ˆT i ˆT i for i, 4. (33) Thus, a convenient way to eliminate the l -scaling constraints in () is to choose ˆT and ˆT 4 as ˆT t t, t t,, t m t m (34) ˆT 4 t 4 t 4, t 4 t 4,, t 4n t 4n t i is an m vector for i,,,m and t 4j is an n vector for j,,,n. With (34), all the T diagonal elements of ˆT ˆT T and ˆT 4 ˆT 4 are found to be unity. Taing (3) and (34) into account, we conclude that the coordinate transformation matrix T of the form T T T 4 (K K 4 ) ( ˆT ˆT 4 ) T (35) automatically satisfies the l -scaling constraints in (). By substituting (35) into S(P ) in (0), the frequency-weighted l -sensitivity measure can be expressed as J o (x) tr ˆT ˆM A ( ˆP ) ˆT T +tr ˆT Ŵ B ˆT T +tr ˆT T ˆKC ˆT (36) with x (t T, tt,, tt m, tt 4, tt 4,, tt 4n )T ˆM A ( ˆP ) (πj) ˆP ˆT T ˆT and Γ Ŷ (z,z ) ˆP Ŷ (z,z ) dz dz Γ z z Ŷ (z,z )(K K 4 ) Y (z,z )(K K 4 ) Y (z,z )W A (z,z )F (z,z )G(z,z ) T Ŵ B (K K 4 ) W B (K K 4 ) ˆK C (K K 4 ) K C (K K 4 ). This shows that the problem of obtaining the bloc-diagonal nonsingular matrix T T T 4 which minimizes S(P ) in (0) subject to the l -scaling constraints in () can be converted into an unconstrained optimization problem of obtaining an (m + n ) vector x which minimizes J o (x) in (36). Applying a quasi-newton algorithm to minimize J o (x) in (36), in the th iteration the most recent point x is updated to point x + as 33 d S J o (x ), S + S + S 0 I m +n, x + x + α d (37) ( + γ T S γ γ T δ α arg min J o(x + αd ) α ) δ δ T γ T δ δ γ T S +S γ δ T γ T δ δ x + x γ J o (x + ) J o (x ). Here, J o (x) is the gradient of J o (x) with respect to x, and S is a positive-definite approximation of the inverse Hessian matrix of J o (x). The algorithm starts with a trivial initial point

6 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ x 0 obtained from an initial assignment ˆT I m+n, and this iteration process continues until J o (x + ) J o (x ) <ε (38) ε>0 is a prescribed tolerance. The implementation of (37) requires the gradient of J o (x), which can be efficiently evaluated using closed-form expressions, see Appendix II for details. IV. CASE STUDIES In this section, we examine the proposed algorithms by applying them to a recursive -D state-space digital filter. In addition, the algorithms are applied to a -D filter utilized in and the results are compared with those obtained by the method of. Example : Consider a -D stable recursive digital filter realization (A o, b o, c o,d), 30, p A o b o T c o d After carrying out the l -scaling for the above realization with a diagonal coordinate matrix T o diag{9.3364, ,.0650, }, we obtain the -D state-space digital filter (A, b, c,d), characterized by A b T c d A T o A o T o, b T o b o,andc c o T o.the -D state-space digital filter (A, b, c,d), now satisfies the l -scaling constraints. The frequency-weighting functions used in this example were given by a -D nonrecursive low-pass digital filter with the following unit-sample response 34, p. 895: w A (i, j) w B (i, j) w C (i, j) exp {(i 4) +(j 4) } for (0, 0) (i, j) (0, 0), and zero else. Using (A.) and (A.) with truncation (0, 0) (i, j) (50, 50) to evaluate the Gramians K C, W B, M A and K, it was found that K C W B M A K In what follows, we evaluate the frequency-weighted l - sensitivity of the -D state-space digital filter (A, b, c,d),. Later on, this sensitivity measure will be used as a benchmar in the examination of the performance of the proposed algorithms. Using (7), the frequency-weighted l -sensitivity of the LSS model (A, b, c,d), was found to be S As will be seen next, the proposed algorithms are able to deduce an equivalent state-space realization with much reduced frequency-weighted l -sensitivity. 4. Application of the Lagrange method Choosing P (0) P (0) P (0) 4 I 4 in (9) as an initial estimate and a tolerance ε 0 8 in (3) as well as in the bisection method, it too the Lagrange-based algorithm 0 iterations to converge to the solution P opt or equivalently, T opt The minimized frequency-weighted l -sensitivity measure in (3) corresponding to the above solution was found to be J(P opt,λ,λ 4 ) with λ and λ , and the optimal state-space filter structure (A, b, c,d), (that

7 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ minimizes (0) subject to the l -scaling constraints in ()) was synthesized by substituting matrix T opt into (9) as A b T c The profile of the frequency-weighted l -sensitivity measure J(P,λ,λ 4 ) and the profiles of the Lagrange multipliers λ and λ 4 for the first 0 iterations are shown in Figs. and 3, respectively. 5 x 0 4 or equivalently, T opt and the minimized frequency-weighted l -sensitivity was found to be J o ( ˆT opt ) We see that the results obtained by this method are identical to those obtained by the Lagrange-based method. The profile of the l -sensitivity measure J o ( ˆT ) during the first 6 iterations is shown in Fig x (P (), λ (+), λ 4 (+) ) J Fig.. Profile of J(P,λ,λ 4 ) during the first 0 iterations. J o (x ) Fig. 4. Profile of J o ( ˆT ) during the first 6 iterations. λ (), λ4 ().5 x λ λ Fig. 3. Profiles of λ and λ 4 during the first 0 iterations. 4. Application of the quasi-newton method By choosing ˆT I I (therefore T (K K 4 ) / in (3)) as an initial estimate and a tolerance ε 0 8 in (38), the quasi-newton algorithm too 6 iterations to converge to the solution ˆT opt We now explain the effectiveness of the optimal realization that minimizes the frequency-weighted l -sensitivity subject to l -scaling constraints. The magnitude response of the original realization (A, b, c,d), is shown in Fig. 5, the maximum value and the l -norm (which corresponds to a square root of the summation of squared values at 0 0 sampling points) were and 33.50, respectively. When all coefficients in the original realization were rounded to power-of-two representation with 8 bits after binary point, the magnitude-response deviation between the original realization and that with rounded coefficients is shown in Fig. 6, the maximum deviation and the l -norm were 4.44 and.5633, respectively. Alternatively, when all coefficients in the optimal realization were rounded in the same manner, the magnitude-response deviation between the original realization and the optimal one with rounded coefficients is shown in Fig. 7, the maximum deviation and the l -norm were and.3380, respectively. From these figures and data, it is observed that the coefficient sensitivity of the optimal realization is considerably lower than that of the original realization. Example : Let a -D stable recursive digital filter realization be specified by (A o, b o, c o,d) 4,4 A o A o A o b o, b o, c o c o A o A o b o c o

8 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ Fig. 5. Magnitude response of the original realization. Fig. 7. Magnitude-response deviation between the original realization and the optimal one with rounded coefficients. and then applying the same frequency-weighted functions as in Example to the resulting realization (A, b, c,d) 4,4,the frequency-weighted l -sensitivity in (7) was found to be S Fig. 6. Magnitude-response deviation between the original realization and that with rounded coefficients. with A o A o b o bo T c o c o d This -D filter was obtained by imbedding the LSS model of Example in into the Roesser LSS model. By performing the l -scaling for the above realization with a diagonal coordinate matrix T o diag{.00000,.00000, , ,.00000,.00000, , } with truncation (0, 0) (i, j) (00, 00) in (A.) and (A.). 4.3 Application of the Lagrange method Choosing P (0) P (0) P (0) 4 I 8 in (9) as an initial estimate and a tolerance ε 0 8 in (3) as well as in the bisection method, it too the Lagrange-based algorithm 04 iterations to converge to the solution P opt or equivalently, T opt The minimized frequency-weighted l -sensitivity measure in (3) corresponding to the above solution was found to be J(P opt,λ,λ 4 ) with λ and λ The profile of the frequency-weighted l -sensitivity measure J(P,λ,λ 4 ) and the profiles of the Lagrange multipliers λ and λ 4 for the first 04 iterations are shown in Figs. 8 and 9, respectively.

9 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ J ( P () (+) (+), λ, λ 4 ) The minimized frequency-weighted l -sensitivity in (36) was found to be J o ( ˆT opt ) which is identical to the minimum value of the frequencyweighted l -sensitivity measure, obtained by the Lagrangebased method. The profile of the l -sensitivity measure J o ( ˆT ) during the first 54 iterations is shown in Fig Fig. 8. Profile of J(P,λ,λ 4 ) during the first 04 iterations. J o (x ) , λ 4 () λ () Fig. 9. Profiles of λ and λ 4 during the first 04 iterations. 4.4 Application of the quasi-newton method By choosing ˆT ˆT ˆT 4 I 8 in (37) as an initial estimate and a tolerance ε 0 8 in (38), the quasi-newton algorithm too 54 iterations to converge to ˆT opt or equivalently, T opt λ λ Fig. 0. Profile of J o ( ˆT ) during the first 54 iterations. To compare the technique reported in with the proposed ones, the optimal realization derived in Example of was imbedded in the Roesser LSS model by using (6). Then the frequency-weighted l -sensitivity of the resulting imbedded model was computed from (7) as S It is observed that this value is.9 times greater than the minimized frequency-weighted l -sensitivity obtained by the proposed techniques. Moreover, the proposed Lagrangebased method attains considerably faster convergence than that reported in, which required 000 iterations for its convergence. V. CONCLUSION We have investigated the problem of minimizing the frequency-weighted l -sensitivity subject to l -scaling constraints for -D state-space digital filters described by the Roesser LSS model. It has been shown that the FM second LSS model can be imbedded in the Roesser LSS model as a special case. Two iterative methods have been developed to solve the problem at hand. The first iterative method is based on the introduction of a Lagrange function and maes use of an efficient bisection method. In our simulation studies, the bisection method was found to be considerably faster than the iterative method proposed in. The second iterative method relies on the conversion of the constrained optimization problem into an unconstrained optimization formulation and utilizes an efficient quasi-newton algorithm. The optimal state-space realization with minimum frequency-weighted l - sensitivity and no overflow has then been constructed by applying an appropriate coordinate transformation. Our computer simulation results have demonstrated the validity and effectiveness of the proposed techniques.

10 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ x h (i +,j+) A A x h (i, j +) x v (i +,j+) 0 0 x v (i, j +) 0 0 x h (i +,j) + A 3 A 4 x v (i +,j) b 0 + u(i, j +)+ u(i +,j) 0 b y(i, j) x h (i, j) c c + du(i, j) x v (i, j) (39) APPENDIX I COMPUTATIONS OF GRAMIANS The matrices K C, W B,andM A can be computed using A (,0) K C W B M A i0 j0 i0 j0 A A 0 0 i0 j0 f C (i, j)f C T (i, j) gb T (i, j)g B(i, j) H A T (i, j)h A(i, j), A (0,) 0 0 A 3 A 4 (A.) A (0,0) I m+n, A ( i,j ) 0 (i ), A (i, j ) 0 (j ) A (i,j) A (,0) A (i,j) + A (0,) A (i,j ) A (i,j) A (,0) + A (i,j ) A (0,), (i, j) > (0, 0) + A (i,j ) 0 b f (i, j) A (i,j) b 0 + ca (i,j ) I n g (i, j) ca (i,j) Im H(i, j) f (, r)g (i, j r) f C (i, j) g B (i, j) H A (i, j) (0,0) (,r)<(i,j) (0,0) (,r)<(i,j) (0,0) (,r)<(i,j) (0,0) (,r)<(i,j) w C (, r)f (i, j r) w B (, r)g (i, j r) w A (, r)h (i, j r) with partial ordering for integer pairs (i, j) 6, p., and w A (, r), w B (, r), andw C (, r) denoting the unit-sample responses of frequency-weighting functions W A (z,z ), W B (z,z ),andw C (z,z ), respectively. The local controllability Gramian K and the other Gramians M A (P ), N A (P ) and ˆM A ( ˆP ) can be computed using K f (i, j)f T (i, j) M A (P ) N A (P ) ˆM A ( ˆP ) i0 j0 i0 j0 i0 j0 i0 j0 HA T (i, j) P H A (i, j) H A (i, j) PH T A(i, j) Ĥ T A(i, j) ˆP Ĥ A (i, j) Ĥ A (i, j) (K K 4 ) HA (i, j)(k K 4 ). APPENDIX II GRADIENT EVALUATION OF J o (x) J o ( ˆT ) t ij (A.) lim Δ 0 J o ( ˆT ij ) J o ( ˆT ) Δ β β +β 3 β 4 (A.3) ˆT ij is the matrix obtained from ˆT ˆT ˆT 4 with a perturbed (i, j)th component, which is given by 35, p. 655 with ˆT ij ˆT + Δ ˆTg ij e T ˆT j Δe T ˆTg, ˆT ij ˆT Δg ij e T j j ij { } tj g ij / t ij t j t j 3 (t ijt j t j e i ) β e T j ˆM A ( ˆT ) ˆTg ij β e T ˆT T j Ĥ A (p, q) ˆT T ˆT Ĥ T A (p, q) g ij p0 q0 β 3 e T j ˆT Ŵ B ˆT T ˆTgij, β 4 e T j ˆT T ˆKc g ij {t, t,, t m+n } {t, t,, t m } {t 4, t 4,, t 4n }. REFERENCES L. Thiele, Design of sensitivity and round-off noise optimal state-space discrete systems, Int. J. Circuit Theory Appl., vol., pp , Jan V. Tavsanoglu and L. Thiele, Optimal design of state-space digital filters by simultaneous minimization of sensitivity and roundoff noise, IEEE Trans. Circuits Syst., vol. CAS-3, pp , Oct L. Thiele, On the sensitivity of linear state-space systems, IEEE Trans. Circuits Syst., vol. CAS-33, pp , May M. Iwatsui, M. Kawamata and T. Higuchi, Statistical sensitivity and minimum sensitivity structures with fewer coefficients in discrete time linear systems, IEEE Trans. Circuits Syst., vol. 37, pp. 7-80, Jan G. Li, B. D. O. Anderson, M. Gevers and J. E. Perins, Optimal FWL design of state-space digital systems with weighted sensitivity minimization and sparseness consideration, IEEE Trans. Circuits Syst. I, vol. 39, pp , May W.-Y. Yan and J. B. Moore, On L -sensitivity minimization of linear state-space systems, IEEE Trans. Circuits Syst. I, vol. 39, pp , Aug. 99.

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