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Precoder Design for Ergodic Rates with Multiplicative Fading

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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 the QoS and RB optimizations with instantaneous rate constraints feature well-established solutions, solving these problems with ergodic rates is demanding.

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Notes

  1. 1.

    These properties follow directly by replacing r k(t) with R k(t) within Sect. A.1 .

  2. 2.

    Thus, the average SINR uses the channels second moments \(\boldsymbol {R}_k=\operatorname {E}[{\boldsymbol {h}}_k{\boldsymbol {h}}_k^{\operatorname {H}}]\) instead of \({\boldsymbol {h}}_k{\boldsymbol {h}}_k^{\operatorname {H}}\).

  3. 3.

    Applying ZF beamforming for only a part of the receivers reduces the computational complexity. This is especially relevant if the number of antennas N and users K are large and N ≥ K.

  4. 4.

    To simplify the notation, the noise variance is without loss of generality set to \(\sigma _k^2=1\).

  5. 5.

    Instead of B k in (3.5), we can alternatively use , x↦A k(x) with

    $$\displaystyle \begin{aligned} A_k(x) = \operatorname{E}[a_k(\zeta_k,x)] = \operatorname{E}[\operatorname{log}_{2}(\zeta_k+x)] = B_k(x)+\operatorname{E}[\operatorname{log}_{2}(\zeta_k)]. \end{aligned}$$

    While (3.5) leads to known closed-form expressions for Gaussian ξ k [see (3.6) and (3.7)], A k could be used if ζ k = (1 + ξ k)−2 were a positive quadratic form Gaussian random variables.

  6. 6.

    This formula for the function \(_0F_1(;b;z)=\varGamma (b)z^{\frac {1}{2}-\frac {1}{2}b}I_{b-1}(2\sqrt {z})\) [220, 16.3] follows with the series expansion of the modified Bessel function I b−1(z) [203, 13.3.1].

  7. 7.

    With , the random variable is \(\zeta _k^{-1}=|1+\xi _k|{ }^{2}=|w_k|{ }^{2}\), where .

  8. 8.

    If ZF is realized only for a subset of the channels, the transmitter either needs perfect knowledge of the remaining channels [139] or applies the bounds in Sect. 3.2.2 to simplify the other constraints.

  9. 9.

    The balancing target is ρ = 1 for the QoS optimization (3.1).

  10. 10.

    The sum power constraint becomes \(\sum _{k=1}^K\|\boldsymbol {t}_k\|{ }_2^2=\sum _{k=1}^K\|\boldsymbol {\tau }_k\|{ }_2^2\leq P\), for example, [cf. (1.11)].

  11. 11.

    This generalized ZF beamformer design has also been considered by [110], but for perfect CSI.

  12. 12.

    See Sect. A.3 for the derivation of these bounds based on Jensen’s inequality and B k.

  13. 13.

    According to Lapidoth and Moser [222, Lemma 10.1] (see also [223, Theorem 3]), the expected value for the logarithm of a non-central chi-square-distributed random variable \(y=\sum _{i=1}^N|x_i+m_i|{ }^2\) of degree 2N, with and i.i.d. , is given by

    $$\displaystyle \begin{aligned} \operatorname{E}[\ln(y)] = \begin{cases} \ln(s^2) + \operatorname{E}_1\mspace{-2mu}(s^2) + \sum_{j=1}^{N-1}(-1)^j \left[ {\operatorname{e}}^{-s^2}(j-1)! -\frac{(N-1)!}{j(N-1-j)!} \right] \left( \frac{1}{s^2} \right)^j, &s^2>0\\ \psi(N), &s^2=0, \end{cases} \end{aligned}$$

    where \(s^2=\sum _{j=1}^N|m_i|{ }^2\), \(\operatorname {E}_1\mspace {-2mu}(\cdot )\) is the exponential integral [203, Chapter 5], and ψ(N) is the Digamma function [203, Section 6.3], e.g., \(\operatorname {E}[\ln (y)]=\ln (s^2)+\operatorname {E}_1\mspace {-2mu}(s^2)\) for N = 1 [224, Appendix B].

  14. 14.

    Using \(R_k^{(\operatorname {B})}\) instead of \(R_{k,+}^{(\operatorname {B})}\) does not change the result as ρ k > 0 requires \(R_k^{(\operatorname {B})}(\boldsymbol {t})>0\).

  15. 15.

    The set \(\mathcal {I}^{\operatorname {c}}\) stands for the complement subset to \(\mathcal {I}\), i.e., \(\mathcal {I}^{\operatorname {c}}=\{1,\ldots ,K\}\setminus \mathcal {I}\).

  16. 16.

    The optimum of (3.22) could even become negative if \(R_{k,+}^{(\operatorname {B})}(\boldsymbol {t})\) was replaced by \(R_k^{(\operatorname {B})}(\boldsymbol {t})\).

  17. 17.

    To simplify exposition, the target function is excluded from the interference function.

  18. 18.

    The projection onto the non-negative orthant itself reads as \([z]^+=\max (0,z)\) for .

  19. 19.

    This indexing is achieved via a reordering, such that \(\rho _1\rho -\mu _1^{(\operatorname {B})}\geq \ldots \geq \rho _K\rho -\mu _K^{(\operatorname {B})}\) for .

  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.

    This precoder can be found numerically, as we have suggested in the previous section.

  22. 22.

    These conditions ensure that none of the powers p k, k = 1, …, K increases unboundedly.

  23. 23.

    Several other interference functions may be found for ergodic rates. The advantage of this version is that it only requires an evaluation of the involved \(\tilde {R}_k\) (3.4) for evaluating I k(p), k = 1, …, K.

  24. 24.

    Similar to the case with instantaneous rates, (3.49) appears to be concave. Its bounds \(p_k/\tilde {R}_k^{(\operatorname {B})}(\boldsymbol {p})\) are concave. Moreover, \(p_k/\tilde {r}_k(\boldsymbol {p})\) is concave and \(\tilde {R}_k\) has similar properties as \(\tilde {r}_k\).

  25. 25.

    It is impossible to further reduce the objective because \(\tilde {\boldsymbol {A}}\geq \mathbf {0}\) and \(\boldsymbol {p}^{\prime }\ngeq \boldsymbol {I}(\boldsymbol {p}^{\prime };\boldsymbol {\rho })\) if \(\exists i:p_i^{\prime }<p_i^\star \).

  26. 26.

    Newton methods would also require the derivative of \(\tilde {R}_k\) with respect to p (n) in each iteration.

  27. 27.

    For example, see [150] for the theory behind the sequential inner approximation strategy.

  28. 28.

    We evaluate \(B_k^{-1}(y)\) with y > 0 numerically, using fixed point methods.

  29. 29.

    The increase in c is approximately exponential due to the close relationship to the SINR.

  30. 30.

    A rigorous mathematical proof for this observation is still missing.

  31. 31.

    Here, we exploited \(\frac {\operatorname {d}}{\operatorname {d} z}{\operatorname {e}}^z\operatorname {E}_1\mspace {-2mu}(z)={\operatorname {e}}^z\operatorname {E}_1\mspace {-2mu}(z)-\frac {1}{z}\) [203, Equation 5.1.2.7] for Rayleigh fading (3.65).

  32. 32.

    We shortened notations by substituting α k := α k(1) and β k := β k(1).

  33. 33.

    A feasible starting point t (0) is found by rescaling the solution of (3.16) based on UB2 (or LB1).

  34. 34.

    Despite the effective noise power β k(ρ)∕α k(ρ) depends on the balancing value ρ, the duality transformation remains the same as with instantaneous rates, e.g., with the dual uplink formulations from [67, 104] for a sum power constraint and from [74, 118, 121] for general power constraints.

  35. 35.

    This is a simplified notation in comparison to the formulation of [143] (cf. Sect. 3.2.3). It is obtained by a substitution of the usual multipliers \(\bar {\lambda }_i\) with \(\lambda _i=\bar {\lambda }_i\alpha _i(\rho )/\beta _i(\rho )\), i = 1, …, K.

  36. 36.

    Here, the matrix Y k(λ (n), μ) is independent of ρ (n+1) in contrast to the version in [143].

  37. 37.

    All approximate interference constraints are active since α k(ρ) is strictly positive if ρ ≥ 0.

  38. 38.

    Here, the effective noise variance is \(\nu _k^{(\operatorname {B})}=1\).

  39. 39.

    The projected subgradient method does not result in a decreasing objective within each iteration.

  40. 40.

    These problems are non-convex because the ergodic rates are neither convex nor concave functions in t and a convex reformulation of R k(t) ≥ ρρ k is missing.

  41. 41.

    This BB problem formulation can be extended such that (p, t) resides in a combination of the generalized transmit power constraint set \(\mathcal {P}\) and the convex reformulations of instantaneous rate requirements if the transmitter is perfectly aware of some of the users’ channels [140].

  42. 42.

    For the bound, we assume that all t i, i ≠ k are collinear with \(\bar {\boldsymbol {h}}_k\) and \(\|\boldsymbol {t}_i\|{ }_2^2=p^{\text{max}}\).

  43. 43.

    In other words, the multiplicative factor is or .

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    Andreas Gründinger

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Gründinger, A. (2020). Precoder Design for Ergodic Rates with Multiplicative 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_3

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