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A new radix 4 fft algorithm
1.
International Journal of
Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME 251 A NEW RADIX-4 FFT ALGORITHM Dr. Syed Abdul Sattar1 , Dr. Mohammed Yousuf Khan2 , and Shaik Qadeer3 1 (Professor & Dean. RITS Chevella. R R Dist. A P . India) 2 (Principal Polytechnic, MANUU, Hyderabad, India) 3 (Department of EED, MJCET, Hyderabad, India) ABSTRACT The Radix-4 Fast Fourier Transform (FFT) is widely accepted for signal processing applications in wireless communication system. Here, we present a new Radix-4 FFT which reduces the operational count by 6% lesser than standard Radix-4 FFT without losing any arithmetic accuracy. Simulation results are also given for the verification of the algorithm. Keywords: Discrete Fourier Transform (DFT); Decimation in Time (DI|T); Fast Fourier Transform (FFT); and Flop counts. I. INTRODUCTION The Discrete Fourier transform (DFT) is among the most fundamental operation in digital signal processing applications [1, 2, 7]. The algorithm to compute DFT is called as FFT. When considering the implementations, the FFT/IFFT algorithm should be chosen keeping in view the execution speed, hardware complexity, flexibility and precision [8, 9]. Most of the above mentioned parameters depend on the exact count of arithmetic operations (real additions and multiplications), herein called flop counts, required for a DFT of a given size N which remains an intriguing unsolved mathematical question. There are various types of FFT algorithms namely Radix-2/4/8 and Split radix including DIT and DIF versions. The flop counts of standard Radix-4 DIT is [5]: NNNT 2log 4 1 4)( = (1) Here we propose a new Radix-4 DIT algorithm which computes FFT in flop counts: INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 4, Issue 3, April 2013, pp. 251-256 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
2.
International Journal of
Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME 252 NNNT 2log 8 1 4)( = (2) Saving of flop counts is almost 6% as compared to the standard Radix-4 FFT. In this paper a simple recursive method is adopted to get the saving. The rest of the paper is organized as follows: Section II covers the new algorithm principle, in section III operation count with the implementation of proposed algorithm is discussed and conclusion in section IV. II. NEW RADIX-4 DIT ALGORITHM In this section we suggest a modification to standard Radix-4 FFT algorithm to reduce the number of multiplications. To derive the algorithm, recall that the DFT is defined by [5] 1..0, 1 0 −=∑= − = NnWxX kn N N k nk (3) where WN K = exp (-j2π/N). Then for N divisible by 4, we perform a decimation in time decomposition of xn into four smaller DFTs, of x4n , x4n+2(the even elements), x4n+1 and x4n-1 (where x-1= xN-1)- this last sub-sequence would be in x4n+3 standard Radix-4 FFT, but here is shifted cyclically by -4 [4]. We obtain: ,4/ 14/ 0 144/ 14/ 0 144/ 14/ 0 24 2 4/ 14/ 0 4 WxWWxWWxWWxX kn N N k n k N kn N N k n k N kn N N k n k N kn N N k nk ∑+∑+∑+∑= − = − − − = + − = + − = (4) Where the Wk N and W-k N are the conjugate pair of twiddle factor. The four recurrent results with subtransforms denoted by Yk, Gk , Hk, and Zk are shown below: ( ) ( ) ( )HWWZ kjGWYX andHWWZ kjGWYX HWWZ kGWYX HWWZ kGWYX k k N k Nk k NkNk k k N k Nk k NkNk k k N k Nk k NkNk k k N k Nk k Nkk − + − + − + − ++−= −−−= +−+= +++= 2 4/3 2 4/ 2 2/ 2 , , , (5) The algorithm for this is shown in figure 1. The total flop count for this is given in (1). Observe that it is equal to standard radix-4 FFT [5]. The key for the new algorithm is observation that algorithm 1, both Hk, and Zk are multiplied by a same trigonometric function (Wk N and W-k N ). Therefore to get saving we can divide Hk, and Zk by a common factor and multiply with the same in the next iteration. Scaling factor ( Wavelet) proposed in [4] can be used as factor for getting saving. It is having the following properties for k4=k mod N/4:
3.
International Journal of
Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME 253 ) ( ) ( ) ( ) ( ) ) ) ) S S WtLet SS skremainingtheforandkofvalueshalffirstfor S S otherwiseSNk NkforSNk forN SDef kN kNk NkN kNNkN kN kN kN kNkN , ,4/ , ,4/, ,4/ , 4,4/ 4,4/, 4 3 ,'cossin2 /42sin 8/4/42cos 41 :1 = = = ≤ ≤ = + θθ π π (6) We use tN,k in our new algorithm which is always tan1 j±± or j±± cot . Computation of tN,k takes 4 flops, whereas WN k takes 6. The new algorithm is shown in figure 2. Here in this new algorithm , we first multiply by SN/4,k in algo1 to both Hk, and Zk which does not take any additional multiplications, however it help us in getting saving for multiplications in algo2 of the new algorithm. Figure:1 Std. Radix-4 FFT algorithm
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International Journal of
Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME 254 Figure:2 New . Radix-4 FFT algorithm III. COMPUTATIONAL COUNT The algorithms shown in figure 1 and figure 2 have same number of additions however due to scaling and recalling operation in algorithm 2-3 we can save multiplications. In algo1 as the numbers of additions as well as multiplications remain same, so we get = ≥=+ + = 4,16 44,17 4 22 4 12 )(1 2 Nif Nif N T N T NT nN (7)
5.
International Journal of
Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME 255 In algo2 we save 4 real multiplications but lost 2. So the net saving is 2 real multiplications in each iterations, = ≥=+ + = 4,16 44,5.16 4 22 4 12 )(2 2 Nif Nif N T N T NT nN (8) Solving (7) and (8) by standard generating function method [3, 6] we get the total flop count T(N) as given in (2). Hence a net saving of 6% is achieved. Figure:3: Comparative Magnitude plot for N=4096. Figure: 4: Comparative angle plot for N=4096.
6.
International Journal of
Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME 256 IV. SIMULATION RESULTS To validate the algorithm a computer simulation using Matlap for N=4096 is performed as shown in figure 3 and figure 4. The results are matching with standard Radix -4 FFT algorithm results. Figure 3 shows a comparative plot of magnitude between proposed FFT and standard FFT whereas figure 4 covers for angle. V. CONCLUSION In this paper a new Radix-4 FFT algorithm is proposed. It is shown that the proposed FFT algorithm is computational efficient without losing any accuracy. The proposed algorithm is suitable for ASIC implementation as it is symmetric, unlike split radix FFT. REFERENCES [1] W.-H. Chang and T. Nguyen, On the Fixed-Point Accuracy Analysis of FFT Algorithms, IEEE Transactions On Signal Processing, 56( 10), 2008, 4673-4682. [2] Wade Lowdermilk and Fred Harris, Finite Arithmetic Considerations for the FFT implemented in FPGA-Based Embedded Processors in Synthetic Instruments, IEEE Instrumentation and Measurement Magazine, August 2007. [3] D. E. Knuth, Fundamental Algorithms, 3rd ed., ser. The Art of Computer Programming (Addison-Wesley, 1997, vol. 1.) [4] ] M. Frigo and S. G. Johnson, A modified Split Radix FFT with fewer arithmetic operations, IEEE Trans. Signal processing 55 (1), 111–119 (2007). [5] Eleanor Chu and Alan George, Inside the FFT Black Box: Serial and Parallel FFT algorithms, (CRC Press) Page No.282, Appendix A. [6] D. E Knuth, Seminumerical Algorithms, 3rd ed., ser. The Art of Computer Programming (Addison-Wesley, 1998, vol. 2.) [7] M.Z.A. Khan and Shaik Qadeer, Streamlined Real Factor EUSIPCO2010, Aalborg, Denmark, August23-27,2010 [8] A. V. Oppenheim and R. Schafer, Digital Signal Processing, .Pearson Education, 2004. [9] J.G. Proakis and D.G. Manolakis,, Digital Signal Processing Principles, Algorithms, and applications ,Pearson 3rd Edition [10] Abhishek choubey, Mayuri Kulshreshtha and Karunesh, “Determination of Optimum FFT for Wi-Max under Different Fading”, International journal of Electronics and Communication Engineering &Technology (IJECET), Volume 3, Issue 1, 2012, pp. 139 - 146, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472. [11] Kamatham Harikrishna and T. Rama Rao, “An Efficient Radix-22 FFT for Fixed & Mobile Wimax Communication Systems”, International journal of Electronics and Communication Engineering &Technology (IJECET), Volume 3, Issue 3, 2012, pp. 265 - 279, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472.
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