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Outage Constrained Beamformer Design

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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

While Chaps. 3 and 4 discuss ergodic robust beamforming, which implies a fast-fading channel model, this chapter focuses on slower fading. The coherence time shall be in the range of the transmit phase, such that the channel is either quasistatic or each transmit code word experiences only a few channel realizations. Transmitter CSI can still be imperfect, e.g., due to limited feedback and delayed CSI usage. In this case, non-robust transmit strategies are unable to ensure reliable data transmission—an unknown number of outages occur when the transmitter sends information at the imposed rates. An outage defines the event that the channel is too bad to decode error-free at the transmit data rate.

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Notes

  1. 1.

    The formulation is short hand for the probability \(\Pr (\mathcal {A}_j({\boldsymbol {x}}))\) with respect to the distribution of ξ.

  2. 2.

    The emphasis is here on probability constraints for scalar stochastic inequalities (5.5), because the downlink beamformer design considers unicast data transmission to independent receivers. An analysis of chance constraints based on vector inequalities is provided by [254, Chapter 4].

  3. 3.

    The references assume a j(x) = x for the Probabilistically constrained linear inequality (5.10).

  4. 4.

    Generalized concavity theory is an utmost important tool for analyzing probability constraints, because the CDF of a random variable with an α-concave PDF is a quasiconcave function under mild restrictions on the concavity parameter α. The theory for this result and further analysis on concave probability measures for stochastic programming is provided by [254, Chapter 4].

  5. 5.

    Wang [36] provides a summary on chance constrained programming for downlink optimization.

  6. 6.

    Lorentz positive maps (LPMs) are linear mappings of vectors from a SOC into vectors of another SOC.

  7. 7.

    A related bound for positive semidefinite \(\boldsymbol {Q}({\boldsymbol {x}})\in \mathcal {S}_+\) is found in [268].

  8. 8.

    A similar 𝜖-outage rate expression (5.18) is also valid for vector PtP channels and additive fading, e.g., see [1, Section 5.4.2] for the multi-antenna receiver and [201] for the multi-antenna transmitter case.

  9. 9.

    Chalise et al. [263] alternatively modeled the sample mean of the SINR with (5.10).

  10. 10.

    Here, \(\Pr (\xi _k=-1)=0\) and the noise variance is without loss of generality set to \(\sigma _k^2=1\).

  11. 11.

    The basis for this integral representation is the relation of the non-central chi-square distribution to the Marcum-Q function [278]. An alternative series expansion in terms of central chi-square distributions in Sect. A.8 follows from [203, Equation 26.4.25].

  12. 12.

    The probability is positive only if B k has a positive eigenvalue and otherwise it is zero.

  13. 13.

    Imhof’s method and references to integral representations are also found in [281, Chapter 4].

  14. 14.

    Adaptive rate control mechanisms are another method to decrease the conservatism in communication systems in favor for performance gains if fast feedback of single bits is available.

  15. 15.

    The set-valued mapping particularly satisfies \(\lim _{n\to \infty }\mathcal {T}(\boldsymbol {p}^{(n)},\sigma _k^2)\to \mathcal {T}(\boldsymbol {p}^\star ,\sigma _k^2)\) if p ⋆ is the limit point of p (n) for n →∞, where convergence is in the Kuratowski–Painlevé sense [287].

  16. 16.

    Generally, continuity in probability means that \(\lim _{n\to \infty }\Pr (\mathcal {A}_n)=\Pr (\mathcal {A}^\ast )\) for any increasing sequence of events \(\mathcal {A}_n\subseteq \ldots \subseteq \mathcal {A}_1\) with convergence set \(\mathcal {A}^\ast \) and \(\lim _{n\to \infty }\Pr (\mathcal {B}_n)=\Pr (\mathcal {B}^\ast )\) for any decreasing sequence of events \(\mathcal {B}_n\supset \ldots \supset \mathcal {B}_1\) with convergence set \(\mathcal {B}^\ast \). If the sequences of sets \(\mathcal {A}_n\) and \(\mathcal {B}_n\) converge in the Kuratowski–Painlevé sense, this results in the above convergence behavior in probability [287].

  17. 17.

    Here, p is normalized to \({\mathbf {1}}^{\operatorname {T}}\boldsymbol {p}=1\) due to the scale-invariance of p k∕I k(p; ρ k).

  18. 18.

    This model is obtained when approximating the Gaussian additive channel error as in Sect. 2.3.

  19. 19.

    A reformulation into the LPM (5.60) misses for the orthogonal channel uncertainty region.

  20. 20.

    This single user rate maximization problem has an equivalent convex reformulation.

  21. 21.

    Even though distinct rate targets do not change this relation between the orthogonal and the spherical uncertainty approximation for the chance constraints. Section 5.7.3 shows that the orthogonal uncertainty approximation outperforms the spherical bound also for ρ 1 = 1.25 and ρ 2 = 0.75 if K = 2 for example.

  22. 22.

    Investigations on a tractable but close to optimal parameter search is an open topic for research.

  23. 23.

    Again, the SDR method for the spherical uncertainty approximation provides higher outage values than the LPM formulation due to the additional approximation (cf. Sect. 5.5).

  24. 24.

    The dominant positive eigenvalue of \(\boldsymbol {A}\in \mathcal {H}^N_+\) is bounded as \(N^{-1/2}\|\boldsymbol {A}_k\|{ }_{\operatorname {F}}\leq \lambda _{\max }^+(\boldsymbol {A}_k)\leq \|\boldsymbol {A}\|{ }_{\operatorname {F}}\).

  25. 25.

    When the RB problem becomes infeasible due to the approximation, ρ is set to zero.

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    Andreas Gründinger

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Gründinger, A. (2020). Outage Constrained Beamformer Design. 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_5

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  • DOI: https://doi.org/10.1007/978-3-030-29578-3_5

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