In this series:

Information theory suggests that the optimum transmit strategy for a multi-user (MU) MIMO broadcast channel (downlink path) requires a pre-interference cancellation technique known as dirty paper coding (DPC)

[1] combined with an implicit user scheduling algorithm. While there is substantial cost to knowing the channel state information at the transmitter, MU-MIMO techniques possess several key advantages: (i) MU-MIMO is immune to propagation channel impairments like channel rank loss and antenna correlation, (ii) there is no need to have multiple antennas at the terminal level to obtain spatial multiplexing gain at the base station, and (iii) MU-MIMO techniques enable capacity improvements proportional to the number of antennas at the base station.

The choice of codebook can significantly affect the overall system performance. Therefore, it needs to be carefully designed. For this blog post, we will limit the discussion to the most-referenced techniques in the literature. Grassmannian line packing (GLP) based codebook design is proven to provide optimal solutions in cases of limited feedback ([2], [3]). Unfortunately, GLP is computationally very demanding and suffers from signal-to-noise ratio (SNR) degradation in correlated channel profiles. In [4], GLP has been modified for correlated channels wherein channel covariance matrix is required. Being used as a signature design for code division multiple access (CDMA) systems, Random Vector Quantization (RVQ) has also been considered for MU-MIMO systems [5]. Unfortunately, RVQ’s performance is limited when the number of feedback bits is lower.

In his work on the systematic design of unitary space-time constellations [6], B. M. Hochwald proposed a codebook structure based on discrete fourier transforms (DFT). This codebook has been adopted in WiMAX (IEEE 802.16) and LTE for the following reasons: (i) compared to GLP and RVQ, DFT-based codebook design is simple enough for a feasible implementation, (ii) the codebook has lower peak to average power ratio (PAPR) as its entries have the same amplitude, and (iii) it is appropriate for correlated channels [7-9].

Recently, several precoding complexity reduction efforts with low-complexity codebooks have been proposed in the SU-MIMO system context [10-13]. Inspired by quadrature amplitude modulation (QAM), the authors in [10] have suggested a codebook whose inline codeword search burden and storage capacity are reduced. Unfortunately, these are still not low enough to be practical for implementation. On the other hand, another codebook structure has been devised using Kerdock coding [12].

In [14], a Kerdock codebook-based precoding scheme has been generalized for a multi-user MIMO broadcast channel with a known channel state at the base station and limited feedback. The proposed scheme exploits the scheduling and precoding methods with the aim to maximize system capacity and pre-cancel multi-user interferences.

Figure 1: Average system capacity

Figure 1: Average system capacity [14]

Figure 1 shows the average system capacity of SU-MIMO, the Gassmannian codebook, and the proposed Kerdock codebook [14]. The number of users to be scheduled at the base station ranges from 4 to 16 for the three schemes. It is clear that the proposed Kerdock scheme exploits the precoding gain and the multi-user diversity gain offered by the scheduler and outperforms the Grassmannian precoder, although the number of precoding vectors (P=6) is lower to ensure minimal complexity.


[1] M. Costa, “Writing on dirty paper,” IEEE Transactions on Information Theory, vol. 29, no.3, May 1983, pp. 439-441.[2] K . K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On Beamforming with Finite Rate Feedback in Multiple-Antenna Systems,” IEEE Trans. Info. Theory, vol. 49, pp. 2562-2579, Oct. 2003.[3] D. J. Love, R. W. Heath, Jr., and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2735-2747, Oct. 2003.[4] D. J. Love, R. W. Heath, Jr., “Grassmannian Beamforming on Correlated MIMO Channels,” Proc. Globecom, Dallas, 2004.[5] N. Jindal, “MIMO broadcast channels with _nite rate feedback,” IEEE Trans.Inform. Theory, vol. 52, no. 11, pp. 5045-5059, Nov. 2006.[6] B. M. Hochwald, T. L. Marzetta, T. L. Richardson, W. Sweldens, and R. Urbanke, Systematic design of unitary space-time constellations,” IEEE Transactions on Information Theory, vol. 46, no. 6, pp. 1962-1973, Sep. 2000.[7] L. Wan, X. Zhong, Y. Zheng and S. Mei, “Adaptive Codebook for Limited Feedback MIMO System,” International Conference on Wireless and Optical Communications Networks, pp. 1-5, 2009.[8] S. Li, H. Jia and J. Kang, “Robust Codebook Design Based on Unitary Rotation of Grassmannian Codebook,” IEEE Vehicular Tech. Conference, pp. 1-5, 2010.[9] D. Yang, L. L. Yang and L. Hanzo, “DFT-based Beamforming Weight-Vector Codebook Design for Spatially Correlated Channels in the Unitary Precoding Aided Multiuser Downlink,” IEEE International Conference on Communications, pp. 1-5, 2010.[10] D. J. Ryan, I. V. L. Clarkson, I. B. Collings, D. Guo, and M. L. Honig, “QAM codebooks for low-complexity limited feedback MIMO beamforming,” in Proc. IEEE Intl. Conf. on Comm., pp. 4162-4167, 2007.[11] T. Inoue and R. W. Heath Jr., “Kerdock Codes for Limited Feedback MIMO Systems,” in Proc. IEEE ICASSP, pp. 3113-3116, 2008.[12] T. Inoue and R. W. Heath, Jr., “Kerdock Codes for Limited Feedback Precoded MIMO Systems,” IEEE Trans. Signal Process., vol. 57, no. 9, pp. 3711-3716, Sept. 2009.[13] F. Cao, M. Sandell, F. Tosato, “A codebook with low complexity for limited feedback precoding,” PIMRC, pp. 1302-1306, 2009.[14] M. Benmimoune and D. Massicotte, “Multi-User MIMO Precoding with Kerdock Codebook,” ISWCS 2010.