The Capacity of Several New Classes of Semi-Deterministic Relay Channels

The relay channel consists of a transmitter input
x
₁
, a relay input
x
₂
, a relay output
y
₂
, and a receiver output
y
₃
. In this paper, we establish the capacity of three new classes of semi-deterministic relay channels: 1) a class of degraded semi-deterministic relay channels, 2) a class of semi-deterministic orthogonal relay channels, and 3) a class of semi-deterministic relay channels with relay-transmitter feedback. For the first class of relay channels, the output of the relay
y
₂
depends on a deterministic function of the transmitter's input
x
₁
, i.e., on
s
=
f
₁
(
x
₁
), rather than on
x
₁
directly. In addition, the relay channels satisfy the condition that
S
→ (
X
₂
,
Y
₂
) →
Y
₃
forms a Markov chain for all input probability distributions
p
(
x
₁
,
x
₂
). Hence, the first class of relay channels includes, but is strictly not limited to, the class of degraded relay channels previously considered by Cover and El Gamal. The partial decode-and-forward strategy achieves the capacity of the class of degraded semi-deterministic relay channels. Next, we consider the class of semi-deterministic orthogonal relay channels where there are orthogonal channels from the relay to the receiver and from the transmitter to the receiver. In addition, the output of the relay
y
₂
is a deterministic function of
x
₁
,
x
₂
and
y
₃
, i.e.,
y
₂
=
f
₄
(
x
₁
,
x
₂
,
y
₃
). The class of semi-deterministic orthogonal relay channels is a generalization of the class of deterministic relay channels considered by Kim. The compress-and-forward strategy achieves the capacity of the class of semi-deterministic orthogonal relay channels. For the third class of relay channels, there is a causal and noiseless feedback from the relay to the transmitter. In addition, similar to the second class of relay channels, the output of the relay
y
₂
is a deterministic function of
x
₁
,
x
₂
, and
y
₃
. Both the generalized strategy of Gabbai and Bross and the hash-and-forward strategy of Kim achieve the capacity of the class of semi-deterministic relay channels with relay-transmitter feedback.