Minimax Design of IIR Digital Filters Using SDP Relaxation Technique

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Minimax Design of IIR Digital Filters Using SDP Relaxation Technique Aimin Jiang, Student Member, IEEE, and Hon Keung Kwan, Senior Member, IEEE Abstract—This paper presents a new algorithm using semidefinite programming (SDP) relaxation to design IIR digital filters in the minimax sense. Unlike traditional design algorithms which try to directly minimize the error limit, the proposed algorithm employs a bisection searching procedure to locate the minimum error limit of the approximation error. Given a fixed error limit at each iteration, the SDP relaxation technique is adopted to formulate the design problem in a convex form. In practice, the true minimax design cannot be always obtained. Thus, a regularized feasibility problem is adopted in the bisection searching procedure. The stability of designed filters can also be guaranteed by adjusting the regularization coefficient. Unlike other iterative design methods, the proposed algorithm tries to find a feasible solution at each iteration of the iterative procedure within a feasible set defined by the relaxed constraints. This feasible set is not restricted within the neighborhood of a given point obtained from the previous iteration. Thus, the proposed method can avoid being trapped in the locally minimum point. Four examples are to be presented in this paper to demonstrate the effectiveness of the proposed method. Index Terms—Bisection searching procedure, infinite impulse response (IIR) digital filter, minimax design, rank minimization, semidefinite programming (SDP) relaxation, trace approximation.

G

I. INTRODUCTION

ENERALLY speaking, there are two types of digital filters, i.e., finite impulse response (FIR) and infinite impulse response (IIR). Since the transfer function of an FIR digital filter is essentially a linear function of filter coefficients, an FIR filter design problem can be strictly cast as a convex optimization problem under various optimization criteria. Thereby, the globally optimal solution can be reliably obtained. Compared with an FIR filter design problem, an IIR filter design problem is more challenging due to its nonconvex property. The nonconvexity is incurred by two aspects: 1) The transfer function of an IIR digital filter has a denominator, which makes it difficult to transform an IIR filter design problem into an equivalent convex form. Hence, the globally optimal solution is difficult to attain. 2) Due to the presence of a denominator, the stability issue must also be considered while Manuscript received October 17, 2008; revised February 16, 2009; revised April 3, 2009. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, Canada N9B 3P4 (e-mail: [email protected]; [email protected]).

both magnitude and phase responses are under consideration in the design problem. Unfortunately, when a denominator order is larger than 2, the stability domain is nonconvex. Recently, a number of design algorithms [1]-[25] have been proposed to solve IIR filter design problems. These algorithms can be classified as iterative design algorithms [1]-[16], noniterative design algorithms [17]-[19], and model reduction algorithms [20]-[25]. The most prevalent design strategy is to employ iterative procedures [1]-[16] to gradually approach an optimal design. The Steiglitz-McBride (SM) scheme [26] is adopted as iterative design algorithms proposed in [1]-[6]. At each iteration, the denominator of an approximation error is replaced by its counterpart obtained at the previous iteration and combined with a weighting function. Then, the approximation error to be minimized can be expressed as a quadratic function of filter coefficients. Different stability constraints have been utilized in these design algorithms, such as the positive-realness [1]-[4], the Lyapunov theory [5], and the argument-principle [6] based stability constraints. The major drawback of the SM design approaches is that the convergence of its iterative procedure cannot be definitely guaranteed. Another design algorithm employing the reweighting technique is proposed by [7], in which a quasi-equiripple design could be achieved by reweighting a least-squares design problem. Unlike the SM scheme, the weighting function at each iteration is determined by the approximation error of the filter obtained at the previous design. The drawback of this design algorithm is that stability constraint cannot be directly incorporated into the design procedure. Thus, the resulting filter may be unstable. In [8], an iterative strategy is employed to design IIR filters in the minimax sense. At each iteration, the frequency response of an IIR filter is approximated by the first-order Taylor series expansion with respect to each of its denominator coefficients. Using the linearized frequency response, the searching direction for a denominator can be first determined. After updating the denominator coefficients, the best numerator can then be computed and taken into the next iteration. The performance of this algorithm is dependent on the selection of the initial point. The stability of the designed filter can be guaranteed by restricting its denominator update within the stable domain. A similar design strategy is utilized in the Gauss-Newton (GN) method proposed in [9] under the weighted least-squares (WLS) criterion. The first-order Taylor series expansion with respect to each of its numerator and

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2 denominator coefficients is utilized to convert the approximation error into a quadratic function. The Rouché’s theorem based stability constraint is used in this algorithm. This GN design method suffers the same drawback as SM design methods regarding non-guaranteed convergence. Another design algorithm using the similar iterative strategy is introduced by [10]. A linearized argument-principle based stability constraint is utilized to guarantee the stability of the designed filters. In [11], a multistage design method is proposed, in which the SM, the GN, and classical descent methods are successively applied to achieve a design in the WLS sense. A liner matrix inequality (LMI) stability constraint in terms of positive realness is developed in [11]. Starting from the WLS design obtained from the multistage design method [11], an equiripple design [12] can be achieved by successively optimize the numerator using the reweighted least-squares method [7]. Another iterative strategy adopted in [13] and [14] is based on a sufficient condition for the optimal rational approximation, which states that an approximation error has a specific number of extreme points over the frequency bands of interest. The Remez multiple exchange algorithm is deployed to identify these extreme points. The initial point should be selected close enough to the optimal solution to guarantee the convergence of the iterative procedure. Recently, the convex relaxation technique is introduced in [15]-[16] to design IIR filters. By solving a relaxed design problem, a lower bound on the optimal value of the original design problem can be obtained. Then, an iterative procedure is developed to reduce the discrepancy between these two design problems. However, since the searching direction at each iteration is restricted to a halfspace, the globally optimal solution may not be obtained. Some design algorithms [17]-[18] try to strictly formulate design problems without any constraint. Then, general-purpose optimization procedures are applied to solve these design problems. Due to the nonconvex nature of IIR filter design problem and its stability issue, the objective function and its initial point should be appropriately selected. In [19], another noniterative design algorithm is proposed to design IIR digital filters under the minimax criterion. For simplicity, the denominator of the approximation error is neglected. However, it is hard to obtain a true minimax design by minimizing the simplified approximation error. Another category of algorithms [20]-[25] design IIR digital filters through model reduction. An FIR digital filter is designed first to satisfy the same specifications. Then, model reduction techniques are utilized to approximate the FIR digital filter by an IIR digital filter. Normally, it is difficult to design filters with accurate cutoff frequencies using this design strategy. Semidefinite programming (SDP) [27]-[28] is a convex optimization problem [29], which unifies several standard optimization techniques, such as linear programming (LP), quadratic programming (QP), and second-order cone programming (SOCP). Methods for designing digital filters using SDP have been reported in [5], [30]-[32]. However, all of these algorithms have been proposed for FIR filter design

except [5] which addresses IIR filters and [31] which addresses 2-D FIR and IIR filters. In this paper, an iterative SDP method is proposed to design IIR digital filters in the minimax sense. By applying the SDP relaxation technique, the original approximation error can be simplified as a linear-fractional function, which eases the subsequent manipulations. The proposed method utilizes a bisection searching procedure to locate the minimum error limit of the approximation error to be formulated. At each iteration, a feasibility problem is to be solved over a feasible set defined by the relaxed constraints. This feasible set is not restricted within the neighborhood of a given point. Thus, the iterative procedure can avoid being trapped in locally minimum points. The rest of this paper is organized as follows. The original design problem is first formulated in Section II. Then, a bisection searching procedure is introduced in Section II-B. In Section II-C, the SDP relaxation technique is applied to formulate a feasibility problem, which is to be solved at each iteration of the bisection searching procedure. In order to achieve a true minimax solution, a regularization term is incorporated in the feasibility problem in Section II-D. Another bisection searching procedure is then introduced to locate the minimum regularization coefficient used in the objective of the regularized feasibility problem. Stability can be ensured by a monitoring strategy introduced in Section II-E. The complete algorithm is presented in Section II-F. Several numerical examples are presented in Section III to demonstrate the effectiveness of the proposed algorithm. Conclusions are made in Section IV.

II. MINIMAX DESIGN OF IIR DIGITAL FILTERS A. Problem Formulation Let D(ω) represent an ideal frequency response over [0, π], and H(z) be the transfer function of an IIR digital filter with real coefficients, i.e., N

P( z ) = H ( z) = Q( z )

∑p z n =0 M

−n

n

1 + ∑ qm z − m

=

pT φN ( z ) 1 + qT φM ( z )

(1)

m =1

where p = [ p0 ,

pN ]

T

(2)

T

q = [ q1 , q2 , " , qM ]

(3)

φN ( z ) = ⎡⎣1, φM ( z ) = ⎡⎣ z −1 ,

p1 , " ,

z −1 , " , z −2 , " ,

z − N ⎤⎦

T

z − M ⎤⎦

(4) T

(5)

The superscript T denotes the transpose of a vector or matrix. Let the (weighted) complex approximation error E(ω) be defined as

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3

E (ω ) = W (ω ) ⎡⎣ D(ω ) − H (e ) ⎤⎦ ,

ω ∈ ΩI

jω

(6)

where ΩI denotes the union of all frequency bands of interest, and W(ω) is a prescribed nonnegative weighting function. Our design task can be expressed as min x

max ω ∈Ω I

(7)

E (ω )

where x = [qT, pT]T. The above formulation assumes there is an implicit stability constraint on x (or q), such that all the poles of obtained IIR filters should lie inside the unit circle. For the sake of explanation, however, the design algorithm is first to be presented without any stability constraint. At the end of this section, the stability issue will then be addressed.

Several remarks on the iterative procedure described above are made here: 1. This bisection searching procedure is different from the usual bisection searching procedure, in which δ(k) is chosen as the arithmetic mean of δ+(k-1) and δ-(k-1), i.e., δ(k) = [δ+(k-1)+δ-(k-1)]/2. When δ* is small, choosing δ(k) as the geometric mean instead of the arithmetic mean can result in a smaller number of iterations required to achieve relative accuracy in locating δ* [33]. Actually, the bisection searching procedure presented above is performed with the arithmetic mean of log10δ+(k) and log10δ-(k). 2. In a design, the bisection searching procedure will be terminated, if the following condition is satisfied:

δ +( k ) − δ −( k ) ≤ κ min δ −( k )

B. Iterative Scheme In order to solve the design problem (7), we constrain |E(ω)|2 within an error limit δ, i.e., 2

E (ω ) =

W (ω ) ⎡⎣ D (ω )Q (e jω ) − P (e jω ) ⎤⎦ jω

Q (e )

2

(9)

where κmin > 0 is a prescribed small number. Let To denote the total number of iterations and it can be verified that

2

≤δ,

⎡ ⎛ log δ (0) − log10 δ −(0) ⎞ ⎤ To ≤ ⎢log 2 ⎜⎜ 10 + ⎟⎟ ⎥ + 1 ⎢ ⎝ log10 (1 + κ min ) ⎠ ⎥

ω ∈ Ω I (8)

Some minimax design algorithms [2], [5], [8], [10], [15], [19] try to find the minimum error limit δ* by directly minimizing δ subject to the constraint (8). However, it is hard to transform (8) into an equivalent convex form. Thus, some algorithms employ iterative procedures to approach the optimal solution. At each iteration, a design problem with an approximate constraint of (8) in a convex form is solved to find a better solution compared with the one obtained at the previous iteration. The iterative procedures continue until there is no significant change on filter coefficients or the performance of the designed filter cannot be further improved. In this paper, a bisection searching procedure is adopted. Instead of directly minimizing δ, a fixed error limit δ(k) is used in (8) at the kth iteration. The iterative scheme is shown below: Step 1: Given a set of design specifications, set k = 0, and then estimate the initial upper bound δ+(0) and lower bound δ-(0) for δ*. Step 2: Set k = k + 1, and choose δ(k) = [δ+(k-1)·δ-(k-1)]1/2, i.e., the geometric mean of δ+(k-1) and δ-(k-1). Then, solve a feasibility problem, in which the constraint (8) is recast by using the fixed error limit δ(k). If a feasible solution is found, which indicates δ+(k-1) ≥ δ(k) ≥ δ* ≥ δ-(k-1), then choose the new upper and lower bounds as δ+(k) = δ(k) and δ-(k) = δ-(k-1). On the contrary, if no feasible solution exists, which means δ+(k-1) ≥ δ* > δ(k) ≥ δ-(k-1), then choose the new upper and lower bounds as δ-(k) = δ(k) and δ+(k) = δ+(k-1). The formulation of the feasibility problem will be presented later. Step 3: If a predetermined accuracy criterion of locating δ* is satisfied, the iterative procedure is terminated; otherwise, go to Step 2 and continue.

3.

(10)

where |¯x¯| denotes the largest integer less than or equal to x. For convenience of the latter discussion, we assume that κmin can be chosen arbitrarily small to accurately locate the minimum error limit. Normally, the initial upper and lower bounds of δ* can be arbitrarily selected as long as the condition 0 < δ-(0) ≤ δ* ≤ δ+(0) is satisfied. However, it can be observed from (10) that if δ-(0) and δ+(0) are closer to each other, the total number of iterations required by such a bisection searching procedure could be reduced. As an attempt to obtain a lower δ+(0), we firstly utilize the LP method [19] to design an IIR filter under the same specifications. Then, δ+(0) can be chosen as the maximum value of |E(ω)|2 over ΩI for the obtained filter. Some other design methods, which may generate smaller δ+(0) for δ*, can also be deployed here. In order to achieve a reasonable δ-(0) for δ*, we utilize the SDP relaxation technique to convert the nonconvex constraint (8) into a convex form. With the relaxed constraint, the design problem can be solved by directly minimizing δ. Since the feasible set defined by the relaxed constraint is larger than that of (8), we always have δ*rel ≤ δ*, where δ*rel denotes the optimal value of the relaxed design problem. Then, δ*rel can be chosen as the lower bound δ-(0). The formulation of such a relaxed design problem using the SDP relaxation technique is presented in the Appendix.

C. Formulation of Feasibility Problem Using SDP Relaxation Technique In this subsection, we will construct a feasibility problem, in which the nonconvex constraint (8) is transformed into a



4 convex form using the SDP relaxation technique. This feasibility problem will be solved in Step 2 of the bisection searching procedure described earlier. In practice, the constraint (8) can be imposed on a set of frequency grid points, i.e., ωi ∈ ΩI for i = 0, 1, ···, L. Given the error limit δ(k) at the kth iteration, by multiplying |Q(ejω)|2 on both sides of (8), we have W (ωi ) ⎡⎣ D (ωi )Q (e jωi ) − P (e jωi ) ⎤⎦

2

= W 2 (ωi ) ⋅ ⎡ D (ωi ) + 2 Re {D (ωi )c H (ωi )} x + x T A(ωi ) x ⎤ ⎣ ⎦ 2

≤ δ ( k ) ⋅ Q (e jωi )

2

{

(11)

}

= δ ( k ) ⋅ ⎡⎣1 + 2 Re φMH (e jωi ) q + qT B (ωi )q ⎤⎦

ωi ∈ Ω I ,

i = 0,1," , L

where ⎡ D(ω )φM (e jω ) ⎤ c (ω ) = ⎢ ⎥ jω ⎣ −φN (e ) ⎦ A(ω ) = c (ω )c H (ω ) jω

(12) (13) (14)

jω

B (ω ) = φM (e )φ (e ) H M

In this paper, the superscript H represents the conjugate transpose operation of a complex vector or matrix, and Re{·} denotes the real part of a complex variable. Although the terms on both sides of (11) are (convex) quadratic functions of x and q, it cannot be directly transformed into an equivalent convex constraint. Consequently, a symmetric matrix is introduced as ⎡ qqT X = xx = ⎢ T ⎣ pq T

qpT ⎤ ⎡ X q ⎥=⎢ T ppT ⎦ ⎣ X q , p

X q, p ⎤ X p ⎥⎦

(15)

D(ωi ) + 2 Re { D (ωi )c H (ωi )} x + Tr { XA(ωi )} 2

i

(16)

where Tr{·} denotes the trace of a matrix. By introducing X, the original nonconvex constraint (8) is transformed into a linear inequality constraint in terms of the entries of x and X. By combining (15) and (16), we can construct a feasibility problem as min

z

2

≤

(17)

δ (k ) ⎡ ⋅ 1 + 2 Re {φMH (e jω )} q + Tr { X q B(ωi )}⎤⎦ + z W 2 (ωi ) ⎣ ωi ∈ Ω I , i = 0,1,", L i

⎡ Xq X = xx T = ⎢ T ⎣ X q, p

X q, p ⎤ ⎡q ⎤ where x = ⎢ ⎥ X p ⎥⎦ ⎣ p⎦

(17.a)

(17.b)

An auxiliary variable z is introduced into (17). It can be verified that a feasible solution (x, X) exists under the constraints (15) and (16) if and only if the minimum value of z obtained by solving (17) is less than or equal to 0. Then, the upper bound δ+(k) can be replaced by δ(k), and taken to be used in the next iteration of the bisection searching procedure. On the contrary, if the minimum value of z is larger than 0, which means given δ(k) the constraints (15) and (16) cannot be simultaneously satisfied. Then, in Step 2 of the bisection searching procedure, the lower bound δ-(k) will be replaced by δ(k), and taken to be used in the next iteration to determine δ(k+1). There is an obstacle to solve the feasibility problem (17). The matrix equality constraint (15) is nonconvex. In order to tackle this obstacle, we relax (15) as X ; xxT, which represents X−xxT is a positive semidefinite (PSD) matrix. The relaxed constraint X ; xxT is equivalent to ⎡1 Z =⎢ ⎣x

xT ⎤ ⎥ ;0 X⎦

(18)

Then, the feasibility problem (17) can be recast as min s.t.

z

D(ωi ) + 2 Re {D(ωi )c H (ωi )} x + Tr { XA(ωi )}

(19)

2

≤

Using X instead of x in the quadratic terms of (11), the constraint (11) can be rewritten in a matrix form as

δ (k ) ⎡ ≤ 2 ⋅ 1 + 2 Re {φMH (e jω )} q + Tr { X q B (ωi )}⎤⎦ W (ωi ) ⎣ ωi ∈ Ω I , i = 0,1," , L

D(ωi ) + 2 Re {D(ωi )c H (ωi )} x + Tr { XA(ωi )}

s.t.

δ (k ) ⎡ ⋅ 1 + 2 Re {φMH (e jω )} q + Tr { X q B(ωi )}⎤⎦ + z W 2 (ωi ) ⎣ ωi ∈ Ω I , i = 0,1,", L

xT ⎤ ⎥ ;0 X⎦ ⎡ Xq where X = ⎢ T ⎣ X q, p

i

⎡1 Z =⎢ ⎣x

X q, p ⎤ ⎡q ⎤ and x = ⎢ ⎥ X p ⎥⎦ ⎣ p⎦

(19.a)

(19.b)

Now (19.a) is a linear inequality constraint in terms of the entries of Z and the auxiliary variable z. Compared with (17.b), the constraint (19.b) defines a larger feasible set. Thus, for a given δ(k), if a feasible solution (x, X) exists for (17), by taking (x, X) into (19.a) and (19.b), the relaxed feasibility problem (19) also has a feasible solution Z, and the corresponding minimum value of z is definitely less than or equal to 0. It should be mentioned that even if a feasible solution Z with z ≤ 0 exists for (19), there is no guarantee that the original feasibility problem (17) also has a feasible solution (x, X). On the contrary, if the minimum value of z for (19) is greater than 0, it implies that there is no feasible solution Z satisfying both the linear



5 inequality constraint (16) and the relaxed LMI constraint (19.b). Accordingly, the original feasibility problem (17) does not have a feasible solution (x, X) for the given error limit δ(k). However, even if there is no feasible solution existing for (17), the relaxed feasibility problem (19) may still have a feasible solution Z with z ≤ 0. Note that if the rank of Z obtained by solving (19) is equal to 1, the relaxed constraint (19.b) is reduced to (17.b). Then, the feasibility problems (17) and (19) are equivalent to each other. Combined with the bisection searching procedure described in Section II-B, a sequence of relaxed feasibility problems (19) with different δ(k) are solved. Based on the analyses above, we arrive at the following sufficient condition for the optimal solution of the original design problem: ∞ ∞ ∞ Proposition 1: Let Z or, equivalently, (x , X ) be the final output of the bisection searching procedure, in which the relaxed feasibility problem (19) is solved at each iteration. The ∞ ∞ corresponding final error limit is denoted by δ . Then, δ is ∞ equal to δ*, and x is the optimal solution of the design problem ∞ (7), if the rank of Z is equal to 1. ∞ ∞ ∞ Proof: Suppose the rank of Z is equal to 1. Then x and X ∞ satisfy the equality constraint (17.b), and x is a minimax solution to the original design problem (7). On the other hand, from the discussion earlier, it follows that by successively solving the relaxed feasibility problem (19), we can always find ∞ ∞ a lower bound of δ*, i.e., δ ≤ δ*. Suppose that δ < δ*, which ∞ means that we could still find a solution x , which can achieve a ∞ lower minimum error limit δ . However, it contradicts the ∞ assumption that δ* is the minimum error limit. Therefore, δ ∞ should be equal to δ*. Accordingly, x is the optimal solution of the original design problem (7). □ This proposition implies that if we can find a rank-1 solution using the bisection searching procedure, then it is an optimal solution for the original design problem. Example 1 will be presented in Section III to demonstrate the capability of the proposed iterative procedure to achieve the optimal solution of a design problem. However, the rank-1 solution cannot always be attained, especially when the denominator order M is large and/or the design specifications are stringent. On the other hand, the stability constraint has not been incorporated in the iterative procedure. It is known that when M > 2, the stability domain cannot be strictly expressed as a convex set. Thus, solving the relaxed feasibility problem (19) during the iterative procedure cannot always lead to an optimal solution. D. SDP Formulation Using Trace Heuristic Approximation In order to obtain a rank-1 solution, we can constrain the rank of Z equal to 1 in the relaxed feasibility problem (19) during the iterative procedure. However, in general, the rank constraint is nonconvex, and incorporating it could make the problem computationally intractable. In practical designs, we employ a trace heuristic method [34] to approximate the design problem with the rank constraint. The idea behind the method is to incorporate a regularization term Tr{Z} = ∑iλi(Z) into the objective of (19), where λi(Z) (i = 1, 2, ···, N+M+2) denote the

real eigenvalues of the symmetric matrix Z. Without loss of generalization, we can assume that λi(Z) are arranged in a non-ascending order, i.e., λ1(Z) ≥ λ2(Z) ≥ ··· ≥ λN+M+2(Z). Since Z is PSD, Tr{Z} is also equal to ||Λ(Z)||1 = ∑i|λi(Z)| = ∑iλi(Z), where Λ(Z) = [λ1(Z), λ2(Z), ···, λN+M+2(Z)]T is a vector composed by all the eigenvalues of Z, and ||·||1 denotes l1-norm of a vector. It is known that in an approximation problem with the objective in the form of l1-norm of the approximation error, the final solution tends to be a sparse vector [29]. Thus, minimizing Tr{Z} renders many of the eigenvalues of Z as zeros, resulting in a low-rank matrix. As the rank of Z is close to 1, Tr{Z} ≈ 2 2 2 λ1(Z) ≈ 1+||q||2 +||p||2 = 1+||x||2 , where ||·||2 denotes the Euclidean (or l2) norm of a vector. Therefore, in practice, Tr{Z} cannot be over-attenuated; Otherwise, ||x||2 could be too small to achieve a satisfactory design. We modify the objective of (19) as the weighted sum of Tr{Z} and z. Then, the relaxed feasibility problem (19) is modified as min s.t.

α Tr { Z } + (1 − α )z

(20)

D(ωi ) + 2 Re { D(ωi )c (ωi )} x + Tr { XA(ωi )} 2

≤

H

δ (k ) ⎡ ⋅ 1 + 2 Re {φMH (e jω )} q + Tr { X q B (ωi )}⎤⎦ + z W 2 (ωi ) ⎣ ωi ∈ Ω I , i = 0,1," , L

xT ⎤ ⎥ ;0 X⎦ ⎡ Xq where X = ⎢ T ⎣ X q, p

i

⎡1 Z =⎢ ⎣x

X q, p ⎤ ⎡q ⎤ and x = ⎢ ⎥ ⎥ Xp ⎦ ⎣ p⎦

(20.a)

(20.b)

where 0 ≤ α ≤ 1. When α = 0, the regularized feasibility problem (20) is reduced to (19). The regularization coefficient α should be chosen as small as possible so as to best approximate the relaxed feasibility problem (19) while avoiding Tr{Z} being over-attenuated. In order to determine the minimum value of α, another bisection searching procedure is introduced. Note that for the complete algorithm, there are two nested bisection searching procedures. They have different effects. The outer bisection searching procedure is used to locate the minimum error limit δ*. Given a fixed error limit δ(k) at the kth outer bisection searching procedure, the inner bisection searching procedure is invoked to find an appropriate α to make the rank of the obtained matrix Z close to 1. Another effect of the inner iterative procedure is to restrict all the poles’ positions for the stability, which will be discussed in detail later. For clarity, in the following, we use l to represent the iteration number of the inner bisection searching procedure, while k for the outer bisection searching procedure. Accordingly, z, α, and Z in (20) are replaced by z(k,l), α(k,l), and Z(k,l) at the iteration step (k, l), respectively. The iterative scheme of the inner bisection searching procedure is shown below: Step 1: Given δ(k), set l = 0, and then choose the initial upper bound α+(k,0) and lower bound α-(k,0), respectively. Step 2: Set l = l + 1, and choose α(k,l) = [α+(k,l-1)·α-(k,l-1)]1/2. Using δ(k) and α(k,l), solve the regularized feasibility



6 problem (20). If λ2(Z(k,l))/λ1(Z(k,l)) < ε, set α+(k,l) = α(k,l) and α-(k,l) = α-(k,l-1); Otherwise, α-(k,l) = α(k,l) and α+(k,l) = α+(k,l-1). Step 3: If the predetermined accuracy of locating the minimum value of α is satisfied, terminate the inner iterative procedure; otherwise, go to Step 2 and continue. Some remarks regarding the inner iterative procedure are made here: 1. In practice, we use λ2(Z(k,l))/λ1(Z(k,l)) < ε to replace the condition that the rank of Z(k,l) is equal to 1. Here, λ1(Z(k,l)) and λ2(Z(k,l)) denote the first and second largest eigenvalues of Z(k,l), and the parameter ε > 0 represents a pre-specified small positive value. 2. Before the inner bisection searching procedure, the relaxed feasibility problem (19) should be solved first. Let (z(k,0), Z(k,0)) denote the result obtained from (19). If z(k,0) > 0, which means there is no feasible solution for the relaxed feasibility problem (19), new upper and lower bounds of δ* are appropriately selected and the design program directly goes to Step 3 of the outer iterative procedure. If z(k,0) ≤ 0 and λ2(Z(k,0))/λ1(Z(k,0)) ≥ ε, the inner bisection searching procedure will be triggered. 3. In our design, the inner iteration will be terminated as [α+(k,l)–α-(k,l)]/α-(k,l) ≤ κmin. Like the outer iterative procedure, the convergence of the inner iterative procedure can be guaranteed. Let Ti(k) (k = 1, 2, ···, To) represent the total number of the inner iterations at the kth outer iteration. Similar to (10), we have ⎡ ⎛ log α ( k ,0) − log10 α −( k ,0) ⎞ ⎤ Ti (k ) ≤ ⎢log 2 ⎜⎜ 10 + ⎟⎟ ⎥ + 1 log10 (1 + κ min ) ⎝ ⎠⎥ ⎢

4.

The initial upper and lower bounds can be arbitrarily chosen as long as the condition 0 < α-(k,0) ≤ α+(k,0) ≤ 1 is satisfied. In order to reduce the total number of the inner iterations, in our design the initial upper and lower bounds of α at the kth outer iteration are chosen as

α +( k ,0) = γα +( k ′,T ( k ′ )) ,

1 ≤ k ′ < k ≤ T0

i

α

5.

( k ,0) −

=γ α −1

( k ′ ,Ti ( k ′ )) −

,

1 ≤ k ′ < k ≤ T0

(21) (22)

where γ ≥ 1, and α+(k',Ti(k')) and α-(k',Ti(k')) denote the final upper and lower bounds of α determined by the inner iterative procedure at the k'th outer iteration. Obviously, the searching range of α can be extended by increasing γ. For the first time the inner iterative procedure is invoked, the initial upper bound α+(k0,0) and lower bound α-(k0,0) (k0 ≥ 1) should be specified by designers. Since there is no prior information to determine α+(k0,0) and α-(k0,0), normally we can choose α+(k0,0) and α-(k0,0) close to 1 and 0, respectively. So far, it has not been strictly proved that there always exists some α for which the rank of Z is equal to 1. Nevertheless, in the extreme situation when α = 1, the

Fig. 1. Flowchart of the complete design algorithm.



7

6.

constraint (20.a) can always be satisfied since z can be chosen as any value without any harm to the objective function of (20). Then, it can be deduced from (20) that the rank of the obtained Z should be equal to 1, and λi{Z} = 0 except λ1{Z} = 1. Therefore, in practice, we can assume that when α is large enough, the rank of the obtained Z is close to 1. Since the regularization term Tr{Z} is incorporated in the objective of (20), even if the rank of the final output Z is equal to 1, it cannot be concluded that the optimal solution is attained. However, as the minimum value of α determined by the inner iterative procedure is small enough, the regularized feasibility problem (20) can serve as a good approximation of the relaxed feasibility problem (19).

E. Stability Issue So far, the proposed iterative method cannot definitely ensure the stability of designed IIR filters. Therefore, stability constraint needs to be incorporated in the design procedure. However, many of stability constraints represent only a sufficient condition for stability, which means that some stable filters could be excluded from the admissible sets defined by such stability constraints. In this paper, we adopt a monitoring strategy to make all the poles lie inside the stability domain. It is known that the positive-realness based stability condition [19] can be expressed as M

Re {Q (e jω )} = 1 + ∑ qm cos(mω ) ≥ ν

(23)

m =1

where ν is a small positive number. This sufficient stability condition can be readily extended to the situation where all the poles are required to lie inside a circle of radius ρmax ≤ 1 for robust stability: M

−m Re {Q ( ρ max e jω )} = 1 + ∑ qm ρ max cos( mω ) ≥ ν

(24)

m =1

From (24), we have M

M

m =1

m =1

−m −m 1 + ∑ qm ρ max cos( mω ) ≥ 1 − ∑ qm ρ max

≥ 1−

M

∑q

2 m

m =1

⋅

(25) M

∑ρ m =1

−2 m max

In (25), the second inequality is obtained by Hölder’s inequality. By combining (24) and (25), we can construct a stability condition as q2=

M

∑q m =1

2 m

≤

1 −ν M

∑ρ m =1

−2 m max

(26)

It can be observed from (26) that if ν is fixed, we can force the poles to move towards the origin (i.e., ρmax → 0) by suppressing ||q||2. When all the poles lie on the origin (i.e., ρmax = 0), we have ||q||2 = 0 and the designed IIR digital filter essentially degenerates to an FIR digital filter. However, the stability condition (26) is too restrictive to be directly applied in a practical design. Instead of employing a fixed upper bound for 2 ||q||2, we can gradually reduce ||q||2 during the iterative design procedure. Note that as the rank of Z is equal to 1, Tr{Xq} = 2 2 ||q||2 . Therefore, we can attenuate ||q||2 by reducing Tr{Xq}, which can be achieved by increasing the regularization coefficient α in the objective of the SDP problem (20). Since a larger α results in Tr{Z} being minimized within a smaller range, a large α could make the poles too close to the origin, which may degrade the performance of designed filters. So the value of α should be carefully selected. Here, we also resort to the inner bisection searching procedure. In Step 2 of the inner bisection searching procedure described earlier, after solving the regularized feasibility problem (20), besides the ratio λ2(Z(k,l))/λ1(Z(k,l)), we also need to check the maximum pole radius of the obtained filter, which is denoted by ρ(q(k,l)). If λ2(Z(k,l))/λ1(Z(k,l)) < ε and ρ(q(k,l)) ≤ ρmax, choose α+(k,l) = α(k,l) and α-(k,l) = α-(k,l-1); otherwise, set α-(k,l) = α(k,l) and α+(k,l) = α+(k,l-1). Similarly, at each outer iteration, we need to check z(k,0), λ2(Z(k,0))/λ1(Z(k,0)), and ρ(q(k,0)) after solving (19) in order to determine whether or not the inner iterative procedure needs to be invoked. In practice, some other constraints can be imposed on x and X to refine the formulation of the feasibility problems (19) and (20). In our design, a set of linear inequality constraints in terms of the denominator coefficients q and the diagonal elements Xq(m,m) of Xq shown respectively in (27)-(28) below have been incorporated into the feasibility problems (19) and (20). m qm ≤ C ( M , m) ρ max ,

m = 1, 2," , M

(27)

2

(28)

m ⎤⎦ , X q (m, m) ≤ ⎡⎣C ( M , m) ρ max

m = 1, 2," , M

where C(M,m) = M!/[m!(M−m)!]. It can be verified that (27) and (28) are necessary conditions for the stability of designed IIR filters. F. Complete Algorithm Fig. 1 shows the flowchart of the complete design algorithm. The dashed box indicates the inner bisection searching procedure described in Section II-D. For the ease of notation, the ratio λ2(Z)/λ1(Z) is replaced by η(Z). It can be seen from Fig. 1 that the major computation involved is for solving the feasibility problem (20).

III. SIMULATIONS In this section, four examples are presented to demonstrate the effectiveness of the proposed algorithm. In all the examples, the parameters ε and κmin are chosen as 5×10-2 and 10-3. The initial upper and lower bounds of α at the outer iteration in



8 which the inner iterative procedure is invoked for the first time are set equal to 10-2 and 10-12, respectively. At the succeeding outer iterations, we choose γ = 5 in (21) and (22) to determine α+(k,0) and α-(k,0). The linear inequality constraints (19.a) and (20.a) are imposed on a set of discrete frequency points taken from 101 equally-spaced grid points over the whole frequency band. If the weighting function W(ω) is not explicitly defined in the specifications, it is always set equal to 1 over ΩI, and 0 otherwise. Similarly, without any explicit declaration, the admissible maximum pole radius is always chosen as ρmax = 1. In order to evaluate the performance of the designed filters, besides the peak and least-squares (L2) errors of the magnitude (MAG) and group delay (GD) responses over ΩI, we also adopt the following (weighted) minimax error: eMM

P(e jω ) = max W (ω ) D(ω ) − ω∈Ω I Q(e jω )

(29)

Fig. 2. Magnitude and group delay responses of the IIR digital filter designed by the proposed method in Example 1.

In our designs, all the SDP problems are solved by the SeDuMi [35] in MATLAB environment. A. Example 1 The first example is to design a lowpass digital filter with the following ideal frequency response ⎧e − j12ω D(ω ) = ⎨ ⎩0

0 ≤ ω ≤ 0.4π 0.56π ≤ ω < π

Filter orders are chosen as N = 15 and M = 4. The design specifications are exactly the same as those adopted by the first example of [9]. Using the proposed algorithm, we design an IIR digital filter, whose magnitude and group delay responses are shown in Fig. 2. The magnitude of the weighted complex approximation error, i.e., |E(ω)|, is shown in Fig. 3. Simulation result reveals that in this design To = 13 and Ti(k) = 0 for k = 1, 2, ···, 13, which means that the inner bisection searching procedure is actually not invoked during the outer iterative procedure. The maximum pole radius of the obtained IIR filter is 0.8589. All the pole locations of the obtained IIR filter are given by 0.0673±j0.8563 and 0.1614±j0.2888, where j denotes the square root of –1. The zero locations are given by 1.6304, 1.4157±j0.7819, 0.8274±j1.3147, –1.2082±j0.2181, –1.0001±j0.6044, –0.6771±j0.8581, –0.2084±j0.9809, and –0.3757±j0.9609. By analyzing the final output Z, we find that except the largest eigenvalue λ1(Z) (= 2.4617), all the other eigenvalues of Z are negligible (≤ 8.9715×10-7). Then, by ignoring λi(Z) (i = 2, 3, ···, N+M+2), the obtained Z can be approximately regarded as a rank-1 matrix. In view of Proposition 1, it can be concluded that the final solution is close to the optimal solution of the original design problem. Note that in this example the denominator order is not too high and the design specifications are not stringent. Thus, a satisfactory design can be achieved by only successively solving the relaxed feasibility problem (19). In general, however, the inner bisection searching procedure has to be utilized to attain the

Fig. 3. Magnitude of complex approximation error |E(ω)| in Example 1. Solid curves: designed by the proposed method; Dashed curves: designed by the SM method [2].

Method Proposed SM [2]

TABLE I MEASUREMENTS OF DESIGN RESULTS IN EXAMPLE 1 Minimax Passband MAG Passband GD Stopband MAG Error eMM (Peak/ L2 in dB) (Peak/ L2) (Peak/ L2 in dB) -45.721/ 2.778e-1/ -45.720/ 5.176e-3 -55.166 2.540e-2 -50.373 -45.998/ 2.933e-1/ -44.807/ 5.747e-3 -54.561 2.604e-2 -50.543

rank-1 solution. For comparison, we also utilize the SM method [2] to design an IIR filter under the same specifications. The initial point is chosen as the optimal FIR design with the filter order equal to N. The design result shows that the SM method can achieve a stable IIR filter even without the positive-realness based stability constraint. The maximum pole radius of the obtained filter is 0.8622. The magnitude of its corresponding complex approximation error is also shown as dashed curves in Fig. 3. All the error measurements are summarized in Table I. Both of design results are close to the optimal point. However, the proposed method can achieve slightly better performance except in peak error of the passband magnitude and L2 error of the stopband magnitude than those obtained by [2].



9 B. Example 2 The second example, which is taken from [13], is to design another lowpass filter. The ideal frequency response is defined as ⎧e− j 5ω D(ω ) = ⎨ ⎩0

0 ≤ ω ≤ 0.2π 0.4π ≤ ω < π

Both numerator order N and denominator order M are set equal to 4. After 14 outer iterations, i.e., To = 14, the iterative procedure converges to the final solution. Only at the second outer iteration, the inner bisection searching procedure is invoked, and Ti(2) = 15. The minimum value of α determined by the inner iterative procedure is 2.3714×10-6. The maximum pole radius of the obtained filter is 0.8975. The first and second largest eigenvalue of the final Z are 19.6301 and 2.1717×10-5. The pole and zero locations are given by 0.6300±j0.6392, 0.6668±j0.2188, and 2.2454, –0.8890, 0.2043±j0.9551. The magnitude and group delay responses are plotted in Fig. 4. The magnitude of the complex approximation error E(ω) is shown in Fig. 5. We also design an IIR filter under the same specifications using the Remez multiple exchange method proposed in [13]. The magnitude of E(ω) is shown as dashed curves in Fig. 7. The maximum pole radius of the obtained filter is 0.8771. All the error measurements summarized in Table II shows that the proposed method achieves better performance.

Fig. 4. Magnitude and group delay responses of the IIR digital filter designed by the proposed method in Example 2.

C. Example 3 The third example is to design two full-band differentiators [8] with the ideal frequency response D(ω ) =

ω j ⎡⎣0.5π −(τ e π

s + 0.5

)ω ⎤⎦

,

0≤ω ≤π

where τs is an integer delay. The first differentiator is of order 8, i.e., N = M = 8. And the filter order in the second design is set to 5. In both designs, τs is chosen as 3. Therefore, the ideal group delay is equal to 3.5 over the whole frequency band. As proposed in [8], the weighting functions in both designs are chosen as ⎧⎪π 0.1π < ω < π W (ω ) ⎨ ω ⎪⎩10 0 ≤ ω ≤ 0.1π

In [8], the differentiator of order 8 is first designed by the modified Ellacott-Williams (EW) algorithm. Nevertheless, the obtained differentiator of order 8 is a degenerate filter. There are three pairs of poles and zeros which are nearly cancelled by each other. After removing these poles and zeros, the remaining poles and zeros are then used to construct the initial filter, from which another differentiator of order 5 with the same ideal group delay is redesigned. The poles and zeros of these two differentiators are given in [8]. The admissible maximum pole radius specified in the design of [8] is 0.98. The maximum pole

Fig. 5. Magnitude of complex approximation error |E(ω)| in Example 2. Solid curves: designed by the proposed method. Dashed curves: designed by the Remez multiple exchange method [13].

Method Proposed Remez [13]

TABLE II MEASUREMENTS OF DESIGN RESULTS IN EXAMPLE 2 Minimax Passband MAG Passband GD Stopband MAG Error eMM (Peak/ L2 in dB) (Peak/ L2) (Peak/ L2 in dB) -33.437/ 5.765e-1/ -33.437/ 2.129e-2 -43.687 7.114e-2 -38.931 -32.697/ 9.573e-1/ -32.617/ 2.340e-2 -43.598 8.654e-2 -36.826

radii of the designed differentiators of order 8 and order 5 are 0.6829 and 0.4400, respectively. For comparison, we choose the admissible maximum pole radii as 0.7 and 0.5 in our two designs, respectively. In the design of differentiator of order 8, it takes 14 outer iterations to reach the final solution, i.e., To = 14. At each outer iteration, the inner iterative procedure is invoked, and simulation result shows that Ti(1) = 15 and Ti(k) = 12 for k = 2, 3, ···, 14. The minimum value of α determined by each inner bisection searching procedure is within the range of [7.2448×10-7, 6.7989×10-5]. The largest eigenvalue of the final Z is 1.3300, and other eigenvalues are less than 9.0490×10-8. The pole and zero locations of the designed differentiator of order 8 are given by –0.6999, 0.5418, 0.0241±j0.5652, –0.2757±j0.2267, 0.2376±j0.2719, and 3.2556, 1.0060, 0.5023,



10

Fig. 6. Design characteristics and errors of the differentiator of order 8 designed by the proposed method in Example 3.

Fig. 8. Magnitudes of complex approximation error |E(ω)| in Example 3. Solid curves: differentiator of order 8; Dashed curves: differentiator of order 5.

Method Proposed Modified EW [8]

Fig. 7. Design characteristics and errors of the differentiator of order 5 designed by the proposed method in Example 3.

–0.6692, –1.1536±j3.0858, 0.0347±j0.5453. In the design of differentiator of order 5, To = 13, and Ti(1) = Ti(2) = 0, Ti(3) = 15, Ti(k) = 12 for k = 4, 5, ···, 13. The minimum value of α determined by each inner iterative procedure is within the range of [7.2385×10-6, 3.5142×10-5]. The obtained Z has eigenvalues λ1(Z) = 1.3651 and λi(Z) ≤ 6.6688×10-7 (i = 2, 3, ···, 12). The pole and zero locations of the obtained differentiator of order 5 are located at –0.4999, –0.1787±j0.3288, 0.2446±j0.2130, and 3.2415, 1.0050, –0.2780, –1.1527±j3.0783. The design characteristics and errors are shown in Figs. 6 and 7. It can be observed that near the origin, the group delay (or phase response) of the designed filter has a large error. However, it can be ignored, since the magnitude responses of the designed differentiator near the origin are almost equal to zero. Therefore, all the peak and L2 errors of group delay are computed within the frequency band [0.05π, π]. The magnitudes of the obtained E(ω) are given in Fig. 8, where solid curves correspond to the differentiator of order 8 and dashed curves correspond to the differentiator of order 5. All the error measurements are summarized in Table III. The modified EW method [8] also adopts an iterative procedure to design IIR filters. However, at each iteration, the searching direction of the denominator is

TABLE III MEASUREMENTS OF DESIGN RESULTS IN EXAMPLE 3 GD within [0.05π, π] Minimax MAG Order (Peak/ L2) Error eMM (Peak/ L2 in dB) 8 1.850e-2 -35.122/ -43.737 3.197e-1/ 6.447e-2 5 2.231e-2 -33.418/ -43.294 2.434e-1/ 6.143e-2 8 2.845e-2 -32.776/ -41.718 3.580e-1/ 7.582e-2 5 4.035e-2 -28.122/ -41.666 3.265e-1/ 7.859e-2

restricted within the neighborhood of the previous denominator. Therefore, the iterative procedure could be trapped in a locally minimum point around the initial point. Although the regularization term is incorporated in our design, the regularization coefficient α determined by the inner iterative procedure is sufficiently small. Consequently, the feasible set defined by the relaxed constraints (20.a) and (20.b) have not been severely restrained. As a result, the proposed method can achieve better performance. D. Example 4 The last example is to design a halfband highpass filter [4]. The ideal frequency responses are given by ⎧e− j12ω D(ω ) = ⎨ ⎩0

0.525π ≤ ω < π 0 ≤ ω ≤ 0.475π

The filter orders are chosen as M = N = 14. First of all, we directly utilize the proposed method to design the IIR filter with ρmax = 0.98. After 14 outer iterations, the design procedure converges to the final solution. The total number of the inner iterations at each outer iteration is Ti(1) = 0, Ti(2) = 15, and Ti(k) = 12 for k = 3, 4, ···, 14. The minimum value α(k,Ti(k)) (k = 2, 3, ···, 14) determined by each inner iterative procedure is within the range of [1.1814×10-6, 3.6685×10-6]. The largest eigenvalue of the final Z is equal to 2.5978 whereas λi(Z) ≤ 7.2489×10-5 for i = 2, ···, 30. The maximum pole radius of the designed IIR filter is 0.9800. The poles and zeros are located at 0.8536±j0.2124, 0.6549±j0.6042, −0.0140±j0.9799, −0.1431±j0.8086, −0.3739±j0.6495, −0.6944±j0.1526, −0.5766±j0.4325, and



11 2.0334, −1.3854, −1.1884±j0.7108, −0.6524±j1.2087, 1.0981±j0.2818, 0.8103±j0.7304, 0.3666±j1.0266, 0.0975±j0.9965. The magnitude and group delay responses, and the magnitude of the complex approximation error are shown as dash-dotted curves in Figs. 9 and 10, respectively. The corresponding error measurements referred as Proposed-1 are given in Table IV. For comparison, the SM method [2] is employed to design an IIR digital filter under the same specifications. The iterative procedure starts from the optimal FIR filter design of order N = 14. The maximum pole radius of the obtained filter is 0.9346. The corresponding magnitude of E(ω) is also shown as dashed curves in Fig. 10. Obviously, the proposed method can achieve better performance. In the above design, in order to ensure the stability, Tr{Z} of the IIR filter obtained above has been over-attenuated. It should be noted that such over-attenuation does not always appear when ρmax < 1. However, if such over-attenuation occurs, we could redesign the IIR filter with a larger admissible maximum pole radius ρmax = 1. In so doing, the final solution can be obtained after 14 outer iterations. Simulation result shows that Ti(2) = 15, Ti(k) = 12 for k = 3, 7, 9, and Ti(k) = 0 for k = 1, 4-6, 8, 10-14. The minimum value α(k,Ti(k)) (k = 2, 3, 7, 9) is within the range of [5.5412×10-8, 1.0228×10-6]. The largest eigenvalue of the obtained Z is 5.1767, and other eigenvalues λi(Z) (i = 2, ···, 30) are less than 0.0888. All the pole locations are given by 0.8939, –0.1045, 0.7585±j0.4787, –0.0089±j0.9997, –0.1036±j0.82694, –0.3276±j0.6641, –0.6680±j0.1585, and –0.5414±j0.4468. The zero locations of the designed filter are given by 1.6520, 1.5332, 1.3297, –1.4417, –1.2359±j0.7402, –0.6754±j1.2521, 0.8817±j0.6503, 0.2836±j0.9983, and 0.0932±j0.9969. It can be observed that the regularization coefficient α utilized in the previous design (Proposed-1) is larger than that utilized in this design. Consequently, in the previous design, the obtained Z is much closer to a rank-1 solution. However, the largest eigenvalue λ1(Z) obtained in the previous design is smaller than that obtained in this design. In other words, the l2-norm of the filter coefficients x obtained in the previous design is over-attenuated. In order to make all the poles inside the circle of the radius ρmax = 0.98, we can simply rescale the denominator coefficients q of the obtained IIR filter (with ρmax = 1) as m

⎡ρ ⎤ qm = qm ⎢ max ⎥ , ⎣ ρ (q ) ⎦

m = 1, 2," M

(30)

where qm denotes the rescaled denominator coefficients. Given T q = [ q1 , q2 , " , qM ] , the optimal numerator coefficients

p = [ p 0 ,

p1 , " ,

T p N ] can be determined by solving the

following SOCP problem: min s.t.

δ G (ωi ) p − g (ωi ) 2 ≤ δ

ωi ∈ Ω I ,

i = 0,1," , L

(31) (31.a)

Fig. 9. Magnitude and group delay responses of the IIR digital filter designed by the proposed method in Example 4. Solid curves: designed by the proposed method (ρmax = 1) followed by rescaling q through (30) and solving (31). Dash-dotted curves: designed by the proposed method (ρmax = 0.98).

Fig. 10. Magnitude of complex approximation error |E(ω)| in Example 4. Solid curves: designed by the proposed method (ρmax = 1) followed by rescaling q through (30) and solving (31). Dash-dotted curves: designed by the proposed method (ρmax = 0.98). Dashed curves: designed by the SM method [2]. TABLE IV MEASUREMENTS OF DESIGN RESULTS IN EXAMPLE 4 Minimax Passband MAG Passband GD Stopband MAG Method (Peak/ L2) (Peak/ L2 in dB) Error eMM (Peak/ L2 in dB) -30.720/ 1.813/ -30.714/ Proposed-1 2.913e-2 -34.970 -38.988 1.689e-1 -32.218/ 1.716/ -32.211/ Proposed-2 2.452e-2 -40.058 1.649e-1 -37.270 -25.333/ 4.062/ -25.051/ SM [2] 6.140e-2 -32.497 4.030e-1 -29.538

where ⎡ ⎧ φTN (e jω ) ⎫⎤ ⎬⎥ ⎢ Re ⎨ T jω ⎩1 + φM (e )q ⎭⎥ ⎢ G (ω ) = W (ω ) ⎢ T jω ⎫⎥ ⎧ ⎢ Im ⎨ φN (e ) ⎬⎥ T jω ⎢⎣ ⎩1 + φM (e )q ⎭⎥⎦


(32)


12 ⎡ Re { D(ω )}⎤ g (ω ) = W (ω ) ⎢ ⎥ ⎣ Im { D(ω )}⎦

(33)

Now, the poles and zeros are located at 0.8763, –0.1024, 0.7435±j0.4692, –0.0087±j0.9799, –0.1016±j0.8130, –0.3212±j0.6510, –0.6548±j0.1554, –0.5307±j0.4380, and 2.4467, –1.4022, –1.1994±j0.7249, –0.6481±j1.2248, 1.2428±j0.2049, 0.8701±j0.6615, 0.2633±j0.9249, 0.0898±j0.9852. The maximum pole radius of the obtained filter is 0.9800 The design results and the magnitude of E(ω) are plotted as solid curves in Figs. 9 and 10, respectively. The corresponding error measurements referred as Proposed-2 are summarized in Table IV. Although by using (30) and (31) the obtained IIR filter is not guaranteed to be (locally) optimal, it can still achieve better performance than the one directly designed by the proposed method with ρmax = 0.98. This narrow transition-band example serves to demonstrate a further improvement can be achieved by applying the Proposed-2 method after the Proposed-1 method. For the Examples 2, 3, and many other filters with a similar level of filter requirements, the Proposed-1 method is able to arrive at a quality design and no further improvement can possibly be achieved by the Proposed-2 method. The Proposed-2 method is also not necessary in those filters with much less stringent filter requirements such as Example 1 as its inner iterative procedure was not even invoked.

IV. CONCLUSIONS In this paper, a new IIR digital filter design algorithm in the minimax sense has been presented. Instead of directly minimizing the error limit of |E(ω)|, a bisection searching procedure is deployed in the algorithm to locate the minimum error limit of |E(ω)|2. At each iteration, a feasibility problem with a fixed error limit is constructed using the SDP relaxation technique. It can be proved that the final output of the iterative procedure is the optimal solution, if the rank of the final Z obtained by successively solving the relaxed feasibility problem (19) is equal to 1. In practice, the rank-1 solution cannot be always obtained. Therefore, a regularization term Tr{Z} is linearly combined with the objective function of (19). Another bisection searching procedure is incorporated to find a minimum regularization coefficient used in the objective function of the regularized feasibility problem (20). To ensure the stability of the designed IIR filter, a monitoring strategy is utilized in the design procedure. Simulation results reveal that in some situations, especially when the denominator order M is not too high and/or the design specifications are not stringent, we can approximately obtain the rank-1 solutions using the proposed design algorithm without the regularization term. In these designs, it can be concluded that each obtained design is close to the optimal solution of its original design problem. However, if the regularization term is incorporated, even if a final output is a rank-1 solution, it cannot be concluded that an optimal design is

attained. In a practical design, the parameters ε and ρmax should be carefully selected in order to avoid Tr{Z} and, accordingly, ||x||2 being over-attenuated. If the obtained regularization coefficient α is sufficiently small, a satisfactory design can be obtained by the proposed algorithm, which has been demonstrated by several filter examples in this paper.

APPENDIX The initial lower bound δ-(0) of δ* can be obtained by using a similar SDP relaxation technique described in Section II-C. The relaxed design problem can be formulated as: min s.t.

δ

D(ωi ) + 2 Re { D(ωi )c (ωi )} x + x A(ωi ) x 2

≤

H

δ ⋅ ⎡1 + 2 Re φMH (e jωi ) q + Tr { X q B (ωi )}⎤⎦ W 2 (ωi ) ⎣

{

(A1)

T

}

(A1.a)

⎡q ⎤ where x = ⎢ ⎥ , and ωi ∈ Ω I , i = 0,1," , L ⎣ p⎦

⎡1 Y =⎢ ⎣⎢q

qT ⎤ ⎥ ;0 X q ⎦⎥

(A1.b)

In (A1), the decision variables are δ, p, and Y. Unlike the linear inequality constraint (16) which is expressed in terms of x and X, now (A1.a) is a hyperbolic constraint with respect to p, q and Xq, which can be further recast as an LMI constraint. The constraints (A1.a) and (A1.b) define a larger feasible set for the problem (A1) compared with (8). Therefore, a lower bound on the optimal value of the original design problem can be obtained by solving (A1), which can be utilized in the subsequent iterations of the bisection searching procedure. A. Some Remarks It should be mentioned that although the relaxed constraints of (19) and (A1) are both obtained by applying the SDP relaxation technique on the nonconvex constraint (8), they are used in different situations and cannot be replaced by each other. In (19) the error limit δ must be fixed. Otherwise, (19.a) is quasiconvex and cannot be directly incorporated. However, in (A1) the objective function to be minimized is chosen as δ with a set of relaxed constraints. Therefore, the relaxed constraints of (19) cannot be applied to find an initial lower bound δ-(0) in (A1). By fixing δ, the relaxed constraints (A1.a) and (A1.b) could be applied to formulate the following feasibility problem, which is similar to (19) min s.t.

z

D(ωi ) + 2 Re { D(ωi )c (ωi )} x + x A(ωi ) x 2

≤

H

(A2)

T

δ (k ) ⎡ ⋅ 1 + 2 Re {φMH (e jω )} q + Tr { X q B(ωi )}⎤⎦ + z W 2 (ωi ) ⎣ i

⎡q⎤ where x = ⎢ ⎥ , and ωi ∈ Ω I , i = 0,1,", L ⎣ p⎦


(A2.a)


13 ⎡1 Y =⎢ ⎣⎢ q

qT ⎤ ⎥ ;0 X q ⎦⎥

(A2.b)

However, this formulation can lead to problematical solutions. Assume that by solving (A2) with a given δ(k), a set of z, x, and Xq (or Y) have been obtained. Since Xq is PSD, we can construct another PSD matrix bXq for any b > 1, which satisfies Tr{bXqB(ω)} > Tr{XqB(ω)} > 0 and bXq ; qqT. By taking bXq into (A2.a) and (A2.b), it can be verified that the scaled matrix bXq can also satisfy these two constraints with the obtained z and x. Thereby, in (A2.a) the value of z can be slightly reduced without changing the inequality sign in (A2.a). This implies that for any sufficiently scaling up Xq (equivalent to increasing b), we can always obtain z < 0. Under this circumstance,δ+(k) will be chosen as δ(k) and eventually reduced to the initial lower bound δ-(0). Obviously, the desired minimum error limit and the corresponding filter coefficients cannot be obtained by applying (A2) in the outer bisection searching procedure to locate δ*.

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