Skip to main content
SpringerLink
Log in

A Structured Dual Split-Radix Algorithm for the Discrete Hartley Transform of Length \(2^{N}\)

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper presents a new split-radix algorithm for DHT of length \({N}=2^{n}\), called the Dual Split-Radix DHT (DSR DHT), that allows an efficient parallel implementation using a dual core system. Moreover, as it is different from existing split-radix algorithms for DHT, it offers an efficient hardware implementation similar to that for FFT. It avoids the so-called retrograde indexing specific to existing DHT algorithms that do not allow an efficient pipeline implementation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. G. Bi, Y. Chen, Y. Zeng, Fast algorithms for generalized discrete Hartley transform of composite sequence length. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 47(9), 893–901 (2000)

    Article  MATH  Google Scholar 

  2. G. Bi, New split-radix algorithm for the discrete Hartley transform. IEEE Trans. Signal Process. 45(2), 297–302 (1997)

    Article  Google Scholar 

  3. S. Bouguezel, M.O. Ahmad, M.N.S. Swamy, New parametric discrete Fourier and Hartley transforms, and algorithms for fast computation. IEEE Trans. Circuits Syst. I Regul. Papers 58(3), 562–575 (2011)

    Article  MathSciNet  Google Scholar 

  4. D.F. Chiper, Fast radix-2 algorithm for the discrete hartley transform of type II. IEEE Signal Process. Lett. 18(11), 687–689 (2011)

    Article  Google Scholar 

  5. D.F. Chiper, Radix-2 fast algorithm for computing discrete Hartley transform of type III. IEEE Trans. Circuits Syst. II Express Briefs 59(5), 297–301 (2012)

    Article  Google Scholar 

  6. D.F. Chiper, M.N.S. Swamy, M.O. Ahmad, An efficient systolic array algorithm for the VLSI implementation of prime-length DHT, in Proceedings of International Symposium on Signals, Circuits and Systems (ISSCS2005), vol. 1, pp. 167–169

  7. D.F. Chiper, A novel VLSI DHT algorithm for a highly modular and parallel architecture. IEEE Trans. Circuits Syst. II 60(5), 282–286 (2013)

    Article  Google Scholar 

  8. R.E. Crochiere, L.R. Rabiner, Multirate Digital Signal Processing (Prentice-Hall, Englewood Cliffs, 1983)

    Google Scholar 

  9. A.C. Erickson, B.S. Fagin, Calculating the FHT in hardware. IEEE Trans. Signal Process. 40(6), 1341–1353 (1992)

    Article  MATH  Google Scholar 

  10. M.T. Heideman, Multiplicative Complexity, Convolution and the DFT (Springer, Berlin, 1988)

    Book  MATH  Google Scholar 

  11. H.S. Hou, The fast Hartley transform algorithm. IEEE Trans. Comput. 147–156 (1987)

  12. J. Hu, J. Deng, J. Wu, Image compression based on improved FFT algorithm. J. Netw. 6(7), 1041–1048 (2011)

    Google Scholar 

  13. T. Le-Ngoc, T.V. Minh, Implementation and performance of the fast Hartley transform. IEEE Micro 9(5), 20–27 (1989)

    Article  Google Scholar 

  14. M. Marchesi, G. Orlandi, F. Piazza, A systolic circuit for fast Hartley transform, in IEEE International Symposium on, Circuits and Systems (1988)

  15. P.K. Meher, J.C. Patra, M.N.S. Swamy, High throughput memory based architecture for DHT using a new convolutional formulation. IEEE Trans. Circuits Syst. II Express Briefs 54(7), 606–610 (2007)

    Article  Google Scholar 

  16. P.K. Meher, T. Srikanthan, J.C. Patra, Scalable and modular memory-based systolic array architectures for discrete Hartley transform. IEEE Trans. Circuits Syst. I Regul. Pap. 53(5), 1065–1077 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. R.C. de Oliveira, R.M. Campello de Souza, H.M. de Oliveira, Matrix expansions for computing the discrete Hartley transform for blocklength \(\text{N}\equiv 0\) (mod4), in IEEE Latin-America Conference on Communications, IEEE, LATINCOM, 7–9 Nov (2012)

  18. M.A. O’Neill, Faster Than Fast Fourier, BYTE, (April 1988), pp. 293–300

  19. S.B. Pan, R.H. Park, Unified systolic array for computation of DCT/DST/DHT. IEEE Trans. Circuits Syst. Video Technol. 7(2), 413–419 (1997)

    Article  Google Scholar 

  20. A. Reeve, Optimized Fast Hartley Transform for the MC68000 with applications in image processing. Master Thesis, Thayer School of Engineering, Dartmouth College, Hanover, March 1990

  21. H. Shu, Y. Wang, L. Senhadji, L. Luo, Direct computation of type-II discrete Hartley transform. IEEE Signal Process. Lett. 14(5), 329–332 (2007)

    Article  Google Scholar 

  22. A.N. Skodras, M.F. Aburdene, A.K. Nandi, Two-band fast Hartley transform. Electron. Lett. 51(1), 57–59 (2015)

    Article  Google Scholar 

  23. H.V. Sorensen, D.L. Jones, C.S. Burrus, M.T. Heideman, On computing the discrete Hartley transform. IEEE Trans. Acoust. Speech Signal Process. ASSP–33, 1231–1238 (1985)

    Article  MathSciNet  Google Scholar 

  24. Z. Wang, Harmonic analysis with a real frequency function, I Aperiodic case. Appl. Math. Comput. 9, 53–73 (1981)

    MathSciNet  MATH  Google Scholar 

  25. Z. Wang, Harmonic analysis with a real frequency function, II. Periodic and bounded cases. Appl. Math. Comput. 9, 153–163 (1981)

    MathSciNet  MATH  Google Scholar 

  26. Z. Wang, Harmonic analysis with a real frequency function, III Data sequence. Appl. Math. Comput. 9, 245–255 (1981)

    MathSciNet  MATH  Google Scholar 

  27. D. Yang, Prime factor fast Hartley transform. Electron. Lett. 26(2), 119–121 (1990)

    Article  MATH  Google Scholar 

  28. W.-C. Yeh, C.-W. Jen, High-speed and low-power split-radix FFT. IEEE Trans. Signal Process. 51(3), 864–874 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electronics, Telecommunications and Information Technology, Technical University “Gheorghe Asachi” Iasi, Iasi, Romania

    Doru Florin Chiper

Authors
  1. Doru Florin Chiper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Doru Florin Chiper.

Appendix

Appendix

1.1 Algorithms for Small DHTs

An efficient implementation of a fast DHT algorithm, with a simple and low complexity software or hardware structure, closely depends on efficient algorithms for a small DHTs. At the following link http://www.etti.tuiasi.ro/index.php?option=com_content&view=article&id=118:dfchiper&catid=37:cadre-didactice&Itemid=78, there are some efficient DHT algorithms for length \(N=4\) and \(N=8\) [7].

1.2 MATLAB Script for the Proposed Split-Radix DHT Algorithm

figure a
figure b

1.3 Solving the Recurrence Relations for the Arithmetic Complexity

(a) Solving \(M(N)=M(N/2)+2\cdot M(N/4)+7N/8-4\)

with \(M(4)=0\), \(M(8)=2\)

We can write

$$\begin{aligned} \left[ \begin{array}{l} {M(N)} \\ *\\ \end{array} \right] =A^{n-3}\cdot b+\left[ I+A/2+\cdots +(A/2)^{n-4}\right] \cdot cN-\left[ I+A+\cdots +A^{n-4}\right] d \end{aligned}$$

where

$$\begin{aligned} A=\left[ {{\begin{array}{ll} 1&{} 2 \\ 1&{} 0 \\ \end{array} }} \right] ; \quad b=\left[ {{\begin{array}{l} 2 \\ 0 \\ \end{array} }} \right] ; \quad c=\left[ {{\begin{array}{l} {7/8} \\ 0 \\ \end{array} }} \right] ; \quad d=\left[ {{\begin{array}{l} 4 \\ 0 \\ \end{array} }} \right] . \end{aligned}$$

We note \(a_{00} (n)\), \(a_{01} (n)\) the elements \(a_{00}\) and \(a_{01}\) of \(A^{n}\).

We have

$$\begin{aligned} a_{00}(n)= & {} \frac{2}{3}2^{n}+(-1)^{n}\frac{1}{3}\\ a_{01}(n)= & {} \frac{2}{3}2^{n}-(-1)^{n}\frac{2}{3}. \end{aligned}$$

We can further write

$$\begin{aligned} M(N)= & {} \left[ \frac{2}{3}\cdot 2^{n-3}+(-1)^{n-3}\frac{1}{3}\right] \cdot 2+\left\{ 1+(n-4)\frac{2}{3}\right. \nonumber \\&\left. +\frac{1}{3}\left[ -\frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{2^{3}}+\cdots +(-1)^{n-4}\frac{1}{2^{n-4}}\right] \right\} \frac{7}{8}N \\&-\left[ 1+\frac{2}{3}2-\frac{1}{3}+\frac{2}{3}2^{2}+\frac{1}{3}+\cdots +\frac{2}{3}2^{n-4}+(-1)^{n}\frac{1}{3}\right] \cdot 4, \end{aligned}$$

where \(n=\log _2 N\).

Results

$$\begin{aligned} M(N)= & {} \frac{1}{3}\cdot \left[ 2^{n-2}+(-1)^{n-3}\right] \cdot 2+\left\{ {\frac{2}{3}(n-3)+\frac{1}{9}\frac{2^{n-3}-(-1)^{n-3}}{2^{n-4}}} \right\} \frac{7}{8}N \\&-\frac{1}{3}\left[ 1+2(2^{n-3}-1)\right] \cdot 4+\frac{4}{3}\cdot <\log _2 N>_2 \\ \end{aligned}$$

and finally

$$\begin{aligned} M(N)=\frac{7}{12}N\cdot \log _2 N-\frac{31}{18}N+\frac{4}{3}+\frac{8}{9}(-1)^{\log _2 N}+\frac{1}{3}<\log _2 N>_2 \cdot 4. \end{aligned}$$

(b) Solving \(A(N)=A(N/2)+2A(N/4)+21N/8-4\)

with \(A(4)=8\); \(A(8)=16.\)

We can write

$$\begin{aligned} \left[ {{\begin{array}{l} {A(N)} \\ *\\ \end{array} }} \right] =A^{n-3}\cdot b+\left[ I+A/2+\cdots +(A/2)^{n-4}\right] \cdot cN-\left[ I+A+\cdots +A^{n-4}\right] d \end{aligned}$$

where

$$\begin{aligned} A=\left[ {{\begin{array}{ll} 1&{} 2 \\ 1&{} 0 \\ \end{array} }} \right] ; \quad b=\left[ {{\begin{array}{l} {16} \\ 8 \\ \end{array} }} \right] ; \quad c=\left[ {{\begin{array}{l} {21/8} \\ 0 \\ \end{array} }} \right] ; \quad d=\left[ {{\begin{array}{l} 4 \\ 0 \\ \end{array} }} \right] . \end{aligned}$$

We note \(a_{00} (n)\), \(a_{01} (n)\) the elements \(a_{00}\) and \(a_{01} \) of \(A^{n}\).

We have

$$\begin{aligned} a_{00}(n)= & {} \frac{2}{3}2^{n}+(-1)^{n}\frac{1}{3}\\ a_{01}(n)= & {} \frac{2}{3}2^{n}-(-1)^{n}\frac{2}{3}. \end{aligned}$$

We can further write

$$\begin{aligned} A(N)= & {} \left[ \frac{2}{3}\cdot 2^{n-3}+(-1)^{n-3}\frac{1}{3}\right] \cdot 16+\left[ \frac{2}{3}\cdot 2^{n-3}-(-1)^{n-3}\frac{2}{3}\right] \cdot 8 \\&+\left\{ {1+(n-4)\frac{2}{3}+\frac{1}{3}\left[ -\frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{2^{3}}+\cdots +(-1)^{n-4}\frac{1}{2^{n-4}}\right] } \right\} \frac{21}{8}N \\&-\left[ 1+\frac{2}{3}2-\frac{1}{3}+\frac{2}{3}2^{2}+\frac{1}{3}+\cdots +\frac{2}{3}2^{n-4}+(-1)^{n}\frac{1}{3}\right] \cdot 4, \end{aligned}$$

where \(n=\log _2 N\).

$$\begin{aligned} A(N)= & {} \frac{1}{3}\cdot \left[ 2^{n-2}+(-1)^{n-3}\right] \cdot 16+\frac{1}{3}\cdot \left[ 2^{n-2}-(-1)^{n-3}2\right] \cdot 8\\&+\left\{ {\frac{2}{3}(n-3)+\frac{1}{9}\frac{2^{n-3}-(-1)^{n-3}}{2^{n-4}}} \right\} \frac{21}{8}N \\&-\frac{1}{3}\left[ 1+2(2^{n-3}-1)\right] \cdot 4+\frac{4}{3}\cdot<\log _2 N>_2 \\ A(N)= & {} \frac{1}{3}\cdot \left[ \frac{N}{4}-(-1)^{\log _2 N}\right] 16+\frac{1}{3}\cdot \left[ \frac{N}{4}+(-1)^{\log _2 N}2\right] 8\\&+\frac{21}{12}N\cdot (\log _2 N-3)+\frac{7}{24}\left( {2N+(-1)^{\log _2 N}16} \right) \\&-\frac{1}{3}\left[ 1+2(N/8-1)\right] \cdot 4+\frac{1}{3}<\log _2 N>_2 \cdot 4 \\ \end{aligned}$$

and finally results

$$\begin{aligned} A(N)=\frac{21}{12}N\cdot \log _2 N-3N+\frac{4}{3}+\frac{14}{3}(-1)^{\log _2 N}+\frac{1}{3}<\log _2 N>_2 \cdot 4. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chiper, D.F. A Structured Dual Split-Radix Algorithm for the Discrete Hartley Transform of Length \(2^{N}\) . Circuits Syst Signal Process 37, 290–304 (2018). https://doi.org/10.1007/s00034-017-0552-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-017-0552-3

Keywords

Search

Navigation