Abstract
The detailed analyses of the weighted sum rate maximization problem in the previous two chapters have revealed that the optimum transmit covariance matrices and the accompanied maximum weighted sum rate are usually not available in closed form for a multi-user MIMO broadcast channel system setup. In fact, the global optimum can be found by means of iterative algorithms in case of dirty paper coding (see Chap. 6) but only few qualitative conclusions can be deduced therefrom. For linear transceivers, the situation is even worse. Up to now, a globally optimum algorithm does not seem to exist although we conjecture that the combinatorial stream allocation algorithm presented in Chap. 7 achieves this global optimum in most of the cases.
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- 1.
The probability of a rank deficient channel matrix must be zero.
- 2.
The solution may require up to \(\min \{M,N\}-1\) comparisons in order to determine the optimum water-level.
- 3.
The weight vector \(\varvec{w}\) reduces to the scalar \(1\) if the users are allowed to cooperate and thus act as a single super-user.
- 4.
For rank deficient \(\varvec{C}_{\varvec{\eta }}\), the sum capacity is infinite for any \(P_{\text{ max}}>0\) if the channel \(\varvec{H}^{\text{ BC}}\) allows for transmission in the respective subspace.
- 5.
To see this, let \(\varvec{A}(x)\) have the eigenvalue decomposition \(\varvec{A}(x) = \varvec{V}(x)\varvec{D}(x)\varvec{V}^\mathrm{H }(x)\) with diagonal \(\varvec{D}(x)=\text{ diag}\{d_k(x)\}_{k=1}^L\) and unitary \(\varvec{V}(x)\). The logarithm of the determinant of \(\mathbf I +\varvec{A}(x)\) then reads as
$$\begin{aligned} \log _2\!\big |\!\mathbf I +\varvec{A}(x)\!\big |\!&= \sum \limits _{k=1}^L \log _2(1+d_k(x)) = \sum \limits _{k=1}^L \log _2 d_k(x) + \sum \limits _{k=1}^L \log _2(1+d_k^{-1}(x))\cong \log _2\!\big |\!\varvec{D}(x)\!\big |\!\\&= \log _2\!\big |\!\varvec{A}(x)\!\big |\!. \end{aligned}$$Strong asymptotic equivalence holds since \(d_k^{-1}(x)\rightarrow 0\ \forall k\) as \(P_{\text{ max}}\rightarrow \infty \).
- 6.
To see this, we have to show that \(\sqrt{P_{\text{ max}}}\big [(P_{\text{ max}}^{-1}\mathbf I +\varvec{A})^{-1}-\varvec{A}^{-1}\big ]\rightarrow \mathbf 0 \) for regular \(\varvec{A}\) as \(P_{\text{ max}}\rightarrow 0\):
$$\begin{aligned} \sqrt{P_{\text{ max}}}\big [(P_{\text{ max}}^{-1}\mathbf I +\varvec{A})^{-1}-\varvec{A}^{-1}\big ]&= \sqrt{P_{\text{ max}}}(P_{\text{ max}}^{-1}\mathbf I +\varvec{A})^{-1}\big [\mathbf I -(P_{\text{ max}}^{-1}\mathbf I +\varvec{A})\varvec{A}^{-1}\big ] \\&= \sqrt{P_{\text{ max}}}(P_{\text{ max}}^{-1}\mathbf I +\varvec{A})^{-1}\big [-P_{\text{ max}}^{-1}\varvec{A}^{-1}\big ] \\&= \frac{-1}{\sqrt{P_{\text{ max}}}}\big [\varvec{A}^2+P_{\text{ max}}^{-1}\varvec{A}\big ]^{-1}\rightarrow \mathbf 0 \end{aligned}$$Convergence to zero follows from the fact that the term inside the last braces goes to \(\varvec{A}^2\) whereas its prefactor goes to zero as \(P_{\text{ max}}\rightarrow 0\).
- 7.
The upper block-triangular structure results from the assumption that the weights satisfy \(w_1\le w_2\le \cdots \le w_K\).
- 8.
We assume \(\varvec{H}\) to have full rank here.
- 9.
This statement remains valid even for an optimized (possibly nonuniform) power allocation, see Sect. 4.2.1.
- 10.
Note that \(\varvec{C}\) is assumed to be symmetric.
- 11.
The rates of those two users grow to infinity that are encoded last. The user that is encoded last does not see any interference at all whereas the user that is encoded next to last sees interference from only one other user. However, it can be rendered harmless by means of filtering since the base station has \(N=2\) antennas.
- 12.
Note that \(L_i=0\) might be possible for some users \(i\). In this case, these users are inactive and do not have any active data streams.
- 13.
For the system with \(L^\prime \) instead of \(L\) streams, \(L_K\) must be replaced by \(L_K^{\prime }\) in (8.59) for the exact description of the achieved rates since user \(K\) now has \(L_K^{\prime }=L_K+1\) data streams.
- 14.
The fraction \(a_i\) of some user \(i\) may only be zero if \(L_i=0\), i.e., if user \(i\) does not have any active stream at all.
- 15.
The constraints on the positivity of \(a_1,\ldots ,a_K\) are not required since they are inactive anyway.
- 16.
More precisely, the introduced inter-user interference in the BC can be canceled at the receivers by means of filtering without an increase of the noise power and without a reduction of the desired signal’s quality.
- 17.
Here, the big-\(O\) notation is with respect to \(K\) instead of \(P_{\text{ max}}\).
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© 2013 Springer-Verlag Berlin Heidelberg
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Hunger, R. (2013). Asymptotic High Power Analysis of the MIMO BC. In: Analysis and Transceiver Design for the MIMO Broadcast Channel. Foundations in Signal Processing, Communications and Networking, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31692-0_8
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DOI: https://doi.org/10.1007/978-3-642-31692-0_8
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