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Statistical Robust Beamforming for Broadcast Channels and Applications in Satellite Communication pp 77–125Cite as

Mean Square Error Transceiver Design for Additive Fading

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

Abstract

For the additive channel model and imperfect CSI at the multi-antenna transmitter, rate based beamformer optimizations are difficult to solve directly. The closed-form expressions for ergodic rates involve numeric integrations and are hardly known for other scenarios than Rayleigh or Rician fading (Kang and Alouini, IEEE Trans Wirel Commun 5:112, 143, 2006; Taricco and Riegler, IEEE Trans Inf Theory 57:4123, 2011). This prevents a reformulation of the QoS and RB optimizations into convex form (cf. Chap. 3).

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Notes

  1. 1.

    In fact, the average SINR results in an upper rate bound for multiplicative fading (see Sect. 3.2).

  2. 2.

    The references about the weighted MSE approximation for weighted sum-rate maximization are only a small subset of the available works. There exist many related publications citing [155].

  3. 3.

    While the MMSE definition in [152] is between the received signal for (interference and) noiseless transmission and the actual received signal with conditional mean estimator, we define the (M)MSE between the intended transmit signal and the corresponding linear MMSE estimate instead.

  4. 4.

    Reference [34] already considers imperfect CSI. Therein, averaging is achieved by a stochastic update rule, which increases the involved samples for the sample average approximations in each iteration. Thereby, the corresponding ACS algorithm still almost surely converges to a KKT point.

  5. 5.

    Each receiver has imperfect knowledge about its corresponding channel, e.g., user k knows the mean and covariance of , and the transmitter has this knowledge for all channels.

  6. 6.

    Approximations for moments of ratios of quadratic forms in random variables are shown in [245].

  7. 7.

    We have exploited [27, Theorem 4] to derive the average MMSE of a point-to-point MIMO channel [157] and for a downlink multi-user MIMO channel [158].

  8. 8.

    Derivatives of eigenvalues are detailed in [66, Appendix A.14] and the references therein.

  9. 9.

    The idea of pre-fixed equalizer functions was also exploited by [235]. Other than [235], f k(h k) are herein the MMSE filters for a pre-defined t ′ and, thereby, approximate the optimal filters.

  10. 10.

    This is in contrast to the QoS optimization with the average SINR from [118], for example.

  11. 11.

    For example, see [246] for a definition of biconvex functions.

  12. 12.

    Alternatively, one may employ a convex Semidefinite program (SDP) formulation of the dual problem.

  13. 13.

    The problems are similar to the SINR constrained QoS optimization (cf. [71, 73]).

  14. 14.

    The primal objective \(\sup \limits _{\boldsymbol {\lambda }\geq 0,\boldsymbol {\mu }}L(p,\boldsymbol {t},\boldsymbol {\lambda },\boldsymbol {\mu })\) upper bounds the dual objective \(\inf \limits _{p\geq 0,\boldsymbol {t}}L(p,\boldsymbol {t},\boldsymbol {\lambda },\boldsymbol {\mu })\).

  15. 15.

    The alternative derivation in Sect. A.6 is up to this point. The remaining steps are the same.

  16. 16.

    This follows from \(\hat {\boldsymbol {h}}_k\in \operatorname {range}\{\boldsymbol {R}_k\}\) and the block entries of the block-diagonal matrix Y are \(\boldsymbol {Y}_k=\sum _{j=1}^M\lambda _j\sum _{k\in \mathcal {G}_j}w_k\boldsymbol {R}_k+\boldsymbol {C}_k\) with the matrix \(\boldsymbol {C}_k=\sum _{\ell =1}^L\mu _\ell \boldsymbol {A}_{k,\ell }\succeq \mathbf {0}\).

  17. 17.

    This restriction is without loss of generality for \(\varepsilon <\sum _{k=1}^Kw_k\) and (λ 1, μ) = (0, 0) for \(\varepsilon \geq \sum _{k=1}^Kw_k\).

  18. 18.

    For the duality results in [68, 161], the power allocation is substituted by λ ′ = P 1λ.

  19. 19.

    The LMI constraint stems from (4.39) within the proof of Theorem 4.1.

  20. 20.

    Berman and Plemmons [229, Chapter 6] provide an extensive discussion about non-singular M-matrices including 50 equivalent statements for their definition [229, Theorem 6.2.3].

  21. 21.

    The scalar projection is onto and reads as \([z]^+=\max (0,z)\) for .

  22. 22.

    This follows from the block-diagonal structure of Y .

  23. 23.

    In [158], we have suggested alternative derivations based on the expectation in (4.18).

  24. 24.

    Using a sum power constraint for this characterization is without loss of generality. Since the transmit power is unbounded, can replace \(\mathcal {P}\) from (1.32).

  25. 25.

    Here, I(λ; ε) is the entrywise minimum of the function I ε(λ, u) = D εM(u)λ + n(u), where \(\boldsymbol {D}_{\boldsymbol {\varepsilon }}=\operatorname {diag}(\varepsilon _1^{-1}-1,\ldots ,\varepsilon _K^{-1}-1)\), \([\boldsymbol {M}(\boldsymbol {u})]_{k,i}=|\hat {\boldsymbol {h}}_k^{\operatorname {H}}\boldsymbol {u}_k|{ }^{-2}\boldsymbol {u}_k^{\operatorname {H}}\boldsymbol {R}_i\boldsymbol {u}_k\), and \([{\boldsymbol {n}}]_k=P^{-1}|\hat {\boldsymbol {h}}_k^{\operatorname {H}}\boldsymbol {u}_k|{ }^{-2}\|\boldsymbol {u}_k\|{ }^{2}\). This reformulation supports the results and algorithms from [89, 90] and many results from [11].

  26. 26.

    Since I ∞(λ ′; ε) =minuD εM(u)λ ′, the fixed point \(\boldsymbol {\lambda }^{{\prime },\star }=\boldsymbol {I}_{\boldsymbol {\varepsilon }}^\infty (\boldsymbol {\lambda }^{{\prime },\star })\) is the eigenvector of D εM(u) with the SIR maximizing equalizers \(\boldsymbol {u}_k=(\sum _{i=1}^K\lambda _i\boldsymbol {R}_i)^{-1}\hat {\boldsymbol {h}}_k\), k = 1, …, K [252, Section 6]. Alternatively, we find λ ′, ⋆  as the limit point of the converging sequence \(\boldsymbol {\lambda }^{{\prime },{(n+1)}}\mspace {-5mu}= \|\boldsymbol {I}_{\boldsymbol {\varepsilon }}^\infty (\boldsymbol {\lambda }^{{\prime },{(n)}})\|{ }_1^{-1}\boldsymbol {I}_{\boldsymbol {\varepsilon }}^\infty (\boldsymbol {\lambda }^{{\prime },{(n)}})\) [92, 226].

  27. 27.

    Both works base their duality derivation on an algorithmic perspective.

  28. 28.

    Only for equal MSE targets, the MSEs for all the users are always equal at the optimum.

  29. 29.

    The minimum for SINR and rate balancing is zero and corresponds to zero transmit power. As soon as the transmit power is positive, transmission to all the users is active, even for unequal targets [11, 166, 167]. This stands in contrast to MSE balancing with unequal MSE targets [164].

  30. 30.

    For the general case, we again consider set of weighted sum MSE constraints (4.6).

  31. 31.

    Equivalence holds for ε > 0. While (4.82) is even valid for ε = 0, this case is excluded in (4.81).

  32. 32.

    This beamformer is also the initialization for equal targets ε k = c and optimal for very low SNR.

  33. 33.

    Assuming fixed v, the optimization problem is convex, because \(\operatorname {AMSE}_k(v_j,\boldsymbol {t})\) is convex quadratic in t for fixed v j and the maximum of convex functions is convex [102, Section 3.2.3].

  34. 34.

    Requiring \(\hat {\boldsymbol {h}}_k^{\operatorname {H}}\boldsymbol {t}_k\) to be real valued does not change the objectives minimum value (cf. Sect. 4.3.1).

  35. 35.

    The same functionality has the test in [79, Algorithm 2], which is for SINR balancing.

  36. 36.

    From (4.98), \(\lambda _j^{(n+1)}\) clearly decreases for increasing ε (n+1) until \(\lambda _j^{(n+1)}=0\) for ε jε (n+1) ≥ m j.

  37. 37.

    The search for M ′ requires less than M ′ iterations for a bisection over the indices i ∈{1, …, M ′′} that halves the number of candidates in each iteration.

  38. 38.

    The LMI constraint is affine in the decision variables μ and ε.

  39. 39.

    The exponential power profile is amongst others used for multi-spotbeam SatCom [255].

  40. 40.

    The alternative updates via a dual SDP formulation provided the same performance.

  41. 41.

    This normalized power iteration is motivated by the complementary slackness conditions for the power constraints, i.e., \(\mu _\ell P_\ell =\mu _\ell \boldsymbol {t}^{\operatorname {H}}\boldsymbol {A}_\ell \boldsymbol {t}\), ℓ = 1, …, L need to be satisfied for any local solution.

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Gründinger, A. (2020). Mean Square Error Transceiver Design for Additive Fading. 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_4

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