Design of efficient circularly symmetric two-dimensional variable digital FIR filters

Circularly symmetric two-dimensional (2D) finite impulse response (FIR) filters find extensive use in image and medical applications, especially for isotropic filtering. Moreover, the design and implementation of 2D digital filters with variable fractional delay and variable magnitude responses without redesigning the filter has become a crucial topic of interest due to its significance in low-cost applications. Recently the design using fixed word length coefficients has gained importance due to the replacement of multipliers by shifters and adders, which reduces the hardware complexity. Among the various approaches to 2D design, transforming a one-dimensional (1D) filter to 2D by transformation, is reported to be an efficient technique. In this paper, 1D variable digital filters (VDFs) with tunable cut-off frequencies are designed using Farrow structure based interpolation approach, and the sub-filter coefficients in the Farrow structure are made multiplier-less using canonic signed digit (CSD) representation. The resulting performance degradation in the filters is overcome by using artificial bee colony (ABC) optimization. Finally, the optimized 1D VDFs are mapped to 2D using generalized McClellan transformation resulting in low complexity, circularly symmetric 2D VDFs with real-time tunability.

Journal of Advanced Research (2016) 7, 336–347 Cairo University Journal of Advanced Research ORIGINAL ARTICLE Design of efficient circularly symmetric two-dimensional variable digital FIR filters Thayyil Bindima *, Elizabeth Elias Department of Electronics and Communication Engineering, National Institute of Technology Calicut, Kerala, India G R A P H I C A L A B S T R A C T A R T I C L E I N F O Article history: Received 11 October 2015 Received in revised form 29 January 2016 Accepted 30 January 2016 Available online 17 February 2016 A B S T R A C T Circularly symmetric two-dimensional (2D) finite impulse response (FIR) filters find extensive use in image and medical applications, especially for isotropic filtering Moreover, the design and implementation of 2D digital filters with variable fractional delay and variable magnitude responses without redesigning the filter has become a crucial topic of interest due to its significance in low-cost applications Recently the design using fixed word length coefficients has gained importance due to the replacement of multipliers by shifters and adders, which reduces the hardware complexity Among the various approaches to 2D design, transforming a * Corresponding author Tel.: +91 9495663242; fax: +91 4952287250 E-mail address: bindima@gmail.com (T Bindima) Peer review under responsibility of Cairo University Production and hosting by Elsevier http://dx.doi.org/10.1016/j.jare.2016.01.005 2090-1232 Ó 2016 Production and hosting by Elsevier B.V on behalf of Cairo University Design of efficient circularly symmetric 2-D variable digital FIR filters Keywords: 2D circularly symmetric FIR Variable digital filters Variable fractional delay filters McClellan transformation Canonic signed digit Artificial bee colony algorithm 337 one-dimensional (1D) filter to 2D by transformation, is reported to be an efficient technique In this paper, 1D variable digital filters (VDFs) with tunable cut-off frequencies are designed using Farrow structure based interpolation approach, and the sub-filter coefficients in the Farrow structure are made multiplier-less using canonic signed digit (CSD) representation The resulting performance degradation in the filters is overcome by using artificial bee colony (ABC) optimization Finally, the optimized 1D VDFs are mapped to 2D using generalized McClellan transformation resulting in low complexity, circularly symmetric 2D VDFs with real-time tunability Ó 2016 Production and hosting by Elsevier B.V on behalf of Cairo University Introduction 2D linear filtering has extensive applications in image and video processing, ranging from highly precise medical imaging to low precision industrial imaging and consumer video applications, as these applications often require filters that assure phase linearity and accurate magnitude response in the pass band As the majority of these applications have tight area and power constraints due to battery lifetime and cost, filters with minimum hardware complexity are desirable Hence, the development of low power, low complexity, and highspeed digital filters has been a captivating topic of research in the recent past Again the recent explosive proliferation of the communication standards and the need for arbitrary sampling rate conversion have paved the way for the demand of low complexity VDFs including variable fractional delay (VFD) [1] and variable cut-off frequency [2] (VCF) FIR systems with online tuning The Farrow structure [3] has been a promising approach for the design of low complexity VDFs Similar to 1D, 2D VDFs also have potential applications in multidimensional signal processing fields such as frame interpolation in video processing, sub-pixel interpolation in still images, pattern recognition, robotic vision and communication systems [4–6] Further, circularly symmetric 2D FIR VDFs are of keen interest in primary applications of image processing especially where there is no preferred spatial frequency axis For the design of low complexity 2D VDFs, transforming the corresponding 1D VDF to its 2D counterpart using any suitable 1D to 2D transformation such as McClellan transformation [7], would be advantageous The transformation-based approach gives a drastic reduction in the design and implementation complexity [8] Moreover, the design of 2D VDFs with infinite precision coefficients has gained wide attention, and several design methods have been proposed [9–12] However, to the best of our knowledge the design using finite precision coefficients has received less attention [2,8] Yeung and Chan [8] proposed the design of elliptical and fan type 2D VDFs and Pun et al [2] proposed the design of circularly symmetric 2D VDFs The first part of the paper gives a brief review of the various design methodologies for 2D variable magnitude response FIR filters and in the second part, we present an efficient approach for the design of low complexity circularly symmetric finite precision 2D VCF filters using CSD VCF filter design using Farrow based polynomial interpolation [2] offers lesser implementation complexity in terms of multipliers than the single stage FIR based design [13] Further reduction in the implementation complexity is attained in this paper when the sub-filter coefficients are represented using CSD with the minimal number of signed power of two (SPT) terms The design of 1D filter in discrete space degrades the overall performance of the filters, which necessitates the use of some nonlinear optimization techniques Since the search space consists of integers, the classical gradient-based methods cannot be used, and meta-heuristic algorithms are preferred In this paper, artificial bee colony (ABC) optimization has been used to optimize the sub-filter coefficients in the CSD space The interpolated impulse responses are finally transformed using generalized McClellan transformation to yield the 2D variable impulse responses The paper is organized as follows: Section ‘‘Overview of 1D variable digital filters and Farrow structure” gives an overview of the Farrow structure for the design of 1D variable digital FIR filter Section ‘‘2D variable digital filter” provides a brief introduction on the 2D VDFs and reviews the different design methods for 2D variable magnitude response FIR filters Sections ‘‘Canonic signed digit representation” and ‘‘Overview of artificial bee colony algorithm” give a brief overview of CSD representation and ABC algorithm The problem statement and the design of continuous coefficient 1D VDF are explained in Sections ‘‘Problem statement” and ‘‘Design of 1D continuous coefficient variable digital filters”, respectively Section ‘‘Design of 2D circularly symmetric VDF” gives the design of 2D circularly symmetric VDFs Results and conclusion are given in Sections ‘‘Results and discussion” and ‘‘Conclusion”, respectively Overview of 1D variable digital filters and Farrow structure Variable digital filters are a class of digital filters whose spectral characteristics can be varied in real-time, without redesigning the filter and with minimum overhead on complexity A detailed review of the different methods for 1D VDF design is given by Laakso et al [1] The spectral parameter approximation method [3,14,15] is a good approach suitable for the design of general VDFs including VFD filters and variable magnitude response (VMR) filters [9] Moreover, the VDF obtained by this method can be implemented efficiently using Farrow structure [3], a promising technique for realizing time varying fractional delay filters [1,16] It enables the filter to be designed offline with easy, accurate online control of its spectral parameters, entirely independent of the filter coefficients In the spectral approximation method, the impulse response of the VDF is assumed to be a linear combination of some /k ðlÞs as in the equation below, 338 hn; lị ẳ T Bindima and E Elias L X cn;k /k lị 1ị kẳ0 where /k lị represents some function of the variable spectral parameter l, which depends on the application [13] The transfer function of the resulting VDF is as in Eq (2), where /k ðlÞ is approximated as lk , such that À0:5 l 0:5 or l [2,3] ! N X L L N X X X Àn Àn Hðz; lị ẳ cn;k /k lịz ẳ cn;k z 2ị /k lị nẳ0 kẳ0 kẳ0 nẳ0 Thus, Eq (2) is a generalization of the Farrow structure, which can be used to design both VFD and VCF digital filters Hence, the transfer function of the resulting VDF takes the form as in Eq (3), ! L N L X X X Àn Hz; lị ẳ cn;k z Ck zịlk 3ị lk ẳ k¼0 n¼0 k¼0 where the impulse response is being approximated as an Lth order polynomial in the variable parameter l The coefficients Ck ðzÞ represent the sub-filter coefficients, which are fixed, and the only variable term is l, which is tuned according to the desired cut-off frequency (for VCF filters) or FD (for VFD filters) The Farrow structure for implementing the variable filter using Eq (3) is shown in Fig 1, where changing the delay parameter l results in filters with a different spectral response 2D variable digital filter jHðx1 ; x2 Þj ds for ðx1 ; x2 Þ Rs : The design approaches for 2D fixed filters can be classified as transformation-based method [7,18–22] and the direct method Direct methods, in turn, include Fourier based and optimization [17,23,24] based approaches 2D VDF design is also based on the above two methods, and the different methodologies are briefly reviewed in this section as summarized in Fig 2D VDFs are also categorized as VMR filters [10–12,25,26] and variable phase or group delay response filters [27–29] The different design approaches for 2D VMR filters are discussed in the following subsection Variable magnitude response filters The initial works [30] in the design of 2D VDFs with variable magnitude response were based on the first order and second order frequency transformations [31] used for 1D variable cut-off linear phase (LP) filters [32] The first order transformation for 1D VCF filter is given by cos x ẳ A0 ỵ A1 cos X, and the second order transformation is given by cos x ẳ A0 ỵ A1 cos X ỵ A2 cos2 X, where xc and Xc represent the cut-off frequencies of the prototype and the P transformed filters, respectively Pk¼0 Ak ¼ with P ¼ for first order and P ¼ for second order Hence, varying the parameter A0 results in a 1D VCF filter This concept is extended to 2D The 1D frequency response HðXÞ is transformed into 2D by applying McClellan transformation given by cosðx1 ; x2 ị ẳ t00 ỵ t10 cos x1 ỵ t01 cos x2 ỵ t11 cos x1 cos x2 , The transfer function of fixed 2D FIR filter takes the form Hðz1 ; z2 ị ẳ N1 X N2 X n2 hn1 ; n2 ịzn z2 4ị n1 ẳ0 n2 ¼0 and the corresponding frequency response will be as in Eq (5) Hx1 ; x2 ị ẳ N1 X N2 X hðn1 ; n2 ÞeÀjx1 n1 eÀjx2 n2 ð5Þ n1 ¼0 n2 ¼0 For a circularly symmetric frequency response, Hðx1 ; x2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi will be a function of x21 þ x22 for x21 þ x22 p [17] and for a circularly symmetric impulse response, hðn1 ; n2 Þ will be a function of n21 ỵ n22 The 2D filter specifications are given by the passband (PB) region Rp , the stop band (SB) region Rs , the PB tolerance dp and the SB tolerance ds and in practice we need the filter to satisfy the following conditions: À dp jHðx1 ; x2 Þj þ dp for ðx1 ; x2 Þ Rp : Fig Farrow structure with fixed sub-filters Ck ðzÞ and variable delay parameter (l) Fig Pictorial abstract of the different approaches in the literature for the design of 2D variable magnitude response filters along with the transformation based approaches used for fixed filter design Design of efficient circularly symmetric 2-D variable digital FIR filters where the coefficients t00 ; t10 ; t01 and t11 control the shape of the contour The final frequency response of the 2D VDF takes the form as given in Eq (6), where cos x1 ẳ A0 ỵ A1 cos X1 and cos x2 ẳ B0 ỵ B1 cos X2 Hx1 ; x2 ị ẳ N X bkịt00 ỵ t10 ẵA0 ỵ A1 cos X1 ỵ t01 ẵB0 kẳ0 ỵ B1 cos X2 ỵ t11 ẵA0 ỵ A1 cos X1 ẵB0 ỵ B1  cos X2 ŠÞ ð6Þ Due to the constraints in the frequency transformation, this approach becomes inefficient for the design of a general 2D VDF or when a complicated set of variable specifications is to be met A general design approach to be used in situations where several spectral parameters control the frequency response of the 2D filter was proposed by Deng and Lian [10] In this approach, an outer product expansion of the 2D variable magnitude characteristics is performed The magnitude response of a VDF given by Hðx1 ; x2 ; W1 ; W2 ; ; Wk Þ with different spectral characteristics given by the vector W ẳ ẵW1 ; W2 ; W3 ; ; Wk Š, is uniformly sampled to yield a set of samples aðm; n; l1 ; l2 ; ; lk Þ These set of samples make the elements of a (k + 2) dimension array A This array is decomposed by applying singular value decomposition to the elements of the array so that, each element in it is expressed P as aðm; n; l1 ; l2 ; ; lk ị % riẳ1 Hi ðm; nÞPi1 ðl1 ÞPi2 ðl2 Þ Á Á Á Pik ðlk Þ Here, Hi ðm; nÞ represents the elements of a non-negative matrix Hi , which in turn forms the magnitude specifications of a 2D constant filter Hi ðz1 ; z2 Þ and the vectors Pi1 ðl1 Þ; Pi2 ðl2 Þ; ; Pik ðlk Þ represent the sampled specifications of the 1D polynomials Pi1 ðw1 Þ; Pi2 ðw2 Þ; ; Pik ðwk Þ This decomposition gives the vectors in one direction, i.e Pi1 The same procedure is applied to gradually reduced dimensional arrays to obtain the vectors in the other directions Thus, the complex problem of designing a variable 2D filter gets reduced to 2D constant filter design and 1D polynomial approximations, which can be done easily using any of the existing methods [17,23,33] But, the decomposition of the 2D variable response with minimum error remains a difficult task Yoshida et al [34] proposed a design using predesigned constant three-dimensional (3D) prototype filter, where obtaining the 3D specifications from the 2D variable filter specifications, again remains a tedious task [12] Deng [12] proposed a simple and flexible approach to express the 2D VDF coefficients as M-dimensional (M-D) polynomials of the spectral parameters W This approach, in turn, avoids the need for specification transformation to that of 2D or 3D constant filters A least square approach for minimizing the total squared error value between the desired and actual 2D response is used to find the optimal polynomial coefficients The overall variable 2D filter is implemented from the polynomial coefficients using a parallel structure similar to the Farrow structure, where the spectral parameters W form the tunable part But, this approach becomes time-consuming when many spectral parameters control the magnitude response since many constant filters are to be designed Deng [25,35] proposed a real complex decomposition approach where the complex valued variable 2D filter specifications are decomposed into the sum of the product of real and complex components, which represent the 2D 339 constant filters and the 1D polynomial coefficients, respectively But, in this approach, the final accuracy is highly affected by the decomposition and approximation errors Deng proposed a modified approach to an improved accuracy [36], where the complex 2D variable frequency response is expressed as the sum of outer products of 2D complex arrays and M-D real arrays The 2D complex arrays correspond to the 2D constant filters, and the M-D real array represents the M-D polynomials of the spectral parameters W Even though this approach breaks the complicated problem of designing the 2D VDFs into simple sub-problems making the design easier, it does not yield filters with complete linear phase A perfect linear phase 2D VDF design method was proposed by Deng [26] where the multidimensional array A was decomposed into the sum of products of 2D real matrices and K-D real arrays This approach is an efficient one for linear phase 2D VDFs and the designed filter is implemented using parallel structures suitable for high-speed applications However, in this method, the design is based on a compromise between the hardware complexity and the design accuracy Among the existing design methods for 2D FIR filter [37– 43], the McClellan transformation remains an efficient and robust technique, suitable for filters with different contour shapes and provides a highly structured implementation architecture Shyu et al [44] extended the conventional McClellan transformation for 2D VDF design, where a cut-off frequency orbit function xc ðlÞ is determined to design the variable 2D transformation kernel and the 1D variable prototype filter The transformation kernel has been precisely designed so that the variable frequency characteristics are achieved by a variable parameter, which in turn continuously matches the cut-off contour mapping with the desired 2D variable filter The design methods discussed above have considered only infinite precision filters The design of finite precision 2D VDFs was considered by Pun et al and Yeung and Chan [2,8] Pun et al [2] have proposed the design of 2D VCF FIR filters using 1D Farrow structure and generalized McClellan transformation 1D filter coefficients considered in the order of their increasing PB edge frequencies were approximated to an Lth order polynomial in the variable delay parameter ðlÞ resulting in a set of 1D linear phase sub-filter coefficients to form the Farrow structure The overall 1D response for each delay is mapped to 2D using generalized McClellan transformation This in turn, is equivalent to transforming the 1D sub-filters Ck ðzÞ in Eq (3) to 2D sub-filters as in Eq (7) b2c X N Ck z1 ; z2 ị ẳ ck nịTn ẵFz1 ; z2 ị nẳ0 ck nị ẳ c0;k for n ẳ ck nị ẳ 2cn;k for n 7ị The transformation results in a set of tunable 2D FIR filters given by Eq (8) Hðz1 ; z2 ; lÞ ¼ L X Ck ðz1 ; z2 Þlk ð8Þ k¼0 Also, the 1D sub-filter coefficients were represented using SOPOT to make the design multiplier-less In SOPOT representation, each sub-filter coefficient cn;k is represented using 340 P aj the expression cn;k ¼ M j¼1 bknj , where bknj ðiÞ À1; and aj ¼ Àl; ; À1; 0; 1; ; l, l is a positive integer representing the range of coefficients and M represents the number of terms required to represent the coefficient approximately To the best of our knowledge, design of 2D circularly symmetric VCF filters in CSD space has not been reported in the literature so far Canonic signed digit representation Multiplier-less representation of the filter coefficients can be realized using any of the SPT representations CSD [45] is one of the SPT representations that is best suitable for performing constant integer multiplication with the least number of adders and subtractors [45] The encoding scheme represents each given constant uniquely in terms of the SPT using the ternary digit set 1, 0, À1 Each infinite precision coefficient c is repP RÀi resented as W using CSD [46–48] encoding scheme i¼1 bi Here, W represents the word length, and R represents the radix, an integer in the range < R < W, which in turn corresponds to the number of digits in the integer part of the number representation In CSD representation, the adjacent bits are never non-zero, and hence, the CSD number with W-bit word length has no more than W ỵ 1ị=2 non-zero bits [46] However, this encoding may lead to deterioration of the filter performance Hence, some optimization techniques are to be deployed to obtain the filter performance as per the specifications In this work, the ABC optimization algorithm is used Overview of artificial bee colony algorithm ABC optimization [49] is a population-based meta-heuristic optimization technique, which is suitable and efficient for a multimodal multi-objective problem [49,50] It works with a set of initial solutions generated either randomly or by an educated guess and tries to improve it The intelligent behavior of honey bee foraging is mimicked for intensification and diversification [51], or exploitation and exploration [52] The artificial colony of bees includes three categories – employed bees, onlooker bees, and scouts depending on how they behave to the food sources Employed bees are those that visit the food sources already visited by them Onlookers wait in the dance area to choose a food source while scouts make a random search for the food source Out of the entire bees in the honey bee colony, one-half is the employed bees, and the other half is the onlookers The employed bees initiate the search process for the food sources by randomly selecting a food source and finding its nectar amount The neighboring areas of the food sources are also searched to find a food source with a better nectar quality If a better food source is found, its location is memorized or else the original food source location itself is retained Once the search by all the employed bees is terminated, the information gathered by them about the food sources is passed on to the onlookers Onlookers choose a food source that is expected to have a higher probability of better nectar quality based on the information collected from the employed bees Similar to the employed bees, onlooker bees also search the neighborhood for a better food source, and if found, its information is memorized, else the original food source information is used Once the food source of a particular employed bee gets abandoned, it becomes the scout, and after that it searches for T Bindima and E Elias food sources randomly and occasionally finds some with a better nectar quality ABC optimization algorithm is used to solve our problem by assuming that the possible solutions in terms of the sub-filter coefficients with a certain number of SPT terms represent the position of the food sources and the fitness values of each solution represent the nectar quality Problem statement Our aim is to design a 2D circularly symmetric variable digital FIR filter with finite precision, suitable for hardware implementation with minimum complexity Since the focus is on a design for minimum implementation complexity, the most appropriate approach would be to design the 1D variable digital FIR filter using 1D Farrow structure in CSD space and then transform it to 2D using any conforming frequency transformation method Generalized McClellan transformation [18] has been used to map the 1D filter to 2D The sub-filter coefficients encoded in the CSD space replace the power-hungry multipliers by shifts and adds, which shoot down the power requirement The rounding off of the infinite precision coefficients using a specific number of SPT terms deviates the overall performance of the variable filters from the desired specifications This deviation emphasizes the use of suitable optimization algorithms to find potential solutions that equally improve the performance of all the filters and reduce the total number of non-zero terms required to represent the sub-filter coefficients Fig gives an abstract description of the design procedure used in this paper, for the design of low complexity 2D VCF filters The design methodology involves – the design of continuous coefficient 1D VCF filters, CSD encoding of the 1D sub-filter coefficients, ABC optimization of the CSD encoded sub-filter coefficients for the performance improvement along with complexity reduction, and finally, the 2D transformation of the optimal 1D VCF filters The various steps are detailed in the following sections Encoding in CSD space The continuous coefficient 1D VDFs are designed based on the Farrow structure using polynomial interpolation with the delay parameter l such that l and increasing values of l correspond to the VDFs with increasing cut-off frequencies [2] Once the infinite precision sub-filter coefficients cn;k are obtained, the next step is to encode them in the CSD space For finite precision CSD representation of the sub-filter coefficients, an indexed look-up table approach is adopted [48,53] rather than the ternary encoding of the coefficients in the CSD space, which in turn eliminates the problem of the candidate solutions turning out to be invalid CSD representations in due course of optimization [54] A 14-bit CSD encoding has been used in this work, with the first two bits from the most significant bit (MSB) side allotted for the integer part and the remaining 12 bits for the fractional part A look-up table corresponding to the CSD representation of the decimal numbers ranging from to a maximum limit is created Each row in the table has four entries: the index, the 14-bit CSD representation, decimal equivalent and the number of SPT terms as shown in Table The sub-filter coefficients are encoded using the signed indices of the look-up table row having decimal equivalent closest to the sub-filter coefficient [48,55,56] If the Design of efficient circularly symmetric 2-D variable digital FIR filters Fig Table 341 Overview of the proposed design approach for low complexity multiplierless 2D VCF filters A typical entry of the CSD look-up table created for CSD encoding of the continuous coefficients Index CSD representation Decimal equivalent Number of SPT terms 4632 0100100010–100–1 1.1306 sub-filter coefficient is negative, then it is encoded as negative of the index corresponding to its positive counterpart [57] The sub-filters of the 1D Farrow structure are of linear phase and exhibit coefficient symmetry Hence, exploiting the coefficient symmetry, only half the coefficients of each sub-filter are concatenated to form the initial seed for the search, so that the sub-filters are jointly optimized Since the solution space is purely integer based, meta-heuristic algorithms are used F1;i xị ẳ jFpo;i xị rp j Objective function formulation The objective function can be formulated as the weighted sum of the absolute approximation errors in the PB deviation and SB attenuation of M filters with respect to the expected PB deviation and SB attenuation, respectively as per the design specifications The maximum PB error and SB error of the ith filter using the optimized sub-filter coefficients with respect to the zero phase ideal filter are given by Eqs (9) and (10), respectively where, i ¼ 1; 2; ; M for the design of M VDFs Fpo;i xị ẳ max jjHx; xịj 1ịj x2ẵ0;xp Fso;i xị ẳ max jHx; xịj x2½0;xs Š  à algorithm and is defined as x ¼ cT0 cT1 Á Á Á cTL where ck ¼ ½c0;k c1;k c2;k Á Á Á cK;k ŠT , L is the interpolation order and Ä Å K ¼ N2 N represents the filter order of the individual filters in the set of VDFs, and each filter is designed using Parks– McClellan method The absolute approximation error between the maximum PB deviation and the expected PB ripple ðrp Þ, for each filter, can be expressed as ð9Þ ð10Þ x represents the vector of the optimized design variables concatenated together that will be obtained from the ABC ð11Þ Similarly, the absolute approximation error in the SB for each filter can be expressed as F2;i xị ẳ jFso;i xị rs j ð12Þ where rs represents the expected SB attenuation for each filter as per the design specification F1;i ðxÞ and F2;i ðxÞ, respectively represent the approximation errors in the PB and SB for the ith filter, i ð1; MÞ Finally, the multi-objective problem is formulated as a single objective optimization problem by applying appropriate weightages to the individual objective function terms as in equation below / ¼ M X a1;i F1;i xị ỵ a2;i F2;i xị iẳ1 s:t nðxÞ nb ð13Þ 342 T Bindima and E Elias Table Performance and complexity analysis for the VDF in the design example for different word lengths and different number of SPT terms Word length Number of SPT terms Best case SB attenuation (dB) Worst case SB attenuation (dB) No of adders for coefficient implementation Continuous coefficient 51.2067 48.5832 16 Maximum precision (8 SPT) 50.68 48.0656 343 14 Maximum precision (7 SPT) 50.58 50.5136 50.7870 39.2454 44.5968 44.035 34.9247 19.1314 282 271 235 173 The constraint restricts nðxÞ, the average number of nonzero SPT terms in the optimized vector x to within a maximum limit nb It ensures an optimal set of discrete sub-filter coefficients using a variable number of non-zero SPT terms for each coefficient, provided the average number of non-zero SPT terms across the vector does not exceed nb This approach is more flexible and gives a better optimum in terms of approximation accuracy along with the minimum number of SPT terms rather than that using CSD approximation with fixed number of non-zero terms [58] The constrained optimization problem can be converted into an unconstrained problem by incorporating the constraint into the objective function using the penalty parameter gxị ẳ maxnxị nb Þ to penalize any constraint violation Hence, the problem can be modeled as / ¼ M X a1;i F1;i xị ỵ a2;i F2;i xị ỵ a3 gxị 14ị iẳ1 a1;i ; a2;i and a3 represent positive weights that can be chosen by trial and error depending on the relative importance of each term, to satisfy the specifications Design of 1D continuous coefficient variable digital filters The initial part of our work involves the design of a reconfigurable 1D variable digital filter for varying cut-off frequencies using Farrow structure The design example considered by Pun et al [2] has been taken Design example Six VCF filters with PB edge frequencies xp equally spaced from 0:2p to 0:4p and transition bandwidth 0:2p have been designed Each of the filters is designed for an order of 31 using Parks–McClellan algorithm, for a minimum attenuation of 50 dB in the SB and a maximum ripple of 0.0045 dB in the PB The filters are chosen in the increasing order of their cut-off frequencies The delay parameter ðlÞ is selected as l where l ¼ corresponds to the filter with the lowest cut-off frequency and l ¼ to that with the highest The impulse response coefficients are approximated using a 5th order polynomial in l This approximation results in a set of sub-filter coefficients of size  32, which implies that the VDF requires 96 multipliers for coefficient implementation along with 191 structural adders and structural multipliers Design of CSD encoded sub-filter coefficients The multipliers required for coefficient implementation are eliminated by quantizing the sub-filter coefficients using 14bit CSD representation by rounding the sub-filter coefficients to the nearest CSD with restricted number of SPT terms Table gives a detailed analysis of the performance of the VDFs along with the implementation complexity for different word lengths and a different number of SPT terms The performance of the filters degrades due to the finite word length CSD rounding Based on an analysis of the performance degradation, the sub-filter coefficients encoded using SPT terms are used for further improvement to attain a worst case SB attenuation of 43 dB [2] Design of VDF using ABC The modified integer coded ABC algorithm [48] is utilized in this paper for optimizing the CSD encoded sub-filter coefficients so as to attain the required performance characteristics for the VDF The various steps of the algorithm are discussed below  Initialization: The initial set of food sources of the algorithm is formed by concatenating the CSD encoded subfilter coefficients Exploiting the coefficient symmetries, only half the set of each of the sub-filter coefficients need to be considered For a wider search space, the initial number of food sources is chosen as an integer multiple of the number of employed bees and hence more food sources are added to the initial set of food sources by random perturbation of the initial set The initial population size and the maximum number of iterations are chosen as 50 and 500, respectively  Employed bee phase: Each employed bee is initially associated with a unique food source and in each iteration the employed bee searches for a better food source in the vicinity of the current food source i by changing the parameter at the randomly selected jth location as xtỵ1 i; jị ẳ xt i; jị ỵ b/xt i; jị xt k; jịịc where / ½À1; 1Š is a random variable, xt ði; jÞ is the jth parameter of the ith food source, xtỵ1 i; jị is the jth parameter of the new food source, which falls within the limits of CSD look-up table by ensuring the following condition if xtỵ1 i; jị < vlb ; if xtỵ1 i; jị > vub ; then xtỵ1 i; jị ẳ vlb then xtỵ1 i; jị ¼ vub Design of efficient circularly symmetric 2-D variable digital FIR filters Table 343 Minimum and maximum values of the weights used for the objective function terms in Eq (14) Parameter weights Minimum value Maximum value Threshold limit a1;i a2;i a3 0.23481 0.3985 1.5999 5.0348 4.7985 4.5999 5.1822eÀ04 0.007 3.4 Fig Implementation transformation of generalized McClellan If d represents the initial set of coefficients being optimized, vlb and vub are initially chosen with a small deviation from d as d À 0:000001d and with a larger deviation as d À 0:01d It has been observed that larger the deviation, the lesser is the number of non-zero terms in the optimal filter coefficients, which in turn results in lesser number of adders but decreases the performance improvement, while, smaller deviation increases the number of non-zero terms and improves the performance in terms of SB attenuation and PB ripple By iteratively changing the deviation and based on a performance complexity trade-off, vlb and vub are finally chosen In this work, vlb and vub are selected as in Eqs (15) and (16), respectively vlb ẳ d 0:0022210999d 15ị vub ẳ d ỵ 0:000018966d 16ị A greedy selection approach is adopted to select the new food source or to discard and retain the old food source depending on its nectar quality or fitness function fitnessi  Onlooker bee phase: Based on the fitness information gathered by the employed bees about the food sources, the onlookers select a food source with a probability i [48], where N is the total number of food P i ¼ Pfitness N1 i¼1  Scout bee phase: If the fitness function of a particular food source does not improve even after a certain number of iterations equal to the ‘limit’ cycle, it gets abandoned, and the corresponding employed bee becomes the scout They also nd random food sources according to v ẳ randiẵlb; ub; dimÞ where randi is a random integer in the range [lb, ub] with the dimension of the food source ‘dim’ and memorizes the best source  Termination: Termination is achieved either when the approximation error is less than a tolerable value or when the number of iterations is greater than or equal to the maximum number of iterations fitnessi sources Hence, the food source with higher fitness function gets more onlookers The onlookers also search for a better food source similar to the employed bees and accept or reject it using the greedy approach The optimization is performed on the CSD encoded subfilter coefficients thereby making them multiplier-less with minimum performance degradation The sub-filters have been made completely multiplier-less, and the number of adders for implementing the sub-filter coefficients has been obtained as 231 Varying weights have been given in Eq (14) for the PB and SB approximation error terms of each filter as well as the penalty term These weights are chosen by comparing the performance parameters of the individual filters and the number of non-zero terms for the coefficient representation with their respective expected values Table shows the minimum and maximum value of the weights used for the PB, SB and penalty parameter terms in Eq (14) Minimum value of the weight of a1;i (i.e., PB approximation error) is applied to the cost function of the ith filter when the corresponding parameter value (i.e PB deviation of the particular filter) is below its expected value, denoted by the threshold limit in Table 3, and the maximum value of the weight is given when the value is higher than the threshold value Similarly, the minimum value of a2;i is applied to the ith filter if its SB attenuation goes below the corresponding threshold limit The weights were automatically adjusted in software, based on the relative variation of the desired parameters Design of 2D circularly symmetric VDF Adding up the products obtained by multiplication of the ABC optimized sub-filter coefficients with the various powered delay parameters ðli ; where i ẳ 0; 1; ; 5ị, followed by interpola- Table Results obtained for the design example using the proposed method based on CSD–ABC in comparison with the SOPOT based approach [2] Filter order Interpolation order Worst case SB attenuation Number of adders for coefficient implementation VDFs (CSD–ABC optimized) VDFs (SOPOT, [2]) 31 43.34 231 31 43.03 241 344 T Bindima and E Elias Fx1 ; x2 ị ẳ 0:3955 ỵ 0:5cosx1 ị þ cosðx2 ÞÞ þ 0:3955  cos x1 cos x2 1 Fz1 ; z2 ị ẳ ẵz1 ỵ z1 ị ỵ z2 ỵ z2 ị ỵ 0:3955ẵz1 þ z1 Þ Â ðz2 þ zÀ1 Þ À 4ŠŠ ð17Þ ð18Þ The implementation of the transformation in Eq (18) is shown in Fig Using the 1D CSD–ABC optimized filter coefficients, the 2D filter is implemented at the expense of one multiplier exclusively for 2D mapping, which in turn is also made multiplier-less The entire design process using the proposed approach has been elaborated in Algorithm 20 ABC optimized Filter CSD rounded Magnitude response in dB tion, gives the filter coefficients of the six 1D VDFs with variable cut-off frequencies The generalized McClellan transformation [18] given by Eq (17) with the transfer function as in Eq (18), has been used to map the 1D filter to the 2D scenario −20 −40 −60 −80 −100 −120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized frequency in radians (ω/π) Fig Frequency response of the VDFs in the design example with CSD rounded and ABC optimized sub-filter coefficients Algorithm Proposed design for finite precision low complexity 2D VCF filters 1: procedure FINITE PRECISION 2D VCF FILTER DESIGN USING ABC ALGORITHM 2: Obtain the sub-filter coefficients of the Farrow filter using polynomial interpolation 3: Encode the sub-filter coefficients using CSD 4: Form the initial food sources by random perturbation of the encoded coefficients and prioritize them 5: repeat 6: Place the employed bees at selected number of food sources and evaluate their nectar quality 7: Search new food sources for the employed bees with better nectar quality than the current food sources 8: Place the onlookers on new food sources and evaluate their nectar quality 9: Search new food sources for the onlooker bees with better nectar quality than the current food sources 10: if (Limit Cycles completed) then 11: Generate scouts and find new food sources for them in place of the abandoned food source 12: end if 13: until (Convergence condition achieved) 14: Obtain the optimal filter by decoding the food sources 15: Reconstruct the 1D filter coefficients corresponding to each VCF using the optimized sub-filter coefficients and the delay parameters 16: Apply generalized McClellan transformation 17: end procedure MODIFIED INTEGER CODED Results and discussion All simulations were performed on core i3 processor operating at 1.7 GHz using MATLAB R2011b The continuous coefficient VCF filters designed using Farrow structure result in a worst case SB attenuation of 48.5832 dB When the subfilter coefficients are encoded using 14-bit CSD representation with maximum precision, it results in a worst case SB attenuation of 44.5968 dB For improved SB attenuation, the word length is to be increased 16-bit word length with maximum precision gives a worst case SB attenuation of 48.0656 dB as shown in Table Since our focus is to attain a minimum SB attenuation of 43.03 dB as in the results of Pun et al [2], 14-bit word length is sufficient Further, when the coefficients are rounded using lesser number of SPT terms, the worst case SB attenuation again decreases as shown in Table Hence, the performance gets degraded by this rounding to the nearest CSD with restricted number of SPT terms Since our aim is to attain a worst case SB attenuation of 43 dB, 14-bit CSD representation with SPT terms has been used, which gives a minimum SB attenuation of 34.9247 dB Thus, there is a degradation of about 10 dB in the SB attenuation from its maximum precision performance The number of adders required to implement the filters with SPT terms is 235 The performance of the filters can be improved up to that of maximum precision using ABC optimization This results in a slight deviation in the number of non-zero terms in the CSD representation of each sub-filter coefficient, but the average number of non-zero terms across the entire set of coefficients remains less than or equal to nb ¼ The optimization algorithm has been performed for 10 iterations and the result closely matching the average has been selected The results obtained for the design of six VCF filters in the design example are shown in Table 4, and the number of adders required for coefficient implementation has been obtained as 231 compared to 241 adders obtained by Pun et al [2] The worst case SB attenuation has been improved to 43.34 dB from 34.9247 dB after using ABC optimization Thus for the same worst case SB attenuation, reduced complexity filters have been obtained by using our approach The magnitude response of the 14-bit CSD rounded filters, and CSD–ABC optimized filters is shown in Fig The 2D magnitude response and the corresponding contour plots for different delay parameters l are shown in Fig 6a–c The contour plots show that as the delay parameter l increases, the cut-off radii of the 2D filter increase Also, 2D VDFs with nearly circular contours, even at wide band radii are attained due to the use of the generalized McClellan transformation Design of efficient circularly symmetric 2-D variable digital FIR filters 345 0.5 0.4 0.2 0.1 0.8 ω2/π Magnitude Response 0.3 1.2 0.6 0.4 −0.1 0.2 −0.2 0.5 −0.3 0.5 −0.4 0 ω2/π −0.5 −0.5 ω /π −0.5 −0.5 (a) μ 0.5 ω /π 0.5 0.4 0.2 0.1 0.8 ω2/π Magnitude Response 0.3 1.2 0.6 0.4 −0.1 0.2 −0.2 0.5 −0.3 0.5 −0.4 ω2/π −0.5 −0.5 ω1/π −0.5 −0.5 (b) μ 0.5 ω1/π 04 0.5 0.4 0.2 0.1 0.8 ω2/π Magnitude Response 0.3 1.2 0.6 0.4 −0.1 0.2 −0.2 0.5 −0.3 0.5 −0.4 0 ω2/π −0.5 −0.5 −0.5 −0.5 ω /π (c) μ 0.5 ω1/π Fig 2D magnitude responses and the corresponding contour plots obtained after transforming the 1D VDF in the design example using generalized McClellan transformation 346 Conclusions In this paper, a review of the different approaches for the design of 2D variable digital FIR filters is done, and a novel approach for designing low complexity 2D circularly symmetric 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power-of-two term allocation scheme for the design of digital filters IEEE Trans Circ Syst II: Analog Digit Signal Process 1999;46(5):577–84  .. .Design of efficient circularly symmetric 2-D variable digital FIR filters Keywords: 2D circularly symmetric FIR Variable digital filters Variable fractional delay filters... statement” and ‘ Design of 1D continuous coefficient variable digital filters”, respectively Section ‘ Design of 2D circularly symmetric VDF” gives the design of 2D circularly symmetric VDFs Results... filter design Design of efficient circularly symmetric 2-D variable digital FIR filters where the coefficients t00 ; t10 ; t01 and t11 control the shape of the contour The final frequency response of