A new radix 4 fft algorithm

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A new radix 4 fft algorithm

  1. 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME251A NEW RADIX-4 FFT ALGORITHMDr. Syed Abdul Sattar1, Dr. Mohammed Yousuf Khan2, and Shaik Qadeer31(Professor & Dean. RITS Chevella. R R Dist. A P . India)2(Principal Polytechnic, MANUU, Hyderabad, India)3(Department of EED, MJCET, Hyderabad, India)ABSTRACTThe Radix-4 Fast Fourier Transform (FFT) is widely accepted for signal processingapplications in wireless communication system. Here, we present a new Radix-4 FFT which reducesthe 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. INTRODUCTIONThe Discrete Fourier transform (DFT) is among the most fundamental operation in digitalsignal processing applications [1, 2, 7]. The algorithm to compute DFT is called as FFT. Whenconsidering the implementations, the FFT/IFFT algorithm should be chosen keeping in view theexecution speed, hardware complexity, flexibility and precision [8, 9]. Most of the above mentionedparameters 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 unsolvedmathematical question.There are various types of FFT algorithms namely Radix-2/4/8 and Split radix including DITand DIF versions. The flop counts of standard Radix-4 DIT is [5]:NNNT 2log414)( =(1)Here we propose a new Radix-4 DIT algorithm which computes FFT in flop counts:INTERNATIONAL JOURNAL OF ADVANCED RESEARCH INENGINEERING 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.aspJournal Impact Factor (2013): 5.8376 (Calculated by GISI)www.jifactor.comIJARET© I A E M E
  2. 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME252NNNT 2log814)( =(2)Saving of flop counts is almost 6% as compared to the standard Radix-4 FFT. In this paper asimple 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 implementationof proposed algorithm is discussed and conclusion in section IV.II. NEW RADIX-4 DIT ALGORITHMIn this section we suggest a modification to standard Radix-4 FFT algorithm to reduce thenumber of multiplications. To derive the algorithm, recall that the DFT is defined by [5]1..0,10−=∑=−=NnWxX knNNknk (3)where WNK= exp (-j2π/N). Then for N divisible by 4, we perform a decimation in timedecomposition of xn into four smaller DFTs, of x4n , x4n+2(the even elements), x4n+1 andx4n-1 (where x-1= xN-1)- this last sub-sequence would be in x4n+3 standard Radix-4 FFT, but here isshifted cyclically by -4 [4]. We obtain:,4/14/0144/14/0144/14/02424/14/04 WxWWxWWxWWxX knNNknkNknNNknkNknNNknkNknNNknk ∑+∑+∑+∑=−=−−−=+−=+−=(4)Where the WkN and W-kN are the conjugate pair of twiddle factor. The four recurrent resultswith subtransforms denoted by Yk, Gk , Hk, and Zk are shown below:( )( )( )HWWZ kjGWYXandHWWZ kjGWYXHWWZ kGWYXHWWZ kGWYXkkNkNkkNkNkkkNkNkkNkNkkkNkNkkNkNkkkNkNkkNkk−+−+−+−++−=−−−=+−+=+++=24/324/22/2,,,(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 thatalgorithm 1, both Hk, and Zk are multiplied by a same trigonometric function (WkN and W-kN ).Therefore to get saving we can divide Hk, and Zk by a common factor and multiply with the same inthe next iteration. Scaling factor ( Wavelet) proposed in [4] can be used as factor for getting saving. Itis having the following properties for k4=k mod N/4:
  3. 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME253) ( ) ( )( ) ( ))))SSWtLetSSskremainingtheforandkofvalueshalffirstforSSotherwiseSNkNkforSNkforNSDefkNkNkNkNkNNkNkNkNkNkNkN,,4/,,4/,,4/,4,4/4,4/,43,cossin2/42sin8/4/42cos41:1===≤≤=+θθππ(6)We use tN,k in our new algorithm which is always tan1 j±± or j±± cot . Computation of tN,ktakes 4 flops, whereas WNktakes 6. The new algorithm is shown in figure 2. Here in this newalgorithm , we first multiply by SN/4,k in algo1 to both Hk, and Zk which does not take any additionalmultiplications, however it help us in getting saving for multiplications in algo2 of the newalgorithm.Figure:1 Std. Radix-4 FFT algorithm
  4. 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME254Figure:2 New . Radix-4 FFT algorithmIII. COMPUTATIONAL COUNTThe algorithms shown in figure 1 and figure 2 have same number of additions however due toscaling and recalling operation in algorithm 2-3 we can save multiplications. In algo1 as the numbersof additions as well as multiplications remain same, so we get=≥=++=4,1644,17422412)(1 2NifNifNTNTNTnN(7)
  5. 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME255In algo2 we save 4 real multiplications but lost 2. So the net saving is 2 real multiplications ineach iterations,=≥=++=4,1644,5.16422412)(2 2NifNifNTNTNTnN(8)Solving (7) and (8) by standard generating function method [3, 6] we get the total flop countT(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. 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME256IV. SIMULATION RESULTSTo validate the algorithm a computer simulation using Matlap for N=4096 is performed asshown in figure 3 and figure 4. The results are matching with standard Radix -4 FFT algorithmresults. Figure 3 shows a comparative plot of magnitude between proposed FFT and standard FFTwhereas figure 4 covers for angle.V. CONCLUSIONIn this paper a new Radix-4 FFT algorithm is proposed. It is shown that the proposed FFTalgorithm is computational efficient without losing any accuracy. The proposed algorithm is suitablefor 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, IEEETransactions On Signal Processing, 56( 10), 2008, 4673-4682.[2] Wade Lowdermilk and Fred Harris, Finite Arithmetic Considerations for the FFTimplemented in FPGA-Based Embedded Processors in Synthetic Instruments, IEEEInstrumentation 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, andapplications ,Pearson 3rdEdition[10] Abhishek choubey, Mayuri Kulshreshtha and Karunesh, “Determination of Optimum FFT forWi-Max under Different Fading”, International journal of Electronics and CommunicationEngineering &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 & MobileWimax Communication Systems”, International journal of Electronics and CommunicationEngineering &Technology (IJECET), Volume 3, Issue 3, 2012, pp. 265 - 279, ISSN Print:0976- 6464, ISSN Online: 0976 –6472.

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