Bull, DR

Redmill, DW

English

This paper considers the design of low complexity FIR filters. Complexity is reduced by constraining the filters to have integer coefficients, which can be efficiently implemented using primitive operator directed graphs (PODG). Genetic algorithms (GAs) are used in conjunction with a heuristic graph design algorithm, to provide a solution set which represents different compromises between performance, complexity and filter order. Example results are presented for both one and two dimensional filters, and are shown to provide both superior performance and complexity, compared to previous methods. The main benefits result from the use of a joint optimization, rather than a separable 2-stage approach. The use of a PODG representation is shown to provide significant improvements over a canonic signed digit (CSD) or signed power-of-two (SPT) representatio

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                          Redmill, D. W., & Bull, D. R. (1998). Automated design of low complexityFIR filters. In Unknown. (Vol. 5, pp. 429 - 432). Institute of Electrical andElectronics Engineers (IEEE). 10.1109/ISCAS.1998.694524Link to published version (if available):10.1109/ISCAS.1998.694524Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.AUTOMATED DESIGN OF LOW COMPLEXITY FIR FILTERS David W Redmill and David R. Bull Image Communications Group, Centre for Communications Research, University of Bristol, Bristol. BS8 1UB UK Tel: [+44] 117 954 5195, Fax: [+44] 117 925 5265 Email: David.Redmil1 @Bristol.ac.uk Dave.Bul1 @Bristol.ac.uk ABSTRACT This paper considers the design of low complexity FIR filters. Complexity is reduced by constraining the filters to have integer coefficients, which can be efficiently imple- mented using primitive operator directed graphs (PODG). Genetic Algorithms (GAS) are used in conjunction with a heuristic graph design algorithm, to provide a solution set which represents different compromises between perfor- mance, complexity and filter order. Example results are presented for both one and two di- mensional filters, and are shown to provide both superior performance and complexity, compared to previous meth- ods. The main benefits result from the use of a joint opti- mization, rather than a separable 2-stage approach. The use of a PODG representation is shown to provide significant improvements over a canonic signed digit (CSD) or signed power-of-two (SPT) representation. 1. INTRODUCTION Finite impulse response (FIR) digital filters have many ap- plications within a wide range of digital signal processing algorithms. Some of these applications (e.g. video pro- cessing) require the filter to operate at very high data rates, which gives rise to the need for dedicated high speed appli- cation specific integrated circuits (ASICs). Many filter design techniques (e.g. Parks and McClel- lan) yield filters with floating point coefficients, which in turn implies the need for high precision multipliers, high hardware complexity and high manufacturing cost. In order to reduce cost, it is desirable to design filters which have similar performance for a significant reduction in complex- ity. One method for achieving this is to round the floating point coefficients to finite precision integers. The integer coefficients can then be represented as the sum of a few signed power-of-two (SPT) terms. Thus float- ing point multipliers are replaced by a few integer adders This work was supported by the engineering and physical sciences research council (EPSRC) grant number GRK25892. 0-7803-4455-3/98/$10.00 1998 IEEE and power-of-two shifting operations. The minimum num- ber of SPT terms required for any given coefficient can be easily determined using a canonic signed digit (CSD) algo- rithm [ 11. The filter design problem can now be stated as that of finding an optimum set of coefficients and a corre- sponding SPT representation which optimize both perfor- mance and complexity. Further improvements can be made by implementing the set of integer coefficient multipliers, using a primitive oper- ator directed graph (PODG) representation [2]. The graph consists of, a set of two-input nodes representing adders, and edges with signed power-of-two (SPT) scaling values. For example, the set coefficients 9 and 13 can be represented as 92 = 8z+a:, 13a: = (9a:) +40. The task of finding a min- imum adder graph for a given set of coefficients is a difficult problem. Various heuristic approaches have been proposed [2, 31, which give optimal graphs for most practical exam- ples.' The above implementation methods lead to the follow- ing design strategy. Firstly an integer coefficient filter is designed, either by rounding a floating point design andor using a optimization technique such as linear programming. Secondly the filter can be implemented using either a CSD or PODG representation. This method yields designs which favor increased performance rather than decreased com- plexity. For the results shown in figures 1, 4 and 5,  this method would yield filters which lie to the right hand end of the various curves. A second design strategy, is to first constrain the im- plementation complexity, and then optimize the various pa- rameters to achieve optimal performance. For example var- ious authors [4, 5 ,  6, 7, 8, 9, 101 have designed filters by constraining them to consists of a finite number of coeffi- cients, each consisting of a finite number of SPT terms. In [ l l ,  121, the conditions were relaxed by constraining only the total number of SPT terms and allowing their distribu- 'For the results presented in figure 1, at least 80% of the filters have a graph which is known to be optimal in the minimum adder sense. The remaining 20% lie towards the less important right hand side of the figure, and are known to be very close to optimal (within 1 or 2 adders). V-429 Authorized licensed use limited to: UNIVERSITY OF BRISTOL. Downloaded on October 1, 2009 at 04:41 from IEEE Xplore.  Restrictions apply. -5 I -20 -25 Gain (W -30 -10 ?Order* %Hx€iOrder 4 -I5 *-Order6 1 " "  " - - - -35 -40 -45 -50 "-Order 12 Order 16 L r d e r  18 - -Order 20 - - - I 0 5 10 15 20 25 30 35 40 45 -55 I Number of adders Figure 1 : Tradeoffs between complexity, order and attenua- tion for an example low pass specification. tion to vary. This strategy is more difficult for filters using a PODG representation [ 131. The difficulties are due to the discrete, non-uniform, and non-linear nature of the search space. 2. PROPOSED METHOD In this paper we propose a joint optimization of both com- plexity and performance. To achieve this we represent each candidate filter in the search space as a set of integer coeffi- cients. Thus for each candidate solution we need to eval- uate both the filter performance, and the implementation complexity. Since the graph design method of [3], is rel- atively quick compared to evaluating the filter performance (in the frequency domain), the approach doesn't require ex- cessive computational effort compared to previous methods (e.g. [ 11, 121). In this paper we use a Genetic Algorithm (GA) to per- form the optimization, since GAS have been proved suc- cessful, on a wide variety of discrete, non-uniform, multi- objective optimization problems. GAS use a population of candidate solutions to evolve towards an effective solution. Each candidate solution (individual) is described by a chro- mosome. The fitness of each individual is measured and used to control which individuals are selected to form the next generation. Between generations, various genetic op- erators (including crossover and mutation) are applied in order to search new regions of the solution space. GAS have been successfully used for low complexity filter de- sign problems including, cascaded FIR filters [14], IIR fil- ters [15] andSPTdesigns [11, 121. Figure 2: Graph structure for implementation of 24th order filter with 26 adders. In this paper, the GA is used to provide a non domi- nated set (NDS) of solutions which represent various com- promises between complexity and performance and filter or- der. Due to the nature of the search space, it is natural to use a GA with integer valued genes (one per coefficient) rather than the more usual binary coding. This approach maintains a close relationship between the chromosome structure, and the problem space. This in turn allows us to custom design the genetic operators to suit the particular problem. The op- erators used consist o f  0 Uniform crossover. e Mutation of coefficients by randomly adding a signed power-of-two (SPT) value. e Mutation of an entire chromosome by scaling and rounding all the coefficients. The last form of mutation is particularly useful, since it allows the GA to easily search filters with similar perfor- mance, but significantly different coefficients. 3. RESULTS To demonstrate the effectiveness of the proposed tech- niques, we have used them to design odd length linear phase filters, using a minimax criterion. The first example is a 1D low-pass filter with normalized pass and stop bands of 0 - 0.15 and 0.25 - 0.5 respectively. This example was chosen to allow comparison with previ- ous work [4,5,6]. V-430 Authorized licensed use limited to: UNIVERSITY OF BRISTOL. Downloaded on October 1, 2009 at 04:41 from IEEE Xplore.  Restrictions apply. -10 -15 -20 -25 Galn (de) -30 -35 -40 -45 -50 0 5 10 15 20 25 30 35 40 -55 Number of adders - - - - - - ~ Directed Graph D - - Figure 3: Comparison between CSD and Directed Graph Design Methods. Figure 1 shows the non dominated set (NDS) obtained for this design. This figure consists of a separate curve for each different filter order. Realizable filters lie on and abovehight of the curves, while an optimal compromise would be towards the lower left of the plot. It can be seen, that the optimal integer filters (no constraint on complexity) lie at the bottom right hand end of these curves, and require much higher complexity than other slightly lower perfor- mance filters. Thus, for a given filter order, the joint opti- mization allows filters with significantly lower complexity and only slightly lower performance, than an optimal inte- ger designs. Secondly, we can see that significant perfor- mance improvements (at a given complexity) can often be made by using a higher order filter. Thirdly we note that some filter orders (e.g. 10 and 20 for this example) are of little use. The reason for this, is that the corresponding coef- ficients are usually very small in the optimum minimax de- signs, and thus have little effect. These coefficients can be usefully approximated to zero. Finally, we note that there is an approximately linear trade-off between complexity and attenuation (measured in dBs). The gradient of this trade- off varies according to the design specifications. For comparison, we shall consider the 24th order exam- ple of [5], using 35 adders, and an attenuation of -43.8 dB. From figure 1, it can be seen that a 24th order filter with slightly improved attenuation (-43.92dB) can be achieved with only 26 adders. Figure 2 shows the graph based struc- ture used to implement this filter. The graph consists of 6 adders to generate the values of 5, 7, 71, 251, 43 and 377. These values are then scaled by signed powers-of-two and summed in the shift register. Note that since 4 of the coe€- ficients are zero, only 20 instead of 24 adders are used in -5 3 I -15 I 5 35 10 15 20 25 30 -50 - -55 Number of adders Figure 4: Tradeoffs between complexity, order and attenua- tion for an example band pass filter specification. the shift register. The rectangular boxes represent possible locations for pipelining registers. Figure 1 also shows that another 24th order filter with 34 adders and an attenuation of -45.45 dB can be designed. Note that this is very close to the optimal (24th order) float- ing point design (obtained using the Parks and McClellan algorithm) which has an attenuation of -46 dB. If we relax the constraint on the filter order, we can design a 28th order filter with 35 adders and an attenuation of -52.26 dB. Thus, for the same complexity we can achieve an improvement of 8.4 dB over the CSD design of [ 5 ] .  Note that the use of the GA does not restrict our con- sideration to designs using directed graphs. By replacing the graph design algorithm with a CSD algorithm, and re- peating the experimentation, we can gain a similar NDS for designs using CSD. Figure 3 shows a comparison of the two methods, from which it can be seen that for more complex filters, the directed graph method yields significantly better results. To demonstrate the flexibility of this approach, a band- pass filter, with transition bands 0.15-0.25 and 0.35-0.45, has been designed. Figure 4 shows the resulting NDS. A 28th order filter with an attenuation of -50.96 dB can be designed with 31 adders. To design a comparable (com- plexity) filter with 2 SPT terms per coefficient (as in [6]) we must consider a maximum of 20th order2, which gives an attenuation of about -36 dB (from [6]). Note that even an optimal floating point 20th order filter only achieves -39dB. The method can also be used to design two-dimensional filters. Figure 5 shows results for the design of 2D linear- 'A 20th order linear phase filter requires up to 20 adders in the shift register and 1 I coefficients with up to one adder each giving a maximum of 31 adders. v-43 1 Authorized licensed use limited to: UNIVERSITY OF BRISTOL. Downloaded on October 1, 2009 at 04:41 from IEEE Xplore.  Restrictions apply. - 1  I 5. REFERENCES i o  20 30 40 50 60 -30 Num adders. Figure 5: Example design for 2-D circularly symmetric low- pass filter. phase circular symmetric low pass filters with an annular transition band between 0.15 and 0.35. These results can be compared with those of [7, 8, 9, 10, 11, 121, and demon- strate filters with both superior complexity and performance for sizes of 5x5, 7x7 and 9x9. For example, in [11, 121, a 53 adder 7x7 filter with a maximum ripple of -28.8dB, was presented. Figure 5 shows that a slightly better perfor- mance (-29.0dB) can be achieved with only 48 adders. The improvement obtained by using larger filters becomes less significant in two dimensions, since an increase in size im- plies a much larger increase in complexity, due to the larger shift register structure. 4. CONCLUSIONS This paper has examined the design of low complexity FIR filters using a primitive operator directed graph (PODG) representation. Genetic algorithms have been used to per- form a joint optimization of both filter performance and complexity. This method provides a range of filters cover- ing the various trade-offs between performance, complexity and filter order. Example minimax designs have been presented for both one and two dimensional linear phase filters, and shown to provide superior results to previous methods. The improve- ments are due to both the use of a PODG representation, and the joint optimization strategy. [ 11 A. Avizienis. Signed-digit number representation for fast parallel arithmetic. IRE Trans. Electon. Comput., 10:389- 400, 1961. [2] D. R. Bull and D. H. Horrocks. Primitive operator digital filters. IEE-Proceedings-G, 138:401412, 1991. [3] A. G. Dempster and M. D. Macleod. Use of minimum-adder multiplier blocks in FIR digital filters. IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Process- ing, 42:569-577, 1995. [4] N. Benvenuto, M. Marchesi, and A. Uncini. Applications of simulated annealing for the design of special digital filters. IEEE Transactions on Signal Processing, 40:323-332, 1992. [5] H. Samueli. An improved search algorithm for the design of multiplierless FIR filters with powers-of-two coefficients. IEEE Transactions on Circuits and Systems, 36: 1044-1047, 1989. [6] Q. Zhao and Y. Tadokoro. A simple design of FIR filters with powers-of-two coefficients. IEEE Transactions on Circuits and Systems, 35566570, 1988. [7] L. Banzato. A design method for 2-D multiplierless FIR dig- ital filters for video applications. IEEE Transactions on Cir- cuits and Systems for Video Technology, 2:273-284, 1992. [8] P. Siohan and A. Benlismane. Finite precision design of op- timal linear phase 2-D FIR digital filters. IEEE Transactions on Circuits and Systems, 36: 11-21, 1989. [9] S. Sriranganathan, D. R. Bull, and D. W. Redmill. Design of 2-D multiplierless FIR filters using genetic algorithms. In First IEWIEEE International Conference on Genetic Al- gorithms in Engineering Systems: Innovations and Applica- tions, 1995 Sheffield UK. 12-14 September 1995. [ I O ]  S. Sriranganathan, D. R. Bull, and D. W. Redmill. The opti- misation of multiplierless two dimensional digital filters. In SPIE VCIP, 1996. [ I  11 S. Sriranganathan, D. R. Bull, and D. W. Redmill. Low com- plexity two-dimensional digital filters using unconstrained SPT term allocation. In Proc IEEE ISCAS 96, Atlanta, USA., pages 762-765, 1996. [ 121 D. R. Bull and D. W. Redmill. Optimisation of image coding algorithms and architectures using genetic algorithms. IEEE Transactions on Industrial Electronics, 43:549-557, 1996. [I31 D. R. Bull and A. Aladjidi. The optimization of multiplier- free directed graphs: An approach using genetic algorithms. In Proceedings IEEE International Symposium on Circuits and Systems, pages 456459, 1994. (141 G. Wade, A. Roberts, and G. Williams. Multiplier-less fir filter design using a genetic algorithm. IEEproceedings, vi- sion, Image and Signal Processing, 81: 175-180, 1994. [15] P. B. Wilson and M. D. Macleod. Low implementation cost IIR filter design using genetic algorithms. In Natural Algo- rithms in Signal Processing Workshop. Chelmsford, Essex: IEE, Nov 1993, pages 4/1-4/8. V-432 Authorized licensed use limited to: UNIVERSITY OF BRISTOL. Downloaded on October 1, 2009 at 04:41 from IEEE Xplore.  Restrictions apply. 

Automated design of low complexity FIR filters

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Automated design of low complexity FIR filters

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