We consider a MISO fading channel whose time-k output Yk∈ C is given by
Yk= HTkxk+ Zk (2.11)
where xk ∈ CnT denotes the time-k channel input vector; where the random vector Hk denotes the time-k fading vector; where HTk denotes the transpose of the vector Hk; and where Zk denotes additive noise. Here C denotes the complex field, CnT denotes the nT-dimensional complex Euclidean space, and nT is the number of transmit antennas. We assume that the additive noise is an IID zero-mean white Gaussian process of variance σ2 > 0.
As for the multi-variate fading process {Hk}, we shall only assume that it is stationary, ergodic, of finite second moment
E£
kHkk2¤
< ∞, (2.12)
and of finite differential entropy rate
h({Hk}) > −∞ (2.13)
(the regularity assumption).
Finally, we assume that the fading process {Hk} and the additive noise process {Zk} are independent and of a joint law that does not depend on the channel input {xk}.
As for the input, we consider two different constraints: a peak-power constraint and an average-power constraint. We use E to denote the maximal allowed instan-taneous power in the former case, and to denote the allowed average power in the latter case. For both cases we set
snr, E
σ2. (2.14)
The capacity C(snr) of the channel (2.11) is given by C(snr) = lim
n→∞
1
nsup I (Xn1; Y1n) (2.15) where we use Xkj to denote Xj, . . . , Xk and where the supremum is over the set of all probability distributions on Xn1 satisfying the constraints, i.e.,
kXkk2≤ E, almost surely, k = 1, 2, . . . , n (2.16) for a peak constraint, or
1 n
Xn k=1
E£
kXkk2¤
≤ E (2.17)
for an average constraint.
Specializing [7, Theorem 4.2] or [2, Theorem 6.10], respectively, to MISO fading, we have
lim
snr↑∞
n
C(snr) − log log snro
< ∞. (2.18)
The fading number χ is now defined as in [7, Definition 4.6] and in [2, Definition 6.13]
by
χ({HTk}), lim
snr↑∞
n
C(snr) − log log snro
. (2.19)
Prima facie the fading number depends on whether a peak-power constraint (2.16) or an average-power constraint (2.17) is imposed on the input. Since a peak-power constraint is more stringent than an average-power constraint, we will derive the upper bound using the average-power constraint and the lower bound using the peak-power constraint. In case of an isotropically distributed fading process we shall see that both constraints lead to identical fading numbers.
Chapter 3
Main Results
3.1 Preliminaries
Before we can state our new results, we need to give some preliminary results.
3.1.1 Escaping to Infinity
We start with a discussion about the concept of capacity achieving input distribu-tions that escape to infinity.
A sequence of input distributions parameterized by the allowed cost (in our case of fading channels the cost is the available power or the SNR, respectively) is said to escape to infinity if it assigns to every fixed compact set a probability that tends to zero as the allowed cost tends to infinity. Loosely speaking, this means that in the limit—when the allowed cost tends to infinity—such a distribution does not use finite-cost symbols.
This notion is of importance since the asymptotic capacity of many channels of interest can only be achieved by input distributions that escape to infinity. As a matter of fact one can show that every input distribution that only achieves a mutual information of identical asymptotic growth-rate as the capacity must escape to infinity. Loosely speaking, for many channels it is not favorable to use finite-cost input symbols whenever the cost constraint is loosened completely.
In the following we will only state this result specialized to the situation at hand.
For a more general description and for all proofs we refer to [7], [2].
Definition 3. Let {QE}E≥0 be a family of input distributions for the memoryless version of the fading channel (2.11), i.e., input distributions of the channel
Y = HTx+ Z (3.1)
where x ∈ CnT. This family is parameterized by the available average power E such that
EQ
E
£kXk2¤
≤ E, E ≥ 0. (3.2)
We say that the input distributions {QE}E≥0 escape to infinity if for every E0> 0
E↑∞limQE¡
kXk2≤ E0¢
= 0. (3.3)
We now have the following:
Lemma 4. Let the memoryless MISO fading channel be given as in (3.1) and let W (·|·) denote the corresponding conditional channel law. Let {QE}E≥0 be a family of input distributions satisfying the power constraint (3.2) and the condition
E↑∞lim
I(QE, W )
log log E = 1. (3.4)
Then {QE}E≥0 escapes to infinity.
Proof. A proof can be found in [7], [2].
Hence, when computing bounds on the fading number (which is part of the capacity in the limit when E tends to infinity, see (2.19)) we may assume that
Pr£
kXk2 ≤ E0¤
= 0. (3.5)
3.1.2 An Upper Bound on Channel Capacity
In [7], [2] a new approach of finding upper bounds to channel capacity has been introduced. Since capacity is by definition a maximization of mutual information, it is implicitly difficult to find upper bounds on it. The new proposed technique bases on a dual expression of mutual information that leads to an expression of capacity as a minimization instead of a maximization. This way it becomes much easier to find upper bounds.
Again, here we only state the upper bound in a form needed in the derivation of Theorem 7, for a more general form, for more mathematical details, and for all proofs we refer to [7], [2].
Lemma 5. Consider a memoryless channel1 with input alphabet CnR and output alphabet C as given in (3.1). Then the mutual information between input and output of the channel is upper-bounded as follows:
I(X; Y ) = −h(Y |X) + log π + α log β + log Γ µ
α,ν β
¶
+ (1 − α)E£ log¡
|Y |2+ ν¢¤
+ 1 βE£
|Y |2¤ + ν
β (3.6)
where α, β > 0 and ν ≥ 0 are parameters that can be chosen freely.
Proof. A proof can be found in [7], [2].
3.1.3 Capacity Achieving Input Distributions and Stationarity One of the main assumption about our channel model is that the fading process and the additive noise are stationary. This assumption is crucial both for the results as well as the derivation, i.e., we don’t believe the results to still be valid in a non-stationary setting.
From an intuitive point of view a stationary channel model should have a capacity achieving input distribution that is stationary. Unfortunately, we are not aware of a rigorous proof of this claim. However, we are able to prove a less strong statement which is basically saying that for our channel model we may limit ourselves to joint input distributions under which the input vectors have the same law for (almost) all time k:
1Actually, the lemma requires some mathematical conditions on the alphabets and the channel law to be satisfied. However, all these conditions are satisfied in our context. For more detail see [7], [2].
Lemma 6. Fix some power E with corresponding snr, E/σ2. Let C(E) denote the corresponding channel capacity. Then for every fixed ǫ > 0 there corresponds some positive integer η = η(E, ǫ) and some distribution QE,ǫ = Q(E, ǫ) on CnT such that for every blocklength n sufficiently large there exists some input Xn1 satisfying the following:
1. The input Xn1 nearly achieves capacity in the sense that 1
nI¡
Xn1; Y1n¢
≥ C(E) − ǫ. (3.7)
2. Except for the first η − 1 vectors Xη−11 and for at most the last 2(η − 1) vectors Xnn−2η+3 the vectors
Xη, Xη+1, . . . , Xn−2η+2 (3.8) all have the same distribution QE,ǫ.
3. This marginal distribution QE,ǫ gives rise to a second moment E:
E£ kXℓk2¤
= E, ℓ = η, . . . , n − 2η + 2. (3.9) 4. The first η −1 symbols and the last 2(η −1) vectors satisfy the power constraint
possibly strictly E£
kXℓk2¤
≤ E, ℓ ∈ {1, . . . , η − 1} ∪ {n − 2η + 3, . . . , n}. (3.10) Proof. See Appendix A.
Note that this lemma and its proof are analogous to a very similar lemma needed in the derivation of the fading number of SIMO fading channels with memory [7], [2].