Minimax Design of Nonnegative Finite Impulse Response Filters

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1 Minimax Design of Nonnegative Finite Imulse Resonse Filters Xiaoing Lai, Anke Xue Institute of Information and Control Hangzhou Dianzi University Hangzhou, 3118 China Zhiing Lin School of Electrical & Electronic Engineering Nanyang Technological University Singaore Chunlu Lai School of Information Science and Engineering Shandong University Jinan, 251 China Abstract Nonnegative imulse resonse (NNIR) filters have found many alications in signal rocessing and information fusion areas. Evidence filtering is one of the examles among others. An evidence filter is required to satisfy a nonnegativity condition and a normalization condition on its imulse resonse coefficients, and thus is basically an NNIR filter. This aer considers the design of nonnegative finite imulse resonse (FIR) filters based on frequency resonse aroximation and rooses a constrained minimax design formulation using the fundamental limitations on the NNIR filter s frequency resonses recently develoed in the literature. The formulation is converted into a linearly constrained ositive-definite quadratic rogramming and then solved with the Goldfarb-Idnani algorithm. The roosed method is alicable to nonnegative FIR lowass as well as other tyes of filters. Design examles demonstrate the effectiveness of the roosed method. Keywords-nonnegative imulse resonse filter; finite imulse resonse filter; minimax design; quadratic rogramming I. INTRODUCTION Distributed sensor networks consisting of a large number of low-cost and low-ower sensors have found many alications in military surveillance, homeland security, traffic surveillance, environment monitoring, etc., for collecting observations and rocessing information. The use of distributed nodes with multile sensing modalities can significantly enhance the robustness and the accuracy of the decision making rocess in such environments [1]. Data collected by nodes are fused at various levels and in different ways in order to make useful inferences at the decision-making center. The inferences made based on the fused information range from distinguishing among different event (or threat) tyes, estimating imortant arameters This work was suorted in art by the National Nature Science Foundation of China under Grants and , and in art by the National Basic Research Program of China under Grants 212CB8212 and 29 CB326. such as location or velocity of an object, target tracking, as well as detecting the resence or absence of events with certain frequency or sectral characteristics [2]. Demster-Shafter belief theory [3] has been extensively used in surveillance and security alications and is excellent in modeling imerfect sensor data by maing into Demster- Shafter theoretic beliefs (evidences) which are nonnegative in nature [4]. By integrating the Demster-Shafter theory with discrete time filtering techniques, an evidence filtering method is established in [2] to fuse temorally ordered information from multile sensor modalities and directly infer on the frequency characteristics of events. Filtering of temorally ordered evidences can be described by a finite imulse resonse (FIR) or infinite imulse resonse (IIR) filter [2]. If an FIR filter is used, the udated (filtered) evidence is a weighted sum of the incoming evidences. If an IIR filter is used, the filtered evidence is a weighted sum of the ast and incoming evidences. In order to ensure the validity of the udated evidence, the imulse resonse coefficients (or, the weights) of an evidence filter should be nonnegative (the nonnegativity condition), and all imulse resonse coefficients should sum to 1. (the normalization condition). These are the inherent constraints should be imosed on the evidence filters when one considers their designs. Because of the nonnegativity of the imulse resonse, an evidence filter does not have arbitrarily secified frequency resonses. Instead, the frequency resonse of an evidence filter has some fundamental limitations [5] [6]. For examle, the frequency resonse of a nonnegative imulse resonse (NNIR) filter inherently has a low-frequency assband, and the maximum roll-off of its frequency resonse that can be achieved in the transition-band is determined by the magnitude resonse at the assband edge frequency [5] [6]. Realizing these fundamental frequency domain limitations, a rocedure is rovided 1441

2 in [6] to check whether a set of secifications are feasible for an NNIR lowass filter and to tune the secifications when they are infeasible for the filter. Although much research has been conducted on the timedomain and frequency-domain characteristics of NNIR filters [5]-[11], very little work has been done on the design of NNIR filters. Since nonnegative FIR filters have some time-domain and frequency-domain limitations, the design of such filters involves time-domain and frequency-domain constraints. The existing eigenfilter aroaches [12] [13] and constrained leastsquares methods [14] [15] can incororate with time-domain and frequency-domain constraints, and thus are rosective methods for nonnegative FIR filter design after exloiting the filters fundamental frequency resonse limitations. In [2], a heuristic rocedure was roosed to design the nonnegative FIR filters for evidence filtering alications. This method first designs a conventional filter using the Hamming window method, and then offset the imulse resonse of the conventional filter to satisfy the nonnegativity and normalization conditions and iteratively smooth the offset filter to imrove the stoband erformance until the stoband degradation becomes satisfactory. This offset-smoothing method is not an otimization method, the designed filter is not otimal in any sense. In this aer, we resent a constrained minimax formulation for nonnegative FIR filters. At this oint, we restrict the designed filters to have linear hases. Then, the constrained minimax roblem is a linear rogramming roblem. In light of Lemma 1 and Theorem 2 of [14], we convert it into a linearly constrained ositive-definite quadratic rogramming roblem, and then use the Goldfarb-Idnani algorithm to solve the quadratic rogramming roblem. Design examles demonstrate the effectiveness of the roosed method. II. LIMITATIONS OF NONNEGATIVE FIR FILTERS An M-th order nonnegative FIR filter can be described by a transfer function H(z) = h + h 1 z h M z M, (1) with an imulse resonse h = [h, h 1,, h M ] T of length (M+1) satisfying the nonnegativity condition h k for k =, 1,, M. (2) For an evidence filter, the imulse resonse of the nonnegative FIR filter should also satisfy the normalization condition h + h h M = 1. (3) The frequency resonse of the filter H(z) is described by H(e jω ) = h + h 1 e jω + + h M e jmω, (4) and the magnitude frequency resonse of the filter is denoted by H(e jω ). From Lemma 6 of [6], if the square magnitude frequency resonse of H(z) at mω, H(e jmω ) 2, is bounded by: H(1) 2 H(e jmω ) 2 H(1) 2 (5) where >, ω [, π/(m+k)], m > and k > are integer, then the decrease in the squared magnitude resonse from mω to (m+k)ω is bounded by H(e jmω ) 2 H(e j(m+k)ω ) 2 m 2 (k 2 +2mk). (6) It follows that the maximum roll-off of the magnitude resonse that can be achieved in the transition band from the assband edge frequency mω to the stoband edge frequeny (m+k)ω is determined by the magnitude resonse at the assband edge frequency mω, which imlies that the assband rile and stoband rile cannot be indeendently secified. Considering the above limitation of the frequency resonse and the requirement of minimum signal-to-noise ratio in ractical alications, Ref. [6] resents a set of frequency domain secifications for NNIR lowass filters as follows: S LP = {ω, ω s, K,, s }, (7) where ω, ω s,, and s resectively reresent the assband and stoband edge frequencies, and the assband and stoband magnitude resonse riles, and K is the minimal transition band attenuation must be maintained. The minimal transition band attenuation is considered as a hard requirement for the NNIR filter. III. FORMULATION OF THE MINIMAX DESIGN PROBLEM In this aer, we will only consider the design of nonnegative FIR filters with exact linear hases. Then an additional constraint exloiting the symmetry of the imulse resonse coefficients should be imosed on the filters as follows: h k = h M k for k =, 1,, M. (8) With the above symmetry constraint, the frequency resonse of the filter can be described by H(e jω ) = P(ω)e jφ(ω), (9) where φ(ω) = M/2ω is the linear hase resonse of the filter, and P(ω) is the magnitude frequency resonse (u to a negative sign) of the filter described by P(ω) = 2h cos(mω/2) + + 2h M/2 1 cos(ω) + h M/2 (1a) for even M, and 1442

3 for odd M. P(ω) = 2h cos(mω/2) + + 2h (M 1)/2 cos(ω/2) (1b) By introducing x = [h, h 1,,h M/2 ] T, (11) where M/2 reresents the integer art of M/2, the magnitude resonse P(ω) can be rewritten as where for even M, and for odd M. P(ω) = c(ω) T x, (12) Mω c ( ω) = 2 cos,, 2 cosω, 1 (13a) 2 Mω 3ω ω c ( ω) = 2 cos,, 2 cos, 2cos (13b) Since the maximum transition band roll-off of the magnitude resonse is determined by the magnitude resonse at the assband edge frequency, as stated in the revious section, the assband and stoband magnitude resonse riles and s cannot be indeendently secified. If we require a design to maintain a minimal transition band attenuation K (in db), then the assband and stoband riles and s should satisfy or 2 log 1 (1 ) 2 log 1 s K, (14a) s 1 (K/2) (1 ). T T (14b) When K is given, we choose the maximum assband magnitude resonse error (magnitude rile) to minimize and let the stoband magnitude resonse rile s satisfy (14), leading to the following constrained minimax design roblem: minimize h,, (15a) where Ω and Ω s denote the assband and stoband of the filter. By using Eqs. (9), (11), (12), and (13), we may rewrite the above minimax roblem as: minimize x, subject to: c(ω) T x 1, for ω Ω, c(ω) T x 1, for ω Ω,, (16a) c(ω) T x + 1 (K/2) 1 (K/2), for ω Ω s, c(ω) T x + 1 (K/2) 1 (K/2), for ω Ω s, x, a T x = 1, (16b) (16c) (16d) (16e) (16f) (16g) where a = [2,, 2, 1] for even M or a = [2,, 2, 2] for odd M. If we relace the assband Ω and stoband Ω s by sufficiently dense discrete frequency sets, say Ω Ω Ω and Ω s Ω s Ω where Ω = {kπ/l, k =, 1,, L} and L is a sufficiently large ositive integer (e.g. L = 1M), the above roblem becomes a finite linear rogramming roblem, the minimum cost value of which is unique. IV. SOLVING THE MINIMAX DESIGN PROBLEM In the revious section, the minimax design roblem of a nonnegative FIR filter has been formulated by a finite linear rogramming. There are several existing algorithms for finite linear rogramming roblems. The linrog rogram in the Matlab otimization tool box is one among others. However, when the numbers of the otimization variables and constraint equations are large, the linrog rogram may fail to obtain the otimal solution. To this end, we convert the finite linear rogramming roblem into an equivalent finite ositivedefinite quadratic rogramming and then solve the quadratic rogramming using the Goldfarb-Idnani algorithm [16]. Noting, the linear rogramming roblem (16) is equivalent to the following quadratic rogramming roblem: minimize x,, (17a) subject to linear constraints (16b) to (16g). 2 (17b) subject to: 1 H(e jω ) 1, for ω Ω, H(e jω ) s 1 (K/2) (1 ), for ω Ω s, h k for k =, 1,, M. (15b) (15c) (15d) Then, by Lemma 1 and Theorem 2 of [14], we may further convert it into another quadratic rogramming roblem as follows: 2 T minimize + λx x, (18a) x, h + h h M = 1. (15e) subject to linear constraints (16b) to (16g), (18b) 1443

4 where λ > is a sufficiently small number. According to Lemma 1 and Theorem 2 of [14], there must exist a sufficiently small ositive number λ * >, such that for any λ satisfying < λ < λ *, the unique solution to roblem (18) is also a solution to roblem (17). In addition, it has been shown in [14] that for FIR filter design, λ = 1 6 can be considered as sufficiently small for the equivalence between roblems (17) and (18). We will choose this value for λ in our designs. Now the cost function in (18a) is ositive definite. Then the Goldfarb-Idnani algorithm [16] can be directly used to solve the linearly constrained quadratic rogramming roblem (18). V. DESIGN OF LOWPASS FILTERS For the minimax design of nonnegative FIR lowass filters by the roosed method, the frequency domain secifications are the filter order M, the assband and stoband edge frequencies ω and ω s, and the minimum transition band magnitude resonse attenuation K in db, i.e., S LP = {M, ω, ω s, K}. (19) The assband and stoband are defined by Ω = [, ω ] and Ω s = [ω s, π]. (2) Examle 1. Design of 8-th order nonnegative FIR lowass filters with a assband edge frequency ω =.1π, a stoband edge frequency ω s =.175π, and different transition band attenuations K ranging from 35 to 65 db. TABLE I. Method DESIGN RESULTS OF THE NONNEGATIVE FIR LOWPASS FILTERS IN EXAMPLE 1 Transition band attenuation K Passband magnitude rile Stoband magnitude rile s 35 db db db 4 db -5.8 db db 45 db db db Magnitue resonse (db) Imulse resonse K = 35dB: K = 45dB: K = 55dB: K = 65dB: K = 35dB: Samle Figure 1. Magnitude resonses and imulse resonse of some nonnegative FIR lowass filters designed in Examle 1 The magnitude frequency resonses of some of the designed nonnegative FIR filters by the roosed method are shown in Fig. 1 and the imulse resonse of the filter with a transition band attenuation of K = 35 db is drawn in Fig. 1. Examle 2. Design of different order nonnegative FIR lowass filters with a assband edge frequency ω =.5π, a stoband edge frequency ω s =.6π, and a minimum transition band attenuation 9 db. Proosed 5 db db db 55 db db db 6 db db db 65 db db db TABLE II. Filter order M DESIGN RESULTS OF THE NONNEGATIVE FIR LOWPASS FILTERS IN EXAMPLE 2 Transition band attenuation K Passband magnitude rile Stoband magnitude rile s 6 9 db db db Offset-smoothing db db -4. db Table I lists the assband and stoband magnitude resonse riles of the resultant filters with K = 35, 4, 45, 5, 55, 6, and 65 db. For a comarison, corresonding results of the filter obtained in [6] using the offset-smoothing method in [2] are also listed in the table. It is seen that the roosed method has obtained much better filters than the offset-smoothing method. It can also be seen that the stoband magnitude rile is much easier to reduce than the assband magnitude rile. 7 9 db db db 8 9 db db db 9 9 db db db 1 9 db db db Table II lists the assband and stoband magnitude resonse riles of the resultant filters with orders M = 6, 7, 8, 9, and 1. It is seen that the magnitude resonse rile is difficult to reduce by increasing the filter order if the transition 1444

5 band magnitude attenuation kees invariant. The magnitude frequency resonse and imulse resonse of the 6-th order filter are shown in Fig. 2. Magnitue resonse (db) Imulse resonse M = 6: M = 6: Samle Figure 2. Magnitude resonse and imulse resonse of the 6-th order nonnegative FIR lowass filter in Examle 2 VI. DESIGN OF HIGHPASS AND BANDPASS FILTERS The minimax design method resented in Sections III and IV is not only able to design nonnegative FIR lowass filters, but also able to design other tyes of nonnegative FIR filters, such as bandass filters and highass filters. However, besides the desired assband, there will be a surious assband [, ω s ] at the low frequency region for these two tyes of filters due to the fundamental frequency domain limitations of the nonnegative FIR filter. The frequency domain secifications for nonnegative FIR bandass filters are the filter order M, the surious assband edge frequency ω s, the low stoband edge frequency ω s1, the assband edge frequencies ω 1 and ω 2, the high stoband edge frequency ω s2, and the minimum transition band magnitude resonse attenuation K in db, i.e., S BP = {M, ω s, ω s1, ω 1, ω 2, ω s2, K}. (21) The assband and stoband are defined by Ω = [ω 1, ω 2 ] and Ω s = [ω s, ω s1 ] [ω s2, π]. (22) It is seen that the stoband is [ω s, ω s1 ] [ω s2, π] rather than [, ω s1 ] [ω s2, π]. For high ass filters, the frequency domain secifications include the filter order M, the surious assband edge frequency ω s, the stoband edge frequency ω s, the assband edge frequency ω, and the minimum transition band magnitude resonse attenuation K in db, i.e., S HP = {M, ω s, ω s, ω, K}. (23) The assband and stoband should be Magnitue resonse (db) Imulse resonse Ω = [ω, π] and Ω s = [ω s, ω s ]. (24) Samle Figure 3. Magnitude resonse and imulse resonse of the 399-th order bandass nonnegative FIR filter in Examle 3 Examle 3. Design of a 399-th order nonnegative FIR bandass filter with a low stoband edge frequency ω s1 =.75π, a assband left edge frequency ω 1 =.77π, a assband right edge frequency ω 2 =.83π, a high stoband edge frequency ω s2 =.85π, and a transition band attenuation K = 85 db. The surious assband edge frequency is ω s =.5π The resultant filter has a assband rile of db and a stoband attenuation of 96.2 db. The magnitude frequency resonse and imulse resonse of the filter are drawn in Fig. 3. Examle 4. Design of a 1-th order nonnegative FIR high- 1445

6 ass filter with a stoband edge frequency ω s =.75π, a assband edge frequency ω =.85π, and a transition band attenuation K = 4 db. The surious assband edge frequency is ω s =.15π The resultant filter has a assband rile of db and a stoband attenuation of db. The magnitude frequency resonse and imulse resonse of the filter are drawn in Fig. 4. Magnitue resonse (db) Imulse resonse Samle Figure 4. Magnitude resonse and imulse resonse of the 1-th order highass nonnegative FIR filter in Examle 4 VII. CONCLUSIONS The design of nonnegative FIR filters has been investigated in this aer, and a constrained minimax formulation has been resented. The constrained minimax formulation has taken into account the nonnegativity and normalization constraints on the NNIR filter s imulse resonse as well as fundamental limitations on the NNIR filter s frequency resonse. By converting the constrained minimax formulation into a linearly constrained quadratic rogramming, the design roblem can be efficiently solved by the Goldfarb-Idnani algorithm. The roosed method has been effectively alied to the design of nonnegative FIR lowass, highass and bandass filters. Design results show that the roosed method has obtained much better filters than the filter obtained in [6] by the offset-smoothing method in [2]. REFERENCES [1] D. L. Hall and J. Llinas, Handbook of Multisensor Data Fusion. Boca Raton, FL: CRC, 21. [2] D. A. Dewasurendra, P. Bauer, and K. Premaratne, Evidence filtering, IEEE Trans. Signal Processing, vol. 55, no. 12, , Dec. 27. [3] G. Shafer, A Mathematical Theory of Evidence. Princeton, NJ: Princeton Univ. Press, [4] R. R. Yager, J. Kacrzyk, and M. Fedrizzi, Advances in the Demster- Shafer Theory of Evidence. New York, NY: John Wiley & Sons, Inc., [5] Y. Liu and P. H. Bauer, Fundamental roerties of non-negative imulse resonse filters, IEEE Trans. Circuits and Systems I, vol. 57, no. 6, , 21. [6], Frequency domain limitations in the design of non-negative imulse resonse filters, IEEE Trans. Signal Processing, vol. 58, no. 9, , 21. [7] L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Alications. New York, NY: Wiley Interscience, 2. [8] J. R. Howell, Some classes of ste-resonse models without extrema, Automatica, vol. 33, no. 7, , [9] B. Anderson, M. Deistler, L. Farina, and L. Benvenuti, Nonnegative realization of a linear system with nonnegative imulse resonse, IEEE Trans. Circuits and Systems I: Fundamental Theory and Alications, vol. 43, no. 2, , Feb [1] A. Rachid, Some conditions on zeros to avoid ste-resonse extrema, IEEE Trans. Automatic Control, vol. 4, no. 8, , Aug [11] B. A. L. de la Barra, On undershoot in SISO systems, IEEE Trans. Automatic Control, vol. 39, no. 3, , Mar [12] S. C. Pei, C. C. Tseng, and W. S. Yang, FIR filter designs with linear constraints using the eigenfilter aroach, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 45, no. 2, , [13] P. P. Vaidyanathan, Eigenfilters: A new aroach to least-squares FIR filter design and alications including Nyquist filters, IEEE Trans. Circuits Syst., vol. 34, no. 1, , [14] X. P. Lai, Otimal design of nonlinear-hase FIR filters with rescribed hase error, IEEE Trans. Signal Processing, 57(9): , 29. [15], Projected least-squares algorithms for constrained FIR filter design, IEEE Trans. Circuits and Systems - I, 52(11): , 25. [16] D. Goldfarb and A. Idnani, A numerically stable dual method for solving strictly convex quadratic rograms, Mathematical Programming, 27,. 1-33,