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Design of Low Complex Linear Precoding Scheme for MU-MIMO Systems

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The block diagonalization (BD) is a linear precoding technique for multiuser interference elimination in multi-user multiple-input multiple-output (MU-MIMO) systems. Though various methods of block diagonalization have been identified, they are convoluted regarding the computations involved. In this paper we have modeled a new paradigm for the performance improvement and complexity reduction in BD based precoding algorithms. This is consummated using principal component analysis (PCA). The traditional highly complex singular value decomposition is replaced by QR decomposition in PCA (QR-PCA) for complexity reduction. The PCA along with QR decomposition and minimum mean squared error (MMSE) channel inversion technique parallelize the MU-MIMO channel into independent and proportionate single user MIMO channel. The simulation result shows that the proposed QR-PCA based precoding algorithm in comparison with the existing algorithm achieves comparatively better sum-rate, lower BER, and lower computational complexity. The PCA along with MMSE channel inversion avoids the decoder structure, which makes the receiver system simple.

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  1. 1.

    Paulraj, A., Nabar, R., & Gore, D. (2003). Introduction to space–time wireless communications. Cambridge: Cambridge University Press.

  2. 2.

    Telatar, I. E. (1999). Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications, 10, 585–595.

  3. 3.

    Tse, D., & Viswanath, P. (2005). Fundamentals of wireless communications. Cambridge: Cambridge University Press.

  4. 4.

    Foschini, G. J., & Gans, M. (1998). On limits of wireless communications in a fading environment when using multiple antennas. Wireless Personal Communications, 6, 311–335.

  5. 5.

    Requirements for further advancements for E-UTRA (LTE-advanced). (2014). 3GPP TR 36.913 V12.0.0 standards.

  6. 6.

    Gesbert, D., Kountouris, M., Heath, R. W., Jr., Chae, C. B., & Salzer, T. (2007). From single user to multiuser communications: Shifting the MIMO paradigm. IEEE Signal Processing Magazine, 24(5), 36–46.

  7. 7.

    Vishwanath, S., Jindal, N., & Goldsmith, A. (2003). Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels. IEEE Transactions on Information Theory, 49, 2658–2668.

  8. 8.

    Viswanath, P., & Tse, D. N. C. (2003). Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality. IEEE Transactions on Information Theory, 49, 1912–1921.

  9. 9.

    Tomlinson, M. (1971). New automatic equalizer employing modulo arithmetic. Electronics Letters, 7(5), 138–139.

  10. 10.

    Harashima, H., & Miyakawa, H. (1972). Matched-transmission technique for channels with intersymbol interference. IEEE Transactions on Communications, 20(4), 774–780.

  11. 11.

    Costa, M. (1983). Writing on dirty paper. IEEE Transactions on Information Theory, 29, 439–441.

  12. 12.

    Peel, C., Hochwald, B., & Swindlehurst, L. (2003). A vector perturbation technique for near-capacity multi-antenna multi-user communication. In Proc. of the 41st Allerton conference on communication, control, and computing.

  13. 13.

    Hassibi, B., & Vikalo, H. (2005). On the sphere decoding algorithm: Part I the expected complexity. IEEE Transactions on Signal Processing, 53(8), 2806–2818.

  14. 14.

    Vikalo, H., & Hassibi, B. (2005). On the sphere decoding algorithm: Part II generalizations, second-order statistics, and applications to communications. IEEE Transactions on Signal Processing, 53(8), 2819–2834.

  15. 15.

    Spencer, Q., Peel, C., Swindlehurst, A., & Haardt, M. (2004). An introduction to the multi-user MIMO downlink. IEEE Communications Magazine, 42(10), 60–67.

  16. 16.

    Peel, C. B., Hochwald, B. M., & Swindelhurst, A. L. (2005). A vector-perturbation technique for near-capacity multi antenna multiuser communication—Part I: Channel inversion and regularization. IEEE Transactions on Communications, 53, 195–202.

  17. 17.

    Spencer, Q. H., Swindelhurst, A. L., & Haardt, M. (2004). Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels. IEEE Transactions on Signal Processing, 52, 461–471.

  18. 18.

    Stankovic, V., & Haardt, M. (2008). Generalized design of multiuser MIMO precoding matrices. IEEE Transactions on Wireless Communications, 7, 953–961.

  19. 19.

    Wang, H., Li, L., Song, L., & Gao, X. (2011). A linear precoding scheme for downlink multiuser MIMO precoding systems. IEEE Communications Letters, 15(6), 653–655.

  20. 20.

    Sung, H., Lee, S.-R., & Lee, I. (2009). Generalized channel inversion methods for multiuser MIMO systems. IEEE Transactions on Communications, 57(11), 3489–3499.

  21. 21.

    Zu, K., de Lamare, R. C., & Haardt, M. (2009). Generalized design of low-complexity block diagonalization type precoding algorithms for multiuser MIMO systems. IEEE Transactions on Communications, 61(10), 4232–4242.

  22. 22.

    Gan, Y. H., Ling, C., & Mow, W. H. (2009). Complex lattice reduction algorithm for low-complexity full-diversity MIMO detection. IEEE Transaction on Signal Processing, 57(7), 2701–2710.

  23. 23.

    Opmeer, M. R. (2012). Model order reduction by balanced proper orthogonal decomposition and by rational interpolation. IEEE Transactions on Automatic Control, 57(2), 472–477.

  24. 24.

    Shlens, J. (2005). A tutorial on principal component analysis.∼elaw/papers/pca.pdf.

  25. 25.

    Smith, L. I. (2002). A tutorial on principal components analysis, Technical report, Cornell University, Ithaca.

  26. 26.

    Sharma, A., Paliwal, K. K., Imoto, S., & Miyano, S. (2013). Principal component analysis using QR decomposition. International Journal of Machine Learning and Cybernetics, 4(6), 679–683.

  27. 27.

    Ascher, U. M., & Greif, C. (2011). A first course in numerical methods. Philadelphia, PA: Society for Industrial and Applied Mathematics.

  28. 28.

    Hassibi, B. (2000). An efficient square-root algorithm for BLAST. In Proc. IEEE intl. conf. acoustic, speech, signal processing (ICASSP) (pp. 5–9). Istanbul.

  29. 29.

    Wubben, D., Bohnk, R., Kuhn, V., & Kammeyer, K. D., (2003). MMSE extension of V-BLAST based on sorted QR decomposition. In Proc. IEEE vehicular technology conf. (VTC). Orlando, FL.

  30. 30.

    Bohnke, R., Wubben, D., Kuhn, V., & Kammeyer, K. D. (2003). Reduced complexity MMSE detection for BLAST architectures. In Proc. IEEE global communications conf. (GLOBECOM). San Francisco, CA.

  31. 31.

    Vishwanath, S., Jindal, N., & Goldsmith, A. J. (2002). On the capacity of multiple input multiple output broadcast channels. In Proc. IEEE int. conf. on communications (ICC) (pp. 1444–1450). New York.

  32. 32.

    Golub, G. H., & Loan, C. F. V. (1989). Matrix computations (3rd ed.). Baltimore: The Johns Hopkins University Press.

  33. 33.

    Kuhn, V. (2006). Wireless communications over MIMO channels: Applications to CDMA and multiple antenna systems. New York: Wiley.

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Correspondence to S. B. M. Priya.


Appendix 1

Let us consider a complex matrix \({\mathbf{X}} \in {\mathbb{C}}^{M \times M}\). Normalizing each row of \({\mathbf{X}}\) to zero mean,

$${\bar{\mathbf{X}}}_{\text{k}} = {\mathbf{X}}_{\text{k}} - {\mathbf{X}}_{\text{k}}^{ '}$$

where \({\mathbf{X}}_{k}^{'}\) represents the mean of kth row of \({\mathbf{X}}\). Now define the unitary matrix \({\mathbf{P}}\) such that

$${\mathbf{Y}} = {\mathbf{PX}}$$

while the covariance of \({\mathbf{Y}}\) matrix is diagonalized. The matrix \({\mathbf{P}}\) denotes the principal components of \({\mathbf{X}}\) and each row of P denotes the Eigen vectors of covariance matrix \({\mathbf{XX}}^{\text{H}}\). If \({\mathbf{z}}_{\text{k}}\) denotes the set of Eigen vector corresponding to Eigen value \(\lambda_{k}\) for the symmetric matrix XX H, then

$$\left( {{\mathbf{XX}}^{\text{H}} } \right){\mathbf{z}}_{\text{k}} = {\varvec{\uplambda}}_{\text{k}} {\mathbf{z}}_{\text{k}}$$

where \(\varvec{z}_{k} = \left( {\varvec{z}_{1} ,\varvec{z}_{2} , \ldots ,\varvec{z}_{m} } \right) \in {\mathbb{C}}^{M \times 1}\) and \(\lambda_{k} = \left( {\lambda_{1} ,\lambda_{2} , \ldots ,\lambda_{m} } \right) \in {\mathbb{R}}^{M \times 1}\)

Let the covariance of \(\bar{\varvec{X}}_{k}\) be \({\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}}\) and \({\bar{\mathbf{X}}}_{\text{k}} = {\mathbf{QR}}\) is the QR decomposition of \({\bar{\mathbf{X}}}_{\text{k}}\) matrix. Now the covariance matrix can be rewritten as,

$${\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = \left( {{\mathbf{QR}}} \right)\left( {{\mathbf{QR}}} \right)^{\text{H}}$$

where \({\mathbf{Q}} \in {\mathbb{C}}^{M \times M}\) is a unitary matrix; \({\mathbf{R}} \in {\mathbb{C}}^{M \times M}\) is an upper triangular matrix.

The matrix \({\mathbf{R}}^{\text{H}}\) can be expanded as \({\mathbf{R}}^{\text{H}} = {\mathbf{UDV}}^{\text{H}}\) using SVD decomposition. So (40) can be rewritten as,

$$\begin{aligned} & {\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = {\mathbf{Q}}\left( {{\mathbf{UDV}}^{\text{H}} } \right)^{\text{H}} \left( {{\mathbf{UDV}}^{\text{H}} } \right){\mathbf{Q}}^{\text{H}} \\ & {\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = {\mathbf{QVD}}\left( {{\mathbf{U}}^{\text{H}} {\mathbf{U}}} \right){\mathbf{DV}}^{\text{H}} {\mathbf{Q}}^{\text{H}} \\ & {\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = \left( {{\mathbf{QV}}} \right){\mathbf{D}}^{2} \left( {{\mathbf{QV}}} \right)^{\text{H}} \left( { \because {\mathbf{U}}^{\text{H}} {\mathbf{U}} = 1} \right) \\ & \left( {{\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} } \right) \left( {{\mathbf{QV}}} \right) = \left( {{\mathbf{QV}}} \right){\mathbf{D}}^{2} \\ \end{aligned}$$

Comparing (39) and (41), the Eigen vector matrix which diagonalizes \(\left( {{\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} } \right)\) is \({\mathbf{QV}}\) and their corresponding Eigen value is \({\mathbf{D}}^{2}\). Thus the principal component,

$${\mathbf{P}} = {\mathbf{QV}}$$

Therefore the PCA extracted matrix is given as, \({\mathbf{Y}} = {\mathbf{PX}}.\)

Appendix 2

The received signal without PCA transform at the receiver is given by,

$${\mathbf{y}} = {\boldsymbol{\mathcal{H}}}{\acute{\mathbf{W}}}{\mathbf{b}} + {\mathbf{n}}$$

Let \({\acute{\mathbf{W}}} = \left({\frac{1}{{\sqrt {\updelta}}}} \right){\acute{\mathbf{W}}}^{1} {\acute{\mathbf{W}}}^{2}\) be the precoding matrix without PCA transform and \(\delta = \frac{{\parallel {\acute{\mathbf{W}}} {\mathbf{b}}\parallel_{F}^{2}}}{{P_{T}}}\) is the scaling factor.

Here we now rearrange (43) in terms of QR-PCA transform. In PCA-MMSE-BD, the PCA is applied only in the manipulation \(\varvec{W}^{2}\). Hence (23) can be rewritten using (22) as,

$${\mathbf{W}}^{2} = {\mathbf{E}} ({\mathbf{PC}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} )^{\text{H}} \left( {\left( {{\mathbf{PC}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} } \right)({\mathbf{PC}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} )^{\text{H}} } \right)^{ - 1}$$

By using the matrix properties [33], \({\mathbf{X}}^{\text{H}} = {\mathbf{X}}^{ - 1}\) for any unitary matrix \({\mathbf{X}}\) and \(\left( {{\mathbf{AB}}} \right)^{ - 1} = \left( {\mathbf{B}} \right)^{ - 1} \left( {\mathbf{A}} \right)^{ - 1}\) for any two matrices \({\mathbf{A}}\) and \({\mathbf{B}}\); (44) can be rewritten as

$$\begin{aligned} {\mathbf{W}}^{2} &= {\mathbf{E}}\left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} \left( {{\mathbf{PC}}^{\text{H}} {\mathbf{PC}}} \right)\left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} } \right)^{ - 1} {\mathbf{PC}}^{\text{H}} } \right) \hfill \\ \mathbf{W}^{2} &= \mathbf{E}\left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} \left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} } \right)^{ - 1} {\mathbf{PC}}^{\text{H}} } \right)\quad \left\{ {\because \mathbf{PCPC}^{\text{H}} = {\mathbf{I}}} \right\} \hfill \\ {\mathbf{W}}^{2} &= {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}^{2} {\mathbf{PC}}^{\text{H}} \hfill \\ \end{aligned}$$

where \({\acute{\mathbf{W}}}^{2} = {\mathbf{E}}\left({{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H}}}_{\text{ext}}^{\text{H}} \left({{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H}}}_{\text{ext}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H}}}_{\text{ext}}^{\text{H}}} \right)^{- 1}} \right)\) . The scaling factor after PCA transform is,

$$\delta = \frac{{\parallel {\mathbf{PCWb}}\parallel_{F}^{2} }}{{P_{T} }}$$

Since \(\parallel {\mathbf{UB}}\parallel_{\text{F}}\, =\, \parallel {\mathbf{B}}\parallel_{\text{F}}\) for any unitary matrix \({\mathbf{U}}\) [33], (46) can be written as,

$$\delta = \frac{{\parallel {\mathbf{W}} {\mathbf{b}}\parallel_{F}^{2} }}{{P_{T} }}$$

The received signal at UE after PCA transform could be obtained by substituting (45) and (47) in (43),

$${\mathbf{y}} = {\boldsymbol{\mathcal{H}}}{\mathbf{PCWb}} + {\mathbf{n}}$$

As \({\mathbf{PC}}\) is a unitary matrix it will not lead any variation in statistical property of \({\mathbf{n}}\) [33]. So (48) can be re-written as,

$${\mathbf{y}} = {\mathbf{PC}}\left( {{\boldsymbol{\mathcal{H}}}{\mathbf{Wb}} + {\mathbf{n}}} \right)$$

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Priya, S.B.M., Kumar, P. Design of Low Complex Linear Precoding Scheme for MU-MIMO Systems. Wireless Pers Commun 97, 1097–1116 (2017).

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  • BD
  • Channel inversion
  • Computational complexity
  • Floating point operation (flop)
  • MMSE
  • Multiuser-MIMO (MU-MIMO)
  • Precoding
  • PCA
  • QR decomposition
  • Regularized block diagonalization (RBD)
  • SVD