A novel design of sparse FIR multiple notch filters with tunable notch frequencies

1 A novel design of sparse FIR multiple notch filters with tunable notch frequencies Wei Xu 1,2, Anyu Li 1,2, Boya Shi 1,2 and Jiaxiang Zhao 3 1 School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China 2 Tianjin Key Laboratory of Optoelectronic Detection Technology and System, Tianjin, 300387, China 3 College of Electronic Information and Optical Engineering, Nankai University, Tianjin 300071, China Correspondence should be addressed to Jiaxiang Zhao; zhaojx@nankaieducn Abstract In this paper, we focus on the design of finite impulse response(fir) multiple notch filters To reduce the computational complexity and hardware implementation complexity, a novel algorithm is developed based on the mixture of the tuning of notch frequencies and the sparsity of filter coefficients The proposed design procedure can be proceeded as follow: first, since sparse FIR filters have lower implementation complexity than full filters, a sparse linear phase FIR single notch filter with the given rejection bandwidth and passband attenuation is designed Second, a tuning procedure is applied to the computed sparse filter to produce the desired sparse linear phase FIR multiple notch filter When the notch frequencies are varied, the same tuning procedure can be employed to render the new multiple notch filter instead of designing the filter from scratch The effectiveness of the proposed algorithm is demonstrated through three design examples I INTRODUCTION The multiple notch filters, which can highly attenuate some frequency components in the input signal while leaving the others relatively unchanged, are widely used in many applications Important examples include radar systems, control and instrumentation systems, communications systems, medical applications, biomedical engineering and indoor localization [1]-[2] Various methods [3]-[8] have been reported to design FIR multiple notch filters In general, the multiple notch filters derived from these algorithms are not sparse Compared with full FIR filters, sparse filters can significantly reduce the implementation complexity in the hardware In [9], we proposed an iterative reweighed OMP algorithm to compute sparse notch filters However, when the notch frequencies are varied, it requires one to design the whole filter from scratch, hence, increasing the computational complexity of this scheme Recently, in [10]-[12], a number of algorithms are proposed to design FIR filters based on LMS minimization or Monte Carlo methods The disadvantage of these approaches is the suboptimality in terms of the filter length related to its selectivity Another disadvantage is that the attenuation at the notch frequency changes during the adaptation process, therefore, a strong attenuation of the disturbing signal at the notch frequency is not guaranteed Moreover, the actual value of the attenuation at notch frequency is cased by the adaptation process In this brief, the design problems of sparse FIR multiple notch filters with tunable notch frequencies are studied To reduce the computational complexity and the hardware complexity, a novel algorithm is developed based on the mixture of the tuning of notch frequencies and the sparsity of filter coefficients The sparse FIR multiple notch filters can significantly reduce the number of the adders and multipliers used in the hardware implementation However, the design of FIR sparse filter always involves iterative procedures and numerical optimization, which results a high computational complexity for the practice system The tuning of notch frequencies is a useful operation for the design of FIR multiple notch filter In the case of variable notch frequencies, the same tuning process is implemented to obtain the new multiple notch filter instead of designing the filter from scratch Therefore, the tuning feature can significantly reduce the computational complexity We demonstrate the effectiveness of this approach through three design examples II PROBLEM FORMULATION Given the design parameters of linear phase FIR multiple notch filter, which include a set of the notch frequencies {ω i } r i=1, rejection bandwidth ω and passbands attenuation α The given notch frequencies {ω i } r i=1 satisfying ω i < ω i+1 for 1 i r are allowed to be non-uniformly distributed in the set [0, π] The ideal multiple notch filter amplitude response H d (ω) satisfies { 0 ω Ω 0 H d (ω)= 1 ω Ω 1, (1) where Ω 0 and Ω 1 are respectively defined as Ω 0 = {ω ω ω i ω/2, 1 i r}, (2) Ω 1 = [0, π] Ω 0 (3) To simplify the presentation, we focus on the design of Type-I linear phase FIR filter H(e jω ) = e jmω H 0 (ω), ie, the filter order N = 2M is even and h(m) = h(n m) for all 0 m N For other types of filter, our design method presented in this letter is feasible For the case of Type-I filter, the zero-phase amplitude response H 0 (ω) can be expressed as M H 0 (ω) = h(m) + 2 h(m m) cos(mω), (4) m=1

2 with M = N/2 III THE PROPOSED SPARSE LINEAR PHASE FIR MULTIPLE NOTCH FILTER DESIGN In this section, a novel design method is presented to produce the sparse FIR multiple notch filter The procedure of computing the linear phase FIR multiple notch filter starts with the estimation of the initial order N of the filter F (e jω ) through N = max N i (5) i {1,,r} From equation [13, eq(20)], N i is computed as N i = max{ ˆN(ω p1 i, F, δ p, δ s ), ˆN(ω p2 i, F, δ p, δ s )}, (6) where F = ω/2 and the function ˆN( ) is determined by equation [13, eq(15)] The arguments of ˆN( ) can be computed as: ω p1 i = (ω i F )/2, (7) ω p2 i = (1 ω i F )/2, (8) δ p = δ s = (1 α)/(1 + α) (9) The following design procedure is mainly comprised of two stages: in the first stage, a sparse linear phase FIR single notch filter F (e jω ) with the given rejection bandwidth and passband attenuation is designed as a fixed sparse filter In the next stage, a tuning process is carried out to compute the desired multiple notch filter with the given notch frequencies base on the filter F (e jω ) A Sparse linear phase FIR single notch filter design In this section, a sparse linear phase FIR single notch filter F (e jω ) of order N with the notch frequency ω 1 = 0 is designed Let F (e jω ) = e jmω F 0 (ω) represents the single notch filter, as shown in the Fig1, the real-valued amplitude response F 0 (ω) satisfies F 0 (ω) = 0, ω = 0, 0 < F 0 (ω) < 1 δ F, 0 < ω < ω/2, (10) F 0 (ω) 1 < δ F, ω/2 ω π The passband ripple δ F of the single notch filter F (e jω ) and the attenuation in the passbands α are related through δ F = 1 α 2r (1 + α) (11) Equation (11) is a conservative choice of δ F which ensures the multiple notch filter yielded from this choice to satisfy the design specifications In most cases, δ F can be chosen between 1 α 2r(1+α) and 1 α (1+α) The design of the sparse single notch filter F (e jω ) can be formulated as min f 0 (12a) f st c(ω)f 1 δ F, ω [ ω/2, π], (12b) c(ω)f = 0, ω = 0, (12c) ω 2 1 δf 1+δF Fig 1 The illustration for the amplitude response of the desired sparse single notch filter where we have c(ω) = [1 cos(ω) cos(mω) cos(mω)], f = [f(m) 2f(M 1) 2f(m) 2f(0)] T, with 0 m M To compute a solution of problem (12), we follow the standard discretization procedure as presented in [14] and replace the continuous parameter ω by L samples (where L 1 is a large positive integer) uniformly distributed in the frequency set [ ω/2, π] Thus, the discretization and normalized formulation of problem (12) is given by min f f 0 (13a) st Af 1 L 1 δ F 1 L 1, (13b) 1 1 L f = 0, (13c) where we have c(ω 1 ) 1 cos(ω 1 ) cos(mω 1 ) c(ω 2 ) 1 cos(ω 2 ) cos(mω 2 ) A = c(ω l ) = 1 cos(ω l ) cos(mω l ), (14) c(ω L ) 1 cos(ω L ) cos(mω L ) with ω l [ ω/2, π] and 1 l L It is known that this optimization problem is in general NPhard due to the existence of l 0 -norm in its objective function To tackle this problem, a great deal of effort has been made to develop efficient algorithms In this paper, we can employed one of these sparse filter algorithms, eg, linear programming [15], iterative second-order cone programming (ISOCP) [16], iterative reweighted l 1 (IRL1) [17], iterative reweighted OMP (IROMP) schemes[9], to attain the desired sparse FIR single notch filter

3 B The design of the desired linear phase FIR multiple notch filter In this section, a tuning process is implemented to derive the desired FIR multiple notch filter base on F (e jω ) of the previous stage For the given notch frequencies set {ω i } r i=1, the multiple notch filter H(e jω ) can be shown that Solve (13) to achieve the sparse FIR single notch filter Derive the sparse FIR Multiple notch filter using (16) r H(e jω ) = e jmω [F 0 (ω+ω i )+F 0 (ω ω i )] (15) i=1 According to the Fourier transform theory, the impulse response h(n) of H(e jω ) can be obtainde as Compute the attenuation in the passbands using (17) Remove one element from z Solve the linear program (18) h(n) = r f(n)cos(nω i ), (16) i=1 where 0 n N Computing the attenuation ˆα in the passbands of the linear phase FIR multiple notch filter H(e jω ) as The computed filter meets our design specifications is changed? ˆα = min(h 0(ω)) max(h 0 (ω)), ω Ω1, (17) If ˆα α, then the computed filter {h(n)} N n=0 is a sparse solution for the given specifications; Otherwise, the following linear program optimization is run to minimizing the attenuation in the passbands of the obtained filter: min µ h,µ (18a) st Bh 1 L 1 (δ + µ) 1 L 1, (18b) c(ω i )h = 0, i = 1, 2,, r, (18c) h(n) = 0, n Z, (18d) where Z represents the set of indices at which h(n) = 0 based on (16), and the matrix B can be written as c(ω 1) c(ω 2) B = c(ω l ), ω l Ω 1 (19) c(ω L ) If the optimal objective value µ of (18) is negative, ie, µ 0, the obtained filter h is a sparse solution for the given specifications; Otherwise, the sparsity pattern Z is infeasible to the given specifications of the multiple notch filter, then the largest element is eliminated from Z and the linear program (18) is solved with the new set Z until µ 0 When the notch frequencies are changed, the same tuning process is implemented to yield the new multiple notch filter instead of designing the filter from scratch Fig 2 outlines the main steps of the proposed algorithm Fig 2 End Flowchart of the proposed design algorithm IV SIMULATION In this section, we confirm the effectiveness of our multiple notch filter design scheme through three examples Example 1: Let us design a multiple notch filter specified by a set of notch frequencies {025π, 049π, 078π}, α = 080 (passbands attenuation) and ω = 005π (the rejection bandwidths) By substituting the design specifications into [9, eq(11)], we obtain the initial order N = 174 In this simulation, we employ the IROMP scheme [9] to design the sparse single notch filter As shown in Fig 3, the amplitude response of the multiple notch filter derived by following steps in Fig 2 It is obvious that the specification is well satisfied The nonzero tap weights of the multiple notch filter yielded from our design method are listed in Table I The filter order, number of nonzero tap, rejection bandwidth, passband attenuation and attenuation at the notch frequency are listed in Table II Example 2: We only change the notch frequencies from {025π, 049π, 078π} of Example 1 to {034π, 043π, 072π} but use the same rejection bandwidth and attenuation in the passbands Since the same rejection bandwidth and attenuation in the passbands as Example 1 are used, the sparse single notch filter F (e jω ) of (13) with N = 174 can be identical to the one computed in Example 1 Following the tuning procedure from (16) to (18), we compute the sparse multiple notch filter with this new set of the notch frequencies Fig 4 show the performance of the sparse multiple notch filter yielded from our scheme The nonzero tap weights of the multiple notch

4 TABLE I NONZERO COEFFICIENTS OF THE DESIGNED FILTER IN EXAMPLE 1 Taps nzero tap weights Taps nzero tap weights 0 174 0004722848554565 62 112 0061953987746997 24 150 0002007423266603 63 111 0047952288148680 33 141 0004871513257007 64 110 0038531961471284 38 136 0030966811403170 65 109 0076026372592104 39 135 0022053492615673 66 108 0015111106082247 40 134 0014951595120278 67 107 0000183867057454 41 133 0021767842979883 68 106 0099331267386019 42 132 0011692029570564 69 105 0012255378374269 43 131 0006950807646344 70 104 0028055658481912 44 130 0044114166070248 71 103 0100962624763570 45 129 0006311408033158 72 102 0038372777955426 46 128 0062066578840798 73 101 0098683027527897 47 127 0005818521570352 74 100 0036089958346951 48 126 0002318412822056 75 99 0029655283702946 49 125 0005091217785500 76 98 0071889586307707 50 124 0020608511742342 77 97 0004739347900282 51 123 0014404626498885 78 96 0002403828886942 52 122 0061394774774335 79 95 0153598792283481 53 121 0008629427173138 80 94 0019548891523249 54 120 0072558769634137 81 93 0089075079456139 55 119 0010787517932644 82 92 0023403440116641 56 118 0022623175477306 83 91 0056274737680820 57 117 0030869196951431 84 90 0018813444978966 58 116 0009932536096148 85 89 0047899881207932 59 115 0014074814764124 86 88 0000409323536052 60 114 0094137780017783 87 0911070614139502 61 113 0004199271003407 TABLE III NONZERO COEFFICIENTS OF THE DESIGNED FILTER IN EXAMPLE 2 Taps nzero tap weights Taps nzero tap weights 0 174 0001484391069741 62 112 0020291841316044 24 150 0015053216345025 63 111 0035719226300918 33 141 0011203465180753 64 110 0069237101344906 38 136 0031930539223127 65 109 0019718488202998 39 135 0011989478832635 66 108 0128901758725893 40 134 0047916512343373 67 107 0025726032859600 41 133 0001724422497417 68 106 0073722260607230 42 132 0012946318823095 69 105 0025836288834871 43 131 0029367872190180 70 104 0045891876538100 44 130 0012950777492872 71 103 0057310758857846 45 129 0058861258880195 72 102 0075489239705611 46 128 0027161496236364 73 101 0063149359304339 47 127 0038124403599415 74 100 0003347070890743 48 126 0005580972074984 75 99 0014446702737718 49 125 0024077586822642 76 98 0058103382956618 50 124 0013761958746995 77 97 0032069195768895 51 123 0036380554634187 78 96 0000558233096476 52 122 0026803049613006 79 95 0006719715851877 53 121 0001700050810428 80 94 0084935075668951 54 120 0023056804511978 81 93 0076509802170425 55 119 0041926654928440 82 92 0097984593409281 56 118 0000428542187068 83 91 0043634372404206 57 117 0005253841608839 84 90 0040410785159360 58 116 0008896975496026 85 89 0089360899339001 59 115 0071272465825430 86 88 0003266951955140 60 114 0037660569869951 87 0915171992267534 61 113 0085639588936268 TABLE II A LIST OF FILTER ORDER, REJECTION BANDWIDTH AND ATTENUATION OF EXAMPLES 1-3 Example Filter order The number of nonzero tap weights Rejection bandwidth Passband attenuation Attenuation at the notch frequency 1 174 105 0050π 07197dB 247dB 2 174 105 0050π 05349dB 259dB 3 174 105 0050π 07687dB 264dB Fig 4 The amplitude response of the filter yielded from our design method for Example 2 Fig 3 The amplitude response of the filter yielded from our design method for Example 1 filter yielded from our design method are listed in Table III The filter order, number of nonzero tap, rejection bandwidth, passband attenuation and attenuation at the notch frequency are listed in Table II Example 3: Change the set of notch frequencies in Example 1 to {025π, 049π, 061π, 078π} while α and ω remain the same Since α and ω are kept constant, we start with the sparse single notch filter F (e jω ) which is same as that derived in Example 1 (N = 174) The sparse multiple notch filter with this new notch frequencies is obtained through the tuning process from (16) to (18) Fig 5 illustrates the amplitude response of this filter It is evident that the specification is satisfied The nonzero tap weights of the multiple notch filter yielded from our design method are listed in Table IV The filter order, number of nonzero tap, rejection bandwidth, passband attenuation and attenuation at the notch frequency are listed in Table II V CONCLUSION In this paper, a novel approach has been presented for the design of sparse FIR multiple notch filters with tunable notch

5 TABLE IV NONZERO COEFFICIENTS OF THE DESIGNED FILTER IN EXAMPLE 3 Taps nzero tap weights Taps nzero tap weights 0 174 0001163545765695 62 112 0033265661189825 24 150 0003165433288996 63 111 0026560471743885 33 141 0002091982284630 64 110 0082576601043643 38 136 0047102411486321 65 109 0085379844129436 39 135 0005481276840672 66 108 0024951523169648 40 134 0019772136963283 67 107 0038897009099235 41 133 0043110955540852 68 106 0088237797320545 42 132 0020979982565140 69 105 0036518046705842 43 131 0015127277492607 70 104 0053707639239860 44 130 0022611789510971 71 103 0134082248075127 45 129 0010481611606251 72 102 0007838779814640 46 128 0039298064713159 73 101 0104434475095505 47 127 0024983870407359 74 100 0081493323669535 48 126 0013239453713585 75 99 0057503261219777 49 125 0022935249461144 76 98 0103032966355301 50 124 0027344765658964 77 97 0046232816359952 51 123 0037957131548364 78 96 0001798639004124 52 122 0075665876343805 79 95 0100284039691277 53 121 0024121474369378 80 94 0057999870693463 54 120 0104718797612301 81 93 0060896839992225 55 119 0014707635070068 82 92 0036785941586311 56 118 0016302396989146 83 91 0041803167312136 57 117 0011567221940061 84 90 0030010449210885 58 116 0010934081396023 85 89 0094188919109719 59 115 0019926233327605 86 88 0018630470290357 60 114 0089656608769556 87 0882383969239141 61 113 0041103476228622 Fig 5 The amplitude response of the filter yielded from our design method for Example 3 frequencies To futher improve the effciency, the proposed algorithm is based on the mixture of the tuning of notch frequencies and the sparsity of filter coefficients In the case of variable notch frequencies, the same tuning procedure can be used to render the new multiple notch filter in place of designing the filter from scratch Therefore, the proposed algorithm can significantly reduce the computational complexity Three examples are given to show the effectiveness of this approach Nature Science Foundation of Tianjin (grant number 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8, pp 4035-4044, 2012 [17] C Rusu and B Dumitrescu, Iterative reweighted l 1 design of sparse FIR filters, Signal Processing, vol 92, no 4, pp 905-911, 2012 VI CONFLICTS OF INTEREST The authors declare that there are no conflict of interest regarding the publication of this article VII ACKNOWLEDGMENTS This research was supported by the Nature Science Foundation of China (grant number 61501324, 61601323), and the