ZHOU et al.: ON OPTIMAL FRONTHAUL COMPRESSION AND DECODING STRATEGIES FOR UPLINK C-RANs 7405
II. ACHIEVABLE RATE REGIONS FOR UPLINK C-RAN
A. Channel Model
This paper considers an uplink C-RAN model, where
K mobile users communicate with a CP through L BSs,
as shown in Fig. 1. The noiseless digital fronthaul link
connecting the BS to the CP has the capacity of C
bits
per complex dimension. The fronthaul capacity C
is the
maximum long-term average throughput of the th fronthaul
link, i.e., lim
n→∞
1
n
n
i=1
C
(i) ≤ C
,whereC
(i) represents the
instantaneous transmission rate of the th fronthaul link at the
ith time slot. Each user terminal is equipped with M antennas;
each BS is equipped with N antennas. Perfect channel state
information (CSI) is assumed to be available to all the BSs
and to the CP. For simple notation, we denote K ={1, ··· , K }
and L ={1, ··· , L} in this paper.
Let X
k
∈ C
M
be the signal transmitted by the kth user,
which is subject to per-user transmit power constraint of P
k
,
i.e. E
X
k
X
†
k
≤ P
k
. The signal received at the th BS can be
expressed as
Y
=
K
k=1
H
,k
X
k
+ Z
,= 1, 2,...,L, (1)
where Z
∼ CN(0,
) represents the additive Gaussian noise
for BS and is independent across different BSs, and H
,k
denotes the complex channel matrix from user k to BS .
We consider the compress-and-forward scheme [25], [26]
applied to the uplink C-RAN system, in which the BSs
compress the received signals Y
, and forward the quantization
bits to the CP for decoding. At the CP, the user messages
are decoded using either joint decoding or some form of
successive decoding. In joint decoding, the quantization code-
words and the message codewords are decoded simultaneously,
whereas, in successive decoding, the quantization codewords
and messages are decoded successively in some prescribed
order. Different orderings can potentially result in different
achievable rates.
B. Achievable Rate-Fronthaul Regions for Joint Decoding,
Successive Decoding, and Generalized Successive Decoding
In the following, we present the achievable rate-fronthaul
regions of compress-and-forward with joint decoding and
different forms of successive decoding.
Proposition 1 ([3, Proposition IV.1]): For the uplink
C-RAN model shown in Fig. 1, the achievable
rate-fronthaul region of compress-and-forward with joint
decoding, P
∗
JD
, is the closure of the convex hull of all
(R
1
, ··· , R
K
, C
1
,...,C
L
) ∈ R
K +L
+
satisfying
k∈T
R
k
<
∈S
C
− I
Y
;
ˆ
Y
|X
K
+ I
X
T
;
ˆ
Y
S
c
|X
T
c
(2)
for all T ⊆ K and S ⊆ L, for some product distribution
K
k=1
p(x
k
)
L
=1
p(ˆy
|y
) such that E
X
k
X
†
k
≤ P
k
for k =
1,...,K .
Note that for the uplink C-RAN model, the rate region (2)
given by compress-and-forward with joint decoding is identi-
cal to the rate region of the noisy network coding scheme [9],
which is an extension of the compress-and-forward scheme
to the general multiple access relay network by using joint
decoding at the receiver and block Markov coding at the
transmitters.
As a more practical decoding strategy, successive decoding
of quantization codewords first, and then the user messages
at the CP can also be used in uplink C-RAN. The following
proposition states the rate-fronthaul region achieved by suc-
cessive decoding.
Proposition 2: ([5, Theorem 1]): For the uplink C-RAN
model shown in Fig. 1, the achievable rate-fronthaul region of
compress-and-forward with successive decoding, P
∗
SD
,isthe
closure of the convex hull of all (R
1
, ··· , R
K
, C
1
,...,C
L
) ∈
R
K +L
+
satisfying
k∈T
R
k
< I
X
T
;
ˆ
Y
L
|X
T
c
, ∀ T ⊆ K, (3)
and
I
Y
S
;
ˆ
Y
S
|
ˆ
Y
S
c
<
∈S
C
, ∀ S ⊆ L, (4)
for some product distribution
K
k=1
p
(
x
k
)
L
=1
p(ˆy
|y
) such
that E
X
k
X
†
k
≤ P
k
for k = 1,...,K .
Note that (3) is the multiple-access rate region, (4) repre-
sents the Berger-Tung rate region for distributed lossy com-
pression [6, Theorem 12.1], while (2) incorporates the joint
decoding of the quantization codewords and the user messages.
Because of its lower decoding complexity, successive decoding
is usually preferred for practical implementation of the uplink
C-RAN systems [21], [22]. Note that in the above strategy,
successive decoding applies only to the vector X
k
(user mes-
sage codewords) and the vector Y
(quantization codewords);
the elements within vectors X
k
and Y
are still decoded
jointly.
It is possible to improve upon the successive decoding
scheme by allowing arbitrary interleaved decoding orders
between quantization codewords and user message codewords.
We call this the generalized successive decoding scheme in
this paper. The generalized successive decoding scheme is
first suggested in [27] under the name of joint base-station
successive interference cancelation scheme. In such a succes-
sive decoding strategy, the set of potential decoding orders
includes all the permutations of quantization and user message
codewords.
Denote π as a permutation on the set of quantization and
user message codewords
ˆ
Y
1
,
ˆ
Y
2
,...,
ˆ
Y
L
, X
1
, X
2
,...X
K
.
For a given permutation π, the decoding order is given by
the index of the elements in π, i.e., π(1) → π(2) →
··· → π(L + K ). For example, consider an uplink C-RAN
model as shown in Fig. 1 with 2 BSs and 2 users. If π =
ˆ
Y
1
, X
1
,
ˆ
Y
2
, X
2
, then the decoding of
ˆ
Y
2
and X
2
can
use both previously decoded user messages and quantization
codewords as side information. The resulting rate region is