COMPARISON OF VARIOUS OPTIMIZATION TECHNIQUES FOR DESIGN FIR DIGITAL FILTERS

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1 NCCI 1 -National Conference on Comutational Instrumentation CSIO Chandigarh, INDIA, 19- March 1 COMPARISON OF VARIOUS OPIMIZAION ECHNIQUES FOR DESIGN FIR DIGIAL FILERS Amanjeet Panghal 1, Nitin Mittal,Devender Pal Singh 3 R.S. Chauhan 4,Sandee K.Arya 5. 1, Electronics & Comm. Dett., MIE, Mohri 3 Central Scientific Instruments Org., Chandigarh 4 Electronics & Comm. Dett., JMI, Radaur. 5 Electronics & Comm.Dett.,GJUS&, Hisar. eced_miet@yahoo.com Abstract- In this aer different method for design of FIR Filters with low grou delay is roosed. he design method is based on Parks McClellan algorithm and Sequential Otimization. In sequential otimization, Newton and Quasi-Newton method is used. In this aer we comare the various otimization techniques and the sequential otimization erforms better in comarison to Parks McClellan algorithm. he reason to formulate and solve the design roblem in a sequential otimization formulation is that the comlementarily conditions associated with the sequential otimization lead to a very small number of non-zero multiliers that need to be udated in a given iteration. his in turn imroves design efficiency as well as the algorithm s numerical stability which is of critical imortance for the design filter. Design examles with comarisons are resented to illustrate the effectiveness of the sequential otimization method. Keywords: FIR, Parks McClellan Algorithm, Newton Method, Sequential Otimization. 1. INRODUCION Over the ast several decades the field of Digital Signal Processing (DSP) has grown to imortant both theoretically and technologically. In DSP, there is two imortant tyes of Systems. he first tye of systems erforms signal filtering in time domain and hence it is known as DIGIAL FILERS. he second tye of systems rovide signal reresentation frequency domain and are known as Sectrum Analyzer. Digital filtering is one of the most owerful tools of DSP. Digital filters are caable of erformance secifications that would, at best, be extremely difficult, if not imossible, to achieve with an analog imlementation. In addition, the characteristics of a digital filter can be easily changed under software control. Digital filters[1,] are classified either as Finite duration unit ulse resonse (FIR) filters or Infinite duration unit ulse resonse (IIR) filters, deending on the form of unit ulse resonse of the system. In the FIR system, the imulse resonse sequence is of finite duration, i.e., it has a finite number of non zero terms. Digital filters are classified as Recursive and Non- Recursive filters. he resonse of Recursive or FIR filters deend only uon Present and revious inut of signal. FIR filters have the following advantages:- hey can have an exact linear hase. hey are always stable. he design methods are generally linear. hey can be realized efficiently in hardware. he filter start u transients has finite duration. Digital filters are integral arts of many digital signal rocessing systems, including control systems, systems for audio and video rocessing, communication systems and systems for medical alications. Due to the increasing number of alications involving digital signal rocessing and digital filtering the variety of requirements that have to be met by digital filters has increased as well. Consequently, there is a need for flexible techniques that can design digital filters satisfying sohisticated secifications. his aer resents methods for the sequential otimization design of digital filters. he Parks-McClellan algorithm and its variant have been most efficient tools for the minimax design of FIR digital filters [].However,these algorithms aly only to the class of linear-hase FIR filters the grou delay introduce by these filters is constant and indeendent of frequency in the entire baseband but it can be quite large. In ractice, a variable grou delay in stoband is of little concern and by allowing the hase resonse to be nonlinear in stobands, FIR filter can be designed with constant grou delay with resect to the assbands which is significantly reduced relative to that achieved with filters that have a constant grou delay the entire baseband. his aer resents a least- th aroach to the design of low delay FIR filters. For FIR filter, the weighted L error function with an even integer can be shown to be globally convex. his roerty, in conjunction with the availability of the gradient and hessian of the objective function in closed form, enable us to develo an otimization method for the design of FIR filter. 177

2 NCCI 1 -National Conference on Comutational Instrumentation CSIO Chandigarh, INDIA, 19- March 1. PROBLEM FORMULAION Consider a FIR digital filter with transfer function N n H( z) = hz (1) n= Such that the weighted L aroximation error n 1 ( ) ( ) ( ) ( ) j fh= W He H d d () is minimized, where W() is weighting function is a ositive integer, and [ 1... h= hh h N ] if it is assumed. H ( ) = H ( ) jh ( ) d dr di [ N ] [ N ] c( ) = 1cos...cos s( ) = sin...sin hen Eq () become { } 1/ dr di fh ( ) = W ( hc H ) ( hsh ) d (3) Where for simlicity the frequency deends of W, c s, Hdr and H di has been omitted. Now if it is consider that. di ( ) e = hc ( ) Hdr ( ) hs ( ) H ( ) (4 then the objective function can be exressed 1/ = f( h) W( ) e ( ) d (5 Using Eq. (5), the gradient and hessian of objective f( h ) function can be readily obtained as 1 1 f( h) = f ( h) W( ) e ( ) q( ) d (6a) where q( ) = hc( ) Hdr ( ) c( ) hs( ) Hdi ( ) s( ) (6b) and f( h) = H1 H H3 (6 c) where 1 1 = H ( 1) f ( h) W( ) e ( ) q( ) q ( ) d (6 d) 1 1 = H f ( h) W( ) e ( ) c( ) c ( ) s( ) s ( ) d (6e) 1 H3 = (1) f ( h) f( h) f( h) (6 f) resectively. Of central imortance to the resent algorithm is the roerty that for each and every ositive integer, the weighted L objective function defined in Eq.() is convex in the entire arameter sace R N1.his roerty can be roved by showing that the Hessian f( h) is ositive semi definite for all h R N1. 3. DESIGN ALGORIHMS An FIR filter whose frequency resonse aroximates a rather arbitrary frequency resonse H d ( ) to within a given tolerance in the minimax sense can be obtained by minimizing ƒ(h) in Eq ().with a sufficient large. For a given, ƒ(h) has a unique global minimizer. herefore, any descent minimization algorithm e.g. Newton and quasi Newton can be used to obtained minimax. he amount comutation required to obtain the design is largely determined by the choice of otimization method as well as the initial oint assumed. A reasonable initial oint can be deduced by using the L -otimal design obtained by minimizing f(h) in Eqn () with =1.It can be written as 1/ f ( h) = ( h Qh h ρ k) (7a) ( ) ( ) Q = W e e ( ) s( ) s ( ) d (7b) P = W( ) [ Hdr ( ) e( ) Hdr ( ) s( ) ] d (7c) Since Q is ositive definite, the global minimizer of f (h) in Eqn (7a) can be obtained as the solution of the linear equation Qh = (8) Since Q in Eq (8) is a symmetric.oelitz matrix [4] for which fast algorithms are available to comute the solution of 8[6]. he minimization of convex objective function f(h) can be accomlished in a number of ways. Since the gradient and Hessian of f(h) are available in closedform and f( h) is ositive semidefinite,the Newton method and the family of Quasi-Newton methods are among the most aroriate. From Eq(5) and (6),we note that f(h), f( h),and f( h) all involve integration which can be carried out using numerical methods. In comuting f( h),the error introduced in the numerical integration can cause the hessian to lose its ositive definiteness but the roblem can be easily fixed by modifying f( h) to f( h) εi where ε is a small ositive scalar. 4. DIREC AND SEQUENIAL OPIMIZAION With a ower, weighting function W(w),and an initial h,chosen,the design can be obtained directly or indirectly. In direct otimization, one of the unconstrained otimization methods is alied to minimize the L objective function in Eq.(5) directly. Based on rather 178

3 NCCI 1 -National Conference on Comutational Instrumentation CSIO Chandigarh, INDIA, 19- March 1 extensive trials, it was found that to achieve a near order FIR filters a value comarable to the filter order N should be used. In sequential otimization, an L P otimization is first carried out with =1.the minimizer thus obtained, h is then used as the initial oint in another otimization with =.the same rocedure is reeated for = 4, 8, 16 until the reduction in the objective function between two successive otimization is less than a rescribed tolerance. 5. DESIGN EXAMPLES he method alied through MALAB is alied to design a low ass FIR filters with low ass band grou delay. Examle 1-Design a low ass FIR filter of order N=45 that would have aroximately constant ass band grou delay of 3s. Assume idealized ass band and sto band gains of 1 and,resectively : normalized samling frequency a = ; assband edge =.5; and sto band edge s =.54; W( )=1 in both the ass band and sto band,and W( )= elsewhere. he magnitude resonse, ole zero resonse of FIR filter using the Parks-McClellan algorithm is deicted in figure 1a, b and c resectively. (b) (c) Figure 1:-Design of low ass FIR filter using Parks McClellan Algorithm for examle 1 (a) Magnitude resonse (b) hase resonse (c) ole zero lot. (a) he magnitude resonse, hase resonse, ole zero lot and grou delay in ass band of FIR filter for examle 1 using sequential otimization are deicted in figure a, b c and d resectively. It is interesting to note that the equirile amlitude resonse is achieved in both ass band and sto band. he ass band grou delay varies from.9 and 3.1 but it is not equirile. his is because the sequential otimization is carried out for the comlex valued frequency resonse, not the hase resonse alone. 179

4 NCCI 1 -National Conference on Comutational Instrumentation CSIO Chandigarh, INDIA, 19- March 1 (a) (d) Figure :-Design of FIR filter with low ass grou delay using Sequential otimization for examle 1(a) magnitude resonse (b) hase resonse (c) ole zero lot (d) grou delay in ass band he design of FIR filter in figure 1a, b and c is imlemented using traditional method of otimization (Parks-McClellan algorithm), while in figure a,b,c and d we discuss the imlementation of sequential otimization. In this aer from fig 1(c) zeroes lies outside the circle therefore the system is less stable in comarison ole zero lot in fig (c). herefore the stability in sequential otimization technique is much higher than in the traditional method of otimization (Parks-McClellan algorithm). (c) (b) 6. CONCLUSION his aer resent various otimization techniques for design of low ass FIR digital filters. In this aer we design low ass fir digital filter with Parks- McClellan algorithm and sequential otimization. he Parks-McClellan algorithm and its variant have been most efficient tools for the minimax design of FIR digital filters. However, these algorithms aly only to the class of linear-hase FIR filters the grou delay introduce by these filters is constant and indeendent of frequency in the entire base band but it can be quite large, but there is a requirement of relatively nonlinear FIR filter. Design examles resented in the aer have indicated that the method can be used to design relatively higher order and nonlinear hase FIR filter that are otimal in the minimax sense. In this aer from ole zero lots that the stability in sequential otimization technique is much higher than in the traditional method of otimization (Parks- McClellan algorithm). 7. REFERENCES [1]. W. Parks and J. H. McClellan, Chebyshev aroximation for nonrecursive digital filters with linear hase, IEEE rans.circuit heory, vol19,, , 197. [] A. Antoniou, Digital Filters: Analysis, Design and Alications, nd ed.mcgraw Hill,

5 NCCI 1 -National Conference on Comutational Instrumentation CSIO Chandigarh, INDIA, 19- March 1 [3] Shak, D.J. and A. Antoniou, "A generalized Remez method for the design of FIR digital filters," IEEE rans. Circuits and Systems, , Feb [4] I. W. Selesnick and C. S. Burrus, Exchange algorithm that comlement the Parks-McClellan algorithm for linear-hase FIR filter design, IEEE rans. on Circuits and Systems [5] J. J. Craig, Introduction to robotics, nd ed. addision wesely, [6] M. Werman and D. Weinshall, Similarity and affine invariant distance between D oint sets, IEEE rans. Pattern Analysis and Machine Intelligence, vol, 17, , August [7] I. W. Selesnick and S. Burrus, Maximally flat low-ass FIR filters with reduced delay,'' IEEE rans. Circuits Syst. II, vol. 45, no. 1, , [8] Nagel, P., Frequency-domain design method for linear-hase and minimum-hase FIR equalizers. IEEE rans. Circuits Syst. v47 i [9] R.Fletcher, Practical Methods of Otimization, nd ed., Wiley, [1] Rabiner, L.R., J.H. McClellan, and.w. Parks, "FIR Digital Filter Design echniques Using Weighted Chebyshev Aroximations," Proc. IEEE 63 (1975). [11] O. Hermann, and H. W. Sch ussler, Design of nonrecursive digital filters with minimum hase, Electron. Lett. 6, , 197. [1] J. H. McClellan,. W. Parks, and L. R. Rabiner, A comuter rogram for designing otimum FIR linear hase digital filters, IEEE rans. Audio Electroacoust. AU-1, 56-56, [13] I. W. Selesnick and C. S. Burrus, Exchange algorithms that comlement the Parks-McClellan algorithm for linear- hase FIR filter design, IEEE rans. on Circuits and Systems II, 44():137-14, February [14] O. Hermann, and H. W. Sch ussler, Design of nonrecursive digital filters with minimum hase, Electron. Lett. 6, , 197. [15] Lightner, M.R.; Erdogmus, D.; Signal rocessing challenges in cognitive assistive technology [Exloratory DSP], Signal Processing Magazine, IEEE Volume 5, Issue 5, Setember 8 Page(s):13 18 [16]B.Dumitrescu, R.Bregovic,.Saramäki - Design of Low-Delay Nonuniform Oversamled Filter banks, Signal Processing, vol.88, no.1, , Oct