Design and implementation of a novel low complexity symmetric orthogonal wavelet filter‐bank

IET Image ProcessingVolume 13, Issue 5 p. 785-793 Research ArticleFree Access Design and implementation of a novel low complexity symmetric orthogonal wavelet filter-bank Pranose J. Edavoor, Corresponding Author Pranose J. Edavoor pranose@gmail.com Department of Electrical & Electronics Engineering, National Institute of Technology Goa, Farmagudi, Goa, 403401 IndiaSearch for more papers by this authorAmol D. Rahulkar, Amol D. Rahulkar Department of Electrical & Electronics Engineering, National Institute of Technology Goa, Farmagudi, Goa, 403401 IndiaSearch for more papers by this author Pranose J. Edavoor, Corresponding Author Pranose J. Edavoor pranose@gmail.com Department of Electrical & Electronics Engineering, National Institute of Technology Goa, Farmagudi, Goa, 403401 IndiaSearch for more papers by this authorAmol D. Rahulkar, Amol D. Rahulkar Department of Electrical & Electronics Engineering, National Institute of Technology Goa, Farmagudi, Goa, 403401 IndiaSearch for more papers by this author First published: 21 March 2019 https://doi.org/10.1049/iet-ipr.2018.6200Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The design of wavelet filter-bank in real continuum domain demands infinite precision to achieve perfect reconstruction (PR). The computations with irrational coefficients in orthogonal filters require a large amount of hardware that result in increased power dissipation, huge memory requirements and reduction in speed of operation. This paper presents a new approach to obtain a complete dyadic (low complexity) 6-tap orthogonal symmetric wavelet filter-bank (FB) with near perfect reconstruction. This is achieved by slightly altering the PR conditions to make orthogonal filters symmetric and to obtain complete dyadic filter coefficients. The proposed wavelet FB has reduced dynamic power dissipation and significantly reduced adder and shifter count. The VLSI architecture of the proposed FB is designed and implemented on Kintex-7 FPGA. The architectural design is carried out using Verilog and functional simulation and synthesis are executed using Vivado 2016.4 software. The proposed FB dissipates 114mW dynamic power and requires only 5 adders and 3 shifters. The effectiveness of the proposed wavelet is verified in three different applications namely image compression, iris recognition system and orthogonal frequency division multiplexing (OFDM). The proposed wavelet FB gives comparable performance in image compression and attains superior performance in iris recognition system (feature extraction) and OFDM. 1 Introduction The discrete wavelet transform (DWT) has been found as a very powerful tool in a wide range of applications such as image and video coding, bio-metrics, pattern recognition, data mining etc. [1, 2]. The performance of DWT-based applications is primarily dependent on the choice of filter-bank (FB). The main focus of the wavelet FB theory is to design critically sampled wavelet FBs with some desirable properties such as regularity [vanishing moments (VMs)], symmetry, orthogonality, frequency selectivity etc. It is found that cdf-9/7 bi-orthogonal [3] and Daubechies orthogonal (db1–db10) [4, 5] FBs are widely used in most of the applications. Both these wavelet FBs have their respective advantages, the orthogonal wavelet FB is preferred due to its orthogonality property (energy preserving) [6]. It is found that db2 and db3 orthogonal wavelets are used in a wide variety of applications. Generally, the db2 wavelet is used in gradually varying signals and db3 wavelet performs well in applications where signals have abrupt changes and spikes at high noise levels [7]. Also, the db3 wavelet extracts better details from medical images than db2. However, orthogonal wavelets (db2, db3 etc.) are not symmetric and suffer from the non-linear phase. Phase linearity plays a very important role in handling boundary distortions in images [6]. The linear phase or symmetry in wavelets is achieved by the design of bi-orthogonal wavelet FBs (cdf-9/7). However, bi-orthogonal wavelets do not preserve energy in their transform coefficients. Hence, it is necessary to design a wavelet FB which satisfies orthogonality as well as symmetry properties. Only Haar or db1 wavelet satisfies both orthogonality and symmetry properties. However, Haar suffers from the following: (i) centre frequency changes from 0 to . (ii) The magnitude of the side lobes gets doubled. (iii) Time-bandwidth (BW) product is infinite. (iv) The scaling function is not confined in the frequency domain. These problems can be solved by increasing the order of wavelet filters. This can be achieved by imposing more number of VMs at under perfect reconstruction (PR) conditions. However, Daubechies orthogonal wavelets designed with these properties give irrational filter coefficients. Hence, infinite precision hardware is required to efficiently represent the irrational filter coefficients. This creates an overhead on hardware and affects the speed of operation. Generally, the higher-order wavelet filters are not suitable for hardware implementation. Hence, it is necessary to design an orthogonal wavelet FB with dyadic filter coefficients. This study presents a new approach to obtain a complete dyadic symmetric six-tap (alternative for the db3 filter by introducing only 0.78% error in PR) near PR orthogonal wavelet FB. In the proposed approach, an error is introduced in the PR condition to achieve symmetry in the design of orthogonal wavelet FB. With this, the proposed six-tap wavelet filter obtains complete dyadic filter coefficients. Next, the designed wavelet has been implemented [very large scale integration (VLSI) architecture] by the use of only shifters and adders and eliminated the need for floating point multiplier in wavelet filter implementation. It is observed that the proposed design reduced hardware usage considerably as compared to the traditional implementation of db3 wavelet filter. The proposed FB has been implemented and verified on Kintex 7 field programmable gate array (FPGA). The performance of the designed wavelet is validated in three different applications, i.e. image compression, iris recognition and orthogonal frequency division multiplexing (OFDM). The structure of the paper is organised as follows. Section 2 presents a brief overview of the existing wavelets. The design of orthogonal wavelet FB is presented in Section 3 that includes a detailed design of the proposed complete dyadic near-PR symmetric orthogonal wavelet FB, VLSI architecture and its implementation on FPGA. Section 4 includes the experimental study of the effectiveness of the proposed filter in different applications such as image compression, iris recognition system (feature extraction) and OFDM. Concluding remarks are presented in Section 5. 2 Related work It is well known that Daubechies orthogonal wavelet FBs have a maximum number of VMs and are not symmetric. These filters are designed by the use of minimum phase spectral factor of maximally flat product filter [8]. Due to its non-symmetric nature, these orthogonal wavelets are not suitable in many signal processing applications such as feature extraction, compression etc. Hence, Daubechies [9] introduced a new class of wavelets named as symmlets in order to obtain near-symmetric wavelets. These filters are obtained from the maximally flat product filter by the modification in phase to obtain an approximately linear phase. This study also mentioned the design of orthogonal wavelet FBs (db2, db3 etc.). Faras and Selesnick [10] designed orthogonal filters with near symmetry property using the Grobner method. Selesnick et al. [11] suggested a nearly symmetric orthogonal wavelet filter by the use of flat group delay and Grobner method. Wei and Bovik [12] presented a new class of compactly supported orthogonal and nearly symmetric wavelet called as generalised coiflet by imposing non-zero-centred VMs on scaling function. Zhao and Swamy [13] proposed a near orthogonal symmetric wavelet filter by the use of non-linear optimisation. Recently, a highly localised near orthogonal wavelet filter bank is designed for graph theory [14]. It is also found that some of the researchers [15-17] proposed techniques to achieve symmetry and orthogonality in case of non-separable multi-dimensional orthogonal wavelet FBs. However, it is found that the aforementioned orthogonal and near symmetric or symmetric and near orthogonal wavelet filter coefficients are irrational numbers [6, 18] and are difficult to represent accurately in standard finite precision number system (two's complement fixed-point format). Due to the presence of irrational numbers, orthogonal wavelets (particularly db3) require a large amount of hardware. This results in increased power dissipation, huge memory requirement, and a reduction in the speed of operation. Most of the researchers have explored different techniques to reduce hardware complexity by the use of either direct rationalisation of filter coefficients [19, 20] or lifting schemes [21-26]. Lifting scheme [27] is utilised to migrate into the binary domain. This results in the reduction of VMs, which impose irregularities on filter response. In these methods, the conventional fixed-point binary representation is used to implement irrational wavelet coefficients. However, these methods suffer from the introduction of round-off errors in computation that leads to degradation in image reconstruction [7]. It is because approximation can result in propagation of errors along the array of filters. The extent of error totally depends on the length (of the array) of the filter. One of the schemes to implement wavelet basis functions is integer wavelet transform [28-30]. However, it requires large adder trees with higher word length. The control circuitry requirement is complex leading to high cost and degraded reconstruction [7]. Madishetty et al. [31] have proposed an algebraic integer (AI)-based technique in Daubechies db2 and db3 orthogonal Wavelet FBs. However, this architecture incurred a huge amount of hardware overhead. In this, the filter coefficients of db2 and db3 remain unaltered. An efficient implementation has been suggested in [7, 32] to eliminate the error caused by rounding-off of irrational wavelet filter coefficients. In their implementation, filter characteristics are achieved by mapping basis functions of irrational transforms with integers using AI quantisation. However, this technique incurs a huge cost in terms of area and hardware utilisation. Also, throughput is very less due to the over utilisation of huge internal computing units. Recently, Le et al. [33] proposed an approach to design rational orthogonal wavelet FBs (db3) using Grobner basis. This basis is used to find better lattice parameters for rationalising db3 irrational filter coefficients. Hasan and Wahid [34] suggested the combination of lifting scheme and integer polynomial mapping to implement db2 and db3 wavelet FBs efficiently. It is found that the aforementioned rationalisation methods for orthogonal wavelet FBs have reduced hardware overhead. However, these methods suffer from non-linear phase (non-symmetry). Hence, this study presents a new class of dyadic orthogonal symmetric wavelet FB with near PR condition. The designed wavelet reduced hardware complexity significantly and achieved good recognition rate and reconstruction quality. 3 Design of proposed wavelet FB 3.1 Review of orthogonal wavelet FB design The block diagram of a two-channel wavelet FB is shown in Fig. 1. Assume and to be the synthesis low pass filter (LPF) and high pass filter (HPF) and and to be the analysis LPF and HPF in a two-channel wavelet FB. The HPFs and can be derived from LPFs by quadrature mirroring as expressed in (1) and (2) (1) (2) Fig. 1Open in figure viewerPowerPoint Two-channel wavelet filter bank The condition for PR in terms of , , and is obtained from (3), where c is a constant (generally c is considered to be 2) (3)In the design of orthogonal FBs, LPFs and are mirror images of each other as given by (4). (4)The two-scale dilation or scaling function is defined as (5)where is a LPF, which contains at least one zero at and satisfies the normalisation condition . Similarly, the corresponding wavelet function is defined as (6)where is a HPF and is obtained from LPF using , where N is the support of . The orthogonal space spanned by scaling function and wavelet function is generated by the use of double-shifting property on LPF . This double shifting property can be written as (7)VMs can be ensured by applying sum rules on filter coefficients and is expressed as (8)where k can be varied from ; where v is the number of VMs (sum rule). It is observed from (8) that v sum rules are required for v VMs. When the value of (means if there are more than one VM), then there is no dc leakage and can be defined as (9)Thus, it ensures that there is a zero dc leakage. 3.2 Proposed class of complete dyadic near-PR symmetric orthogonal wavelet FB This study emphasised on the design of six-tap orthogonal wavelet FB. The proposed design is able to achieve dyadic filter coefficients by slightly altering the PR condition. Consider six-tap filter coefficients of analysis LPF as . The PR condition for orthogonal wavelet FB is derived from the use of double shifting property (7) as follows: (10) (11)Equations (10) and (11) are called as first and second double shifting equations, respectively. The shift invariance and variable resolution-based analysis give a better representation of signals using dyadic transformation on filter coefficients at all scales. Hence, it enables the implementation of filters by the use of only shifters and adders which occupies less area on a chip rather than the area and power consuming multipliers. Thus, this study presents a dyadic transformation by introducing very small errors and in first and second double-shifting equations (10) and (11), respectively. It is found from the extensive analysis that when the total error , a fully dyadic (efficient VLSI implementation) and efficient orthogonal symmetric wavelet FB can be obtained. Also, this dyadic and symmetric wavelet FB satisfies near PR condition. The design of the proposed wavelet FB is as follows: Start with the design of db3 (six-tap) orthogonal wavelet FB. It is well known that db3 indicates three VMs and support . Using this information, analysis LPF can be written as (12)This can be expanded as (13)This implies (14)The parameters p and q of are to be designed in such a way to get dyadic coefficients of (analysis LPF). It is well known that orthogonal wavelets cannot achieve linear phase (symmetry) along with PR condition. Some of the researchers [19, 32] tried to achieve linear phase with orthogonality and PR by relaxing orthogonality conditions (leads to bi-orthogonal wavelet designs) and by relaxing symmetry conditions (which leads to symmlets [9] design). It is found that both these designs satisfy PR condition. Also, it is observed from the literature that the rationalisation of filter coefficients can be achieved either by removing VMs or by slightly altering the PR condition [35]. In this study, a new approach is suggested to obtain rational or dyadic coefficients along with orthogonality and symmetry by slightly altering PR condition. The first and second double shifting (PR condition equations) expressed in (10) and (11) are modified to obtain linear phase (symmetric) with dyadic coefficients as (15) (16)Here, a small number of errors (ideally = 0) and (ideally = 0) are induced in the PR condition equations to obtain dyadic transformation. First, the symmetry condition is achieved in the proposed wavelet by assuming in analysis LPF . With this, the resultant analysis LPF expressed in (14) can be rewritten as (17)It is observed that the first and last coefficients of the analysis LPF are the same by assuming . Now, it is necessary to decide the parameters p and q in such a way that the filter achieves complete symmetry with dyadic coefficients. Let , , , , , and . With these assumptions, the modified first and second double shifting equations (15) and (16) are expressed in terms of the parameters p and q as (18)and (19) It is observed from the aforementioned (12)–(19) that the resultant coefficients may be rational or irrational. Most of the existing designs have obtained PR condition with better frequency response by the appropriate selection of p and q values. With this, the resultant coefficients are irrational and require infinite precision in hardware implementation. In this study, the issue of irrationality is solved by a small amount of compromise in PR condition. This is achieved by selecting the combination of the term as a dyadic number. Thus, the proposed dyadic transformation is expressed as (20)By substituting this condition in (13) with , the resultant LPF can be expressed as (21)where is used as a normalisation constant to get dyadic coefficient and . Next, this dyadic condition (20) is substituted in modified first and second double shifting equations (18) and (19) to obtain dyadic coefficients with symmetry as (22)and (23) Now, the aim is to find the values of such that both the errors and are as small as possible (approximately zero). It means total error in PR condition has to be minimised. This can be done by either minimising two individual errors ( and ) or two combined errors . The best way to formulate this is by applying the minimisation constraints on and such that total error (24) In this study, a well-known root mean squared error approach is used to optimise the values of by the minimisation of constraint . Thus, the objective function is defined as and is expressed as (25) (26)where is a normalisation factor. The value of is found by adding all the coefficients of given in (21) and is expressed as: . Thus, has been expressed in terms of a and . The continuous regression minimisation method is used to obtain the optimised values of a and . The steps to find the values a, , and are as follows: Decide the range of a as . Initialise randomly , ideally should be varied from to . Compute error . Next, decide the values of a and for the minimum error. Repeat these steps for the different iterations. Decide the minimum error from different iterations and select the values of a and . Finally, obtain the value of from the optimised values of a and . With this extensive experimentation (with different iterations), the values of a, and are selected as −1, 2 and 16, respectively. The results with two different selected iterations (i.e. –−100:0 and −50:50) and the zoomed version of −50:50 are shown in Figs. 2-4, respectively, in terms of error (), a and . It is observed from these graphs that error is minimum (0.0078) in Fig. 3 and its zoomed version in Fig. 4. The values of a and corresponding to this minimum error are −1 and 2 and are selected as optimum values in order to achieve dyadic and symmetry properties with near PR in the designed orthogonal wavelet FB. Thus, the dyadic transformation is written as (27)and (28)With this, the value of induced error in the proposed design is . Fig. 2Open in figure viewerPowerPoint Plot for error () versus a and for ranging from −100 to 0 and a ranging from −1 to 1 Fig. 3Open in figure viewerPowerPoint Plot for error () versus a and for ranging from −50 to 50 and a ranging from −1 to 1 Fig. 4Open in figure viewerPowerPoint Plot for error () versus a and for ranging from −5 to 5 and a ranging from −1 to 1 (zoomed version of −50 to 50) Finally, the values of p and q are re-substituted in the resultant analysis LPF and is expressed as (29) Thus, the proposed design has achieved completely dyadic coefficients and symmetry with 0.78% error in PR condition. With this, the proposed wavelet FB preserved 99.22% PR property. Hence, the designed wavelet named as complete dyadic near-PR symmetric orthogonal wavelet FB. The frequency responses of the proposed analysis LPF and HPF are compared with the same length original db3 [9] and recently published FBs [18, 33, 34] as shown in Fig. 5. The zoomed version is shown in Fig. 6. It is observed that the frequency response of the proposed FB gives less energy of error as compared to existing FBs [18, 33, 34]. Fig. 5Open in figure viewerPowerPoint Comparison of filter characteristics of the proposed filter with existing filter banks Fig. 6Open in figure viewerPowerPoint Zoomed version of filter characteristics of proposed FB with existing FBs 3.3 Energy distribution of the designed wavelets It is known that energy distribution in the wavelet sub-bands after decomposition decides the suitability of wavelet for any application [35]. Table 1 shows the comparison of the percentage energy distribution of the designed wavelet with original db3 [9] and existing wavelet FBs [18, 33, 34]. The values in Table 1 are obtained after first level wavelet decomposition of standard Baboon, Parrot, Lena and Barbara images. It is observed that the proposed FB preserves energy distribution when compared with existing FBs [9, 18, 33, 34]. Table 1. Comparison of the energy distribution of proposed FBs with existing FBs for different images Images Energy distribution Original db3 [9] Murugesan and Tay [18] Le et al. [33] Hasan and Wahid [34] Proposed wavelet FB Baboon 99.9744 99.9451 99.9532 99.9672 99.9754 0.0112 0.0270 0.0234 0.017294 0.0094 0.0121 0.0257 0.02125 0.01332 0.0134 0.0022 0.0021 0.002148 0.002186 0.0018 Parrot 99.9410 99.9132 99.9218 99.9327 99.9441 0.0281 0.03899 0.03567 0.0341 0.0274 0.02851 0.04508 0.03991 0.03073 0.0263 0.00239 0.00273 0.00262 0.00247 0.0022 Lena 99.0031 98.8370 98.9213 98.9560 99.0910 0.4550 0.5934 0.5346 0.4689 0.4103 0.5136 0.56659 0.54122 0.57224 0.49589 0.00283 0.00301 0.00288 0.00286 0.00281 Barbara 98.7233 97.9001 98.3121 98.6711 98.7824 0.6327 0.9961 0.7964 0.6479 0.6012 0.64093 1.10062 0.88838 0.67791 0.61336 0.00307 0.00318 0.00312 0.00309 0.00304 Table 2 shows the comparison of property measures in terms of Sobolev regularity, time frequency product and energy in error for the proposed FB with existing FBs [9, 18, 33, 34]. Sobolev regularity [36] is a measure of the regularity (smoothness) of wavelet and scaling functions. The trade-off between better time and frequency localisation is bounded by uncertainty principle [37]. Ideally, the time and frequency spread are bounded below by 0.25. Energy in error is the measure of the deviation from ideal filter characteristics. Lower the value of energy in error, closer the filter characteristics to the ideal filter characteristics. It is observed that the proposed FB achieved slightly better Sobolev regularity, time frequency product and has lower energy in error than existing FBs [9, 18, 33, 34]. Thus, the proposed FB is suitable for any application with low complexity due to its complete dyadic coefficients. Table 2. Comparison of property measure of proposed FBs with existing FBs Filter banks Sobolev regularity Time frequency product Energy in error original db3 [9] 1.415 0.5667 29.1011 Murugesan and Tay [18] 1.409 0.5458 29.1081 Le et al. [33] 1.411 0.5471 29.1065 Hasan and Wahid [34] 1.4137 0.5548 29.1034 proposed wavelet FB 1.4408 0.3607 28.1009 3.4 Design of VLSI architecture for proposed wavelet FB Since, the denominator of the designed filter coefficient is integer powers of 2, the division operation can be implemented using right shifters. In addition, numerator values of the proposed filter coefficients are either 1 or 8. Hence, these coefficients can be implemented using left shifters. Therefore, the proposed FB can be executed only by the use of adders and shifters. With this, the use of hardware consuming multipliers can be completely removed. Hence, the proposed FB reduces the hardware considerably. The architecture of the proposed wavelet FB is shown in Fig. 7. As the values of designed filters and are , so inputs and are passed to the adder circuit as it is. The division by 16 for all filter coefficients is carried out after computing the sum of all products. In order to compute the negative coefficients of the filters ( and are ), two's complement of and are passed into the adder. To realise and whose values are , left shift register (which shifts the input three times to the left) followed by a two's complement is used to facilitate the negative sign in the filter coefficient. Fig. 7Open in figure viewerPowerPoint Hardware architecture for proposed FB This proposed VLSI architecture is implemented using the FPGA platform with a KC705 evaluation kit (Kintex 7 FPGA from Xilinx). It is because FPGA implementation enables the functional testing of the proposed filter on a hardware platform. The modular design methodology in digital design enables the reuse of hardware and thereby reduces the hardware consumption. Also, VLSI architectures have been implemented for five different existing same length orthogonal FBs using the same platform (Kintex 7 FPGA from Xilinx). These are original db3 [9], Murugesan and Tay [18], Madishetty et al. [31], Le et al. [33], and Hasan and Wahid [34]. The performance of the proposed FB is compared with the existing FBs [9, 18, 31, 33, 34] in terms of dynamic power dissipation, adders, multipliers, and shifters. The results are tabulated in Table 3. It is observed that the proposed dyadic FB dissipates 114 mW dynamic power and requires only five adders and three shifters as hardware resources. Thus, the proposed wavelet FB reduced dynamic power dissipation, adders, and shifters significantly than existing FBs [9, 18, 31, 33, 34]. Hence designed wavelets can be used efficiently in real-time multimedia applications. Table 3. Comparison of the proposed filter with the existing FBs with adder counts Filter banks Dynamic power dissipation, mW Adder count Shifter count multiplier Count Digital Signal Processing (DSP) count db3 with 16bit precision [9] 214 7 — 1 1 Murugesan and Tay [18] 189 12 18 0 0 Madishetty et al. [31] 236 25 42 1 1 Le et al. [33] 187 5 5 1 1 Hasan and Wahid [34] 173 15 32 0 0 proposed wavelet FB 114 5 3 0 0 4 Application of proposed orthogonal wavelet FB In this study, the effectiveness of the proposed orthogonal wavelet FB was verified in three different applications namely image compression, iris recognition system, and OFDM. 4.1 Image compression The proposed wavelet FB has been tested for its possible application in image compression. The performance of the proposed wavelet has been tested on four standard images – Lena, Barbara, Parrot, and Baboon. The peak-signal to noise ratio (PSNR) is used as a performance measure for different word-lengths of filter coefficients (4-bits and 8-bits). The image compression algorithm by the use of the proposed and existing FBs [9, 18, 31, 33, 34] has been implemented in FPGA architecture with a Kintex7 KC-705 evaluation kit. The selected images have been decomposed into two levels by the use of the proposed wavelet FB. Next, all the detail sub-band coefficients are encoded as zero and only approximate sub-band has been considered for the reconstruction. Table 4 shows the comparison of hardware resources utilised by image compression algorithm using the proposed and existing FBs [9, 18, 31, 33, 34] for different word lengths in terms of monitored resources (number of slice registers, look-up tables (LUTs), and number of configurable logic blocks (CLBs)) and figure of merits (critical path delay (CPD), area time product (AT), area time product square (AT)2 and maximum operating frequency (Max. Freq)) [31]. The LUTs and maximum operating frequency have been selected as performance measures for comparison of hardware resource utilisation. In addition, Table 5 shows the comparison of PSNR obtained for four standard testing images with two different word-lengths (4-bits and 8-bits). Table 4. Comparison for the effect of word length on the performance of FB in terms of hardware utilisation Parameters Original db3 [9] Murugesan and Tay [18] Madishetty et al. [31] Le et al. [33] Hasan and Wahid [34] Proposed Wavelet FB 4 bit 8 bit 4 bit 8 bit 4 bit 8 bit 4 bit 8 bit 4 bit 8 bit N/A slices 2342 2774 1796 2364 2174 2482 1423 2177 1902 2424 1978 LUTs 6548 8564 5002 7683 6098 8119 4103 7218 5672 7924 5136 CLB 3786 4204 2996 3375 3279 3916 2312 2998 3120 3697 3122 CPD, ns 3.