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Canonic Composite Length Real-Valued FFT

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Abstract

This paper presents a novel algorithm to compute real-valued fast Fourier transform (RFFT) that is canonic with respect to the number of signal values. A signal value can correspond to a purely real or purely imaginary value, while a complex signal consists of 2 signal values. For an N-point RFFT, each stage needs not compute more than N signal values, since the degrees of freedom of the input data are N. Any more than N signal values computed at any stage is inherently redundant. In order to reduce the redundant samples, a sample removal lemma, and two types of twiddle factor transformations are proposed: pushing and modulation. We consider 4 different cases for an N = P × Q point canonic RFFT: 1) P is odd, Q is odd; 2) P is odd, Q is even; 3) P is even, Q is odd; and 4) P is even, Q is even. No twiddle factor transformation is required when P is odd. It is shown that the number of twiddle factors can be reduced by performing modulation transformation when P is even and Q is odd, while 2 real twiddle factor operations are pushed to 1 complex twiddle factor operation when P and Q are both even. Canonic RFFT for any composite length can be computed by applying the proposed algorithm recursively. Performances of different RFFTs are also discussed in this paper. The major advantages of the canonic RFFTs are that they require the least number of butterfly operations, lead to more regular sub-blocks in the data-flow, and only involve real datapath when mapped to architectures. However, we also show that canonic RFFTs may not be desirable when taking the cost of twiddle factor operations as the major consideration.

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References

  1. Ayinala, M., Brown, M., & Parhi, K.K. (2012). Pipelined parallel FFT architectures via folding transformation. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 20(6), 1068–1081.

    Article  Google Scholar 

  2. Ayinala, M., Lao, Y., & Parhi, K.K. (2013). An in-place FFT architecture for real-valued signals. IEEE Transactions on Circuits and Systems II: Express Briefs, 60(10), 652–656.

    Article  Google Scholar 

  3. Ayinala, M., & Parhi, K.K. (2013). FFT architectures for real-valued signals based on radix- 23 and radix- 24 algorithms. IEEE Transactions on Circuits and Systems I: Regular Papers, 60(9), 2422–2430.

    Article  MathSciNet  Google Scholar 

  4. Baas, B.M. (1999). A low-power, high-performance, 1024-point FFT processor. IEEE Journal of Solid-State Circuits, 34(3), 380–387.

    Article  Google Scholar 

  5. Bergland, G.D. (1968). A fast Fourier transform algorithm using base 8 iterations. Mathematics of Computation, 22(102), 275–279.

    MathSciNet  MATH  Google Scholar 

  6. Bowers, K.J., Lippert, R.A., Dror, R.O., & Shaw, D.E. (2010). Improved twiddle access for fast Fourier transforms. IEEE Transactions on Signal Processing, 58(3), 1122–1130.

    Article  MathSciNet  MATH  Google Scholar 

  7. Chi, H.F., & Lai, Z.H. (2005). A cost-effective memory-based real-valued FFT and hermitian symmetric IFFT processor for DMT-based wire-line transmission systems. In IEEE International symposium on circuits and systems (ISCAS) (pp. 6006–6009).

  8. Cooley, J.W., & Tukey, J.W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of computation, 19(90), 297–301.

    Article  MathSciNet  MATH  Google Scholar 

  9. Duhamel, P., & Hollmann, H. (1984). Split radix FFT algorithm. Electronics Letters, 20(1), 14–16.

    Article  Google Scholar 

  10. Garrido, M., Parhi, K.K., & Grajal, J. (2009). A pipelined FFT architecture for real-valued signals. IEEE Transactions on Circuits and Systems I: Regular Papers, 56(12), 2634–2643.

    Article  MathSciNet  Google Scholar 

  11. He, S., & Torkelson, M. (1996). A new approach to pipeline FFT processor. In: Proceedings of the 10th international parallel processing symposium (IPPS’96) (pp. 766–770).

  12. Hu, Z., & Wan, H. (2005). A novel generic fast Fourier transform pruning technique and complexity analysis. IEEE Transactions on Signal Processing, 53(1), 274–282.

    Article  MathSciNet  MATH  Google Scholar 

  13. Lao, Y., & Parhi, K.K. (2015). Canonic real-valued radix- 2n FFT computations. In 49th Asilomar conference on signals, systems and computers (pp. 441–446).

  14. Lao, Y., & Parhi, K.K. (2016). Data-canonic real FFT flow-graphs for composite lengths. In Proceedings of IEEE International workshop on signal processing Systems (SiPS) (pp. 189–194).

  15. Medina-Melendrez, M., Arias-Estrada, M., & Castro, A. (2009). Input and/or output pruning of composite length FFTs using a DIf-DIT transform decomposition. IEEE Transactions on Signal Processing, 57(10), 4124–4128.

    Article  MathSciNet  MATH  Google Scholar 

  16. Meher, P.K., Mohanty, B.K., Patel, S.K., Ganguly, S., & Srikanthan, T. (2015). Efficient VLSI architecture for decimation-in-time fast Fourier transform of real-valued data. IEEE Transactions on Circuits and Systems I: Regular Papers, 62(12), 2836–2845.

    Article  MathSciNet  Google Scholar 

  17. Milder, P.A., Franchetti, F., Hoe, J.C., & Püschel, M. (2010). Hardware implementation of the discrete fourier transform with non-power-of-two problem size. In 2010 IEEE International conference on acoustics speech and signal processing (ICASSP) (pp. 1546–1549).

  18. Nussbaumer, H.J. (2012). Fast Fourier transform and convolution algorithms (Vol. 2). Springer Science & Business Media.

  19. Oppenheim, A.V., Schafer, R.W., & Buck, J.R. (1989). Discrete-time signal processing. Prentice-hall Englewood Cliffs.

  20. Parhi, M., Lao, Y., & Parhi, K.K. (2014). Canonic real-valued FFT structures. In 48th Asilomar conference on signals, systems and computers (pp. 1261–1265).

  21. Salehi, S.A., Amirfattahi, R., & Parhi, K.K. (2013). Pipelined architectures for real-valued FFT and hermitian-symmetric IFFT with real datapaths. IEEE Transactions on Circuits and Systems II: Express Briefs, 60(8), 507–511.

    Article  Google Scholar 

  22. Singleton, R.C. (1969). An algorithm for computing the mixed radix fast Fourier transform. IEEE Transactions on Audio and Electroacoustics, 17(2), 93–103.

    Article  Google Scholar 

  23. Sorensen, H.V., Jones, D.L., Heideman, M., & Burrus, C.S. (1987). Real-valued fast Fourier transform algorithms. IEEE Transactions on Acoustics Speech and Signal Processing, 35(6), 849–863.

    Article  Google Scholar 

  24. Tolimieli, R., An, M., & Lu, C. (2013). Algorithms for discrete Fourier transform and convolution. Springer Science & Business Media.

  25. Van Loan, C. (1992). Computational frameworks for the fast Fourier transform (Vol. 10). SIAM.

  26. Voronenko, Y., & Puschel, M. (2009). Algebraic signal processing theory: Cooley–Tukey type algorithms for real DFTs. IEEE Transactions on Signal Processing, 57(1), 205–222.

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, L., Zhou, X., Sobelman, G.E., & Liu, R. (2012). Generic mixed-radix FFT pruning. IEEE Signal Processing Letters, 19(3), 167–170.

    Article  Google Scholar 

  28. Yin, X., Yu, F., & Ma, Z. (2016). Resource-efficient pipelined architectures for radix-2 real-valued FFT with real datapaths. IEEE Transactions on Circuits and Systems II: Express Briefs.

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Authors and Affiliations

  1. Electrical and Computer Engineering, Clemson University, Clemson, SC, USA

    Yingjie Lao

  2. Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA

    Keshab K. Parhi

Authors
  1. Yingjie Lao
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  2. Keshab K. Parhi
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Correspondence to Yingjie Lao.

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Lao, Y., Parhi, K.K. Canonic Composite Length Real-Valued FFT. J Sign Process Syst 90, 1401–1414 (2018). https://doi.org/10.1007/s11265-017-1296-9

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  • DOI: https://doi.org/10.1007/s11265-017-1296-9

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