Multiplicationless DFT Calculation Using New Algorithms

Transcription

1 Multiplicationless DT Calculation Using ew Algorithms Jaya Krishna Sunkara, Chiranjeevi Muppala PG Scholar, SVUCE, Tirupati, IDIA. Asst. Prof., TJS College of Engineering, Anna University, IDIA. Abstract ew telecommunication systems are based more than ever before on digital signal processing. High speed digital telecommunication systems such as ODM and DSL need real-time high-speed computation of the Discrete ourier Transform. Thus there is a need of innovative methods to improve the speed. In this paper, we propose new methods to implement DT in an efficient way. The proposed schemes come with the simplest and efficient way to solve the complex DT calculation with less number of multiplications and additions. The methods proposed change the DT and T algorithms by considering the values of input signal. The simulation results have verified that the number of complex additions and multiplications have reduced drastically when compared to the direct evaluation and T.. Introduction rom a theoretical point of view, the complexity issue of the discrete ourier transform has reached a certain maturity. ote that Gauss, in his time, did not even count the number of operations necessary in his algorithm. In particular, Winograd's work on DTs whose lengths have coprime factors both sets lower bounds (on the number of multiplications) and gives algorithms to achieve these []-[3], although they are not always practical ones. Similar work was done for length-" DTs, showing the linear multiplicative complexity of the algorithm [4]-[7] but also the lack of practical algorithms achieving this minimum (due to the tremendous increase in the number of additions [8][9]). Considering implementations, the situation is of course more involved since many more parameters have to be taken into account than just the number of operations. evertheless, it seems that both the radix-4 and the split-radix algorithm are quite popular for lengths which are powers of, while the PA, thanks to its better structure and easier implementation, wins over the WTA for lengths having coprime factors []- []. Recently, however, new questions have come up because in software on the one hand, new processors may require different solutions (vector processors, signal processors), and on the other hand, the advent of VLSI for hardware implementations sets new constraints (desire for simple structures, high cost of multiplications versus additions). ew algorithms are proposed in [3]-[9] recently. In this paper four methods are proposed for the implementation of DT with less number of additions and multiplications. The method- method will require less number of complex multiplications and complex additions as compared with the direct evaluation method of DT. method- method relays on the property of complex conjugation and it is applicable to real sequence only. The method- method will require less number of complex multiplications and complex additions as compared with the method- method. In method- method, the property of twiddle factor was exploited. T is superior to the first two methods. But, the method-3 method avoids all trivial multiplications in T. Hence, it reduces the total number of complex multiplications to lesser than that of T. By using method-4 method, in cases the number of multiplications and additions can be zero. This can be done by checking the input. The rest of the paper is structured as follows. The next two sections discuss the first two methods proposed. Section IV presents standard T. Section V and VI presents the remaining two methods proposed. Section VII concludes the paper.. Method - A notable reduction in number of multiplications and additions is possible by applying the following property: * X ( k) X ( k) Here we need to calculate first coefficients using direct evaluation DT. The remaining coefficients can be evaluated with no extra multiplications or additions. By using method- method will require less number of complex multiplications and complex additions as compared IJCTA Mar-Apr 5 3

2 with the direct evaluation method of DT. In method- method, the number of multiplications for point DT is and the number of additions is ( ), where direct evaluation of DT requires number of complex multiplications and ( ) complex additions. TABLE I: UMBER O COMPLEX MULTIPLICATIOS AD COMPLEX ADDITIOS I METHOD- ig. Evaluation of 4-point DT using method- method umber of umber of or example, in 8 point DT can evaluated first 5 (i.e. + (/) =+8/ = 5) terms are evaluated and then next 3 terms are evaluated directly by using the conjugation property. X (5) = x*(8-5) = x*(3) X (6) = x*(8-6) = x*() X (7) = x*(8-7) = x*() 3. Method- The method- method will require less number of complex multiplications and complex additions as compared with the method- method. In method- method, the property of twiddle factor was exploited. T is superior to the first two methods. In method- by applying decimation operation in time, we can add the multiplicands of W p, W p+, etc., hence reducing the total number of. The 4-point DT calculation with respect to figure is given below. ig. Evaluation of 8-point DT using method- method orm the above diagram, the number of complex multiplications as well as complex additions are log = 8log 8 = 4 TABLE II: UMBER O COMPLEX MULTIPLICATIOS AD COMPLEX ADDITIOS I METHOD- umber of umber of Stage-I Stage-II () x() () x() () x() () x() x() x() x(3) x(3) X () () W X () () W X (3) () W () () Here the number of multiplications and additions are reduced as compare to method. In this method there are log multiplications and additions, in each butterfly diagram, for each one there are two multiplications. The evaluation of 8-point DT using method- is shown in figure. X () () W 3 () () 4. Standard ast ourier Transform DT is one of the most important tools in the field of digital signal processing. Due to its computational complexity, several T algorithms have been developed over the years. The most popular T algorithms are the Cooley-Tukey algorithms. It has been shown that the decimation-in-time (DIT) algorithms provide better signal-to-noise-ratio than the decimation-in-frequency algorithms when finite word length is used. The T butterfly structure is shown in the figure 3. IJCTA Mar-Apr 5 3

3 ig. 3 Butterfly structure of standard T ig. 5 Evaluation of 8-point DT using Method-3 TABLE III: UMBER O COMPLEX MULTIPLICATIOS AD COMPLEX ADDITIOS I METHOD-3 umber of umber of ig. 4 DIT-T calculation DIT-T calculation is shown in the figure 4. DIT is used to calculate the DT of a point sequence. The idea is used to break the point sequence into two sequences, the DT of which can be combined to give the DT of the original point sequence. Initially the -point sequence divided into (/) point sequences X e (n) and X o (n), which have the even and odd members of X(n) respectively. The (/) point DT of these two sequences are evaluated and combined to give the - point DT. 5. Method - 3 In this method-3 all trivial multiplications, i.e., multiplications with W are reduced. Hence, it reduces the total number of complex multiplications to lesser than that of T. In the calculation of -point DT using T (-) such multiplications are there. So, the number of complex multiplications in this method-3 method is log. The evaluation of DT using method-3 is shown in figure 5. The table 3 gives the number of complex additions and multiplications required in method Method-4 In method-4, depending on the inputs to every stage, the multiplications and additions will be performed as and when required. If the one of the input sample is say, then the first multiplication need not be performed, and if the input sample is, both the addition and multiplications are not required. Like this, in cases, DT can be found with zero multiplications also. The table 4 shows the comparison of number of multiplication of various techniques is given. The comparison is plotted in the figure 6. Obviously, the method-4 requires at most the number of additions and multiplications as required by the method-3. Hence the worst-case performance of method-4 is that of method- 3. ow, the best and better cases of method-4 is given in tables 5 and 6 one for 4-point and other for 8-point. TABLE IV: UMBER O MULTIPLICATIOS I THE CALCULATIO O DT DT Method- Method- T Method IJCTA Mar-Apr 5 3

4 References ig. 6 Comparison of number of multiplications required in different techniques TABLE V: UMBER O MULTIPLICATIOS REQUIRED I VARIOUS TECHIQUES OR DIERET SEQUECES O LEGTH 4 Sequence DT M- M- T M-3 M-4 {,,,} {,,,} {,,3,4} {,,,} {,,,} TABLE VI: UMBER O MULTIPLICATIOS REQUIRED I VARIOUS TECHIQUES OR DIERET SEQUECES O LEGTH 8 Sequence DT M- M- T M-3 M-4 {,,,,,,,} {,,,,,,,} {,,,,,,,} {,,,,,,,} {,,,3,4,5,6,7} Conclusions The speed of calculation of DT is crucial in many applications including medical, military and space applications. The hardware required is other concern. In this paper, an attempt has been made to reduce the number of complex additions and multiplications required to perform DT using various methods. The method- uses the property of twiddle factor and tries to substitute the calculated DT coefficients to find the remaining coefficients. The method- modifies the butterfly structure to add the common multiples of twiddle factor but found to involve more calculations than T. The method-3 skips the trivial multiplications and method-4 skips both trivial multiplications and additions. The algorithms are simulated in language C on Windows 7 machine. The suitable hardware implementations particularly the comparators have to be studied, and has to be verified for longer sequence lengths, say = 4. [] A Steven G. Johnson and Matteo rigo, A modified split-radix T with fewer arithmetic operations, IEEE Transactions on Signal Processing, 55 (), 9, 7. [] G. Plonka and M. Tasche, ast and numerically stable algorithms for discrete cosine transforms, Linear Algebra Appl., vol. 394, pp , 5. [3] M. P uschel, J. M.. Moura, J. R. Johnson, D. Padua, M. M. Veloso, B. W. Singer, J. Xiong,. ranchetti, A. Gaˇci c, Y. Voronenko, K. Chen, R. W. Johnson, and. Rizzolo, SPIRAL: Code generation for DSP transforms, Proc. IEEE, vol. 93, no., pp. 3 75, 5. [4] S. Bouguezel, M. O. Ahmad, and M.. S. Swamy, A new radix-/8 T algorithm for length-q m DTs, IEEE Trans. Circuits Syst. I, vol. 5, no. 9, pp , 4. [5] M. rigo and S. G. Johnson, The design and implementation of TW3, Proc. IEEE, vol. 93, no., pp. 6 3, 5. [6] P. Duhamel and H. Hollmann, Split-radix T algorithm, Electron. Lett., vol., no., pp. 4 6, 984. [7] M. Vetterli and H. J. ussbaumer, Simple T and DCT algorithms with reduced number of operations, Signal Processing, vol. 6, no. 4, pp , 984. [8] J. B. Martens, Recursive cyclotomic factorization a new algorithm for calculating the discrete ourier transform, IEEE Trans. Acoust., Speech, Signal Processing, vol. 3, no. 4, pp , 984. [9] H. Sorensen, M. Heideman, and C. Burrus, On computing the split-radix T, Acoustics, Speech and Signal Processing, IEEE Transactions on, 34():5 56, 3. [] K.A. Sakallah, Symmetry and satisfiability, Handbook of Satisfiability, pages , 9. [] S. Ranise and C. Tinelli, The SMT-LIB standard: Version., Department of Computer Science, The University of Iowa, Tech. Rep, 6. [] R. ieuwenhuis and A. Oliveras, On sat modulo theories and optimization problems, Theory and Applications of Satisfiability Testing-SAT 6, pages 56 69, 6. [3] R. ieuwenhuis, A. Oliveras, and C. Tinelli, Solving SAT and SAT Modulo Theories: rom an abstract Davis Putnam Logemann Loveland procedure to DPLL (T), Journal of the ACM (JACM), 53(6): , 6. [4] G. ordin, P.A. Milder, J.C. Hoe, and M. Püschel, Automatic generation of customized discrete ourier transform IPs, In Proceedings of the 4nd annual Design Automation Conference, pages ACM, 5. [5] T. Lundy and J. Van Buskirk, A new matrix approach to real Ts and convolutions of length k, Computing, 8():3 45, 7. IJCTA Mar-Apr 5 33

5 [6] V. Manquinho, O. Roussel, and M. Deters, Pseudo- Boolean Competition, [7] T. Mateer, ast ourier transform algorithms with applications, PhD thesis, Clemson University, 8. [8] A. Mishchenko. ABC: A System for Sequential Synthesis and Verification. [9] M.W. Moskewicz, C.. Madigan, Y. Zhao, L. Zhang, and S. Malik, Chaff: Engineering an efficient SAT solver, Design Automation Conference, Proceedings, pages , IEEE..B. Smith, C.D. Jones, and E.. Roberts, Article Title, Journal, Publisher, Location, Date, pp. -. Author s Profile Jaya Krishna Sunkara received B.Tech in Electronics and Communication Engineering from GKCE, Sullurpet (JTU HYD) in 6 and M.E in Information Technology from UVCE, Bangalore (Bangalore University Campus) in 9. He has held various positions in Priyadarshini college of Engineering, Sullurpet including Asst. Prof., Head of the Dept., Training & Placement Officer during 9-4. Currently he is doing his Masters in Signal Processing in SVUCE, Tirupati (Sri Venkateswara University Campus). He stood University first in ME and also qualified in GATE for 7 times, four times in EC and thrice in CS. He has published eight research papers in national and international journals. His research interests include Signal and Image Processing. Chiranjeevi Muppala received B.Tech in Electronics and Communication Engineering from KMCET, JTU HYD in 8 and M.Tech in VLSI System Design from Anuraag Engineering College, JTU HYD in. He is working as Asst. Prof., in TJS Engineering College, Anna University. Previously, he worked as Asst. Prof., in Priyadarshini college of Engineering, Sullurpet during -3. His research interests include Signal and Image Processing, VLSI Design. IJCTA Mar-Apr 5 34