DIGITAL FILTER DESIGN OF IIR FILTERS USING REAL VALUED GENETIC ALGORITHM

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1 DIGITAL FILTER DESIGN OF IIR FILTERS USING REAL VALUED GENETIC ALGORITHM MIKAEL NILSSON, MATTIAS DAHL AND INGVAR CLAESSON Bekinge Institute of Technoogy Department of Teecommunications and Signa Processing Box 52, SE Ronneby SWEDEN E-mai: Abstract: - This paper presents a new paradigm for infinite impuse response (IIR) fiter design using genetic agorithms (GA). By encode or transform the fiter design probem into the z-pane the GA optimization procedure wi be simpified. Additionay, given the z-pane encoding new mutation techniques are introduced, with the intention to ocate promising regions in the search space. With proper design of the fitness function, the proposed agorithm can be used to evove both fu precision or quantized fiter structures. Key-Words: - Digita Fiter Design, IIR, FIR, Quantization, Genetic Agorithm 1 Introduction Cassica design methods of infinite impuse response (IIR) fiters are normay restricted to specific norms such as minimax or east square [1, 2]. Additionay, the quantization effects of the coefficients are normay not possibe to consider during the design process. To design fiters with specia requirements such as a tradeoff in norms or concerning quantization effects, there is a need of more genera optimization techniques. A common approach to sove hard fiter design probem is to use stochasticay based search procedures. Some paper proposes variants of simuated anneaing (SA) [3, 4], other different genetic agorithms (GAs) [5, 6]. Simuated anneaing in particuar is known to provide good resuts, but is very time consuming due to the stochastic search. A genetic agorithm is a too for optimization in compex mutidimensiona spaces. The inspiration for a genetic agorithm originates in Darwin s ideas of evoution and surviva of the fittest. The agorithm simuates the evoutionary process where the goa is to evove soutions by means of seection, crossover and mutation [7, 8]. This paper proposes a fiter design paradigm based on a rea vaued genetic agorithm. By incuding speciaized fiter mutation operators the stochastic search procedure wi take part of known cassica design properties. 2 The z-pane Encoding In the genera case an IIR fiter can be described by: y(n)+ p a iy(n i) = q b jx(n j) (1) j= where x(n) is the discrete input signa, y(n) the output signa, b j are the coefficients for the Finite Impuse Response (FIR) part, a i are the coefficients for the recursive IIR part, q is the number of FIR coefficients and p is the number of recursive IIR coefficients. Equation (1) can be transformed into the z-domain: q b jz j j= H(z) = 1+ p (2) a iz i By finding the roots of the numerator and denominator poynomias, the transfer function can be expressed with poes and zeros H(z) =b q ( 1 βjz ) j=1 p (1 α iz ) where β j and α i are the zeros and the poes respectivey. With the constraint that a of the fiter coefficients are rea, i.e. b j Rand a i Rthe rea zeros and poes (type 1) or compex conjugated pairs of zeros and zeros (type 2) can be divided into two different categories: Type 1: poar representation of singe zeros and/or poes with the ange degrees, i.e. there is no imaginary part. The radius of the zeros wi be denoted r β1 and r α1 for the poes. The number of zeros in the z-pane wi be denoted β1 and the number of poes α1. Note that r β1 and r α1 can be positive or negative radius. (3)

2 H(z) =b TYPE 1 {}}{{}}{ β1 ( 1 rβ1 z ) β2 (1 r β2 e j 18 π θ β2 z )( 1 r β2 e j 18 π θ β2 z ) j β1 =1 α1 j α1 =1 ( 1 rα1z ) } {{ } TYPE 1 j β2 =1 α2 j α2 =1 TYPE 2 (1 r α2e j π 18 θ α2 z )( 1 r α2e j π 18 θ α2 z ) } {{ } TYPE 2 (4) Type 2: poar representation of compex conjugated zeros and/or poes i.e. two zeros or two poes are needed. Due to the symmetry property of compex conjugated pairs the optimization process ony uses the zero or poe in the anguar interva 8 degrees ( π rad). The radius, anges, the number of zeros and poes in 18 wi be denoted r β2 and r α2, θ β2 and θ α2, β2 and α2, respectivey. Note that r β2 and r α2 is positive and haf of tota number of zeros and poes are taken into consideration. r β1 r β2 r max r α1 r max r α2 r max θ β2,θ α2 18 b where r max 1 to ensure stabiity of an IIR fiter. 3 Evoving the Popuation (6) Given the encoding and the search space defined by equation (6), the initia popuation of size P corresponding to the fiter candidates is created. The initia popuation (fiters) can for exampe be randomized initiaized. Given the popuation, genetic operators are appied to evove the popuation and the evoution steps made can be found in Fig. 3. Fig. 1. Zero and poe pattern exampes, eft Type 1 - right Type 2. Initia popuation Evauate popuation The reationship between the number of coefficients and the number of zeros and poes can be expressed as: Check popuation Seection q = β1 +2 β2 +1 p = α1 +2 α2 +1 The transfer function can now be described by (4). The GA encoding of the fiter is a string (chromosome, individua), denoted x, containing the type 1 and type 2 variabes, see Fig. 2. r 1 r 2 1 r 1 r Fig. 2. The z-pane encoding of the fiter and the ength for the different parameters. The search space for the encoding in Fig. 2 is in genera found as: b (5) Mutation Crossover Fig. 3. The evoving steps 3.1 Evauate popuation Evauation is done by assigning a fitness measure to each individua in the popuation by using a fitness function. The fitness function maps an individua from the popuation to a singe judging vaue (the cost). The fitness function is based on the design goas for the fiter (e.g. the ampitude deviations or constraints, the cost of non-zero bits in quantized fiters, etc.). The fitness function shoud avoid to be extremey rugged which wi ead to sow or poor convergence of the GA [9].

3 The encoding scheme in this paper empoying a fitness function appied to (4). The goa for the GA is to find a H (z) which fufi certain constraints on this function. This is probaby the most convenient and straight forward soution to create a proper fitness function. Extension to fitness functions for quantized fiters can for exampe be created by using the foowing procedure: 1. Convert the string (chromosome) to b j and a i 2. Transform b j and a i to a desired structure 3. Quantize the coefficients in the desired structure 4. Transform back to b j and a i 5. Check stabiity of the fiter 6. Check the constraints on H (z) by using (2) and cacuate the fitness vaue 7. Penaize unstabe fiters Note that the quantization operations wi require a check of stabiity. By adjusting some settings, e.g. r max, the amount of instabe fiters due to quantization can be reduced. The penaty for unstabe fiters can be achieved by: { f (xquant) if x eva (x) = quant is stabe E worst + stabiity measure otherwise (7) where x quant is the quantized coefficients in some chosen structure, f (x quant ) is the fitness criteria of the fiter given the quantized coefficients in the chosen structure, E worst the worst found individua vaue found so far (from a previous generations unti now) and the stabiity measure is a measure of how far from stabiity the fiter is (e.g. the maximum radius for the poes found from a j in step 4). 3.2 The Seection Severa seections techniques using GA has been proposed: steady state, tournament, proportiona, rank, etc. In this paper the rank seection is used in combination with k-eitism and a kind of immigrants scheme [7, 8, 9]. In k-eitism scheme the k most fitted individuas in the popuation wi be directy seected to the next generation. A number λ of randomy created immigrants wi be inserted into the next generation as we. The remaining P k λ individuas, which are used for crossover and mutation, are seected according to the probabiity: p rank = rank (xi) P rank (x i) where rank ( ) wi rank the individuas from [P, P 1,...,1] according to the fitness. (8) The described combination of seection operators has experimentay been found usefu for maintaining the diversity of the popuation as we as preserving good genetic materia. 3.3 Crossover Given two seected parents according to (8), crossover are used to create new offsprings. Crossover wi be performed if a uniform distributed random number wi be ess than the crossover probabiity p c. Four standard techniques for crossover are considered singe point crossover, two point crossover, uniform crossover and arithmetic crossover [7, 8, 9]. The choice of the crossover operation appied depends on the probabiity vector: p c,vec = 3.4 Mutation P (singe point crossover) P (two point crossover) P (uniform crossover) P (arithmetic crossover) (9) After a crossover operation the resuting individuas wi undergo mutation. The probabiity of mutation p m is used to decide if mutation shoud take pace. Five different mutation operations are considered: the Gaussian mutation is commony used in rea coded GAs, the other four are new mutation paradigms introduced by considering cassica digita fiter and their poe and zeros patterns in the z-pane. The seection of actua mutation operation is performed according to the probabiity vector: p m,vec = Gaussian mutation P (Gaussian mutation) P (minimum phase mutation) P (unit circe mutation) P (inear phase mutation) P (magnitude mutation) (1) The Gaussian mutation modifies the string by adding a randomy generated vector from the gaussian distribution [8]. x = x + N (, σ) (11) where x is the individua to to mutated, x the mutated individua and σ the vector which contros the shaped of the gaussian distribution. Here σ wi contain σ rβ1, σ rβ2, σ θβ2, σ rα1, σ rα2, σ θα2 and σ b, in this way the movements of the different parts in x can be controed minimum phase mutation In minimum phase mutation the zeros outside the unit circe are wrapped into the unit circe by inverting the radius, see Fig. 4. This operation combined with a scaing of b yieds

4 a fiter with the same magnitude with minimum phase property [1]. Fig. 6. Exampe of inear phase mutation Fig. 4. Exampe of minimum phase mutation unit circe mutation Zeros on the unit circe in the z-pane commony appear then fiters are designed by cassica approaches [1]. With this in mind the unit circe mutation change the radius to one for a zeros (i.e. keep the anges but move a zeros in the z-pane to the unit circe), see Fig. 5. Fig. 5. Exampe of unit circe mutation inear phase mutation A inear phase FIR fiter can be achieved by moving a zeros to the unit circe as described in unit circe mutation or mirror the zero in the unit circe ( mirrored positions). Assume that there is a zero with radius r when it exists another zero with radius 1 r. Note that this mutation wi ony give exact inear phase if the fiter is stricty FIR. The inear phase mutation is appied by repacing the upper haf of r β1 and r β2 with the mirrored zero position of the ower haf. If β1 or β2 is odd, wi the remaining radius be set to one, see Fig magnitude mutation The fiter magnitude in digita fiter design is commony cose to one in the passband regions or some frequency regions. The magnitude mutation seects a singe ange between and 18 degrees randomy with uniform distribution and adjusts b so that the magnitude of the fiter wi be one at this ange. 3.5 Check popuation After the crossover and mutation, some individuas can be outside the desired search space (e.g. considering the Gaussian mutation on r apha1 ). This wi be avoided by the check popuation step after the genetic operations. If the parameters are outside the desired search space they wi be transformed back into the desired search space according to: r β2 = r { β2 36 θβ2 mod 36 if θ θ β2 = β2 mod 36 > 18 { θ β2 mod 36 ese 1 r α1 = r α1 if r α1 >r max { r α1 ese 1 r α2 = r α2 if r α2 >r max { r α1 ese 36 θα2 mod 36 if θ θ α2 = α2 mod 36 > 18 θ α2 mod 36 ese b = b (12) where a mod b are the moduus operator working on rea numbers (e.g..1 mod 36 = and mod 36 = 12.1). 4 Design Exampe As a design exampe using the proposed agorithm, a fiter with both magnitude 1 and group deay constraints is considered, see Fig. 7. The compete fitness function used for optimization is a sum of three specia max-norms. Consider the upper imit function for the magnitude in decibes, M u (f) where f is normaized frequency. A norm for this constraint is defined as: E 1 =max[(m(f) M u(f)) (M(f) >M u(f))] f.5 f (13) where M(f) = 2 og 1 H(f) is the magnitude of the fiter in decibes. This norm wi give the maximum deviation in decibes from the upper imit in the magnitude. If M(f) is stricty beow M u (f) the norm wi yied in E 1 =. 1 The fiter magnitude is found from [3] and compemented with a group deay constraint

5 Group deay [sampes] and one IIR with q =2( β1 =1, β2 =9) and p =8 ( α1 =1, α2 =3). The settings used for both runs can be found in Tab. 1 and Tab. 2. Variabe Description Vaue(s) N Number of generations 5 P Popuation size 1 k Number of eitists 2 λ Number of immigrants 1 r max Max radius for poes.99 p c Probabiity of crossover.7 p m Probabiity of mutation.6 p c,vec See (9) [ ] p m,vec See (1) [ ] Tabe 1. Settings for GA Fig. 7. Fiter specification. Upper pot: a zoom of the passband imits for the magnitude. Midde pot: the upper and ower imits for the magnitude on the fu frequency range. Lower pot: the group deay constraints in the passband. The norm for the ower imit function for the magnitude in decibes is defined as: E 2 =max[(m (f) M(f)) (M(f) <M (f))] f.5 f (14) where M u (f) is the ower imit for the magnitude. This norm wi give the maximum deviation in decibes from the ower imit. If M(f) >M (f) f the norm wi yied in E 2 =. The norm for the upper imit function for the group deay in sampes is defined as: E 3 =max[(g(f) G u(f)) (G(f) >G u(f))] f.5 f (15) where G(f) is the group deay in sampes for the fiter and G u (f) is the upper imit for the group deay. This norm wi give the maximum deviation in sampes from the upper imit of the group deay. If G(f) <G u (f) f the norm wi yied in E 3 =. In the practica impementation of the three norms the normaized frequency is samped yieding in f k [,.5] for K =1, 2,...,K, for the simuations made here is K = 256 used. Given these three norms the sum of these wi yied in the compete norm for the fiter design: f (x) =E 1 + E 2 + E 3 (16) Note that 1 db deviation in the magnitude wi punish a fiter with the same order as 1 sampe deviation in the group deay. Two fiters are considered for fu precision optimization, one FIR fiter with fiter ength q =35( β1 =, β2 =17), Variabe Start vaue Stop vaue Type of decay σ rβ1 2.5 exponentia σ rβ2 2.5 exponentia σ θβ2 5.5 exponentia σ rα1.5.5 exponentia σ rα2.5.5 exponentia σ θα2 5.5 exponentia σ b 1.1 inear Tabe 2. Settings for Gaussian mutation in GA The resut for the FIR optimization can be found in Fig. 8 and the IIR resut is found in Fig Group deay [sampes] Fig. 8. Resut for FIR fiter of ength 35, optimized with proposed method The fu precision fiters are found to fufi the specification. However no quantization is taken into consideration. As a fina optimization the design probem of an IIR fiter is considered to be impemented in direct form with quantization of the coefficients in two-compement using 8 bits. A direct quantization of the fu precision fiter, yied a fiter which vioates the desired specification. By using the

6 Group deay [sampes] Fig. 9. Resut for IIR fiter of ength 2 (FIR) and 8 (IIR), optimized with proposed method fu precision fiter as a popuation initiation and use the fitness cacuation according to (7) a resut which fufis the specification is reached, see Fig Group deay [sampes] References: [1] T.W. Parks and C.S. Burrows, Digita Fiter Design, Wiey & Sons Inc., 1987, ISBN [2] Sanjit K. Mitra, Digita Signa Processing, McGraw- Hi, 1998, ISBN [3] P. Persson, S. Nordebo, and I. Caesson, Mutimode mean fied anneaing technique to design recursive digita fiters, in IEEE Transactions on Circuits and Systems II: Anaog and Digita Signa Processing, December 21, vo. 48, pp [4] P. Persson, S. Nordebo, and I. Caesson, Hardware efficient digita fiter design by mutimode mean fied anneaing, in Machine Learning and Cybernetics 22, Juy 21, vo. 8, pp [5] Jin Ouyang and Wei-Dong Qu, Robust poe pacement using genetic agorithms, in Machine Learning and Cybernetics 22, 22, vo. 2, pp [6] M. C. Lang, Least-squares design of iir fiters with prescribed magnitude and phase responses and a poe radius constraint, IEEE Transactions on Signa Processing, vo. 48, no. 11, pp , November 2. [7] David E. Godberg, Genetic Agorithms in Search Optimization and Machine Learning, Addison-Wesey, 1989, ISBN [8] Zbigniew Michaewicz, Genetic Agorithms + Data Structures = Evoution Programs, Springer-Verag, 1996, ISBN [9] Rasmus K. Ursem, Mode for Evoutionary Agorithms and Their Appications in System Identification and Contro Optimization, Ph.D. thesis, University of Aarhus, 23. Fig. 1. Direct form quantized, two-compement with 8 bits, IIR fiter of ength 2 (FIR) and 8 (IIR), optimized with proposed method 5 Concusions A new genetic genetic agorithm with encoding in the z- pane and specia mutation operations is proposed. The proposed agorithm takes into account properties which has been found interesting in cassica design of fiters by means of the new mutation operations. Design exampes iustrating the abiity of designing specia FIR and IIR fiters are presented. The exampes takes into account both magnitude and group deay in the optimization procedure.