Design of fractional delay filter using discrete Fourier transform interpolation method
Accelerated plane-wave destruction
Accelerated plane-wave destructionZhonghuan Chen 1,Sergey Fomel 2,and Wenkai Lu 1INTRODUCTIONLocal slope fields have been widely used in geophysical applications,such as wave-field separation and denoising (Harlan et al.,1984;Fomel et al.,2007),antialiased seismic interpolation (Bardan,1987),seislet transform (Fomel and Liu,2010),velocity-independent NMO correction and imaging (Fomel,2007b ;Cooke et al.,2009),predictive painting (Fomel,2010),seismic attribute analysis (Marfurt et al.,1999),etc.Several tools exist for local slope estimation:local slant stack (Ottolini,1983;Harlan et al.,1984),complex trace analysis (Barnes,1996),multiwindow dip search (Marfurt,2006),local structure tensor (Fehmers and Höcker,2003;Hale,2007),and plane-wave destruction (Claerbout,1992;Fomel,2002).Plane-wave destruction (PWD)approximates the local wavefield by a local plane wave,and models it using a linear differential equation (Claerbout,1992).When plane-wave destruction is applied on discrete sampled seis-mic signals,the corresponding differential equation needs to be dis-cretized by finite differences.Claerbout (1992)used explicit finite differences.In this method,plane-wave approximation of the wave-field can be seen as applying a linear finite impulse response (FIR)filter to the wavefield.Slope estimation is equivalent to estimating a parameter of the FIR filter.A least-squares estimator of the local slope can be obtained by minimizing the prediction error of the fil-ter.To improve estimation performance of the explicit finite differ-ence scheme,Schleicher et al.(2009)proposed total least-squares estimation.The implicit finite difference scheme was applied to the differ-ential equation by Fomel (2002).Using an infinite impulse response (IIR)filter,known as the “Thiran allpass filter ”(Thiran,1971),to approximate the phase-shift operator,the plane-wave destruction equation becomes a nonlinear equation of the slope.An iterative algorithm was designed to estimate the slope.To improve stability in the iterative algorithm,a smoothing regularization (Fomel,2007a )of the increment can be applied at each iteration.Iterations of regular-ization can be time-consuming,however,particularly in the 3D case.In this paper,we prove that the plane-wave destruction equation is a polynomial equation of an unknown slope.In the case of a three-point approximation of Thiran ’s filter,the convergence results of the iterative algorithm can be analytically analyzed.In this case,we obtain an analytical estimator of the local slope and show that the smoothing regularization can be applied on the final estimator only once.This approach reduces the computational time signifi-cantly.We present 2D and 3D examples,which demonstrate that the proposed algorithm can obtain a slope-estimation result faster than the iterative algorithm,with a similar or even better accuracy.THEORYReview of PWDThe local plane wave can be represented by the following differ-ential equation (Claerbout,1992)Manuscript received by the Editor 23April 2012;revised manuscript received 7August 2012;published online 25January 2013.1Tsinghua University,Department of Automation,State Key Laboratory of Intelligent Technology and Systems,Tsinghua National Laboratory for Informa-tion Science and Technology,Beijing,China.E-mail:zhonghuanchen@;lwkmf@.2The University of Texas at Austin,Bureau of Economic Geology,Jackson School of Geosciences,Austin,Texas,USA.E-mail:sergey.fomel@beg.utexas .edu.©2013Society of Exploration Geophysicists.All rights reserved.V1GEOPHYSICS,VOL.78,NO.1(JANUARY-FEBRUARY 2013);P.V1–V9,9FIGS.,1TABLE.10.1190/GEO2012-0142.1D o w n l o a d e d 09/22/13 t o 221.226.175.186. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /∂u ∂x þσ∂u ∂t¼0;(1)where σis the local slope in continuous space,with dimension of time/length.The wavefields observed at the two positions x 1,x 2have a time delay which is proportional to their distance,σj x 1−x 2j .In the sampled system with space and time intervals Δx and Δt ,we define the discrete space slope in the unit of Δt ∕Δx as p ¼σΔx ∕Δt .Because p is independent of the sampling interval,it can be directly used in irregular data set (in this case,the unit of the slopes becomes space-variant).The time delay between two adjacent positions is then the slope p Δtu ðx;t Þ¼u ðx þΔx;t þp Δt Þ:(2)With the Z transform applied along time and space directions,the above equation becomesð1−Z x Z p t ÞU ðZ x ;Z t Þ¼0;(3)where Z t is the unit time-shift operator,Z x denotes the unit space-shift operator,and U ðZ x ;Z t Þis the Z transform of u ðx;t Þ.Theoperator 1−Z x Z p t is the plane-wave ing Thiran ’sfractional delay filter H ðZ t Þ¼B ð1∕Z t ÞB ðZ t Þ(Thiran,1971)to approximate the time-shift operator Z p t ¼e j ωp ,where ωis the circular frequency,the plane-wave destructor can be expressed as (Fomel,2002),C ðp Þ¼B ðZ t Þ−Z x B 1Z t;(4)whereB ðZ t Þ¼X N k ¼−Nb k ðp ÞZ −k t ;(5)N is the order of the noncausal temporal filter,and b k ðp Þare func-tions of the local slope p .Equation 4is a 2D filter.Applying the filter at an arbitrary point in the wavefield,the plane-wave destruction equation 3becomes a nonlinear equation for the local slope pC ðp;Z x ;Z t ÞU ðZ x ;Z t Þ≈0:(6)An iterative method,such as Newton ’s method,can be applied to find the slope.In practice,wavefields are polluted by noise and the plane wave assumption may not hold true where faults and conflict-ing boundaries exist.To obtain a stable slope estimation,an addi-tional smoothing regularization process (Fomel,2007a )is needed at each step.The total computational cost of slope estimation by plane-wave destruction becomes O ðN d N f N l N n Þ,where N d is the size of the data,N f ¼2N þ1is the size of the filter,N l is the num-ber of linear iterations for regularization,and N n is the number of nonlinear iterations for solving equation 6.Typical values are N f ¼3,5,N l ¼10–50,and N n ¼5–10.Accelerated PWDGauss-Newton ’s iteration searches the solutions for nonlinearequation 6as follows:Let p k be the estimated slope at step k ,with estimating error (or destructive error)e k ¼C ðp k ÞU ðZ x ;Z t Þ.To findthe correct solution p k þ1that minimizes e k þ1,we need to find theincrement Δp k from the local linearizatione k þ1¼C ðp k ÞU ðZ x ;Z t ÞþC 0ðp k ÞU ðZ x ;Z t ÞΔp k ≈0;(7)where C 0ðp k Þis the derivative of C ðp Þat p k with respect to p .The iterative algorithm stops when a stationary point or a root of C ðp ÞU is reached.They are:•Points where C 0ðp k ÞU ¼0:When p k satisfies C 0ðp ÞU ¼0,then e k þ1¼e k ,and the Δp k dependency in equation 7is removed,stopping further iterations on p k .•Points where C ðp k ÞU ¼0and C 0ðp ÞU ≠0:In this case,Δp k ¼0,thus p k þ1¼p k ,eliminating the need for further improvements on p k .The iterative algorithm for equation 6may converge at different points,depending on the initial point that we chose;p 0¼0is a common practical choice for the initial solution.In this case,the iterative algorithm may converge to the least absolute root,which denotes the event with smallest dip angle.To analyze the convergence results,the maximally flat fractional delay filter (Thiran,1971;Zhang,2009)is designed with polyno-mial coefficientsb k ðp Þ¼ð2N Þ!ð2N Þ!ð4N Þ!ðN þk Þ!ðN −k Þ!Y N −1−km ¼0ðm −2N þp Þ×Y N −1þk m ¼0ðm −2N −p Þ:(8)Details on how to design the filter can be found in Appendix A .Because b k ðp Þis a polynomial of p ,expanding it,we get b k ðp Þ¼P 2N i ¼0c ki p iandB ðZ t ;p Þ¼X N k ¼−N X 2N i ¼0c ki Z −k t p i:(9)From equation 8,it is obvious that b k ðp Þ¼b −k ð−p Þ,thereforec ki ¼ð−1Þi c −k;i andB ðZ t ;p Þ¼B1Z t;−p :(10)Substituting the above two equations,the nonlinear equation 6becomes a 2N th°polynomial equation for pX 2N i ¼0a i p i ¼0;(11)and the coefficients of the polynomial plane-wave destruction canbe expressed asa i ¼½1−ð−1ÞiZ xX N k ¼−Nc ki Z −kt U;(12)which says that the coefficients of the polynomial PWD can be ob-tained by applying a 2D filter on the wavefield u .Moreover,the 2D filter can be decoupled into the cascade of two 1D directional filters;the temporal filter P N k ¼−N c ki Z −k t and the spatial filter 1−ð−1Þi Z x .V2Chen et al.D o w n l o a d e d 09/22/13 t o 221.226.175.186. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /In the special case of N ¼1,we get a three-point approximation of B ðZ t Þ.It takes the following form (Fomel,2002)B ðZ t Þ¼ð1þp Þð2þp Þ12Z −1t þð2þp Þð2−p Þ6þð1−p Þð2−p Þ12Z t :(13)The plane-wave destruction equation 11is a quadratic equation.The coefficients a i ði ¼0;1;2Þcan be solved for and expressed asa i ¼112½1−ð−1Þi Z x v i ;(14)where v i are outputs of the following three-point temporal filtersv 0¼2ðZ −1t þ4þZ t ÞU;(15)v 1¼3ðZ t −Z −1t ÞU;(16)andv 2¼ðZ −1t −2þZ t ÞU:(17)In this case,the quadratic plane-wave destruction equation 11has one stationary point and two roots,which can be analytically ex-pressed as:f −a 12a 2;−a 1ÆffiffiffiDp 2a 2g ,where D ¼a 21−4a 0a 2.The plots in Figure 1show the convergence process of the itera-tive algorithm when we choose p 0¼0as the starting value.Geo-metrically when D ≤0,the iteration converges to the stationarypoint −a 12,as shown in Figure 1a .When D >0,it converges to the least absolute solution of equation 11.Figure 1b and 1c shows the convergence process to the least absolute solution in different cases.We can summarize the convergence result of the iterative algorithm as followsp ¼8>><>>:−a12a 2D ≤0−2a0a 1−ffiffiffiD p D >0;a 1<0−2a 0a 1þffiffiffiDpD >0;a 1>0:(18)As −2a 0a 1ffiffiffiD p ¼−a 1ÆffiffiffiDp 2a 2inthe above equation,we use−2a 0a 1ffiffiffiDp instead of −a 1ÆffiffiffiDp 2a 2to obtainbetter numerical stability.When the data is polluted by noise,to obtain a robust slope es-timation,we can combine the equations in a local window into the following equation setFp ≈g ;(19)where F is a normalized diagonal matrix and g is a vector.Their elements are denominators and numerators of equation 18,respec-tively.When we are solving the above equation set by least squares,we can use Tikhonov ’s regularization (Fomel,2002)or the shaping regularization (Fomel,2007a ,equation 13)to obtain a smooth solu-tion as followsp ¼H ½I þH T ðF T F −I ÞH −1H T F T g ;(20)where H is an appropriate smoothing operator.In this case,the reg-ularization runs only once;therefore,the computational cost isreduced to O ðN d N f N l Þ.In 3D applications,there are two polynomial PWD equations for inline and crossline slopes separately.Note that,using the decou-pling,inline,and crossline slope estimations can share the temporal filtering results in equations 15–17.We can obtain the coefficients of the crossline plane-wave destruction equation asa i ¼112½1−ð−1Þi Z y v i .(21)The five-point or longer approximations of B ðZ t Þcan achieve higher accuracy.Equation 6in this case becomes a higher-order polynomial equation (see details in Appendix A ),which can be solved numerically.However,there are multiple stationary points,and it is difficult to determine the right one analytically.For appli-cations that need five-point or higher accuracy,we suggest obtain-ing an initial slope estimation by the proposed three-point method and using it to make the iterative algorithm converge faster (to de-crease N n ).EXAMPLESSynthetic examplesTo test the performance of the proposed slope-estimation method,we generated a harmonic wave field with constant slope p 0¼0.3,shown in Figure 2.We added different scales of additive white Gaussian noise (AWGN)to the wave field,and estimated the slope by the proposed method.To compare with the iterative algorithm by Fomel (2002),the mean square error (MSE)is used as the criterionMSE ðp Þ¼E fðp −p 0Þ2g :(22)We use five iterations in the iterative method,N n ¼5.For the constant slope model,using a large smoothing window,theFigure 1.The iterative results of Newton ’s algorithm.The initial point is p 0¼0(solid circle),and the convergence result is marked by an empty circle:(a)when D ≤0,(b)when D >0and a 1a 2>0,(c)when D >0,and a 1a 2<0.Accelerated plane-wave destructionV3D o w n l o a d e d 09/22/13 t o 221.226.175.186. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /smoothing regularization can converge faster (with less N l )and we can obtain a better estimation accuracy.For each method,we try the smoothing windows from 2to 200and show the mean square errors in Figure pared with the iterative method,the proposed method has better accuracy at the left upper (low SNR and small smoothing window)and worse accuracy at right bottom corner (high SNR and large smoothing window).We show the total runtime of all the noise scale data in Figure 4.For all smoothing windows in the regularization,the proposed method (solid line)only uses about one-fifth of the run time of the three-point (N ¼1)iterative method (dash line).A more complicated model from Claerbout (1999)and Fomel (2002)is shown in Figure 5a .It has variable slopes in synclines,anticlines,and faults.The slope estimated by the proposed method is shown in Figure 5b .The estimation takes about 20ms.The three-point (N ¼1)iterative method can obtain a similar estimation (shown in Figure 5c )by five iterations,which takes about 130ms.Bothmethods use a four-point smoothing window in the regularization,but the proposed method obtains a slightly smoother estimation.In Figure 5d ,we show the faults detected by the residuals of the pro-posed plane-wave destruction.ApplicationThe 2D seislet transform (Fomel and Liu,2010)uses local slopes to predict and update even and odd traces in the wavelet lifting scheme.The seislet transform itself is fast,but the slope estimation step is comparatively slow.The 3D seislet transform can be con-structed in the same way by using 3D slopes and cascading 2D transforms in inline and crossline directions.So that an efficient transform can be built,the proposed accelerated plane-wave de-struction is applied to slope estimation.Figure 6shows a part of the Teapot Dome image from Wyoming.The 3D slope estimation yields two slope fields along twospaceFigure 2.Harmonic waves with constant slope p 0¼0.3.a)b)Figure 3.Mean square errors of the slope estimations by the proposed method (a)and the three-point (N ¼1)iterative method(b).Figure 4.Run time of the proposed method (solid line)and the three-point (N ¼1)iterative method (dashed line).V4Chen et al.D o w n l o a d e d 09/22/13 t o 221.226.175.186. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /directions.Figure7shows the local inline and crossline slope fields estimated by the proposed algorithm.The two slopes are used by the lifting scheme in the seislet trans-form to obtain the seislet transform coefficients.The coefficients of the2D seislet transform are concentrated at the planes near the zero inline plane,as shown in Figure8a,whereas in Figure8b,the3D transform coefficients are concentrated in the corner region near the origin.To illustrate the compressive performance of the3D seislet trans-form,Figure9shows the reconstruction results at different percen-tages of the compression.Most of the data can be reconstructed using only1%of the seislet coefficients(1∶100compression ratio). To compare with the iterative methods quantitatively,we define the following normalized crosscorrelation(NCC)NCCðx;yÞ¼x T yk x k2k y k2:(23)a)c)b)d)Figure5.2D slope estimation example:(a)synthetic data,(b)slope field estimated by the proposed algorithm,(c)slope field estimated by the iterative algorithmafter five iterations,(d)faults detection by plane-wave destruction with the estimated slope.Figure6.A portion of the3D data from the Teapot data set.Accelerated plane-wave destruction V5 Downloaded9/22/13to221.226.175.186.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/Figure7.3D slopes estimated by the proposed algorithm:(a)inline slope(b)crossline slope.Figure8.Coefficients of:(a)the2D seislet transform along inline direction only,(b)the3D seislettransform.Figure9.Reconstruction of3D-seislet-compressed data using:(a)5%of the seislet coefficients,(b)1%of the seislet coefficients.V6Chen et al.Downloaded9/22/13to221.226.175.186.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/In the seislet transform,the more accurate dip we use,the better compression we can obtain.The NCC between the reconstructed data and the original data can be used to quantify the compressive performance.In Table 1,the proposed method is compared with three-point (N ¼1)and five-point (N ¼2)iterative methods.In all three methods,we use 10-point smoothing windows in in-line and crossline dimensions,and use five-point smoothing win-dow in time direction.To obtain a similar slope estimation as the proposed method,the three-point (N ¼1)and five-point (N ¼2)iterative methods need about six and five iterations,respectively.The five-point method can obtain a better compression ratio than the three-point method because of its better accuracy in slope es-timation.However,although the iterative methods have smaller PWD residuals,neither of them achieves a better NCC than the proposed method.In this case,using the noniterative estimation as the initial in five-point estimation can save at least 250s.That is to say,the computational time cost is reduced by a factor of about six.CONCLUSIONSIn this paper,we derived an analytical estimator of the local slope in three-point implicit plane-wave destruction.On the basis of this result,we built an accelerated slope-estimation algorithm.Exam-ples show that the proposed method can produce a result that is similar to that of the iterative algorithm,at a reduced computation time.Two or more conflicting slopes can be estimated simultaneously by the iterative algorithm.In this case,the PWD equation is a multi-variable polynomial,and the convergence analysis becomes com-plicated.The proposed noniterative method is not yet suitable for multislope estimation.However,we believe that it can find many applications in situations where one dominant slope is suf-ficient.ACKNOWLEDGMENTSWe thank Alexander Klokov for his helpful technical sugges-tions.We also thank the reviewers and the editors for their carefulreview of the paper.The Teapot Dome data is provided by the Rocky Mountain Oilfield Testing Center,sponsored by the U.S.Department of Energy.Zhonghuan Chen ’s joint research is sponsored by the China Scholarship Council and China State Key Science and Technology Projects (2011ZX05023-005-007).APPENDIX APOLYNOMIAL FORM OF PWDIf all the coefficients of B ðZ t Þare polynomials of p ,equation 4is also a polynomial of p ,and the plane-wave destruction equation becomes in turn a polynomial equation of p .The problem is to de-sign a 2N þ1-points filter B ðZ t Þwith polynomial coefficients suchthat the all pass system H ðZ t Þ¼B ð1∕Z t ÞB ðZ t Þcan approximate the phase-shift operator Z p t ¼e j ωp .Denoting the phase response of the system as θðωÞ,that is H ðe j ωÞ¼e j θðωÞ,the group delay of the system isτðωÞ¼∂θðωÞ∂ω:(A-1)The maximally flat criteria designs a filter with a smoothest phase response.There are 2N unknown coefficients in H ðZ t Þ,so we can add 2N flat constraints for the first 2N -th order deviratives of the phase response.It becomes (Zhang,2009,equation 7)τðωÞ¼p ∂n τðωÞ∂ωn ¼0n ¼1;2;:::;2N;(A-2)which is equivalent to the following linear maximally flat condi-tions (Thiran,1971)X N k ¼−Nðd −k Þ2n þ1b k ¼0;(A-3)where n ¼0;1;:::;2N −1and d ¼p ∕2is the fractional delay of B ð1∕Z t Þor 1∕B ðZ t Þ.To solve b k from the above equations,Thiran (1971)used an ad-ditional condition b 0¼1,which leads b k ðk ≠0Þto be a fractional function of p .Differently from that,we use the following condition,X N k ¼−Nb k ¼1;(A-4)where b k can be proved to be polynomials of p .Let vector b ¼½b 0;b N ;:::;b 1;b −1;:::;b −N T .Combining equations A-3and A-4,we rewrite them into the following matrix formTable 1.Performance of different dip estimation methods in 3D seislet transform:Runtime is the runtime of the slope estimation process;RES-inline is the inline residual;RES-xline is the crossline residual;NCC-1is the normalized crosscorrelation between the original data and the reconstruction using 1%of the seislet coefficients;NCC-5uses 5%of the coefficients.The run time for 2D and 3D seislet transform are about 22.45s and 45.0s,respectively.Method N IterationsRuntime (s)RES-inline RES-xline NCC-1NCC-5Noniterative 1054.630.23140.22420.88580.9618Iterative 16334.50.22670.21720.88140.961Iterative25301.60.21940.20750.88380.9615Accelerated plane-wave destruction V7D o w n l o a d e d 09/22/13 t o 221.226.175.186. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /266666411:::11:::1d d −N :::d −1d þ1:::d þN d 3ðd −N Þ3:::ðd −1Þ3ðd þ1Þ3:::ðd þN Þ3......::::::::::::...d 2N −1ðd −N Þ4N −1:::ðd −1Þ4N −1ðd þ1Þ4N −1:::ðd þN Þ2N −13777775×b ¼2666664100 03777775:The matrix on the left side,denoted as V ,can be split into four blocksA B C Das shown above.Following the lemma of matrix inversion,V −1¼ðA −BD −1C Þ−1−ðA −BD −1C Þ−1D −1−D −1C ðA −BD −1C Þ−1D −1þD −1ðA −BD −1C Þ−1BD −1;(A-5)therefore,the coefficientsb ¼V −1½1;0;:::;0 T¼ ðA −BD −1C Þ−1−D −1C ðA −BD −1C Þ−1:(A-6)Let subindex i ¼−N;−N þ1;:::;−1;1;2;:::;N andx i ¼d þi .Submatrix D can be expressed asD ¼EX¼26666641:::11:::1ðd −N Þ2:::ðd −1Þ2ðd þ1Þ2:::ðd −N Þ2ðd −N Þ4:::ðd −1Þ4ðd þ1Þ4:::ðd þN Þ4...:::......:::...ðd −N Þ4N −2:::ðd −1Þ4N −2ðd þ1Þ4N −2:::ðd þN Þ4N −23777775×diag 2666666664x −N ...x −1x 1...x N3777777775;so D −1¼X −1E −1.Denoting U ¼E −1with elements u ij ,j ¼1;2;:::;2N ,as E is a Vandermonde matrix,u ij and Lagrange intepolating polynomials have the following relationshipX 2N j ¼1u ij x 2j −2¼l i ðx Þ;(A-7)where i ¼−N;:::;−1;1;:::;N ,and l i ðx Þis the Lagrange poly-nomial related to the basis d þi ,l i ðx Þ¼Ym ≠i;m ≠0−N ≤m ≤Nx 2−ðd þm Þ2ðd þi Þ2−ðd þm Þ2:(A-8)Substituting the above equation,u ij and x into equation A-7,we can prove equation A-7.It follows that½E −1C i ¼d l i ðd Þ;(A-9)½D −1C i ¼½X −1E −1C i ¼dd þi l iðd Þ;(A-10)withl i ðd Þ¼ð−1Þi þ1N !N !ðN þi Þ!ðN −i Þ!d þi d Y Nm ¼−N 2d þm2d þm þi:(A-11)Thus,A −BD −1C ¼XN i ¼−Nð−1ÞiN !N !ðN þi Þ!ðN −i Þ!Y N m ¼−N p þmp þm þi ¼ð4N Þ!N !N !1Q m ¼N þ1(A-12)and½A −BD −1C −1¼ð2N Þ!ð2N Þ!ð4N Þ!N !N !Y 2Nm ¼N þ1ðm 2−p 2Þ:(A-13)It is the coefficient b 0,a 2N th°polynomial of p .Substituting it into equation A-6,the coefficients at k ¼Æ1;Æ2;:::ÆN are expressed asb k ¼−½D −1C k ½A −BD −1C −1¼ð2N Þ!ð2N Þ!ð4N Þ!ðN þk Þ!ðN −k Þ!Y N −1−km ¼0ðm −2N þp Þ×Y N −1þk m ¼0ðm −2N −p Þ:(A-14)With the additional condition A-4in 2N þ1points approxima-tion,all the coefficients are polynomials of p of 2N th°.Thus,theplane-wave destruction equation 6is proved to be a polynomial equation of 2N th°.REFERENCESBardan,V .,1987,Trace interpolation in seismic data processing:Geo-physical Prospecting,35,343–358,doi:10.1111/j.1365-2478.1987.tb00822.x .Barnes,A.E.,1996,Theory of 2-D complex seismic trace analysis:Geo-physics,61,264–272,doi:10.1190/1.1443947.Claerbout,J.F.,1992,Earth soundings analysis:Processing versus inver-sion:Blackwell Scientific Publications,/sep/prof/index.html ,accessed 10October 2012.V8Chen et al.D o w n l o a d e d 09/22/13 t o 221.226.175.186. R e d i s t r i b u t i o n s u b j e c t t o SE G l i c e n s e o r c o p y r i g h t ; s e e T e r m s o f U s e a t h t t p ://l i b r a r y .s e g .o r g /。
farrow结构的分数延迟滤波器
farrow结构的分数延迟滤波器
"Farrows 结构" 是一种用于实现分数延迟滤波器（Fractional Delay Filter）的一种结构。
分数延迟滤波器是一类用于在信号处理中引入精确和可变延迟的滤波器。
Farrows 结构常常被用于实现这种类型的滤波器。
Farrows 结构的关键特点是能够以分数的精度提供延迟，这对于许多信号处理应用非常重要，例如音频处理、通信系统等。
这个结构通常使用插值技术，例如线性插值，以提供连续的、精确的延迟。
Farrows 结构的基本思想是使用一个滤波器来产生一个带有延迟效果的信号，并通过插值来获得所需的分数延迟。
这个结构一般包含了一组延迟单元和一个插值滤波器。
具体的Farrows 结构的实现可能会涉及到一些数学和信号处理的细节，因此具体的算法和代码可能会有所不同，具体取决于应用的需求和设计。
matlab工具箱安装教程
1.1 如果是Matlab安装光盘上的工具箱，重新执行安装程序，选中即可；1.2 如果是单独下载的工具箱，一般情况下仅需要把新的工具箱解压到某个目录。
2 在matlab的file下面的set path把它加上。
3 把路径加进去后在file→Preferences→General的Toolbox Path Caching里点击update Toolbox Path Cache更新一下。
4 用which newtoolbox_command.m来检验是否可以访问。
如果能够显示新设置的路径，则表明该工具箱可以使用了。
把你的工具箱文件夹放到安装目录中“toolbox”文件夹中，然后单击“file”菜单中的“setpath”命令，打开“setpath”对话框，单击左边的“ADDFolder”命令，然后选择你的那个文件夹，最后单击“SAVE”命令就OK了。
MATLAB Toolboxes＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝/zsmcode.htmlBinaural-modeling software for MATLAB/Windows/home/Michael_Akeroyd/download2.htmlStatistical Parametric Mapping (SPM)/spm/ext/BOOTSTRAP MATLAB TOOLBOX.au/downloads/bootstrap_toolbox.htmlThe DSS package for MATLABDSS Matlab package contains algorithms for performing linear, deflation and symmetric DSS. http://www.cis.hut.fi/projects/dss/package/Psychtoolbox/download.htmlMultisurface Method Tree with MATLAB/~olvi/uwmp/msmt.htmlA Matlab Toolbox for every single topic !/~baum/toolboxes.htmleg. BrainStorm - MEG and EEG data visualization and processingCLAWPACK is a software package designed to compute numerical solutions to hyperbolic partial differential equations using a wave propagation approach/~claw/DIPimage - Image Processing ToolboxPRTools - Pattern Recognition Toolbox (+ Neural Networks)NetLab - Neural Network ToolboxFSTB - Fuzzy Systems ToolboxFusetool - Image Fusion Toolboxhttp://www.metapix.de/toolbox.htmWAVEKIT - Wavelet ToolboxGat - Genetic Algorithm ToolboxTSTOOL is a MATLAB software package for nonlinear time series analysis.TSTOOL can be used for computing: Time-delay reconstruction, Lyapunov exponents, Fractal dimensions, Mutual information, Surrogate data tests, Nearest neighbor statistics, Return times, Poincare sections, Nonlinear predictionhttp://www.physik3.gwdg.de/tstool/MATLAB / Data description toolboxA Matlab toolbox for data description, outlier and novelty detectionMarch 26, 2004 - D.M.J. Taxhttp://www-ict.ewi.tudelft.nl/~davidt/dd_tools/dd_manual.htmlMBEhttp://www.pmarneffei.hku.hk/mbetoolbox/Betabolic network toolbox for Matlabhttp://www.molgen.mpg.de/~lieberme/pages/network_matlab.htmlPharmacokinetics toolbox for Matlabhttp://page.inf.fu-berlin.de/~lieber/seiten/pbpk_toolbox.htmlThe SpiderThe spider is intended to be a complete object orientated environment for machine learning in Matlab. Aside from easy use of base learning algorithms, algorithms can be plugged together and can be compared with, e.g model selection, statistical tests and visual plots. This gives all the power of objects (reusability, plug together, share code) but also all the power of Matlab for machine learning research.http://www.kyb.tuebingen.mpg.de/bs/people/spider/index.htmlSchwarz-Christoffel Toolbox/matlabcentral/fileexchange/loadFile.do?objectId=1316&objectT ype=file#XML Toolbox/matlabcentral/fileexchange/loadFile.do?objectId=4278&object Type=fileFIR/TDNN Toolbox for MATLABBeta version of a toolbox for FIR (Finite Impulse Response) and TD (Time Delay) NeuralNetworks./interval-comp/dagstuhl.03/oish.pdfMisc.http://www.dcsc.tudelft.nl/Research/Software/index.htmlAstronomySaturn and Titan trajectories ... MALTAB astronomy/~abrecht/Matlab-codes/AudioMA Toolbox for Matlab Implementing Similarity Measures for Audiohttp://www.oefai.at/~elias/ma/index.htmlMAD - Matlab Auditory Demonstrations/~martin/MAD/docs/mad.htmMusic Analysis - Toolbox for Matlab : Feature Extraction from Raw Audio Signals for Content-Based Music Retrihttp://www.ai.univie.ac.at/~elias/ma/WarpTB - Matlab Toolbox for Warped DSPBy Aki Härmä and Matti Karjalainenhttp://www.acoustics.hut.fi/software/warp/MATLAB-related Softwarehttp://www.dpmi.tu-graz.ac.at/~schloegl/matlab/Biomedical Signal data formats (EEG machine specific file formats with Matlab import routines)http://www.dpmi.tu-graz.ac.at/~schloegl/matlab/eeg/MPEG Encoding library for MATLAB Movies (Created by David Foti)It enables MATLAB users to read (MPGREAD) or write (MPGWRITE) MPEG movies. That should help Video Quality project.Filter Design packagehttp://www.ee.ryerson.ca:8080/~mzeytin/dfp/index.htmlOctave by Christophe COUVREUR (Generates normalized A-weigthing, C-weighting, octave and one-third-octave digital filters)/matlabcentral/fileexchange/loadFile.do?objectType=file&object Id=69Source Coding MATLAB Toolbox/users/kieffer/programs.htmlBio Medical Informatics (Top)CGH-Plotter: MATLAB Toolbox for CGH-data AnalysisCode: http://sigwww.cs.tut.fi/TICSP/CGH-Plotter/Poster: http://sigwww.cs.tut.fi/TICSP/CSB2003/Posteri_CGH_Plotter.pdfThe Brain Imaging Software Toolboxhttp://www.bic.mni.mcgill.ca/software/MRI Brain Segmentation/matlabcentral/fileexchange/loadFile.do?objectId=4879Chemometrics (providing PCA) (Top)Matlab Molecular Biology & Evolution Toolbox(Toolbox Enables Evolutionary Biologists to Analyze and View DNA and Protein Sequences) James J. 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Chen/support/books/book5295.jsp?category=new&language=-1MATLAB M-files and Simulink models/matlabcentral/fileexchange/loadFile.do?objectId=3834&object Type=file•OPNML/MATLAB Facilities/OPNML_Matlab/Mesh Generation/home/vavasis/qmg-home.htmlOpenFEM : An Open-Source Finite Element Toolbox/CALFEM is an interactive computer program for teaching the finite element method (FEM)http://www.byggmek.lth.se/Calfem/frinfo.htmThe Engineering Vibration Toolbox/people/faculty/jslater/vtoolbox/vtoolbox.htmlSaGA - Spatial and Geometric Analysis Toolboxby Kirill K. Pankratov/~glenn/kirill/saga.htmlMexCDF and NetCDF Toolbox For Matlab-5&6/staffpages/cdenham/public_html/MexCDF/nc4ml5.htmlCUEDSID: Cambridge University System Identification Toolbox/jmm/cuedsid/Kriging Toolbox/software/Geostats_software/MATLAB_KRIGING_TOOLBOX.htmMonte Carlo (Dr Nando)http://www.cs.ubc.ca/~nando/software.htmlRIOTS - The Most Powerful Optimal Control Problem Solver/~adam/RIOTS/ExcelMATLAB xlsheets/matlabcentral/fileexchange/loadFile.do?objectId=4474&objectTy pe=filewrite2excel/matlabcentral/fileexchange/loadFile.do?objectId=4414&objectTy pe=fileFinite Element Modeling (FEM) (Top)OpenFEM - An Open-Source Finite Element Toolbox/NLFET - nonlinear finite element toolbox for MATLAB ( framework for setting up, solving, and interpreting results for nonlinear static and dynamic finite element analysis.)/GetFEM - C++ library for finite element methods elementary computations with a Matlabinterfacehttp://www.gmm.insa-tlse.fr/getfem/FELIPE - FEA package to view results ( contains neat interface to MATLA/~blstmbr/felipe/Finance (Top)A NEW MATLAB-BASED TOOLBOX FOR COMPUTER AIDED DYNAMIC TECHNICAL TRADINGStephanos Papadamou and George StephanidesDepartment of Applied Informatics, University Of Macedonia Economic & Social Sciences, Thessaloniki, Greece/fen31/one_time_articles/dynamic_tech_trade_matlab6.htm Paper: :8089/eps/prog/papers/0201/0201001.pdfCompEcon Toolbox for Matlab/~pfackler/compecon/toolbox.htmlGenetic Algorithms (Top)The Genetic Algorithm Optimization Toolbox (GAOT) for Matlab 5/mirage/GAToolBox/gaot/Genetic Algorithm ToolboxWritten & distributed by Andy Chipperfield (Sheffield University, UK)/uni/projects/gaipp/gatbx.htmlManual: /~gaipp/ga-toolbox/manual.pdfGenetic and Evolutionary Algorithm Toolbox (GEATbx)/Evolutionary Algorithms for MATLAB/links/ea_matlab.htmlGenetic/Evolutionary Algorithms for MATLABhttp://www.systemtechnik.tu-ilmenau.de/~pohlheim/EA_Matlab/ea_matlab.html GraphicsVideoToolbox (C routines for visual psychophysics on Macs by Denis Pelli)/VideoToolbox/Paper: /pelli/pubs/pelli1997videotoolbox.pdf4D toolbox/~daniel/links/matlab/4DToolbox.htmlImages (Top)Eyelink Toolbox/eyelinktoolbox/Paper: /eyelinktoolbox/EyelinkToolbox.pdfCellStats: Automated statistical analysis of color-stained cell images in Matlabhttp://sigwww.cs.tut.fi/TICSP/CellStats/SDC Morphology Toolbox for MATLAB (powerful collection of latest state-of-the-art gray-scale morphological tools that can be applied to image segmentation, non-linear filtering, pattern recognition and image analysis)/Image Acquisition Toolbox/products/imaq/Halftoning Toolbox for MATLAB/~bevans/projects/halftoning/toolbox/index.htmlDIPimage - A Scientific Image Processing Toolbox for MATLABhttp://www.ph.tn.tudelft.nl/DIPlib/dipimage_1.htmlPNM Toolboxhttp://home.online.no/~pjacklam/matlab/software/pnm/index.htmlAnotherICA / KICA and KPCA (Top)ICA TU Toolboxhttp://mole.imm.dtu.dk/toolbox/menu.htmlMISEP Linear and Nonlinear ICA Toolboxhttp://neural.inesc-id.pt/~lba/ica/mitoolbox.htmlKernel Independant Component Analysis/~fbach/kernel-ica/index.htmMatlab: kernel-ica version 1.2KPCA- Please check the software section of kernel machines.KernelStatistical Pattern Recognition Toolboxhttp://cmp.felk.cvut.cz/~xfrancv/stprtool/MATLABArsenal A MATLAB Wrapper for Classification/tmp/MATLABArsenal.htmMarkov (Top)MapHMMBOX 1.1 - Matlab toolbox for Hidden Markov Modelling using Max. Aposteriori EM Prerequisites: Matlab 5.0, Netlab. Last Updated: 18 March 2002./~parg/software/maphmmbox_1_1.tarHMMBOX 4.1 - Matlab toolbox for Hidden Markov Modelling using Variational Bayes Prerequisites: Matlab 5.0,Netlab. Last Updated: 15 February 2002../~parg/software/hmmbox_3_2.tar/~parg/software/hmmbox_4_1.tarMarkov Decision Process (MDP) Toolbox for MatlabKevin Murphy, 1999/~murphyk/Software/MDP/MDP.zipMarkov Decision Process (MDP) Toolbox v1.0 for MATLABhttp://www.inra.fr/bia/T/MDPtoolbox/Hidden Markov Model (HMM) Toolbox for Matlab/~murphyk/Software/HMM/hmm.htmlBayes Net Toolbox for Matlab/~murphyk/Software/BNT/bnt.htmlMedical (Top)EEGLAB Open Source Matlab Toolbox for Physiological Research (formerly ICA/EEG Matlabtoolbox)/~scott/ica.htmlMATLAB Biomedical Signal Processing Toolbox/Toolbox/Powerful package for neurophysiological data analysis ( Igor Kagan webpage)/Matlab/Unitret.htmlEEG / MRI Matlab Toolbox/Microarray data analysis toolbox (MDAT): for normalization, adjustment and analysis of gene expression_r data.Knowlton N, Dozmorov IM, Centola M. Department of Arthritis and Immunology, Oklahoma Medical Research Foundation, Oklahoma City, OK, USA 73104. We introduce a novel Matlab toolbox for microarray data analysis. This toolbox uses normalization based upon a normally distributed background and differential gene expression_r based on 5 statistical measures. The objects in this toolbox are open source and can be implemented to suit your application. AVAILABILITY: MDAT v1.0 is a Matlab toolbox and requires Matlab to run. MDAT is freely available at:/publications/2004/knowlton/MDAT.zipMIDI (Top)MIDI Toolbox version 1.0 (GNU General Public License)http://www.jyu.fi/musica/miditoolbox/Misc. (Top)MATLAB-The Graphing Tool/~abrecht/matlab.html3-D Circuits The Circuit Animation Toolbox for MATLAB/other/3Dcircuits/SendMailhttp://carol.wins.uva.nl/~portegie/matlab/sendmail/Coolplothttp://www.reimeika.ca/marco/matlab/coolplots.htmlMPI (Matlab Parallel Interface)Cornell Multitask Toolbox for MATLAB/Services/Software/CMTM/Beolab Toolbox for v6.5Thomas Abrahamsson (Professor, Chalmers University of Technology, Applied Mechanics,Göteborg, Sweden)http://www.mathworks.nl/matlabcentral/fileexchange/loadFile.do?objectId=1216&objectType =filePARMATLABNeural Networks (Top)SOM Toolboxhttp://www.cis.hut.fi/projects/somtoolbox/Bayes Net Toolbox for Matlab/~murphyk/Software/BNT/bnt.htmlNetLab/netlab/Random Neural Networks/~ahossam/rnnsimv2/ftp: ftp:///pub/contrib/v5/nnet/rnnsimv2/NNSYSID Toolbox (tools for neural network based identification of nonlinear dynamic systems) http://www.iau.dtu.dk/research/control/nnsysid.htmlOceanography (Top)WAFO. Wave Analysis for Fatigue and Oceanographyhttp://www.maths.lth.se/matstat/wafo/ADCP toolbox for MATLAB (USGS, USA)Presented at the Hydroacoustics Workshop in Tampa and at ADCP's in Action in San Diego /operations/stg/pubs/ADCPtoolsSEA-MAT - Matlab Tools for Oceanographic AnalysisA collaborative effort to organize and distribute Matlab tools for the Oceanographic Community /Ocean Toolboxhttp://www.mar.dfo-mpo.gc.ca/science/ocean/epsonde/programming.htmlEUGENE D. GALLAGHER(Associate Professor, Environmental, Coastal & Ocean Sciences)/edgwebp.htmOptimization (Top)MODCONS - a MATLAB Toolbox for Multi-Objective Control System Design/mecheng/jfw/modcons.htmlLazy Learning Packagehttp://iridia.ulb.ac.be/~lazy/SDPT3 version 3.02 -- a MATLAB software for semidefinite-quadratic-linear programming .sg/~mattohkc/sdpt3.htmlMinimum Enclosing Balls: Matlab Code/meb/SOSTOOLS Sum of Squares Optimi zation Toolbox for MATLAB User’s guide/sostools/sostools.pdfPSOt - a Particle Swarm Optimization Toolbox for use with MatlabBy Brian Birge ... A Particle Swarm Optimization Toolbox (PSOt) for use with the Matlab scientific programming environment has been developed. PSO isintroduced briefly and then the use of the toolbox is explained with some examples. A link to downloadable code is provided.Plot/software/plotting/gbplot/Signal Processing (Top)Filter Design with Motorola DSP56Khttp://www.ee.ryerson.ca:8080/~mzeytin/dfp/index.htmlChange Detection and Adaptive Filtering Toolboxhttp://www.sigmoid.se/Signal Processing Toolbox/products/signal/ICA TU Toolboxhttp://mole.imm.dtu.dk/toolbox/menu.htmlTime-Frequency Toolbox for Matlabhttp://crttsn.univ-nantes.fr/~auger/tftb.htmlVoiceBox - Speech Processing Toolbox/hp/staff/dmb/voicebox/voicebox.htmlLeast Squared - Support Vector Machines (LS-SVM)http://www.esat.kuleuven.ac.be/sista/lssvmlab/WaveLab802 : the Wavelet ToolboxBy David Donoho, Mark Reynold Duncan, Xiaoming Huo, Ofer Levi /~wavelab/Time-series Matlab scriptshttp://wise-obs.tau.ac.il/~eran/MATLAB/TimeseriesCon.htmlUvi_Wave Wavelet Toolbox Home Pagehttp://www.gts.tsc.uvigo.es/~wavelets/index.htmlAnotherSupport Vector Machine (Top)MATLAB Support Vector Machine ToolboxDr Gavin CawleySchool of Information Systems, University of East Anglia/~gcc/svm/toolbox/LS-SVM - SISTASVM toolboxes/dmi/svm/LSVM Lagrangian Support Vector Machine/dmi/lsvm/Statistics (Top)Logistic regression/SAGA/software/saga/Multi-Parametric Toolbox (MPT) A tool (not only) for multi-parametric optimization. http://control.ee.ethz.ch/~mpt/ARfit: A Matlab package for the estimation of parameters and eigenmodes of multivariate autoregressive modelshttp://www.mat.univie.ac.at/~neum/software/arfit/The Dimensional Analysis Toolbox for MATLABHome: http://www.sbrs.de/Paper: http://www.isd.uni-stuttgart.de/~brueckner/Papers/similarity2002.pdfFATHOM for Matlab/personal/djones/PLS-toolbox/Multivariate analysis toolbox (N-way Toolbox - paper)http://www.models.kvl.dk/source/nwaytoolbox/index.aspClassification Toolbox for Matlabhttp://tiger.technion.ac.il/~eladyt/classification/index.htmMatlab toolbox for Robust Calibrationhttp://www.wis.kuleuven.ac.be/stat/robust/toolbox.htmlStatistical Parametric Mapping/spm/spm2.htmlEVIM: A Software Package for Extreme Value Analysis in Matlabby Ramazan Gençay, Faruk Selcuk and Abdurrahman Ulugulyagci, 2001.Manual (pdf file) evim.pdf - Software (zip file) evim.zipTime Series Analysishttp://www.dpmi.tu-graz.ac.at/~schloegl/matlab/tsa/Bayes Net Toolbox for MatlabWritten by Kevin Murphy/~murphyk/Software/BNT/bnt.htmlOther: /information/toolboxes.htmlARfit: A Matlab package for the estimation of parameters and eigenmodes of multivariate autoregressive models/~tapio/arfit/M-Fithttp://www.ill.fr/tas/matlab/doc/mfit4/mfit.htmlDimensional Analysis Toolbox for Matlab/The NaN-toolbox: A statistic-toolbox for Octave and Matlab®... handles data with and without MISSING VALUES.http://www-dpmi.tu-graz.ac.at/~schloegl/matlab/NaN/Iterative Methods for Optimization: Matlab Codes/~ctk/matlab_darts.htmlMultiscale Shape Analysis (MSA) Matlab Toolbox 2000p.br/~cesar/projects/multiscale/Multivariate Ecological & Oceanographic Data Analysis (FATHOM)From David Jones/personal/djones/glmlab (Generalized Linear Models in 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分数时延滤波器设计
Abstract
Abstract
The fractional delay filter is a kind of digital filter whose sampling interval of delay is the fractional form. Generally, the way of design for the fractional delay filter is to approach the characteristics of the ideal fractional delay filter by scheme of designing suitable filter, and this filter is the expected fractional delay filter of the input signal. This paper will mainly introduce two kinds of design methods. The first kind is to design the fractional delay filter using the FIR digital filter, including that the methods of delay parameters is fixed, such as the least squared design method, Lagrange interpolation design method, equiripple design method and the methods of delay parameters is variable, such as Fourier Transform method and Farrow Structure method. Another kind is to design the fractional delay filter using the all-pass digital filter, including that the methods of delay parameters is fixed, such as the least squared design method, equiripple design method and the methods of delay parameters is variable, such as the Matrix Transform Method and Farrow Structure method. At the end of this paper, with the fractional delay filter for the wideband digital array beamforming of the wideband phased array radar for application cases. It will try to use two methods (multiphase structure method and FARROW structure method) to achieve the fractional delay filter, and these two kinds of methods also will be compared and summarized.
基于Farrow结构的宽带信号分数延时滤波器的FPGA设计与测试
火 控雷达技术
Ｆｉｒｅ Ｃｏｎｔｒｏｌ Ｒａｄａｒ Ｔｅｃｈｎｏｌｏｇｙ
Ｖｏ１．４５ Ｎｏ．２（Ｓｅｒｉｅｓ １７６）
Ｊｕｎ． ２０１６
z z z PsgiolePfrp
基 于 Ｆａｒｒｏｗ结 构 的宽 带 信 号 分 数 延 时 滤 波 器 的 ＦＰＧＡ设 计 与测 试
０ 引 言
宽带 相控 阵雷 达是 目前 雷达 技术 发展 的重 要方 向之一 。宽带信 号 可 以提 高 相控 阵雷 达 的抗 干 扰能 力 ，具 有更 高 的测 距 精度 及分 辨率 ，有 利于 目标 的成 像 、识别 与 分类 。但 传 统 的 相 位 控 制 方 法 由于 孑Ｌ径渡 越 问题会 导 致 波 束 的指 向偏 移 及 扫 描 不 准 ， 需要 采用 真 实 时 间延 时 线 （Ｔｒｕｅ Ｔｉｍｅ Ｄｅｌａｙ，ＴＴＤ） 来代 替移 相 器 。而 模 拟 延 迟 线 的成 本 高 、体 积 庞大 、功 耗 大 、稳 定 性 差 ，不 利 于 阵 列 雷 达 的应 用 。 数字 波 束 形 成 （Ｄｉｇｉｔａｌ Ｂｅａｍ Ｆｏｒｍｉｎｇ，ＤＢＦ）技 术 是
实现延 时补 偿具 有 成本低 、精 度 高、实现 简单 、稳 定性 好 等 众 多优 点 。 而基 于 Ｆａｒｒｏｗ结构 的 分数 延
时滤波 器 结构 简单 、延 时变化 灵活 快捷 ，且 易于构 建 宽带信 号 的分数 延 时滤 波 器。在 ＦＰＧＡ 系统 中
实现 宽带信 号 的分数 延 时 ，并 对其 延 时性 能 的 测 试 可 为将 来 宽 带信 号 的 ＤＢＦ工程 应 用 提 供技 术
基 础 。
关 键 词 ：数 字 波 束 形 成 ；Ｆａｒｒｏｗ结 构 ；分数 延 时
基于拉格朗日插值的分数延时滤波器研究
舰船电子对抗
SHIPBOARD ELECTRONIC COUNTERMEASURE
Oct.2018
Vol.41 No.5
基于拉格朗日插值的分数延时滤波器研究
黄 伟,周其超,陶存炳
(中国船舶重工集团公司第七二三研究所,江苏 扬州 225101)
摘要:分数延时滤波器可以在数字域实现对信号小数倍采样间隔时延,比传统模拟延迟线稳定、精度高,在数字 波 束
形成中应用广泛。基于拉格朗日插值的分 数 延 时 滤 波 器 实 现 简 单,低 频 具 有 很 好 的 幅 度 响 应 和 相 位 延 时 响 应。 介
绍了基于拉格朗日插值的分数延时滤波器的实现原理,仿真分析了 滤 波 器 的 幅 频 特 性 和 相 频 特 性,仿 真 结 果 表 明 拉
格朗日插值的分数延时滤波器用于信号的延时处理时具有良好的延时效果。
ResearchintoTheFractionalDelayFilterBasedonLagrangeInterpolation
HUANG Wei,ZHOU Qi-chao,TAO Cun-bing
(The723InstituteofCSIC,Yangzhou225101,China)
Abstract:Thefractionaldelayfiltercanrealizethefractionaltimessamplingintervaldelayofsignal indigitaldomain,whichismorestableandmoreaccuratethanthetraditionalanalogdelayline,and iswidelyusedindigitalbeamforming.ThefractionaldelayfilterbasedonLagrangeinterpolationis simpleandhasagoodamplituderesponseandphasedelayresponseinlowfrequency.ThispaperintroducestherealizationprincipleoffractionaldelayfilterbasedonLagrangeinterpolationmethod, simulatesandanalyzestheamplitude-frequencypropertyandthephase-frequencypropertyofthe filter.SimulationresultsshowthatthefractionaldelayfilterbasedonLagrangeinterpolationmethodhasagoodeffectintheapplicationofsignaldelayprocessing. Keywords:fractionaldelay;digitalbeamforming;digitalfilter;Lagrangeinterpolation method
DSP System Toolbox 80
Acoustic noise cancellation algorithm using System objects in MATLAB (above left). Filter coefficients can be plotted to display their values before adaptation (top right) and after adaptation (bottom right).DSP Algorithms for System Design and PrototypingDSP System Toolbox lets you mathematically model the behavior of your system and then simulate the model to accurately predict and optimize system performance. Using the system toolbox, you can simulate digital systems in MATLAB and Simulink. When you use the system toolbox in Simulink, you can also model advanced systems such as mixed-signal and multidomain systems.Algorithms in DSP System Toolbox serve as building blocks of signal processing systems in communications, audio, speech, RADAR, control systems, image and video processing, medical, and industrial applications. Algorithm Libraries for DSPAll algorithms in the system toolbox—whether implemented as MATLAB functions, MATLAB System objects or Simulink blocks—support double-precision and single-precision floating-point data types. Most also support integer and fixed-point data types (requires Fixed-Point Toolbox™or Simulink Fixed Point™).Algorithm categories in the system toolbox include:▪Signal operations such as convolution, windowing, padding, modeling delays, peak finding, and zero-crossing ▪Signal transforms such as fast Fourier transform (FFT), discrete cosine transform (DCT), short-time Fourier transform, and discrete wavelet transform (DWT)▪Filter design and implementation methods for digital FIR and IIR filters▪Statistical signal processing tools for signal analysis and spectral estimation▪Signal management methods such as buffering, indexing, switching, stacking, and queuing▪Linear algebra routines, including linear system solvers, matrix factorizations, and matrix inverses▪Scalar and vector quantizer encoding and decodingPartial list of System objects available in MATLAB (top) and category view of blocks available in Simulink (middle), with expanded views of the Signal Processing Sources and Transforms block libraries (bottom).Modeling Multirate SystemsDSP System Toolbox supports multirate processing for sample rate conversion and the modeling of systems in which different sample or clock rates need to be interfaced. Multirate functionality includes multirate filters and signal operations such as upsampling, downsampling, interpolation, decimation, and resampling.Sigma-delta A/D converter model in Simulink showing signals operating at multiple sample rates.Variable-Size SignalsDSP System Toolbox supports signal inputs that can change in size and value at run time. A subset of System objects and Simulink blocks provide support for variable-size signals that change size during the simulation or during distinct mode-switching events that occur in the initialization of conditionally executed subsystems. Support for variable-size signals enables you to model systems with varying resources, constraints, and environments.Adaptive, Multirate, and Specialized Filter Design MethodsDSP System Toolbox provides many methods for designing and implementing digital filters. You can design filters with lowpass, highpass, bandpass, bandstop, and other response types and realize them using filter structures such as direct-form FIR, overlap-add FIR, direct-form II with second-order sections, cascade allpass, and lattice structures.You can design filters in several ways:at the MATLAB command line, interactively using FDA Tool or Filterbuilder, or in Simulink using the filter design block library.The system toolbox supports a number of design methods, including:Advanced equiripple FIR filters,including minimum-order, constrained-ripple, minimum-phase designs Nyquist and halfband FIR and IIR filters,providing linear phase, minimum-phase, and quasi-linear phase (IIR) designs, as well as equiripple, sloped-stopband, and window methodsOptimized multistage designs,enabling you to optimize the number of cascaded stages to achieve the lowest computational complexityFractional-delay filters,including implementation using Farrow filter structures well-suited for tunable filteringapplicationsAllpass IIR filters with arbitrary group delay, enabling you to compensate for the group delays of other IIR filters to obtain an approximate linear phase passband responseLattice wave digital IIR filters,for robust fixed-point implementationArbitrary magnitude and phase FIR and IIR filters,enabling design of any filter specificationSpecialized filter designs in MATLAB showing LMS adaptive filter applied to a noisy music signal (top left), arbitrary magnitude filter design (top right), direct-form FIR filter responses for fixed-point data types (bottom left), and octave filter design (bottom right).Adaptive FiltersDSP System Toolbox provides several techniques for the design of adaptive filters: LMS-based, RLS-based, affine projection, fast transversal, frequency-domain, and lattice-based. The system toolbox also includes algorithms for the analysis of these filters, including tracking of coefficients, learning curves, and convergence.Multirate FiltersDSP System Toolbox provides functions for the design and implementation of multirate filters, including polyphase interpolators, decimators, sample-rate converters, and CIC filters and compensators, as well as support for multistage design methods. The system toolbox also provides specialized analysis functions to estimate the computational complexity of multirate filters.Interactive design of a lowpass filter in the Filterbuilder tool (left) and visualization of magnitude response (right).Specialized Filters for DSP ApplicationsDSP System Toolbox lets you design and implement specialized digital filters, including:▪Audio weighting filters, octave filters, and parametric equalizer filters for audio, speech, and acoustic applications▪Pulse shaping, peak or notch, and multirate filters for communications systems▪Kalman filters for aerospace and navigation systemsUsing Filters in Simulink System ModelsThe digital filters you design in DSP System Toolbox can also be used in system-level models in Simulink. MATLAB functions and System objects enable you to generate bit-true Simulink models from MATLAB filter designs. You can also use filter design block libraries in DSP System Toolbox to design, simulate, and implement filters directly in Simulink.Streaming and Frame-Based Signal ProcessingDSP System Toolbox enables the efficient simulation of real-time signal processing systems by supporting streaming signal processing and frame-based processing in MATLAB and Simulink.Streaming and frame-based processing techniques accelerate simulations by buffering input data into frames and processing multiple samples of data at a time. Faster simulations are achieved due to the distribution of the fixed process overhead across many samples. Although these techniques introduce a certain amount of latency in the system, in many instances you can select frame sizes that improve throughput without creating unacceptable latencies.In MATLAB, streaming signal processing is enabled by using System objects to represent data-driven algorithms, sources, and sinks. System objects implicitly manage many details of stream processing, such as data indexing,buffering, and algorithm state management. You can mix System objects with standard MATLAB functions andoperators. MATLAB programs that use System objects can be incorporated into Simulink models via the MATLAB Function block. Most System objects have corresponding Simulink blocks with the same capabilities.In Simulink, DSP System Toolbox blocks process input signals as frames when the specified input processing mode on the block dialog is set to frame-based processing. DSP System Toolbox supports sample-based processing for low latency processes and for applications that require scalar processing. Many blocks support both sample-based and frame-based processing modes.Envelope detection algorithm illustrating stream processing in MATLAB with System objects. Simulation results are shown for both the Hilbert transform and amplitude modulation methods of envelope detection.Signal Generation, I/O, and VisualizationGenerating and Importing SignalsThe signals that you work with can be acquired from a variety of sources. You can:▪Import audio signals from multimedia files▪Record audio data from soundcards▪Acquire multichannel audio data in real time▪Receive UDP packets from an IP network portSimulation results can be exported to audio files, audio devices, or transmitted as UDP packets over an IP network.You can also generate binary signals, random signals, and common waveforms such as sine waves and chirp signals using functions in DSP System Toolbox.Visualizing Signals in Time and Frequency DomainsDSP System Toolbox provides several tools for time-domain and frequency-domain visualization: Time Scope, Spectrum Scope, Vector Scope, and Waterfall Scope.Visualizing time-domain signals in the Time Scope tool. Simulation controls enable starting, pausing and stopping simulations from within the Time Scope.The Time Scope displays signals in the time-domain and supports a variety of signals—continuous and discrete, fixed-size and variable-size, floating and fixed-point data, and N-dimensional signals. You can also display multiple signals on the same axes, where each input signal has different dimensions, sample rates, and data types. Simulation controls on the Time Scope let you start, pause, continue, take a snapshot, or stop the simulation without having to switch windows.The Spectrum Scope estimates the spectrum of a time-domain input signal and displays its frequency spectrum on a linear or log scale. Scope parameters enable you to specify FFT length, buffer size and overlap, and spectrum units.The Vector Scope is a comprehensive display tool similar to a digital oscilloscope. You can use it to plot consecutive time samples from a vector or to plot vectors containing data such as filter coefficients or spectral magnitudes.The Waterfall Scope displays multiple vectors of data at one time, where each vector represents the input data at consecutive sample times. This tool only displays real-valued, double-precision data.Fixed-Point Implementation and Code Generation for DSP System ModelsYou can use DSP System Toolbox with Fixed-Point Toolbox or Simulink Fixed Point to model fixed-point signal processing algorithms and analyze the effects of quantization on system behavior and performance.Fixed-point support in the system toolbox includes:▪Word sizes from 1 to 128 bits▪Overflow handing and rounding methods▪Logging overflows, maxima, and minima of internal variables▪Manual or automatic scaling▪Data type override options to control system-level data type settingsFixed-Point Modeling and SimulationYou can configure System objects and blocks in the system toolbox for fixed-Point modes of operation, enabling you to perform design tradeoff analyses by running simulations with different word lengths, scaling, overflow handling, and rounding method choices before you commit to hardware.Fixed-point modes are supported for several DSP algorithms, including:▪FFT, DCT, IFFT, IDCT, and other signal transforms▪Digital Filter, Biquad Filter, LMS Filter, and other filter implementations▪Mean, Variance, Autocorrelation, Histogram, and other statistics▪Levinson-Durbin, Forward Substitution, Backward Substitution, and other linear system solvers▪Matrix Multiply, Matrix Product, Matrix Sum, Matrix 1-Norm, and other matrix operations▪Cumulative Product, Cumulative Sum, Difference, Normalization, and other math operationsIn Simulink, DSP System Toolbox automates the configuration of blocks for fixed-point operation. For example:▪Accumulator and multiplier sizes are specified to ensure compatibility for specific hardware targets.▪Binary point of a filter’s coefficient is automatically located based on user-defined word length, precision, and actual values.▪Product output retains all bits in the products between filter coefficients and input values.▪Accumulator is configured to avoid overflows.Block dialog for FFT block in DSP System Toolbox. The dialog provides options for fixed-point data type specification of accumulator, product, and output signals (requires Simulink Fixed-Point).Fixed-Point Filter DesignFilter design functions in DSP System Toolbox enable you to design floating-point filters that can be easily converted to fixed-point data types with Fixed-Point Toolbox. This design flow simplifies the design of fixed-point filters and lets you easily analyze quantization effects.Generating C and HDL CodeUsing DSP System Toolbox with MATLAB Coder and Simulink Coder, you can generate C code from your algorithms and system models. The generated code can be used for verification, rapid prototyping, and implementation of your system during the product development process.Using DSP System Toolbox with Filter Design HDL Coder™, you can generate HDL code from digital filter designs. In Simulink, DSP System Toolbox blocks provide support for HDL code generation when used withSimulink HDL Coder™.Product Details, Demos, and System Requirements/products/dsp-systemTrial Software/trialrequestSales/contactsalesTechnical Support/support ResourcesOnline User Community /matlabcentral Training Services /training Third-Party Products and Services /connections Worldwide Contacts /contact© 2011 The MathWorks, Inc. MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See /trademarksfor a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders.11。
Matlab的第三方工具箱大全
Matlab的第三方工具箱大全(按住CTRL点击连接就可以到达每个工具箱的主页面来下载了)Matlab Toolboxes∙ADCPtools - acoustic doppler current profiler data processing∙AFDesign - designing analog and digital filters∙AIRES - automatic integration of reusable embedded software∙Air-Sea - air-sea flux estimates in oceanography∙Animation - developing scientific animations∙ARfit - estimation of parameters and eigenmodes of multivariate autoregressive methods∙ARMASA - power spectrum estimation∙AR-Toolkit - computer vision tracking∙Auditory - auditory models∙b4m - interval arithmetic∙Bayes Net - inference and learning for directed graphical models∙Binaural Modeling - calculating binaural cross-correlograms of sound∙Bode Step - design of control systems with maximized feedback∙Bootstrap - for resampling, hypothesis testing and confidence interval estimation ∙BrainStorm - MEG and EEG data visualization and processing∙BSTEX - equation viewer∙CALFEM - interactive program for teaching the finite element method∙Calibr - for calibrating CCD cameras∙Camera Calibration∙Captain - non-stationary time series analysis and forecasting∙CHMMBOX - for coupled hidden Markov modeling using max imum likelihood EM ∙Classification - supervised and unsupervised classification algorithms∙CLOSID∙Cluster - for analysis of Gaussian mixture models for data set clustering∙Clustering - cluster analysis∙ClusterPack - cluster analysis∙COLEA - speech analysis∙CompEcon - solving problems in economics and finance∙Complex - for estimating temporal and spatial signal complexities∙Computational Statistics∙Coral - seismic waveform analysis∙DACE - kriging approximations to computer models∙DAIHM - data assimilation in hydrological and hydrodynamic models∙Data Visualization∙DBT - radar array processing∙DDE-BIFTOOL - bifurcation analysis of delay differential equations∙Denoise - for removing noise from signals∙DiffMan - solv ing differential equations on manifolds∙Dimensional Analysis -∙DIPimage - scientific image processing∙Direct - Laplace transform inversion via the direct integration method∙DirectSD - analysis and design of computer controlled systems with process-oriented models∙DMsuite - differentiation matrix suite∙DMTTEQ - design and test time domain equalizer design methods∙DrawFilt - drawing digital and analog filters∙DSFWAV - spline interpolation with Dean wave solutions∙DWT - discrete wavelet transforms∙EasyKrig∙Econometrics∙EEGLAB∙EigTool - graphical tool for nonsymmetric eigenproblems∙EMSC - separating light scattering and absorbance by extended multiplicative signal correction∙Engineering Vibration∙FastICA - fixed-point algorithm for ICA and projection pursuit∙FDC - flight dynamics and control∙FDtools - fractional delay filter design∙FlexICA - for independent components analysis∙FMBPC - fuzzy model-based predictive control∙ForWaRD - Fourier-wavelet regularized deconvolution∙FracLab - fractal analysis for signal processing∙FSBOX - stepwise forward and backward selection of features using linear regression∙GABLE - geometric algebra tutorial∙GAOT - genetic algorithm optimization∙Garch - estimating and diagnosing heteroskedasticity in time series models∙GCE Data - managing, analyzing and displaying data and metadata stored using the GCE data structure specification∙GCSV - growing cell structure visualization∙GEMANOVA - fitting multilinear ANOVA models∙Genetic Algorithm∙Geodetic - geodetic calculations∙GHSOM - growing hierarchical self-organizing map∙glmlab - general linear models∙GPIB - wrapper for GPIB library from National Instrument∙GTM - generative topographic mapping, a model for density modeling and data visualization∙GVF - gradient vector flow for finding 3-D object boundaries∙HFRadarmap - converts HF radar data from radial current vectors to total vectors ∙HFRC - importing, processing and manipulating HF radar data∙Hilbert - Hilbert transform by the rational eigenfunction expansion method∙HMM - hidden Markov models∙HMMBOX - for hidden Markov modeling using maximum likelihood EM∙HUTear - auditory modeling∙ICALAB - signal and image processing using ICA and higher order statistics∙Imputation - analysis of incomplete datasets∙IPEM - perception based musical analysisJMatLink - Matlab Java classesKalman - Bayesian Kalman filterKalman Filter - filtering, smoothing and parameter estimation (using EM) for linear dynamical systemsKALMTOOL - state estimation of nonlinear systemsKautz - Kautz filter designKrigingLDestimate - estimation of scaling exponentsLDPC - low density parity check codesLISQ - wavelet lifting scheme on quincunx gridsLKER - Laguerre kernel estimation toolLMAM-OLMAM - Levenberg Marquardt with Adaptive Momentum algorithm for training feedforward neural networksLow-Field NMR - for exponential fitting, phase correction of quadrature data and slicing LPSVM - Newton method for LP support vector machine for machine learning problems LSDPTOOL - robust control system design using the loop shaping design procedure LS-SVMlabLSVM - Lagrangian support vector machine for machine learning problemsLyngby - functional neuroimagingMARBOX - for multivariate autogressive modeling and cross-spectral estimation MatArray - analysis of microarray dataMatrix Computation- constructing test matrices, computing matrix factorizations, visualizing matrices, and direct search optimizationMCAT - Monte Carlo analysisMDP - Markov decision processesMESHPART - graph and mesh partioning methodsMILES - maximum likelihood fitting using ordinary least squares algorithmsMIMO - multidimensional code synthesisMissing - functions for handling missing data valuesM_Map - geographic mapping toolsMODCONS - multi-objective control system designMOEA - multi-objective evolutionary algorithmsMS - estimation of multiscaling exponentsMultiblock - analysis and regression on several data blocks simultaneously Multiscale Shape AnalysisMusic Analysis - feature extraction from raw audio signals for content-based music retrievalMWM - multifractal wavelet modelNetCDFNetlab - neural network algorithmsNiDAQ - data acquisition using the NiDAQ libraryNEDM - nonlinear economic dynamic modelsNMM - numerical methods in Matlab textNNCTRL - design and simulation of control systems based on neural networks NNSYSID - neural net based identification of nonlinear dynamic systemsNSVM - newton support vector machine for solv ing machine learning problems NURBS - non-uniform rational B-splinesN-way - analysis of multiway data with multilinear modelsOpenFEM - finite element developmentPCNN - pulse coupled neural networksPeruna - signal processing and analysisPhiVis- probabilistic hierarchical interactive visualization, i.e. functions for visual analysis of multivariate continuous dataPlanar Manipulator - simulation of n-DOF planar manipulatorsPRT ools - pattern recognitionpsignifit - testing hyptheses about psychometric functionsPSVM - proximal support vector machine for solving machine learning problems Psychophysics - vision researchPyrTools - multi-scale image processingRBF - radial basis function neural networksRBN - simulation of synchronous and asynchronous random boolean networks ReBEL - sigma-point Kalman filtersRegression - basic multivariate data analysis and regressionRegularization ToolsRegularization Tools XPRestore ToolsRobot - robotics functions, e.g. kinematics, dynamics and trajectory generation Robust Calibration - robust calibration in statsRRMT - rainfall-runoff modellingSAM - structure and motionSchwarz-Christoffel - computation of conformal maps to polygonally bounded regions SDH - smoothed data histogramSeaGrid - orthogonal grid makerSEA-MAT - oceanographic analysisSLS - sparse least squaresSolvOpt - solver for local optimization problemsSOM - self-organizing mapSOSTOOLS - solving sums of squares (SOS) optimization problemsSpatial and Geometric AnalysisSpatial RegressionSpatial StatisticsSpectral MethodsSPM - statistical parametric mappingSSVM - smooth support vector machine for solving machine learning problems STATBAG - for linear regression, feature selection, generation of data, and significance testingStatBox - statistical routinesStatistical Pattern Recognition - pattern recognition methodsStixbox - statisticsSVM - implements support vector machinesSVM ClassifierSymbolic Robot DynamicsTEMPLAR - wavelet-based template learning and pattern classificationTextClust - model-based document clusteringTextureSynth - analyzing and synthesizing visual texturesTfMin - continous 3-D minimum time orbit transfer around EarthTime-Frequency - analyzing non-stationary signals using time-frequency distributions Tree-Ring - tasks in tree-ring analysisTSA - uni- and multivariate, stationary and non-stationary time series analysisTSTOOL - nonlinear time series analysisT_Tide - harmonic analysis of tidesUTVtools - computing and modifying rank-revealing URV and UTV decompositions Uvi_Wave - wavelet analysisvarimax - orthogonal rotation of EOFsVBHMM - variation Bayesian hidden Markov modelsVBMFA - variational Bayesian mixtures of factor analyzersVMT- VRML Molecule Toolbox, for animating results from molecular dynamics experimentsVOICEBOXVRMLplot - generates interactive VRML 2.0 graphs and animationsVSVtools - computing and modifying symmetric rank-revealing decompositions WAFO - wave analysis for fatique and oceanographyWarpTB - frequency-warped signal processingWAVEKIT - wavelet analysisWaveLab - wavelet analysisWeeks - Laplace transform inversion via the Weeks methodWetCDF - NetCDF interfaceWHMT - wavelet-domain hidden Markov tree modelsWInHD - Wavelet-based inverse halftoning via deconvolutionWSCT - weighted sequences clustering toolkitXMLTree - XML parserYAADA - analyze single particle mass spectrum dataZMAP - quantitative seismicity analysis。
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1 Àkn ; 0rnrN À 1 X ðkÞWN N k¼0
ð3Þ
à Corresponding author.
E-mail addresses: tcc@.tw (C.-C. Tseng), lilee@.tw (S.-L. Lee). 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.10.016
1. Introduction In many signal processing applications, there is a need for a delay which is a fraction of the sampling period. These applications include beam steering of antenna arrays, time adjustment in digital receivers, modeling of music instruments, speech coding and synthesis, comb filter design and analog digital conversion, etc. [1–6]. An excellent survey of fractional delay filter design is presented in tutorial paper [1]. Recently, the relationships among fractional delay, differentiator, Nyquist filter, lowpass filter and diamond-shaped filter have been established such that fractional delay becomes a versatile building block in the design of these practical filters [7–9]. The ideal frequency response of a fractional delay filter is given by DðoÞ ¼ e
abstract
In this paper, the design of fractional delay FIR filter is investigated. First, the interpolation formula of a discrete-time sequence is derived by using discrete Fourier transform (DFT). Then, the DFT-based interpolation formula is applied to design fractional delay FIR filter by using suitable index mapping. The filter coefficients are easily computed because a closed-form design is obtained. Next, design examples are demonstrated to show that the proposed DFT method has a smaller design error than those of the conventional Lagrange and window fractional delay FIR filters when using the same design parameters. Finally, the designed DFT-based fractional delay FIR filter is applied to design a digital differentiator and half-band filter. & 2009 Elsevier B.V. All rights reserved.
ÀjoðIþdÞ
where I is a positive integer and d is a fractional number in the interval [0,1). The transfer function of the FIR filter of length N used to approximate this specification is given by HðzÞ ¼
N À1 X r ¼0
hðr ÞzÀr
ð2Þ
So far, several methods of designing an FIR filter H(z) to fit fractional delay specification DðoÞ as closely as possible have been developed. Two typical approaches are Lagrange interpolation method and window method [1]. On the other hand, the discrete Fourier transform (DFT) is defined as follows: X ðkÞ ¼
0
in the time domain, so we should be able to use zero padding in the high frequency range of the DFT domain as a means of interpolating finite-duration, discrete-time signals. The details can be found in the literature [11–17]. So far, the DFT interpolation and fractional delay filter design are two independent research topics in signal processing area. The relation between them has not been investigated. The purpose of this paper is to study this relation such that DFT interpolation concept can be directly used to design fractional delay FIR filters. The main advantage is that the filter coefficients h(r) are easily computed because a closed-form design is obtained. This paper is organized as follows. In Section 2, the interpolation formulas for odd and even length N are derived. In Section 3, the interpolation formula is applied to design fractional delay FIR filter. There are two kinds of fractional delay filters to be designed. One concerns the case of even-length N, the other is the case of odd-length N. In Section 4, numerical examples are included in order to compare the filters designed using the proposed DFT method with conventional Lagrange and window fractional delay FIR filters. In Section 5, the designed DFTbased fractional delay FIR filter is also applied to design a digital differentiator and half-band filter. Finally, a conclusion is made. 2. DFT interpolation formula In this section, the DFT interpolation formula with even-length sequence is first derived. Then, the oddlength case is discussed. Given DFT X(k) of an even-length real-valued sequence x(n), let us define the zero-padded DFT as 8 N > > 0rkr À 1 > LX ðkÞ; > 2 > > > > 1 N N > > > LX ð Þ; k¼ > > 2 2 2 > > < N N þ 1rkrM À À 1 0 ; Xd ðkÞ ¼ ð4Þ 2 2 > > > > 1 N N > > > LX ð Þ; k¼MÀ > > 2 2 2 > > > > N > > þ 1rkrM À 1 LX ð k À M þ N Þ ; M À : 2 Also, we assume M is an integer multiple of N, say M=LN, where L is called interpolation factor. The above DFT has zero values in the high frequency range and satisfies the following conjugate symmetry condition: Xd ðM À kÞ ¼ Xd ðkÞÃ ; N 1rkr 2 ð5Þ