We assume that Xkk−κ is distributed according to the quasi-stationary distribution Qκ+1E,ǫ . We bound as follows:
Here, in (4.51) we split the vectors Xℓ up into magnitude and direction; in (4.52) we add the additional term HkXˆk to the argument of mutual information; in (4.53) we drop Yk because given HkXˆk it is independent of the other random quantities;
then in (4.54) we remove the conditioning on kXℓk because it does not provide any useful information; and in the last step (4.55) we made use of the stationarity of {Hk}.
Similar to the derivation of the upper bound, in the following we will again introduce a shorthand and rename ˆXkk−κ by ˆX0−κ. Note that since the upper bound that is derived in this appendix will not depend on { ˆXk}, we lose the dependence on k in the end anyway. Here in (4.57) we add more terms to the argument of mutual information; and (4.59) follows from the third statement of Theorem 7.
Now note that for ˆX0−κ being quasi-stationary and for all i ∈ {−κ, . . . , −1} we where (4.60) follows from the stationarity of {Hk} and the quasi-stationarity of ˆX0−κ (note that i < 0 so that i + 1 ≤ 0); in (4.61) we add ˆX−κ which, conditional on ˆXi+1−κ+1, is independent of the other random quantities; then in (4.62) we add H−κXˆ−κ to the conditioning which does not change anything as it is a function of
the given terms H−κ and ˆX−κ; and the inequality (4.63) then follows by dropping H−κ which cannot reduce entropy.
Therefore,
Here, (4.65) follows from the chain rule; in (4.66) we drop ˆX0i+1because conditioned on ˆXi−κthey are independent of the other random quantities; and in (4.67) we apply (4.64) several times to each term of the sum.
Hence we have Using this in (4.59) and (4.55) we finally get
I Hk−κ−11 ; Yk conditional on H0−κ it is independent of H−κ−1−∞ ; and (4.77) follows from stationarity.
Note that δ1(κ) does neither depend on k nor on the input {Xk} and that by Theorem 6 it monotonically tends to zero as κ tends to infinity.
Chapter 5
Discussion & Conclusion
We have proven three results about stationary fading channels: the first two are fundamental results that are true in a much more general context and are in no way restricted to the situation of MIMO fading channels: the first results states that a stationary channel has a capacity-achieving input distribution that is stationary.
This is intuitively very pleasing and confirms our belief that stationary processes in general behave “the way engineers expect them to behave.”
The second result concerns entropy rates of various forms. We have shown that entropy rates are also well-defined for processes with a continuous alphabet as long as the processes are stationary (and the entropy rate is finite). Moreover, we also generalize this definition to include various sorts of conditional entropy rates and show that they are also well-defined. These conditional entropy rates are strongly matched to the situation of MIMO fading channels, however, this is not necessary and similar type of results can be derived in many other situations, too. Even though we have shown the proof only for two cases, it is clearly seen that the fundament of the proof is the stationarity assumption only.
As third result we prove that a certain mutual information expression of the form I X−∞−κ; X0
X−κ+1−1
tends to zero as κ tends to infinity. This corresponds to the intuition that a process is supposed to forget about things that happened an infinite amount of time before.
This result is more strongly bound to the context of MIMO fading channel. Never-theless it contains some general message: it is a further example for the nice behavior of stationary processes. Note, however,that in order to prove this mathematically, it is crucial not to leave the scope of stationarity, i.e., steps introducing a supremum or similar might open the door to unrealistic, but mathematically dangerous processes, which will destroy the nice properties of the family of stationary distributions.
We believe that all three results are of interests in many different situations.
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