In this section, we show that the lower bound derived in Theorem 1 is tight for a class of semi-deterministic orthogonal relay channel, where, the output Y of the destination is a deterministic function of XD, Xr and SD, i.e., Y = f(XD, Xr, SD), and the output Yr of the relay node is controlled only by X
R
and SR, i.e., the channel from the source to the relay is governed by the conditional distribution . This assumption is reasonable in many cases, e.g., when the two orthogonal channels use two different frequency bands, the received signal Yr at the relay node will not be affected by its input signal Xr. The channel can be expressed as
(18)
where, f(·) is a deterministic function and 1{·} denotes the indicator function. The channel state information on SR and SD is known to both the source and the relay non-causally. The capacity of this class of semi-deterministic orthogonal relay channel is characterized as shown in the following theorem.
Theorem 3 The capacity of the channel ( 18) with the channel state information known non-causally to the source and the relay is characterized as
(19)
where the maximization is over all measures on
of the form
(20)
is an auxiliary random variables with
(21)
and 1{·} denotes the indicator function.
Proof The achievability follows from Theorem 1. First note that the joint distribution of (20) can also be written as
(22)
with additional requirement that
(23)
Note that, when is fixed, all the items on the right-hand side (RHS) of (19) are fixed except for I(XR; Yr|SR), which is independent of . Therefore, the maximization over all joint distributions of the form (20) can be replaced by the maximization only over those distributions, where xr and xD are two deterministic functions of (sD, ur, y), i.e., of the form
(24)
for some mappings gr: (ur, sD) → xr, gd: (y, ur, sD) → xD and subject to (23). Thus, we only have to prove the achievability of the rate that satisfies (19) for some distribution of the form (24).
The achievability follows directly from Theorem 1 by taking U = Y since Y = f(XD, Xr, SD), letting XR be independent of Ur and Xr considering the fact that Yr is only determined by XR and SR, and by setting xr = gr(ur, sD), xD = gd(y, ur, sD). Note that with these choices of the random variables, if we chose stochastic kernels and , two deterministic mappings gr:(ur, sD) → xr and gd:(y, ur, sD) → xD, combined with and the channel law, the joint distribution (24) for which (23) is satisfied will be determined.
The proof of the converse is as follows.
Consider an (ϵ
n
, n, R) code with an average error probability P
e
(n) ≤ ε
n
. By Fano's inequality, we have
(25)
where δ
n
→ 0 as n → + ∞. Thus,
(26)
Defining the auxiliary random variable , we have
(27)
where the second inequality follows from the fact that and are independent of W.
Calculate the two terms in (27) separately as follows:
(28)
where (a) holds since XR,i is a function of ; (b) follows from the fact that conditioning reduces entropy and the Markov chain .
(29)
where (a) holds since Xr,i is a function of ; (b) follows from the fact that conditioning reduces entropy.
From (26) to (29), we have
(30)
The proof of the bound I(W; Yn) given in the second term in (19) is as follows:
(31)
where (a) holds due to Csiszar and Korner's sum identity; (b) follows since SD,i is independent of , and (c) follows from the fact that .
By (26) and (31),
(32)
From the above, we have
(33)
Introduce a time-sharing random variable T, which is uniformly distributed over {1, 2, …, n} and denote the collection of random variables
Considering the first bound in (33), we have
(34)
where the last step follows from the fact that T is independent of all the other variables and the Markov chain T ↔ (XR, SR) ↔ Yr.
Similarly, considering the second bound in (33), we have
(35)
Defining , we get
(36)
Therefore, for a given sequence of (ϵ
n
, n, R) code with ϵ
n
going to zero as n goes to infinity, there exists a measure of the form , such that the rate R essentially satisfies (19).
Considering the facts that I(XR; Yr|SR) is determined by the joint distribution and the other three items on the RHS of (19) is independent of , the maximum in (19) taken over all joint probability mass functions is equivalent to that taken over all joint probability mass functions of the form
The bound of the cardinality of can be proven in a similar way as that proven in Theorem 1. It is omitted here for brevity.
This concludes the proof.