Publication in the conference proceedings of EUSIPCO, Toulouse, France, 200

Fujie, Tetsuya

Hirabayashi, Ryuichi

Ito, Rika

Suyama, Kenji

English

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NEW DESIGN METHOD OF FIR FILTERS WITH SP2 COEFFICIENTS BASED ON A NEWLINEAR PROGRAMMING RELAXATION WITH TRIANGLE INEQUALITIESRika ITO† Tetsuya FUJIE†† Kenji SUYAMA† Ryuichi HIRABAYASHI††Faculty of Engineering ††School of Economics and Business AdministrationScience University of Tokyo Kobe University of CommerceABSTRACTIn this paper, we propose new design methods for linearphase FIR filters with signed power-of-two (SP2) coeffi-cients based on a semi-definite programming (SDP) relax-ation method. The proposed methods include a linear pro-gramming (LP) relaxation and a relaxation by adding tri-angle inequalities. Although such the design problems areknown as one of the NP-hard problems, these methods cansolve the design problems in a low computational cost incomparison with a traditional SDP relaxation method. It isshown by several numerical experiments that those methodare superior to the simple SDP relaxation method.1. INTRODUCTIONRecently, many studies on a design method for linear phaseFIR filters with discrete coefficients have been published,in which, a numerical representation by a sum of signedpower of two(SP2) has been used in several methods [1],[2], [3], [5]. It is a reason that a small number of non-zerodigits is often required to include for a representation of thecoefficients in a VLSI implementation of the filters.However, it is difficult to design such the filters since itreduces to an integer programming problem(IP), which iswell-known as one of the NP-hard problems [8]．Lu [9] proposed a new design method of filters with SP2coefficients, in which, the design problem was formulatedas a semi-definite programming(SDP) relaxation problemand solved by the interior point method. The SDP relaxationis one of the techniques for approximation of a non-convexprogramming problem including the IP [10]. Although theSDP relaxation is attractive to be solved in a polynomialtime under some constraints qualification [11], it also in-volves a large computational complexity with a proportionof the designed filter’s order [12].It is well known that SDP problem is a convex program-ming problem and can be solved in polynomial time undersome constraint qualification [11]．In this paper, we pro-pose two new relaxation methods based on the SDP relax-ation, i.e., a simple linear programming (LP) relaxation anda relaxation by adding triangle inequalities. It is shown bysome numerical experiments that these relaxations are su-perior to the simple SDP relaxation.2. DESIGN METHOD OF DIGITAL FILTERS BYUSING {−1, 1}-OPTIMIZATION PROBLEMIn this section, we introduce the design method of digital fil-ters with SP2 coefficients using SDP problem based on [9].This design method is constructed by two steps: (1) solvethe design problem of digital filters with desired frequencycharacteristics by using continuous variables, (2) formulatethe design problem of digital filters with SP2 coefficientsas a {−1, 1}-optimization problem. To consider the struc-ture of the {−1, 1}-optimization problem obtained, we willconvert the {−1, 1}-optimization problem to a minimumcut problem with negative coefficients which belongs to theclass NP-hard.2.1. Design problem of FIR digital filters with continu-ous coefficientsWe design the FIR filters with SP2 coefficients so as to min-imize the square error defined as,e =∫ π0W (ω)[H(ejω)−Hd(ω)]2dω (1)where W (ω) ≥ 0 is a weight function and Hd(ω) is the de-sired frequency response function. In the first, we considerthe continuous coefficient case. Then the transfer functionof the FIR filter is:Hc(z) =N−1∑k=0hkz−k (2)where hk (k = 0, 1, . . . , N−1) are real numbers. Now, weassume M is the total number of SP2 terms that can be usedin H(z) and mk is the number of SP2 terms used in the kterm of the frequency response H(ejω), i.e.,∑N−1k=0 mk =M . Then we denoteH(z) =N−1∑k=0dkz−k. (3)The allocation of SP2 terms is determined, for example, by[13]．We assume that the absolute value of each SP2 term|{dk}| is in the interval [20, 2−U ] where U is a natural num-ber. Then, by (3),dk =mk∑i=1b(k)i 2−q(k)i . (4)Here, we have b(k)i ∈ {−1, 1} and q(k)i ≤ U, (1 ≤ i ≤mk, 0 ≤ k ≤ N − 1).For given {mk, k = 0, . . . , N − 1} and U , when anoptimal continuous solution Hc(z) =∑N−1k=0 hkz−k is ob-tained, it is easy to find the maximum SP2 number dk andthe minimum SP2 number d¯k that satisfy d¯k ≤ hk ≤ dkwhose dk and d¯k satisfy (4) for the given mk.Let dmk = (dk + d¯k)/2 be the middle point of the inter-val [dk, d¯k] and δk = (dk − d¯k)/2 be the half length of theinterval. Then, dk and d¯k are expressed as dmk+xkδk(xk =−1) and dmk + xkδk (xk = 1), respectively. Hence, thetransfer function H(z) with discrete coefficient function be-comesH(ejω) =N−1∑k=0d(s)k e−jkω (5)= Hm(ejω) + xT [cδ(ω)− jsδ(ω)] (6)by usingdk = dmk + xkδk, (7)andHm(ejω) = dTm[c(ω)− js(ω)],dm = (dm0, dm1, . . . , dm,N−1)T ,c(ω) = (1, cosω, . . . , cos(N − 1)ω)T ,s(ω) = (0, sinω, . . . , sin(N − 1)ω)T ,cδ(ω) = (δ0, . . . , δN−1 cos(N − 1)ω)T ,sδ(ω) = (0, . . . , δN−1 sin(N − 1)ω)T ,x = (x0, x1, . . . , xN−1)T , xi ∈ {−1, 1}.(8)By (6), we can easily verify that the objective function (1)becomese = xTQx+ 2xTq + const (9)whereQ =∫ π0W (ω)[cδ(ω)cTδ (ω)]dωq =∫ π0W (ω)[Er(ω)cδ(ω) + Ei(ω)sδ(ω)]dωEr(ω) = dTmc(ω)−Hdr(ω),Ei(ω) = dTms(ω)−Hdi(ω),Hd(ω) = Hdr(ω)− jHdi(ω).(10)Now, design problem of H(z) with SP2 coefficients forminimizing weighted least square becomes a {−1, 1}-quadraticinteger programming problems [9]:min xTQx+ 2qTxsub. to x ∈ {−1, 1}N . (11)2.2. Derivation to the minimum cut problemTo know the structure of the optimization problem is al-ways important. In the following, we will point out that theproblem (11) can be easily converted to the minimum cutproblem with negative coefficients. Minimum cut problemwith negative coefficients belongs to the class of NP-hard.Hence, this shows (11) is a hard problem to solve.Let G = (V,E) be a perfect graph with a vertex set V ={0, 1, . . . , N} and an edge set E = {ij | 0 ≤ i < j ≤ N}.And, a weight we(e ∈ E) of the each edge be{wjN = −qj (j = 0, . . . , N − 1),wij = −Qij (0 ≤ i < j ≤ N − 1). (12)Then the minimum cut problem of the graph G is:min 4∑e∈Eweye + eTQe+ 2qTesub. to y : a 0-1 cut vector of G,(13)or equivalentlymin∑ij∈Ewij(xi − xj)2 + eTQe+ 2qTesub. to x ∈ {−1, 1}N+1.(14)It is easy to see that problem (11) and (14) are equivalenteach other. Here, we denote, without loss of generality,xN = 1 in (14). Since, as we pointed out that minimum cutproblem with negative coefficients belongs to the class ofNP-hard, there will no hope to develop efficient algorithmsto solve (11) directly.3. SDP RELAXATION AND LP RELAXATIONFollowing Lu [9], we reformulate (11) asmin Q •X + 2qTxsub. to X − xxT = O,x ∈ {−1, 1}N(15)where Q•X =∑i,jQijXij . Then, since Xii = x2i = 1 (i =0, . . . , N − 1) we obtain an relaxation problem of (11) bymin Q •X + 2qTxsub. to Xii = 1 (i = 0, . . . , N − 1),X − xxT  O.(16)Here, A  O denotes that A is positive semidefinite. Be-tween a cut vector y ∈ {0, 1}E of the graph G and (x,X)that satisfies (15), the following equations hold:{xj = −2y0j + 1 (j = 0, . . . , N − 1),Xij = −2yij + 1 (0 ≤ i < j ≤ N − 1). (17)We can use software to solve SDP in polynomial times,for example, SeDuMi [14].It is easily verified that the next triangle inequalities hold[15]:xi + xj + Xij ≥ −1,xi − xj −Xij ≥ −1,−xi − xj + Xij ≥ −1,−xi + xj −Xij ≥ −1.(18)Here, we denote 0 ≤ i < j ≤ N − 1. And, for 0 ≤ i < j <k ≤ N − 1,Xij + Xik + Xjk ≥ −1,Xij −Xik −Xjk ≥ −1,−Xij −Xik + Xjk ≥ −1,−Xij + Xik −Xjk ≥ −1(19)hold. (18) is the triangle inequalities for the vertex set {i, j,N},and (19) is the triangle inequalities for the vertex set {i, j, k}.Therefore, the next optimization problemmin Q •X + 2qTxsub. to Xii = 1 (i = 0, . . . , N − 1),(18), (19),X − xxT  O(20)strengthen the problem (16).Since it is easier to solve LP problems than SDP prob-lem, we will relax the SDP relaxation problem to LP relax-ation problem. The following minimization problemmin 2∑i<jQijXij + 2qTx+N−1∑i=0Qiisub. to (18),−1 ≤ xi ≤ 1 (i = 0, . . . , N − 1)(21)is a LP relaxation problem with a bounded optimal solution.Adding (19) as constraints,min 2∑i<jQijXij + 2qTx+N−1∑i=1Qiisub. to (18), (19),−1 ≤ xi ≤ 1 (i = 0, . . . , N − 1)(22)we have a strengthened problem for (21).4. COMPARISON OF THE RELAXATIONPROBLEMSFrom the theoretical view point, SDP relaxation with trian-gle inequalities are stronger than SDP relaxation, LP relax-ation, or LP relaxation with triangle inequalities. However,SDP relaxation with triangle inequalities is a rather heavyrelaxation for programming techniques and computation.Hence, we simply compare the SDP relaxation, LP relax-ation and, LP relaxation with triangle inequalities throughnumerical experiments.In the numerical experiments, the specification of thefilter design problem is basically same as Lu [9], hence,FIR filter is an odd degree and even symmetric linear phaselowpass filter. The design specification is as follows: thenormalized passband is [0, ωp] = [0, 0.225], stopband is[ωs, 1] = [0.275, 1], W (ω) = 1 on [0, ωp], W (ω) = 500 on[ωs, 0.5], L = 12. And, we set each mk = 2. The CPU usedis mobile Pentium III 650 MHz, memory is 192 Mbytes. Allproblems are solved by SeDuMi (Ver.1.03) [14]. The CPUtime contains only the execution time of SeDuMi.By the numerical experiments, we found (1) all the so-lutions of LP with triangle inequalities (22) are optimal so-lutions of the original problem (11), that is, all the solutionsof (22) are {−1, 1}-integer solutions and automatically opti-mal solutions for (11). This reveals that the triangle inequal-ities are very important. At now, all triangle inequalities areincluded in the relaxation problems, we can also develop analgorithm that has triangle inequalities as cutting planes.How to obtain an {−1, 1}-solution from the solution ofthe relaxation problems is as follows:(1) For the SDP relaxation problem, let the solution of therelaxation problem (x˜, X˜),(1-1) sign(x˜).(1-2) let vi (i = 0, . . . , N − 1) be the eigen vectors of X˜ ,and set sign(vi) (i = 0, . . . , N − 1).(1-3) Use the Goemans-Williamson’s randomized algorithm[16]．Select the best solution of the above solutions. (Lu [9] ex-ploits (1-1) and (1-2) for the maximal eigen vectors, How-ever, we recommend the above, since it does not take somuch CPU time, and may improve the solution.)(2) For LP relaxation problems, we exploited the sign vectorvi of the solution of the relaxation problems x˜.5. CONCLUSIONIn this paper, we proposed LP based relaxation techniquesto solve the design problem of FIR filters with SP2 coeffi-cients under MLE criterion. And, we compared this relax-ation technique and the SDP relaxation through numericalexperiments. By the numerical example, LP based relax-ation technique seems to work fairly good.Table 1. Comaparison of upper bounds (∗ is the best solu-tion).N WLS LP LP+Tri SDP7 0.29413 0.42797∗ 0.42797∗ 0.42797∗13 0.10269 0.10543∗ 0.10543∗ 0.10543∗19 0.02473 0.08771∗ 0.08771∗ 0.08771∗25 0.01169 0.02655 0.02606∗ 0.02606∗31 0.00288 0.01258∗ 0.01258∗ 0.0141537 0.00141 0.06321 0.06231∗ 0.0626343 0.00035 0.15770 0.08531∗ 0.08531∗49 0.00018 0.05953 0.05781∗ 0.05781∗55 0.00004 0.04949 0.04946∗ 0.0497957 0.00004 0.05035∗ 0.05035∗ 0.05035∗59 0.00002 0.05359 0.05345∗ 0.0535061 0.00002 0.05425 0.05409∗ 0.05442Table 2. Computational Time (sec)n LP LP+Tri SDP7 0.04 0.05 0.0613 0.08 0.09 0.0819 0.08 0.15 0.1125 0.10 0.31 0.1131 0.19 0.67 0.1337 0.19 1.34 0.1643 0.25 2.51 0.1949 0.31 4.80 0.2055 0.43 8.85 0.2657 0.45 9.78 0.2659 0.48 14.46 0.2861 0.50 14.94 0.306. REFERENCES[1] D.M. Kodek and K. Steiglitz, ”Finite-length word-length tradeoffs in FIR digital filter design,” IEEETrans. Acoustics, Speech, Signal Processing, vol.ASSP-28, pp.739-744, December 1980.[2] Y.C. Lim and S.R. 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New design method of FIR filters with SP2 coefficients based on a new linear programming relaxation with triangle inequalities

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New design method of FIR filters with SP2 coefficients based on a new linear programming relaxation with triangle inequalities

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