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Multi-User Downlink Communication

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Statistical Robust Beamforming for Broadcast Channels and Applications in Satellite Communication

Part of the book series: Foundations in Signal Processing, Communications and Networking ((SIGNAL,volume 22))

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Abstract

Simultaneous wireless transmission from a single multi-antenna transmitter, e.g., a base station or a satellite, to K receivers—the users—is a standard model for terrestrial and satellite communication (SatCom) (Tse and Viswanath, Fundamentals of wireless communications. Cambridge University Press, New York, NY, 2008; Arapoglou et al., IEEE Commun Surv Tutorials 13:27, 2011). This multi-user downlink model is also known as a broadcast channel (BC) (e.g., see Tse and Viswanath (Fundamentals of wireless communications. Cambridge University Press, New York, NY, 2008)). The transmitter forms its transmit signal from independent data that are simultaneously conveyed to the users (Cover and Thomas, Elements of information theory. Wiley, Hoboken, NJ, 2006). Therefore, a terminal’s received signal not only includes the intended data signal, but also interfering signals that are destined to other terminals.

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Notes

  1. 1.

    This is in contrast to multicast setups, where all users receive the same information.

  2. 2.

    Lately, a DPC type scheme was proposed for block fading single antenna channels [8]. However, as extensions to multi-antenna setups are missing, this approach is beyond the scope of this work.

  3. 3.

    Further literature on multi-user downlink beamformer optimizations with ergodic rates and the related MSE or outage probability metrics is provided in the contributions section, i.e., Sect. 1.5, and at the beginning of Chaps. 3 and 4 or Chap. 5, respectively.

  4. 4.

    See the book by Li and Stoica [40], the book by Dietrich [41], and Chaps. 3–5 for more early references on robust beamforming for wireless communications.

  5. 5.

    The source for the plot is IEEEXplore and available online via ieeexplore.ieee.org. The following journals have been selected: IEEE Transactions on Signal Processing, IEEE Transactions on Communications, IEEE Transactions on Wireless Communications, IEEE Transactions on Vehicular Technology, IEEE Selected Topics in Signal Processing, IEEE Selected Topics in Communications, IEEE Signal Processing Letters, IEEE Communications Letters.

  6. 6.

    Interference becomes severe if the frequency reuse is one, i.e., the same bands are used in all cells.

  7. 7.

    Alternatively, the inputs may also be modeled as complex circularly symmetric Gaussian vectors , k = 1, …, K. However, the optimal transmit covariance matrices Q k of the intended problems are rank-one if sufficiently accurate CSI is available at the transmitter [18].

  8. 8.

    The advantageous use of the MSE and the MMSE compared to the rate and the SINR is especially for QoS feasibility and imperfect transmitter CSI as will be detailed in Sect. 1.2 and Chap. 4.

  9. 9.

    A precoder t is feasible if there is a p ≥ 0 such that \(p^{-1}\boldsymbol {t}\in \mathcal {P}\) and \(\boldsymbol {r}(\boldsymbol {t})\in \mathcal {R}\).

  10. 10.

    See Sect. A.1 for more detailed discussions to the basic properties of (1.12).

  11. 11.

    Detailed reviews on this topic are also provided by Schubert and Boche [11, 67, 83].

  12. 12.

    This fixed point map is also known as Picard iteration [86, Secion 1.2]. This reference also shows other iteration procedures with superior convergence that focus more on stability and contraction. The restriction to (1.18) for this work is due to its simplicity and variability for power allocation.

  13. 13.

    The mathematical basis for conic optimization, the generalized inequality formulations, and the corresponding interior point methods is amongst others due to Nesterov and Nemirovski [100]. A brief tutorial on conic optimization can be found in [101], for example, and more detailed introductions and applications are provided by Boyd and Ben-Tal in their study books [102, 103].

  14. 14.

    This corresponds to a relaxation of the rank-one conditions to \(\operatorname {rank}(\boldsymbol {Q}_i)\leq N\).

  15. 15.

    This stochastic interpretation of the above substitution and relaxation for the quadratic terms results from \(\operatorname {tr}(\boldsymbol {Q}_k\boldsymbol {R})=\operatorname {E}_{\boldsymbol {\tau }_k}[\boldsymbol {\tau }_k^{\operatorname {H}}\boldsymbol {R}\boldsymbol {\tau }_k]\) for Gaussian vectors (e.g., see [109]).

  16. 16.

    This is in contrast to the non-linear DPC case, where any finite rate targets are reachable [83].

  17. 17.

    Whether (1.18) is contractive or not depends on the system realization (cf. [112, Theorem 2]).

  18. 18.

    If \(\exists \mathcal {I}\subseteq \{1,\ldots ,K\}:\operatorname {rank}\{\boldsymbol {H}_{\mathcal {I}}\}<\min \{N,|\mathcal {I}|\}\), the matrix H is singular [115, Section III.C].

  19. 19.

    This table is not meant to be complete. Similar dualities may be found in other works as well.

  20. 20.

    We obtain these average SINRs by independently taking the mean of the nominator and the denominator if only second order channel information is available. These measures are commonly employed in the literature, even though they provide neither a lower nor an upper bound for R k.

  21. 21.

    Surrogate constrained programming is an approach to employ multiplier and duality theory for quasiconvex optimizations with multiple inequality constraints [123], where the standard Lagrangian duality theory does not generally apply. For the surrogate primal problem, the convex inequality constraints are linearly combined to a single scalar inequality constraint (cf. [122, 124]). The corresponding weights are the surrogate multipliers. This formulation allows for a simplified notion of duality.

  22. 22.

    The downlink is especially interesting due to the imposed correlations between the intended signal power and the interference within the users’ SINRs. Both signal parts are distorted by the same fading channels. This is in contrast to the Multiple access channel (MAC) and Interference channel (IFC), where the distortions of the interference and the useful signal are independent (e.g., see [110,111,127]).

  23. 23.

    ZF beamforming for only a part of the receivers can help to reduce the computational complexity, which is especially interesting if the number of antennas N and users K are large and N ≥ K.

  24. 24.

    These closed-form expressions were originally derived for the related MIMO models [157, 158].

  25. 25.

    Even though the shown ACS considers only imperfect transmitter CSI, i.e., the receivers employ perfect CSI equalizers, the duality is also valid for imperfect transmitter and receivers CSI (cf. [116]).

  26. 26.

    Examples of such stringent limitations are low power constraints and interference temperature constraints that set a threshold on the generated interference in a cognitive radio system [165].

  27. 27.

    Interference becomes severe for frequency reuse one as the same bands are used in all the cells.

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Gründinger, A. (2020). Multi-User Downlink Communication. In: Statistical Robust Beamforming for Broadcast Channels and Applications in Satellite Communication. Foundations in Signal Processing, Communications and Networking, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-29578-3_1

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