Block-Circulant Inverse Orthogonal Jacket Matrices and Its Applications to the Kronecker MIMO Channel

  • Han Hai
  • Moon Ho LeeEmail author
  • Xiao-Dong Zhang


This paper presents a note on the block-circulant generalized Hadamard matrices, which is called inverse orthogonal Jacket matrices of orders \(N=2p, 4p, 4^kp, np\), where k is a positive integer for the Kronecker MIMO channel. The class of block Toeplitz circulant Jacket matrices not only have many properties of the circulant Hadamard conjecture but also have the construction of block-circulant, which can be easily applied to fast algorithms for decomposition. The matrix decomposition is with the form of the products of block identity \(I_2\) matrix and block Hadamard \(H_2\) matrix. In this paper, a block fading channel model is used, where the channel is constant during a transmission block and varies independently between transmission blocks. The proposed block-circulant Jacket matrices can also achieve about 3db gain in high SNR regime with MIMO channel. This algorithm for realizing these transforms can be applied to the Kronecker MIMO channel.


Circulant Hadamard conjecture Center-weighted Hadamard matrix Reciprocal transpose Block-circulant Jacket transform DFT matrix Toeplitz matrix Kronecker MIMO channel 



The second author visited Concordia University from July 4 to July 25, 2008, in Canada, and talked about Jacket Matrix many times, thanks to Professor M. Omair Ahmad and Professor M.N.S. Swamy.


  1. 1.
    N. Ahmed, K.R. Rao, Orthogonal Transforms for Digital Signal Processing (Springer, New York, 1975)CrossRefGoogle Scholar
  2. 2.
    S. Bouguezel, M.O. Ahmad, M.N.S. Swamy, An efficient algorithm for the computation of the reverse Jacket transform, in 2006 IEEE International Symposium on Circuits and Systems, ISCAS 2006, Proceedings, 21–24 May 2006, Island of Kos, pp. 1–4. (2008).
  3. 3.
    S. Bouguezel, M.O. Ahmad, M.N.S. Swamy, A new class of reciprocal-orthogonal parametric transforms. IEEE Trans. Circuits Syst. I Reg. Pap. 56(4), 795–804 (2009). MathSciNetCrossRefGoogle Scholar
  4. 4.
    Z. Chen, M.H. Lee, G. Zeng, Fast cocyclic Jacket transform. IEEE Trans. Signal Process. 56(5), 2143–2148 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Y.S. Cho, J. Kim, W.Y. Yang, C.G. Kang, MIMO-OFDM Wireless Communications with MATLAB (Wiley-IEEE Press, New York, 2010)CrossRefGoogle Scholar
  6. 6.
    J.J. Ding, S.C. Pei, P.H. Wu, Jacket Haar transform, in ISCAS (Rio de Janeiro, Brazil, 2011), pp. 1520–1523.
  7. 7.
    J.J. Ding, S.C. Pei, P.H. Wu, Arbitrary-length Walsh-Jacket transforms, in APSIPA Annual Summit and Conference (Xi’an, China, 2011).
  8. 8.
    C.P. Fan, J.F. Yang, Fast center weighted Hadamard transform algorithms. IEEE Trans. Circuits Syst.-II Analog Digit. Signal Process. 45(3), 429–432 (1998). CrossRefGoogle Scholar
  9. 9.
    L. Fan, R. Zhao, F. Gong, N. Yang, G.K. Karagiannidis, Secure multiple amplify-and-forward relaying over correlated fading channels. IEEE Trans. Commun. 65(7), 2811–2820 (2017). CrossRefGoogle Scholar
  10. 10.
    A.V. Geramita, J. Seberry, Orthogonal Designs: Quadratic forms and Hadamard Matrices (Marcel Dekker Inc, New York, 1979)zbMATHGoogle Scholar
  11. 11.
    G.H. Golub, Charles F. van Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1996)zbMATHGoogle Scholar
  12. 12.
    R.M. Gray, Toeplitz and circulant matrices: a review. Found. Trends Commun. Inf. Theory 2(3), 155–239 (2006). CrossRefzbMATHGoogle Scholar
  13. 13.
    J. Gutierrez-Gutierrez, P.M. Crespo, Block Toeplitz matrices: asymptotic results and applications. Found. Trends Commun. Inf. Theory 8(3), 179–257 (2011). CrossRefzbMATHGoogle Scholar
  14. 14.
    J. Gutierrez-Gutierrez, P.M. Crespo, Asymptotically equivalent sequences of matrices and Hermitian block Toeplitz matrices with continuous symbols: applications to MIMO systems. IEEE Trans. Inf. Theory 54(12), 5671–5680 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University, Cambridge, 1991)CrossRefGoogle Scholar
  16. 16.
    K.J. Horadam, Hadamard Matrices and Their Applications (Princeton University Press, Princeton, 2007)zbMATHGoogle Scholar
  17. 17.
    H. Huh, A.M. Tulino, G. Caire, Network MIMO with linear zero-forcing beamforming: large system analysis, impact of channel estimation and reduced-complexity scheduling. IEEE Trans. Inf. Theory 58(5), 2911–2934 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    X.-Q. Jiang, M. Wen, J. Li, W. Duan, Distributed generalized spatial modulation based on Chinese remainder theorem. IEEE Commun. Lett. 21(7), 1501–1504 (2017). CrossRefGoogle Scholar
  19. 19.
    X.-Q. Jiang, M. Wen, H. Hai, J. Li, S. Kim, Secrecy-enhancing scheme for spatial modulation. IEEE Commun. Lett. 22(3), 550–553 (2018). CrossRefGoogle Scholar
  20. 20.
    M.H. Lee, Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing (LAP LAMBERT Publishing, Saarbrücken, 2012)Google Scholar
  21. 21.
    M.H. Lee, The center weighted Hadamard transform. IEEE Trans. Circuits Syst. Analog Digit. Signal Process 36(9), 1247–1249 (1989). CrossRefGoogle Scholar
  22. 22.
    M.H. Lee, A new reverse Jacket transform and its fast algorithm. IEEE Trans. Circuits Syst.-II Analog Digit. Signal Process 47(1), 39–47 (2000). CrossRefzbMATHGoogle Scholar
  23. 23.
    M.H. Lee, H. Hai, X.-D. Zhang, MIMO Communication Method and System using the Block Circulant Jacket Matrix, USA Patent. no. 9,356,671, 05/31/2016Google Scholar
  24. 24.
    M.H. Lee, J. Hou, Fast block inverse Jacket transform. IEEE Signal Process. Lett. 13(8), 461–464 (2006). CrossRefGoogle Scholar
  25. 25.
    M.H. Lee, M. Kaveh, Fast Hadamard transform based on a simple matrix factorization. IEEE Trans. Acoust. Speech Signal Process ASSP–34(6), 1666–1667 (1986). CrossRefGoogle Scholar
  26. 26.
    M.H. Lee, M.H.A. Khan, K.J. Kim, D. Park, A fast hybrid Jacket–Hadamard matrix based diagonal block-wise transform. Signal Process. Image Commun. 1(29), 49–65 (2014). CrossRefGoogle Scholar
  27. 27.
    M.H. Lee, F. Szollosi, A note on inverse-orthogonal Toeplitz matrices. Electron. J. Linear Algebra 26, 898–904 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    M.H. Lee, F. Szollosi, Hadamard Matrices Modulo 5. J. Comb Des. 22(4), 171–178 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    M.H. Lee, X.-D. Zhang, W. Song, X.-G. Xia, Fast reciprocal Jacket transform with many parameters. IEEE Trans. Circuits Syst. I Reg. Pap. 59(7), 1472–1481 (2012). MathSciNetCrossRefGoogle Scholar
  30. 30.
    K.H. Leung, B. Schmidt, New restrictions on possible orders of circulant Hadamard matrices. Des. Codes Cryptogr. 64, 143–151 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    J. Li, M. Wen, X. Cheng, Y. Yan, S. Song, M.H. Lee, Generalized precoding-aided quadrature spatial modulation. IEEE Trans. Veh. Technol. 66(2), 1881–1886 (2017). CrossRefGoogle Scholar
  32. 32.
    J. Li, M. Wen, X. Jiang, W. Duan, Space-time multiple-mode orthogonal frequency division multiplexing with index modulation. IEEE Access 5, 23212–23222 (2017). CrossRefGoogle Scholar
  33. 33.
    F.J. Macwilliams, N.J.A. Sloane, The theory of error correcting codes (North-Holland, New York, 1977)zbMATHGoogle Scholar
  34. 34.
    S.C. Pei, J.J. Ding, Generalizing the Jacket transform by sub orthogonality extension, in EUSIOCO (European Signal Processing Conference), pp. 408–412 (2009)Google Scholar
  35. 35.
    B.S. Rajan, M.H. Lee, Quasi-cyclic dyadic codes in the Walsh–Hadamard transform domain. IEEE Trans. Inf. Theory 48(8), 2406–2412 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    K.W. Schmidt, E.T.H. Wang, The weight of Hadamard matrices. J. Comb. Theory A23, 257–263 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    G. Strang, Linear Algebra and Its Applications, 4th edn. (Cengage Learning, Boston, 2005)zbMATHGoogle Scholar
  38. 38.
    G. Strang, Essays in Linear Algebra (Wellesley-Cambridge Press, Wellesley, 2012)zbMATHGoogle Scholar
  39. 39.
    E. Telatar, Capacity of multi-antenna Gaussian channels. Eur. Trans. Telecommun. 10(6), 585–595 (1999). MathSciNetCrossRefGoogle Scholar
  40. 40.
    D. Tse, P. Viswanath, Fundamentals of Wireless Communication (Cambridge University Press, Cambridge, 2005)CrossRefGoogle Scholar
  41. 41.
    M. Wen, E. Basar, Q. Li, B. Zheng, M. Zhang, Multiple-mode orthogonal frequency division multiplexing with index modulation. IEEE Trans. Commun. 65(9), 3892–3906 (2017). CrossRefGoogle Scholar
  42. 42.
    R.K. Yarlagadda, J.E. Hershey, Hadamard Matrix Analysis and Synthesis With Applications to Communications Signal/Image Processing (Kluwer Academic Publishers, Norwell, 1997)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Information Science and TechnologyDonghua UniversityShanghaiPeople’s Republic of China
  2. 2.Engineering Research Center of Digitized Textile & Apparel TechnologyMinistry of EducationShanghaiPeople’s Republic of China
  3. 3.Division of Electronic Engineering, IT Convergence Research CenterChonbuk National UniversityJeonjuKorea
  4. 4.School of Mathematical Sciences, MOE-LSCShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

Personalised recommendations