INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) A NEW RADIX-4 FFT ALGORITHM

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1 ITERATIOAL JOURAL OF ADVACED REEARC I EIEERI AD TECOLO (IJARET Iteratioal Joural of Advaced Research i Egieerig ad Techology (IJARET I (Prit I (Olie Volume Issue 3 April (3 IAEME I (Prit I (Olie Volume Issue 3 April 3 pp IAEME: Joural Impact Factor (3: (Calculated by II IJARET I A E M E A E RADI- FFT ALORITM Dr. yed Abdul attar Dr. Mohammed ousuf Kha ad hai Qadeer 3 (Professor & Dea. RIT Chevella. R R Dist. A P. Idia (Pricipal Polytechic MAUU yderabad Idia 3 (Departmet of EED MJCET yderabad Idia ABTRACT The Radi- Fast Fourier Trasform (FFT is widely accepted for sigal processig applicatios i wireless commuicatio system. ere we preset a ew Radi- FFT which reduces the operatioal cout by 6% lesser tha stadard Radi- FFT without losig ay arithmetic accuracy. imulatio results are also give for the verificatio of the algorithm. Keywords: Discrete Fourier Trasform (DFT; Decimatio i Time (DI T; Fast Fourier Trasform (FFT; ad Flop couts. I. ITRODUCTIO The Discrete Fourier trasform (DFT is amog the most fudametal operatio i digital sigal processig applicatios [ 7]. The algorithm to compute DFT is called as FFT. he cosiderig the implemetatios the FFTIFFT algorithm should be chose eepig i view the eecutio speed hardware compleity fleibility ad precisio [8 9]. Most of the above metioed parameters deped o the eact cout of arithmetic operatios (real additios ad multiplicatios herei called flop couts required for a DFT of a give size which remais a itriguig usolved mathematical questio. There are various types of FFT algorithms amely Radi-8 ad plit radi icludig DIT ad DIF versios. The flop couts of stadard Radi- DIT is [5]: T ( log ere we propose a ew Radi- DIT algorithm which computes FFT i flop couts: ( 5

2 Iteratioal Joural of Advaced Research i Egieerig ad Techology (IJARET I (Prit I (Olie Volume Issue 3 April (3 IAEME 5 T log 8 ( ( avig of flop couts is almost 6% as compared to the stadard Radi- FFT. I this paper a simple recursive method is adopted to get the savig. The rest of the paper is orgaized as follows: ectio II covers the ew algorithm priciple i sectio III operatio cout with the implemetatio of proposed algorithm is discussed ad coclusio i sectio IV. II. E RADI- DIT ALORITM I this sectio we suggest a modificatio to stadard Radi- FFT algorithm to reduce the umber of multiplicatios. To derive the algorithm recall that the DFT is defied by [5].. (3 where K ep (-jπ. The for divisible by we perform a decimatio i time decompositio of ito four smaller DFTs of (the eve elemets ad - (where this last sub-sequece would be i 3 stadard Radi- FFT but here is shifted cyclically by - []. e obtai: ( here the ad - are the cojugate pair of twiddle factor. The four recurret results with subtrasforms deoted by ad Z are show below: ( ( ( Z j ad Z j Z Z 3 (5 The algorithm for this is show i figure. The total flop cout for this is give i (. Observe that it is equal to stadard radi- FFT [5]. The ey for the ew algorithm is observatio that algorithm both ad Z are multiplied by a same trigoometric fuctio ( ad -. Therefore to get savig we ca divide ad Z by a commo factor ad multiply with the same i the et iteratio. calig factor ( avelet proposed i [] ca be used as factor for gettig savig. It is havig the followig properties for mod :

3 Iteratioal Joural of Advaced Research i Egieerig ad Techology (IJARET I (Prit I (Olie Volume Issue 3 April (3 IAEME 3 Def : ( cos π ( si( π ( Let t siθ for first half values of ad cosθ for the remaiig ' s for for 8 otherwise (6 e use t i our ew algorithm which is always ± ± j ta or ± cot ± j. Computatio of t taes flops whereas taes 6. The ew algorithm is show i figure. ere i this ew algorithm we first multiply by i algo to both ad Z which does ot tae ay additioal multiplicatios however it help us i gettig savig for multiplicatios i algo of the ew algorithm. Figure: td. Radi- FFT algorithm 53

4 Iteratioal Joural of Advaced Research i Egieerig ad Techology (IJARET I (Prit I (Olie Volume Issue 3 April (3 IAEME Figure: ew. Radi- FFT algorithm III. COMPUTATIOAL COUT The algorithms show i figure ad figure have same umber of additios however due to scalig ad recallig operatio i algorithm -3 we ca save multiplicatios. I algo as the umbers of additios as well as multiplicatios remai same so we get T T( 6 T 7 if if (7 5

5 Iteratioal Joural of Advaced Research i Egieerig ad Techology (IJARET I (Prit I (Olie Volume Issue 3 April (3 IAEME I algo we save real multiplicatios but lost. o the et savig is real multiplicatios i each iteratios T T ( 6 T 6.5 if if olvig (7 ad (8 by stadard geeratig fuctio method [3 6] we get the total flop cout T( as give i (. ece a et savig of 6% is achieved. (8 Figure:3: Comparative Magitude plot for 96. Figure: : Comparative agle plot for

6 Iteratioal Joural of Advaced Research i Egieerig ad Techology (IJARET I (Prit I (Olie Volume Issue 3 April (3 IAEME IV. IMULATIO REULT To validate the algorithm a computer simulatio usig Matlap for 96 is performed as show i figure 3 ad figure. The results are matchig with stadard Radi - FFT algorithm results. Figure 3 shows a comparative plot of magitude betwee proposed FFT ad stadard FFT whereas figure covers for agle. V. COCLUIO I this paper a ew Radi- FFT algorithm is proposed. It is show that the proposed FFT algorithm is computatioal efficiet without losig ay accuracy. The proposed algorithm is suitable for AIC implemetatio as it is symmetric ulie split radi FFT. REFERECE [].-. Chag ad T. guye O the Fied-Poit Accuracy Aalysis of FFT Algorithms IEEE Trasactios O igal Processig 56( [] ade Lowdermil ad Fred arris Fiite Arithmetic Cosideratios for the FFT implemeted i FPA-Based Embedded Processors i ythetic Istrumets IEEE Istrumetatio ad Measuremet Magazie August 7. [3] D. E. Kuth Fudametal Algorithms 3rd ed. ser. The Art of Computer Programmig (Addiso-esley 997 vol.. [] ] M. Frigo ad.. Johso A modified plit Radi FFT with fewer arithmetic operatios IEEE Tras. igal processig 55 ( 9 (7. [5] Eleaor Chu ad Ala eorge Iside the FFT Blac Bo: erial ad Parallel FFT algorithms (CRC Press Page o.8 Appedi A. [6] D. E Kuth emiumerical Algorithms 3rd ed. ser. The Art of Computer Programmig (Addiso-esley 998 vol.. [7] M.Z.A. Kha ad hai Qadeer treamlied Real Factor EUIPCO Aalborg Demar August3-7 [8] A. V. Oppeheim ad R. chafer Digital igal Processig.Pearso Educatio. [9] J.. Proais ad D.. Maolais Digital igal Processig Priciples Algorithms ad applicatios Pearso 3 rd Editio [] Abhishe choubey Mayuri Kulshreshtha ad Karuesh Determiatio of Optimum FFT for i-ma uder Differet Fadig Iteratioal joural of Electroics ad Commuicatio Egieerig &Techology (IJECET Volume 3 Issue pp I Prit: I Olie: [] Kamatham aririsha ad T. Rama Rao A Efficiet Radi- FFT for Fied & Mobile ima Commuicatio ystems Iteratioal joural of Electroics ad Commuicatio Egieerig &Techology (IJECET Volume 3 Issue 3 pp I Prit: I Olie:

Chapter 3 Classification of FFT Processor Algorithms

Chapter 3 Classification of FFT Processor Algorithms Chapter Classificatio of FFT Processor Algorithms The computatioal complexity of the Discrete Fourier trasform (DFT) is very high. It requires () 2 complex multiplicatios ad () complex additios [5]. As