New Linear-Programming-Based Filter Design - CiteSeerX

276

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 5, MAY 2005

New Linear-Programming-Based Filter Design Hoang Duong Tuan, Member, IEEE, Tran Thai Son, Hoang Tuy, and Truong Nguyen, Fellow, IEEE

Abstract—It is well known that the filter-design problem with mask constraints can be formulated as a semi-infinite program. There are two approaches toward the solution of this semi-infinite program. The first griding approach relaxes the semi-infinite constraint by refining it in the finite griding domain so it does not always guarantee global optimal and feasible solution. The second semi-definite programming (SDP)-based approach does guarantee the global optimal solution and excellently handles positive real constraints. However, the magnitude constraints are still persistent and not yet handled by SDP tool in an efficient manner. In this brief, a new tight polyhedral approximation for semi-infinite constrained domain is proposed. Based on it, we present a new linear-programming-based solution method for filter design, which unlike the griding approach yields global solution and unlike SDP based approach is practical for even long-tap filters. Simulation results confirm the viability of the proposed method. Index Terms—Finite-impulse response (FIR) filter, linear programming, semi-definite programming.

I. INTRODUCTION

O

NE of the most fundamental and classical problems in signal processing is the finite-impulse response (FIR) filter design problem, whose optimization-based formulation is well known (see, e.g., [2], [3], [7], [10], [11] and some details in Section IV) (1)

The main interest is how to handle the semi-infinite problem (SIP) in (2) which often expresses the desired magnitude mask constraints for the filter at the stopband and passband. The simplest and traditional treatment [7] of SIP constraints in (2) is to check them at a sufficiently dense set of grid points of , i.e., to replace (2) by a finite number of linear constraints. Consequently, the optimal solution of the resulting linear/convex program is not guaranteed to be a feasible solution of (2). Quite recently, it has been discovered in [4] that SIP constraints (2) are characterized by semi-definite programming (SDP) constraints additional variables. As a result, by introducing the dimension of the corresponding SDP may be very high (with more than several hundreds of additional variables), preventing them from being efficiently and practically solvable by existing SDP solvers such as [5], [9]. Another SDP formulation [6], [1] scalar variables but is ill-posed is of moderate size of about even for very moderate . Consequently, it cannot be practically implemented for in [6] and is actually implemented in a simple case. in [1] for moderate The above discussion motivates the need for developing sensible and practical algorithms for solving the class of problems (1)–(2) without introducing additional variables while guaranteeing a global optimal solution. The aim of this paper is to propose a new method for solving the SIP problem (1)–(2) based on a new inner polyhedral approximation of the feasible set of the following equivalent SIP of (2):

(2)

(3)

is a quadratic convex where functional of the filter coefficient is a convex compact set defined by linear/convex constraints, are affine vector-valued functions of and the inequalities (2) are component-wise understood.

The new linear-programming-based method which gives a global optimal solution, allows us to design FIR filter with large length. Furthermore, another advantage of using the alternative SIP constraint (3) over the SIP constraint (2) is that it allows in an effective manner without loss us to reduce the order of essential information, and thereby substantially improve the computational efficiency. The paper is organized as follows. The new method for solving SIPs (1) and (3) is presented in Section II. Section III uses the proposed method to design FIR filter and discusses, in particular, several effective implementation tools. The simulation is provided to demonstrate the viability of the proposed method. The conclusions are drawn in Section IV. Finally, for the reader convenience, detailed derivations are shown in the Appendixes. The notation is standard. In particular, for a vector , the notation means its smallest component, . i.e.,

Manuscript received January 7, 2003; revised April 6, 2004. This paper was recommended by Associate Editor Y. Lian. H. D. Tuan is with the School of Electrical Engineering and Telecommunication, University of New South Wales, Australia (e-mail: [email protected]). T. T. Son is with the Department of Electrical and Computer Engineering, Toyota Institute of Technology, Japan (e-mail: [email protected]). H. Tuy is with the Institute of Mathematics, Hanoi, Vietnam (e-mail: [email protected]). T. Nguyen is with the Department of Electrical and Computer Engineering, University of California in San Diego, USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2005.846880

1057-7130/$20.00 © 2005 IEEE

TUAN et al.: NEW LINEAR-PROGRAMMING-BASED FILTER DESIGN

277

II. EFFICIENT POLYHEDRAL APPROXIMATION METHOD The focus of this section is solving the SIP problems (1), (3). First, note that constraint (3) can be rewritten as

of

Proposition 3: Suppose that such that and

is a sequence of partitions . Then (10)

and (4) where for every (5) The challenge is how to handle the SIP constraint (4) because are not polytopes. The main result in this section the sets defined by (5) can be approximated is to show that the set in an efficient, simple and elegant way by an enclosing polyvertices defined by the relations (6) below, tope with just . Our computational experience i.e., an -simplex is a tight approximation of confirms that this simplex provided that . Let denote the th unit vector of , i.e., an -vector with the th component equal to 1 and every other component equal to 0. The following result of independent interest on polyplays a crucial role in our aphedral approximation of proach. is contained in the simplex with vertex Lemma 1: set

Note that the tightness of for approximating depends not only on the difference - but also on the value . For it is obvious that the SIP constraint of is equivalent to (12) where (12) is obtained by expanding each term by Newton polynomial formula and then grouping the terms involving the same power of . Since becomes very small as increases and can thus be practically neglected for when carrying the computation [see Appendix II for detailed analysis of the case of (12) arising from the trigonometric constraint (2)] to get a practione can replace the order in (12) as cally equivalent but much simplified SIP constraint (13) Thus,

can be computed as follows, instead of using (9):

(6) For our purpose, the following properties of tant. Proposition 1: For every

are impor-

(14)

(7) where is the Hausdorff distance between the two sets and . whenever Proposition 2: We have . be any partition of Let , i.e., with and define , then iff every member of is a member of . Let be the optimal value of the SIP optimization problem (1), (3). Clearly where

(8)

(9) Note that both (8) and (9) are linear/convex optimization problems. A direct consequence of Propositions 1 and 2 is.

where , and is the th unit vector . The above development leads to the following algorithm which generates a sequence of feasible solutions converging to a global optimal solution of problems (1) and (3). of

Global optimization algorithm Step 1) Let be any prescribed relative tolerance. Take and for solve (8) and (9) any partition and an to obtain an initial lower bound initial upper bound respectively. Set . generate a finer partition Step 2) While by adding a new point to where . solve (8) and (9) to obtain Step 3) For and respectively. Go to Step 2. Instead of generating by adding just one point to as in Step 2, one can also add at once a point for . The algorithm can be sped up by choosing every

278


dense enough at the initial step to obtain tight initial upper and lowerbounds. Remark: In certain applications, instead of the constraint in (3) we may have constraints of the following more general form:

(15) where . corresponds to Then, the definition of the set fied as follows:

(the case in (3)

TABLE I TRUNCATION RESULT FOR q p 0:02 COMPUTATIONAL ACCURACY 10

0

AND

There are two possibilities. 1) , i.e., either or . In this case, setting when , and rearranging, we can rewrite (21) exactly in the form (3). . Then (21) can be rewritten as 2)

[see (5)] should be modi(16)

Accordingly, we should approximate the curve with vertices tope

by the poly-

(17) [instead of (6)]. It can easily be shown that all the results earlier obtained for remain valid with expressions in (16) and (17), replacing those in (5) and (6). III. FIR FILTER DESIGN: SIP FORMULATION AND SIMULATION We will show later how several FIR filter design problems can be formulated as SIP problems (1) and (2). Before that, let us point out how a SIP constraint (2) can be transformed into an equivalent SIP constraint (3) and discuss the advantage of such a transformation. is the th-order Chebyshev polyIt is well known that [8]. However, for our later purpose, we need the nomial of following interesting closed form

(18)

(22) which is again of the form (3). Let us now mention the advantage of using the alternative SIP constraint in (3) for SIP constraint in (2). As noticed above, by writing (3) as in (12), we can truncate the order to obtain a simplified SIP constraint in (13) with much lower order of which would not be possible with the SIP constraint in (2). Table I shows some results of this truncation technique as given in the Appendix II. Now, let us discuss the filter-design problem. The peak-constrained least-squares errors (PCLS) filters introduced in [2], [3] are regarded as a filter class with good signal-to-noise ratio (SNR) while maintaining an acceptable error in the passband and stopband. For simplicity, let us consider only the PCLS low-pass filter design since other cases and their multiband extensions can be formulated and solved in a similar manner. Generally, a PCLS low-pass filter design problem can be formulated , passas follows: given a desired frequency response and stopband , we wish to design a linearband with impulse response phase filter of a given length that meets the following requirements. • The weighted-square error is minimized:

Therefore, a SIP constraint (2) can be converted to the following form: (23)

where define

are still affine functions of

(19) . Now,

where by

(20) • so that by setting

. Since is a continuous function, the (SIP) constraint (19) is equivalent to

is the frequency response of with

defined

. Here is the maximal integer not exceeding . The peak-error constraints in the passband and stopband are satisfied (24) (25)

(21)

for given

.


DATA

OF

279

TABLE II PROBLEM (23), (24), (25) IN SIMULATION EXAMPLES WITH

! = 2k

AND

! = 2k

Fig. 2. Frequency response of the linear phase FIR filter with order M = 76; 77 taps(N = [M=2] = 38; 39 variables), found with truncation N = 23.


Clearly, (23)–(25) can be rewritten as (26) (27) (28) where (so is positive definite). The objective (26) is thus a convex quadratic function while even the constraints (27)–(28) readily have the form (2). for On the other hand, for odd , by formula (18) the constraints . (27)–(28) can be equivalently expressed as (15) with In our simulations, we solved problem (23)–(25) with different data given in Table II and the desired frequency response . The value of and is set to 2 and 2000, respectively. The number of covering polytopes (the number of points taken for partition) iteratively increases with step 50 and the global optimization algorithm stops when the relative tolerance begins to be slowly improved (less than 5% per iteration). Fig. 1 shows the frequency responses of , correspondent to 35 taps) by the designed filter ( using different optimization problems (9) and (14) where the total number of covering polytopes is 302. In this case, both (9) and (14) involve 18 variables but the number of linear constraints in (14) is 14 448 versus that 32 508 constraints in (9), i.e., the number of constraints decreases considerably with the truncation technique. As theoretically expected, these two filters are identical and the corresponding relative tolerance is 0.020 998 but the size of the linear program


(14) is much less than that of (9). Further, the truncated optimization problem (14) is used to solve (23), (24), and (25) for several cases of long length filters. Fig. 2 (Fig. 3, respectively) shows the frequency response of the designed filter with order , correspondent to 77 taps ( , correspondent to 101 taps, respectively) and the number of polytopes in (14) is 1501 (1650, respectively) at which the relative tolerance is 0.087 252 4 (0.003 003, respectively). Corresponding to the polytope number 1501 in case (1650 in , respectively), we have 39 (51, respectively) case of variables and 96 000 (132 200, respectively) constraints when using truncation instead of 351 000 (504 900, respectively) and tolerance constraints. Table III shows the value of . corresponding to the number of polytopes in case of and tolerance in case of Table IV presents the result of and . Note that the last case cannot be handled by the LMI approach since the resulting

280


TABLE III RESULT OF ; AND RELATIVE TOLERANCE CORRESPONDING TO THE POLYTOPE NUMBERS IN CASE OF M = 34

By (6), the vertices of

are

and

where condition similar to (30). Therefore, which implies (31). By changing proved RESULT OF ;

AND

satisfy a , in (31) we thus have

TABLE IV TOLERANCE IN CASE OF M = 76 AND M = 100

(32) Next, we will show that for

LMI size is very large and (the size of the additional variable is 2550) cannot be successfully solved by existing LMI software such as [9]. IV. CONCLUSION In this brief, we presented a new linear-programming-based global optimization method for solving a general SIP problem. This method is practical for solving FIR filter design problems of large dimension that cannot be handled by the convex LMI optimization method. The application of our new method to the design of biorthogonal cosine-modulated filter bank with channels and with length for asymmetric digital subscriber line (ADSL) in high-bandwidth communication is under way.

(33)

Observe that , where, as in (30), . Therefore, the satisfy , proving

of vertices (33). Proposition 2 now follows from (32) and (33): . APPENDIX II TRUNCATION TECHNIQUES

We describe here the truncation technique for handling the , which, typical trigonometrical constraint by formula (18), can be rewritten as (34)

APPENDIX I PROOFS OF BASIC RESULTS A. Proof of Lemma 1

, while all

with

are bounded by

, it suffices to show that any To prove admits the following convex combination of vertices of

(35) (29)

, LHS of (34) is indeed

Now, for

with

(30) and, so To this end, notice that is of the form , where . For such ities (30) can be shown based on the fact that

and so any , the inequal. (36)

B. Proof of Proposition 2 First we will show that (31)

which is imal value

with the max.


281

Therefore, taking account of (35) and the estimate

the coefficients of the variables

REFERENCES

in (36) are bounded by

We can thus truncate the order of from to if the numerical value of is practically zero, i.e., if , where is the limitation power of the computational accuracy. Such is found by solving this inequality in value of

(37) ACKNOWLEDGMENT The authors would like to thank Dr. H. Q. Ngo, SUNY, Buffalo, NY for providing the interesting closed-loop formula (18).

[1] B. Alkire and L. Vandenberge, “Interior-point methods for magnitude filter design,” in Proc. 26th IEEE Int. Conf. Acoustics, Speed, Signal Processing, vol. 6, May 2001, pp. 3821–3824. [2] J. W. Adams, “A new optimal window,” IEEE Trans. Signal Process., vol. 39, no. 8, pp. 1753–1769, Aug. 1991. [3] J. W. Adams and J. L. Sullivan, “Peak-constrained least squares optimization,” IEEE Trans. Signal Process., vol. 46, no. 2, pp. 306–321, Feb. 1998. [4] T. Davidson, T. Luo, and J. Sturm, “Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design,” IEEE Trans. on Signal Process., vol. 50, no. 11, pp. 2702–2715, Nov. 2002. [5] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: The MathWorks Inc.. [6] Y. Genin, Y. Hachez, Yu. Nesterov, and P. Van Dooren, “Convex optimization over positive polynomials and filter design,” in Proc. 2000 UKACC Int. Conf. Control, 2000, CD-ROM paper SS-41. [7] P. Moulin, M. Anitescu, K. Kortanek, and F. A. Potra, “The role of linear semi-infinite programming in signal adapted QMF bank design,” IEEE Trans. Signal Process., vol. 45, no. 9, pp. 2160–2174, Sep. 1997. [8] J. G. Proakis and D. G. Manolakis, Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1996. [9] J. F. Sturm, SeDuMi: A Matlab Toolbox for Optimization Over Symmetric Cones. [10] B. Vo, A. Cantoni, and K. L. Teo, “Envelope constrained filter with linear interpolator,” IEEE Trans. Signal Process., vol. 45, no. 6, pp. 1405–1414, Jun. 1997. , Filter Design with Time Domain Mask Constraints: Theory and [11] Applications. Norwell, MA: Kluwer, 2001.