Complex Algorithms for Lattice Adaptive IIR Notch Filter

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1 4th Iteratioal Coferece o Sigal Processig Systems (ICSPS ) IPCSIT vol. 58 () () IACSIT Press, Sigapore DOI:.7763/IPCSIT..V58. Complex Algorithms for Lattice Adaptive IIR Notch Filter Hog Liag +, Nig Jia ad Chagsheg Yag College of Marie, Northwester Polytechical Uiversity Xi a, ShaaXi, Chia Abstract. The problem of detectig the complex arrowbad sigal usig complex adaptive IIR otch filter is ivestigated. I this paper, three complex coefficiet adaptive IIR otch filters usig gradiet-based algorithms are proposed. These three methods are compared amog themselves i terms of detectig performace ad estimatio accuracy. Simulatio results show that these proposed algorithms ca detect ad estimate the frequecy of siusoid, but a improved simplified lattice complex algorithm outperforms the other two i the covergece speed, tracig speed ad steady-state mea square error at low SNR. Keywords: arrowbad sigal, adaptive IIR otch filter, algorithms.. Itroductio Adaptive IIR otch filter ca elimiate arrowbad or siusoidal iterferece while leavig broadbad sigals uchaged, i other had, it ca ehace arrowbad or siusoidal sigal embedded i broadbad oise. I recet years, real coefficiet adaptive IIR otch filter has bee widely used i the areas such as biomedical egieerig, commuicatios, cotrol, ad radar system. But, whe the sigal cosists of iphase ad quadrature-phase compoets, its sample values are complex umbers, such as radar ad commuicatios, the complex coefficiet adaptive filter must be developed. The adaptive otch filter (ANF) ad adaptive lie ehacer (ALE) are geerally realized by usig FIR filters. For FIR filter, correspodig to polyomial trasfer fuctios, typically require large filter orders to obtai satisfactorily sharp cutoff characteristics. O the other had, a IIR otch filter is oe whose magitude respose vaishes at a particular value (the otch frequecy w ) o the uit circle, ad whose magitude respose is early costat at other poits o the uit circle. Excellet approximatios may be obtaied usig secod-order filter sectios. Geerally speaig, the IIR filter is more computatioally efficiet ad has better statistical performace tha the FIR filter. Although so may complex ANF ad ALE algorithms have bee studied, oly a few adaptive IIR filters have bee cosidered [] [] [3]. The purpose of this paper is to preset class of complex IIR filters with lattice formulatios to costruct the ANF, for the lattice formulatio allows idepedet tuig of the otch frequecy ad atteuatio badwidth. Simulatio results are provided to show the detectio ad frequecy estimatio performace of three ovel gradietbased adaptive IIR complex algorithms otch filter. The rest of this paper is orgaized as follows. Sectio itroduces the structure of secod order IIR lattice otch filter. Sectio 3 presets three proposed complex algorithms. I Sectio 4 describes the umerical results of three algorithms which demostrate the frequecy estimatio ad siusoid sigal detectio performace of the three proposed complex algorithms at lower SNR. Sectio 5 is a coclusio of the paper. + Correspodig author. address: liaghog@wpu.edu.c. 68

2 . Secod Order IIR Lattice Notice Filter Cosider a oisy siusoidal with uow frequecy w, amplitude A, ad phase φ that is uiformly distributed betwee ad π, that is: where w( ) is a zero-mea complex Gaussia oise. j( w f ) y ( ) = Ae + + w( ) () Fig. depicts the basic bloc diagram of a secod order IIR otch filter with lattice form structure. y() is the output sigal of otch filter. The trasfer fuctio of a adaptive IIR otch filter with lattice form structure [4] from iput to otch output is expressed as H(z): - - si + q si + q z + z H( z) = si q(+ si q ) z + siq z with w deotig the otch frequecy ad B the 3dB atteuatio badwidth, oe ca show that () q= w - p/ ω [, π] (3) - ta( B ) si q = < q < p ta( + B ) (4) So, the idepedet tuig of the otch frequecy ad atteuatio badwidth ca be realized. The θ proposed algorithm to tue the otch frequecy parameter θ uder held the badwidth parameter costat will be itroduced i the followig sectio. Fig.. Secod order lattice IIR otch filter. 3. Proposed Three New Complex Algorithms The gradiet descet real algorithm [4] for lattice form adaptive otch filter to tue the otch frequecy parameter θ is itroduced i tableⅠ. Where ϕ( ) = y( ) θ, y ( ) is the output of otch filter, ( ) is a variable step-size parameter, > is a iitial step-size, is a forgettig factor. Table : The lattice gradiet real algorithm g cosθ si θ x ( ) ω = siθ cos θ x ( ) y ( ) = [ x ( ) + ω ] g cosθ si θ x( ) ω = siθ cos θ ξ( ) θ ( + ) = θ ( ) ( ) y( ) ϕ( ) 69

3 x( + ) cos θ( + ) si θ( + ) g = x ( + ) si θ( + ) cos θ( + ) x( ) ξ( + ) cos θ( + ) si θ( + ) g = ξ ( + ) si θ( + ) cos θ( + ) ξ( ) ( ) = = [ ϕ ( )] Based o the lattice gradiet real algorithm (LGRA), the lattice gradiet complex algorithm (LGCA) is proposed. For the adaptive otch filter, θ is chose to miimize the otch filter output power, where deotes a complex cojugate, y( ) y ( ) = Re y( ) ϕ ( ) θ ( ). For LGCA, the updatig formula becomes: y( ) y ( ), θ( + ) = θ( ) Re[ y( ) ϕ ( )] (5) ( ) = < = ϕ( ) The simplified lattice real algorithm(slra) [4] for adaptive otch filter to tue the otch frequecy parameter θ ad are as follows: (6) θ ( + ) = θ ( ) ( )[ y( ) x ( )] (7) ( ) = < [ x ( )] = So, the simplified lattice complex algorithm(slca) based o SLRA i this paper is give by (8) θ ( + ) = θ ( ) Re[ y( ) x ( )] (9) ( ) = < x ( ) = where x ( ) is regressor sigal obtaied from a trasfer fuctio cos θ cos θz H () z = B. + si θ( + si θ ) + siθ z z I order to cotrol the steady of algorithm at lower SNR, Referece [5] proposed a steady variable stepsize algorithm, called improved simplified lattice real algorithm (ISLRA): () () + [ ( )] = ( ) = << < c x where < c<< x ( ) is a factor cotrols the steady of algorithm at lower SNR. = Accordig to ISLRA, the improved simplified lattice complex algorithm (ISLCA) is also proposed: θ ( + ) = θ ( ) Re[ y( ) x ( )] () ( ) = < c x ( ) + = (3) 7

4. The simulatio results For some active soar waveforms which are badpass with high frequecy, the algorithm ca ot be realize at real-time i microcomputer if it sample at Nyquist rate.

Thus a typical meas for obtaiig digital samples of a soar waveform is to demodulate the waveform to dc, low-pass filter the waveform ad sample.

4 4. The simulatio results For some active soar waveforms which are badpass with high frequecy, the algorithm ca ot be realize at real-time i microcomputer if it sample at Nyquist rate. However, it is show that all the iformatio cotet of a badpass sigal is cotaied i the complex evelope. This results i sigificat computatioal ad storage savigs. Thus a typical meas for obtaiig digital samples of a soar waveform is to demodulate the waveform to dc, low-pass filter the waveform ad sample. The samples are obtaied by quadrature samplig or delay samplig. I the simulatios, the ceter frequecy of iput sigal is 3Hz, f = Hz s is a samplig rate, 3ms ad ms are observatio time ad duratio of sigal respectively, zero mea Gaussia oise. Demodulate frequecy is 5Hz, f = 3.5 Hz is a evelope samplig rate. We ca obtai the complex date usig quadrature samplig: xr() i = [ x(8)cos(6 i π fi/ fs)] LP ( i =,, 3 N) xi( i) = [ x(8 i)si(6 π fi/ fs)] LP ( i =,, 3 N) where [] LP is a low-pass operator, xr () i ad xi () i are the real ad imagiary parts of the desired complex sigal respectively. I order to study the performace of detectio ad estimatio a sigle siusoid usig these three complex algorithms, the results of the output evelope (Fig.(c),(d),(e))ad estimated frequecy after demodulated (Fig.(f)) (correct demodulated frequecy is 5Hz) usig Mote Carlo simulatios at SNR=-dB are illustrated i Fig.. It is show that the proposed ISLCA is clearly outperforms the other two algorithms i detectio ad frequecies estimatio accuracy at lower SNR. Fig. (a). The origial sigal. Fig. (b). The demodulated real part of iput oisy sigal. Fig. (c). The output evelope of LGCA. Fig. (d). The output evelope of SLCA. Fig. (e). The output evelope of ISLCA. Fig. (f). The estimated frequecy of three complex algorithm. 7

5 5. Coclusios I this paper, we preseted three gradiet-based complex algorithms for adaptive IIR otch filter to ehace ad estimate a sigle siusoid i Gaussia oise. The results of the computer simulatio have show the SLCA has the lowest complexity, ad the ISLCA has superiority of detectig performace ad estimatio accuracy over the other two complex algorithms at lower SNR. We believe that the complex algorithms proposed i this paper will have more applicatios i other areas. 6. Refereces [] S. Nishimura, H. Y. Jiag, ad T. Hiamoto, Covergece aalysis of complex adaptive IIR otch filters with colored oisy sigal, IEEE Pacific Rim Coferece o Commuicatios, Computers ad Sigal Processig, pp:35-38, 999. [] S. C. Pei, C. C. Tseg, Complex adaptive IIR otch filter algorithm ad its applicatios, IEEE Trasactios o Circuits ad Systems Ⅱ : Aalog ad Digital Sigal Processig, 4():58-63, 994. [3] S. Nishimura, H. Y. Jiag, Gradiet-based complex adaptive IIR otch filters for frequecy estimatio, Proceedig of IEEE Asia Pacific Coferece o Circuits ad Systems 96, pp: 35-39, 996. [4] P. A. Regalia, Adaptive IIR filterig i sigal processig ad cotrol, New Yor: Marcel Deer, ic, 995. [5] H. Liag, Y. P. Hog, Z. S. Li. A steady variable step-size for adaptive IIR otch filter ad its usigi movig target detectio, Techical Acoustics, 4(4):97-, 5. 7

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