# Uplink Multicell Processing with Limited Backhaul via Per-Base-Station
Successive Interference Cancellation ††thanks: Manuscript submitted to IEEE
Journals on Selected Areas of Communications on Oct 13, 2012, revised on March
20, 2013, accepted on May 2, 2013. This work was supported by the Natural
Science and Engineering Research Council (NSERC) of Canada. The materials in
this paper have been presented in part at IEEE Global Communications
Conference (Globecom), Anaheim, CA, U.S.A., Dec 2012.  ††thanks:  L. Zhou was
with the The Edward S. Rogers Sr. Department of Electrical and Computer
Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada. He is now with
Qualcomm Technologies Inc., Santa Clara, CA 95051 USA (email:
zhoulei@comm.utoronto.ca). ††thanks:  W. Yu is with The Edward S. Rogers Sr.
Department of Electrical and Computer Engineering, University of Toronto,
Toronto, ON M5S 3G4 Canada (email: weiyu@comm.utoronto.ca). Phone:
416-946-8665. FAX: 416-978-4425. Kindly address correspondence to Wei Yu.

Lei Zhou, and Wei Yu

###### Abstract

This paper studies an uplink multicell joint processing model in which the
base-stations are connected to a centralized processing server via rate-
limited digital backhaul links. Unlike previous studies where the centralized
processor jointly decodes all the source messages from all base-stations, this
paper proposes a simple scheme which performs Wyner-Ziv compress-and-forward
relaying on a per-base-station basis followed by successive interference
cancellation (SIC) at the central processor. The proposed scheme significantly
reduces the implementation complexity of joint decoding while resulting in an
easily computable achievable rate region. Although suboptimal in general, this
paper proves that the proposed per-base-station processing SIC scheme can
achieve the sum capacity of a special cellular Wyner model to within a
constant gap. Under the proposed SIC framework, this paper also studies the
capacity scaling with limited backhaul. It is established that in order to
achieve to within constant gap to the maximum SIC rate with infinite backhaul,
the limited-backhaul system must have backhaul link capacities that scale
logarithmically with the signal-to-interference-and-noise ratios (SINR) at the
base-stations. This paper further studies the optimal backhaul rate allocation
problem for an uplink multicell joint processing model with a total backhaul
capacity constraint. The analysis reveals that the optimal rate allocation
that maximizes the overall sum rate should have backhaul link rates that also
scale logarithmically with the SINR at each base-station. Finally, the
proposed per-base-station SIC scheme is evaluated in a practical multicell
orthogonal frequency division multiple access (OFDMA) network to quantify the
performance gain brought by the centralized processor.

###### Index Terms:

Coordinated multi-point (CoMP), interference channel, limited backhaul network
multiple-input multiple-output (MIMO), relay channel, successive interference
cancellation, Wyner-Ziv coding

##  I Introduction

Traditional wireless cellular networks operate on a cell-by-cell basis where
out-of-cell interference is treated as part of the noise. However, as cellular
base-stations become more densely deployed and heterogeneous networks become
commonplace where small cells overlap heavily with macro-cells, modern
cellular networks are becoming increasingly interference limited. Because of
the intercell interference, the per-cell achievable rates in traditional
cellular deployments are typically much smaller than that of a single isolated
cell.

Multicell joint processing is a promising technique that has the potential to
significantly improve the cellular throughput by taking advantage of its
capability for intercell interference mitigation. When base-stations share the
transmitted and received signals, the codebooks, and the channel state
information (CSI) with each other through high-capacity backhaul, it is
theoretically possible to perform joint transmission in the downlink and joint
reception in the uplink to eliminate out-of-cell interference completely. This
paper deals with the information theoretical capacity analysis of the joint-
processing architecture in the uplink. In this architecture, antennas from
multiple base-stations essentially become a virtual multiple-input multiple-
output (MIMO) array capable of spatial multiplexing multiple user terminals.

One way to implement a multicell joint processing system is to deploy a
centralized processing server which is connected to all the base-stations via
high-capacity backhaul links. When the capacities of the backhaul links are
sufficiently large, the joint processing system implemented across the
different cells in the network can be modeled as a multiple-access channel in
the uplink and a broadcast channel in the downlink, giving rise to the concept
of network MIMO [1, 2].

The practical implementation of a network MIMO system must also consider the
effect of finite capacity in the backhaul. While the capacity of the network
MIMO system with infinite backhaul is easy to compute, when finite backhaul is
considered, the information theoretical analysis of multicell processing
becomes significantly more complicated. This paper focuses on an uplink
network MIMO model as shown in Fig. 1. From an information theoretical
perspective, the overall system can be thought of as a combination of a
multiple-access channel (with remote terminals acting as the transmitters and
the centralized processor as the receiver) and a relay channel (with the base-
stations acting as relays).

\begin{overpic}[width=231.26378pt]{./fig/cloud_model_new}
\centering\put(50.0,47.5){\small$Y_{1}:\hat{Y}_{1}$}
\put(50.0,33.5){\small$Y_{2}:\hat{Y}_{2}$}
\put(49.5,4.5){\small$Y_{L}:\hat{Y}_{L}$} \put(-5.0,47.5){\small$X_{1}$}
\put(-5.0,33.5){\small$X_{2}$} \put(-5.0,4.5){\small$X_{L}$}
\put(36.0,53.0){\small$Z_{1}$} \put(36.0,39.0){\small$Z_{2}$}
\put(35.0,-1.0){\small$Z_{L}$} \put(13.0,50.0){\small$h_{11}$}
\put(21.5,44.5){\small$h_{21}$} \put(4.0,18.0){\small$h_{L1}$}
\put(30.0,18.0){\small$h_{1L}$} \put(11.0,11.0){\small$h_{L2}$}
\put(24.0,11.0){\small$h_{2L}$} \put(15.0,35.5){\small$h_{22}$}
\put(13.0,2.0){\small$h_{LL}$} \put(69.0,44.0){\small$C_{1}$}
\put(68.0,34.0){\small$C_{2}$} \put(68.0,7.0){\small$C_{L}$}
\put(80.0,34.0){\small$\hat{Y}_{1}$} \put(74.0,26.0){\small$\hat{Y}_{2}$}
\put(79.0,16.0){\small$\hat{Y}_{L}$} \@add@centering\end{overpic} Figure 1:
Uplink multicell joint processing via a centralized processor

###  I-A Related Work

The uplink network MIMO model considered in this paper, where the joint
processing takes place in the central server, is related to but is different
from the multicell joint processing model with finite-capacity uni- or bi-
direction backhaul links deployed between the base-stations. In this latter
setting, the uplink multicell model becomes an interference channel with
partial receiver cooperation. Although a complete characterization of capacity
for such a model is still an open problem, information theoretical studies
have been carried out for the case of two-user Gaussian interference channel
with receiver cooperation [3, 4, 5, 6, 7]. Likewise, the analysis of uplink
cellular network under this setup has been carried out in [8, 9], where
strategies such as decode-and-forward and compress-and-forward are proposed,
and analytical results under the Wyner model are obtained. We mention briefly
here that the downlink counterpart of this model has been studied in [10].

For the uplink multicell joint processing model considered in this paper,
where the limited-capacity backhaul links are deployed between the base-
stations and a central processor as depicted in Fig. 1, although a complete
characterization of its information theoretical capacity region is also an
open problem, much work has been done on finding its achievable rate region
and the gap to the outer bound [11, 12, 13, 14, 15, 16]. The early work by
Sanderovich et al. [11] considers a joint decoding scheme in which the base-
stations quantize the received signals and forward them to the centralized
processor, which then performs _joint decoding_ of both the source messages
and quantized codewords. Instead of joint decoding, [15, 14] describe schemes
which _successively_ decode the quantization codewords first, then the source
messages. A comparison of successive decoding vs. joint decoding is contained
in [16]. Alternatively, base-stations can also _decode_ part of the messages
of users in their own cell, then forward the decoded data along with the
remaining part to the centralized processor [11, 12, 17]. The counterpart of
these results for the downlink with finite backhaul has been described in
[18]. A review of some of these results is available in [2].

The uplink multicell joint processing model is actually a special case of a
general multiple multicast relay network studied in [19, 20], for which a
generalization of quantize-and-forword can be shown to achieve to within a
constant gap to the information theoretical capacity of the overall network.
This is proved in [19] using a quantize-map-and-forward technique, and in [20]
using noisy network coding, both of which require the joint decoding of the
source messages and the quantized codewords. Thus in retrospect, the joint
processing scheme of [11] already achieves the approximate capacity of the
uplink finite-backhaul multicell model, provided that the quantization noise
level is appropriately chosen and joint decoding is done at the central
processor.

It should be emphasized that joint decoding is challenging to implement. Not
only is the network-wide CSI required at the central processor, the
computational complexity of joint decoding typically scales exponentially with
the total number of nodes in the network. Moreover, even a mere evaluation of
the achievable rate can be computationally prohibitive. The achievable rate
region of the joint decoding scheme as presented in [11, Proposition IV.1]
involves $2^{L}-1$ rate constraints, each requiring a minimization of $2^{L}$
terms, where $L$ is the number of users in the uplink multicell model. To
obtain analytical results, [11] and likewise [9, 21] assume certain symmetry
and obtain rate expressions under a Wyner cellular model. But the rate
expression for an arbitrary network remains difficult to evaluate. On top of
this, the achievable rate region needs to be optimized over all choices of $L$
quantization noise levels, which is a difficult task, although recent progress
has been made in [14, 16, 22].

###  I-B Main Results

To fully overcome the aforementioned difficulty yet still take full advantage
of the centralized processing is expected to be challenging. This paper
instead aims to design schemes with simple receiver structures that result in
computationally feasible achievable rate regions for the uplink multicell
joint processing problem. Toward this end, this paper focuses on the uplink
multicell model with finite-capacity backhaul to the centralized processor,
and proposes suboptimal schemes based on _successive interference
cancellation_ (SIC). Instead of performing joint decoding of the source
messages and quantization codewords [11], or successive decoding of all the
quantization codewords first, then the source messages [14, 15], this paper
applies the Wyner-Ziv compress-and-forward relaying scheme on a per-base-
station basis and performs single-user decoding with SIC at the centralized
processor. The Wyner-Ziv coding here also uses the previously decoded messages
as side information. The main advantage of this proposed scheme is that it
leads to a receiver architecture in which only the decoded transmit signals
are shared between the successive stages, so it is more amenable to
implementation. Further, the quantization noise levels can now be easily
optimized, resulting in an achievable rate region computable with complexity
that scales linearly with $L$.

Although the proposed per-base-station SIC scheme is no longer the best
possible for a general uplink joint processing model, this paper shows that
for a special Wyner soft-handoff model [23], in which the mobile terminals and
the base-stations are placed along a line, the proposed per-base-station SIC
scheme achieves a sum rate that is at most a constant gap away from capacity.
This paper shows that the constant gap is a linear function of the number of
transmitter-receiver pairs in the uplink multicell joint processing model.

Under the proposed per-base-station SIC framework for the finite-backhaul
uplink model, we also ask the following question: How much backhaul capacity
is needed to approach the theoretical infinite-backhaul successive-decoding
rate? The results of this paper show that the backhaul rates need to scale
logarithmically with the received signal-to-interference-and-noise ratio
(SINR) in order to achieve to within $\frac{1}{2}$ bit of the successive
decoding rates attainable with unlimited backhaul capacity.

Further, this paper addresses the question of how to allocate the backhaul
rates optimally across the different backhaul links. Under a sum rate
constraint on the total backhaul capacity, this paper shows that in order to
maximize the sum rate over the entire network, the allocation of individual
backhaul link capacity should again scale logarithmically with the SINR under
the proposed per-base-station SIC framework.

This paper focuses on capacity analysis with the assumption that the CSI for
the entire network is available at the central processor. This is reasonable
as the base-stations may estimate the channel in the uplink, then send CSI via
the backhaul links to the central processor. For convenience, this paper
neglects the additional backhaul capacity needed to support the sharing of
CSI. We mention that the capacity of network MIMO under channel estimation
error has been investigated in [8]. Study on the use of processing techniques
that are robust against channel uncertainty has been reported in [15].

Finally, under the perfect CSI assumption, this paper evaluates the
performance of the proposed per-base-station SIC scheme in an OFDMA network.
Numerical simulations show that the proposed per-base-station SIC scheme can
almost double the per-cell sum-rate over a baseline scheme where no
centralized processor is deployed and each base-station simply treats
interference as noise. Further, with optimized backhaul capacity allocation,
most of such gains can already be obtained when the backhaul sum capacity is
at about 1.5 times the achieved per-cell sum rate.

###  I-C Organization of the Paper

The rest of the paper is organized as follows. Section II presents the
multicell joint processing model. Section III introduces the per-base-station
SIC scheme and compares it with other achievability schemes such as noisy
network coding and joint-base-station processing. The optimality of the per-
base-station SIC scheme for the Wyner model is provided in Section IV. The
backhaul rate allocation problem with a total backhaul constraint is solved in
Section V. Finally, Section VI numerically evaluates the per-base-station SIC
scheme for a two-user symmetric setting and for a realistic OFDMA network.

##  II Channel Model

Consider the uplink of a multicell network with a central processor. Assuming
that there is only one user operating in each time-frequency resource block in
each cell, the multicell network can be modelled by $L$ users each sending a
message to their corresponding base-stations. Base-stations serve as
intermediate relays for the centralized server, which eventually decodes all
the transmit messages.

As depicted in Fig. 1, the uplink joint processing model consists of two
parts. The left half is an $L$-user interference channel with $X_{i}$ as the
input signal from the $i$th user, $Y_{i}$ as the output signal, $Z_{i}$ as the
additive white Gaussian noise (AWGN), and $h_{ij}$ as the channel gain from
user $i$ to base-station $j$, where $i,j=1,2,\cdots,L$. The right half can be
seen as a noiseless multiple-access channel with capacities $C_{i}$,
$i=1,2,\cdots,L$, modelling the backhaul links between the base-stations and
the central processor. The input signal $X_{i}$ is power constrained, i.e.,
$E[|X_{i}|^{2}]\leq P_{i}$; the receiver noises are assumed to be independent
and identically distributed with $Z_{i}\sim\mathcal{N}(0,N_{0})$,
$i=1,2,\cdots,L$. The signal-to-noise ratios (SNRs) and the interference-to-
noise ratios (INRs) are defined as follows:

$\mathsf{SNR}_{i}=\frac{h_{ii}^{2}P_{i}}{N_{0}},\quad\mathsf{INR}_{i,j}=\frac{h_{ij}^{2}P_{i}}{N_{0}},\quad
i,j=1,\cdots,L$ (1)

In addition, define the vectors $\mathbf{X}=\\{X_{1},X_{2},\cdots,X_{L}\\}$
and $\mathbf{Y}=\\{Y_{1},Y_{2},\cdots,Y_{L}\\}$.

##  III Multicell Processing Schemes

This paper focuses on uplink multicell processing schemes in which the base-
stations quantize the received signals, then subsequently transmit a function
of the quantized signal to the central processor via noiseless backhaul links.
Quantize-and-forward is a natural strategy in the uplink setting. This is
because in order to reap the benefit of multicell processing, the user
terminals must transmit at rates higher than that decodable at its own base-
station alone. This prevents the use of the decode-and-forward strategy. In
Fig. 1, the received signal is denoted as $Y_{i}$; its quantized version is
denoted as $\hat{Y}_{i}$. This paper also assumes that the base-stations
either transmit $\hat{Y}_{i}$ directly, or performs a binning operation on
$\hat{Y_{i}}$. The task at the central processor is to decode
$\\{X_{1},X_{2},\cdots,X_{L}\\}$ based on either
$\\{\hat{Y_{1}},\hat{Y_{2}},\cdots,\hat{Y_{L}}\\}$ or their bin indices. In
the following, we examine the information theoretical achievable rates for
various quantization and decoding strategies.

###  III-A Joint Decoding

Consider a coding scheme in which each base-station quantizes its received
signal using a Gaussian codebook at certain distortion or quantization noise
level $\mathsf{q}_{i}$, bins the quantized message, then forwards the bin
index to the central processor. In the joint decoding strategy, the central
processor, upon receiving all the bin indices, decodes
$\\{X_{1},X_{2},\cdots,X_{L}\\}$ and
$\\{\hat{Y}_{1},\hat{Y}_{2},\cdots,\hat{Y}_{L}\\}$ jointly based on joint
typicality. This strategy is first proposed by Sanderovich et al. [11,
Proposition IV.1]. (Note that there is no requirement that
$\\{\hat{Y}_{1},\hat{Y}_{2},\cdots,\hat{Y}_{L}\\}$ are decoded correctly; only
$\\{X_{1},X_{2},\cdots,X_{L}\\}$ need to.)

The key parameter here is the setting of the quantization noise levels
$\mathsf{q}_{i}$. For example, $\mathsf{q}_{i}$’s can be set to be such that
$\\{\hat{Y}_{1},\hat{Y}_{2},\cdots,\hat{Y}_{L}\\}$ can be correctly decoded
based on the bin indices. The values of $\mathsf{q}_{i}$’s can then be
determined based on the capacities of the backhaul links
$(C_{1},C_{2},\cdots,C_{L})$. But, such a setting of $\mathsf{q}_{i}$ may not
be optimal. Indeed, as shown in [19] and also later in [20], for a much larger
class of general relay networks, setting the quantization noise levels to be
at the background noise level, i.e., $\mathsf{q}_{i}=N_{0}$, leads to an
achievable rate that is within at most a constant gap from the capacity outer
bound. This gap scales linearly with the number of nodes in the network, but
is independent of the channel gains and the backhaul capacities. Applying this
result to the uplink multicell model in Fig. 1, we immediately have the
following joint decoding achievable rate. The achievable rate expression can
be derived based on the noisy-network-coding theorem [20], and is equivalent
to the expression in [11].

######  Lemma 1.

For the multicell joint processing model as shown in Fig. 1, the following
rate region is achievable

$R(\mathcal{S})\leq\min_{\mathcal{T}\subseteq\mathcal{L}}\left\\{\frac{1}{2}\log\left|I+\Lambda^{-1}_{\mathcal{\mathcal{T}}^{c}}(N_{0}+\mathsf{q})\mathbf{H}_{\mathcal{ST}^{c}}\Lambda_{\mathcal{S}}(P)\mathbf{H}_{\mathcal{ST}^{c}}^{T}\right|\right.\\\
+\left.\sum_{i\in\mathcal{T}}\left(C_{i}-\frac{1}{2}\log(1+\frac{N_{0}}{\mathsf{q}_{i}})\right)\right\\},\quad\forall\;\mathcal{S}\subseteq\mathcal{L}\triangleq\\{0,1,\cdots,L\\}$
(2)

where $R(\mathcal{S})=\sum_{i\in\mathcal{S}}R_{i}$, $\mathsf{q}_{i}$ is a
positive real number representing the quantization noise level at base-station
$i$, $\mathbf{H}_{\mathcal{ST}^{c}}$ denotes for the transfer matrix from
input $\mathbf{X}(\mathcal{S})$ to output $\mathbf{Y}(\mathcal{T}^{c})$,
$\Lambda^{-1}_{\mathcal{\mathcal{T}}^{c}}(N_{0}+\mathsf{q})$ is a diagonal
matrix of $\frac{1}{N_{0}+\mathsf{q}_{i}}$ with $i\in\mathcal{T}^{c}$, and
$\Lambda_{\mathcal{S}}(P)$ is a diagonal matrix of $P_{i}$ with
$i\in\mathcal{S}$. This rate region is within a constant gap to the capacity
region of the multicell joint processing model if we set
$\mathsf{q}_{i}=N_{0},i=1,2,\cdots,L$.

The above achievable rate region is valid for any choice of $\mathsf{q}_{i}$,
and is approximately optimal when $\mathsf{q}_{i}=N_{0}$. But, the evaluation
of such an achievable rate region can be difficult. For the uplink joint
processing model, the achievable rate region as derived in Lemma 1 requires a
minimization of $2^{L}$ terms for each rate constraint, and there are
$2^{L}-1$ different rate constraints describing the rate region. Even when the
size of the network is in a reasonable range, for example as in a $19$-cell
topology, it is computationally prohibitive to minimize over $2^{19}$ terms
each involving $2^{19}-1$ different rate constraints. Further, the
implementation of the joint decoding scheme itself tends to have exponential
complexity, so the achievable rate given in Lemma 1 is not practically
implementable for a reasonably-sized network.

In order to derive more tractable performance analysis of the multicell joint
processing scheme, [11] resorts to a modified Wyner model (see [23]), where
each transmitter-receiver pair interferes only with one other neighboring
transmitter-receiver pair, and is subject to interference from only one
neighboring transmitter-receiver pair. Further, certain symmetry is introduced
so that all the direct channels are identical, and so are all interfering
channels. With this symmetric cyclic structure, computation of the sum rate
becomes tractable under joint decoding [11].

###  III-B Per-Base-Station SIC

This paper focuses on the general multicell model (instead of the symmetric
Wyner model) and proposes suboptimal schemes based on the successive decoding
of source messages. Based on the observation that the exponential complexity
of the rate expression in Lemma 1 is due to the joint decoding step at the
destination, this paper proposes a per-base-station SIC approach (in contrast
to joint-base-station SIC to be described in detail later) at the centralized
processor that significantly reduces the complexity of computation of the
achievable rate region. In addition, the proposed scheme involves the use of
compress-and-forward relaying technique [24] at each base-station
independently, which also significantly reduces the complexity of its eventual
implementation in a practical system.

Specifically, we use an SIC approach assuming a _fixed_ order of decoding
first $X_{1}$, then $X_{2},X_{3},\cdots,X_{L}$. A central feature of the
proposed scheme is that only the decoded signals are used as side information
in subsequent stages. This simplifies the sharing of information among the
base-stations and facilitates practical implementation.

At the $k$th stage, the base-station $k$, upon receiving $Y_{k}$, quantizes
$Y_{k}$ into $\hat{Y}_{k}$ using the compress-and-forward technique and sends
bin index of $\hat{Y}_{k}$ to the destination via the noiseless link of
capacity $C_{k}$. Note that the quantization process at the base-station $k$
includes interference from all other users. The quantization noise level at
each base-station $k$ is chosen so as to fully utilize $C_{k}$ such that
$\hat{Y}_{k}$ can be immediately recovered at the central processor. This
quantization process can be done either with Wyner-Ziv coding that takes
advantage of the already-decoded user messages $(X_{1},\cdots,X_{k-1})$ as
side information, or without Wyner-Ziv coding for ease of implementation.

At the central processor, to decode user $k$’s message $X_{k}$ the central
processor first decodes the quantization message $\hat{Y}_{k}$ upon receiving
its description from the digital link $C_{k}$. It then decodes the message of
user $k$ using joint typicality decoding between $X_{k}$ and the quantized
message $\hat{Y}_{k}$. The decoding of $X_{k}$ also takes advantage of the
knowledge of previously decoded messages $(X_{1},\cdots,X_{k-1})$ at the
centralized processor. In this way, the impact of interference from
$X_{1},\cdots,X_{k-1}$ eventually disappears and the effective interference is
only due to users not yet decoded, i.e., $X_{j}$, for $j>k$. After decoding
$X_{k}$, the central processor moves to the next decoding stage, adding
$X_{k}$ to the set of known side information.

####  III-B1 Per-Base-Station SIC with Wyner-Ziv

The following theorem gives the achievable rate of the proposed per-base-
station SIC scheme with Wyner-Ziv compress-and-forward relaying.

######  Theorem 1.

For the uplink multicell joint processing channel depicted in Fig. 1, the
following rate is achievable using Wyner-Ziv compress-and-forward relaying at
the base-stations followed by SIC at the centralized processor with a fixed
decoding order:

$R_{k}=\frac{1}{2}\log\frac{1+\mathsf{\overline{SINR}}_{k}}{1+2^{-2C_{k}}\mathsf{\overline{SINR}}_{k}},$
(3)

where

$\mathsf{\overline{SINR}}_{k}=\frac{\mathsf{SNR}_{k}}{1+\sum_{j>k}\mathsf{INR}_{j,k}}.$
(4)

\begin{overpic}[width=231.26378pt]{./fig/channel_k_new}
\centering\put(-5.0,7.0){\small$X_{k}$} \put(9.0,10.0){\small$h_{kk}$}
\put(58.0,11.0){\small$C_{k}$} \put(16.0,18.0){\small$Z_{k}+\sum_{j\neq
k}h_{jk}X_{j}$} \put(39.0,7.0){\small$Y_{k}:\hat{Y}_{k}$}
\put(70.0,7.0){\small$\hat{Y}_{k}$} \put(69.0,19.0){\small$X_{1},X_{2},\cdots
X_{k-1}$} \@add@centering\end{overpic} Figure 2: Equivalent channel of user
$k$ in the $k^{th}$ decoding stage

###### Proof:

In the $k$th stage of the SIC decoder, $X_{1},\cdots,X_{k-1}$ decoded in the
previous decoding stages serve as side information for stage $k$. The
equivalent channel of user $k$ is depicted in Fig. 2. This is a three-node
relay channel without the direct source-destination link. Specifically, the
source signal $X_{k}$ is sent from the transmitter to the relay, which
receives $Y_{k}$, quantizes it into $\hat{Y}_{k}$ and forwards its description
to the centralized processor via the noiseless digital link of capacity
$C_{k}$. At the centralized processor, $X_{1},\cdots,X_{k-1}$ serve as side
information and facilitate the decoding of $\hat{Y}_{k}$ and $X_{k}$.
According to [24, Theorem 6], the achievable rate of user $k$ using Wyner-Ziv
compress-and-forward can be written as

$R_{k}=I(X_{k};\hat{Y}_{k}|X_{1},\cdots,X_{k-1})$ (5)

subject to the constraint

$I(Y_{k};\hat{Y}_{k}|X_{1},\cdots,X_{k-1})\leq C_{k}.$ (6)

We constrain ourselves to Gaussian input signals
$X_{k}\sim\mathcal{N}(0,P_{k})$, and the Gaussian quantization
scheme111Gaussian input and Gaussian quantization are used for convenience and
for simplicity only. They are not necessarily optimal; see [12] for an
example., i.e.,

$\hat{Y}_{k}=Y_{k}+e_{k},$ (7)

where $e_{k}\sim\mathcal{N}(0,\mathsf{q}_{k})$ is the Gaussian quantization
noise independent of everything else. To fully utilize the digital link, it is
natural to set

$I(Y_{k};\hat{Y}_{k}|X_{1},\cdots,X_{k-1})=C_{k}.$ (8)

Now, substituting $Y_{k}=\sum_{j=1}^{L}h_{jk}X_{j}+Z_{k}$ and
$\hat{Y}_{k}=Y_{k}+e_{k}$ into (8), we have

$\displaystyle C_{k}$ $\displaystyle=$ $\displaystyle
I(Y_{k};\hat{Y}_{k}|X_{1},\cdots,X_{k-1})$ (9) $\displaystyle=$ $\displaystyle
h(\hat{Y}_{k}|X_{1},\cdots,X_{k-1})-h(e_{k})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{N_{0}+\sum_{j\geq
k}h_{jk}^{2}P_{j}}{\mathsf{q}_{k}}\right),$

which results in the following quantization noise level that fully utilizes
the digital links $C_{k}$:

$\mathsf{q}_{k}^{*}=\frac{N_{0}+\sum_{j\geq k}h_{jk}^{2}P_{j}}{2^{2C_{k}}-1}.$
(10)

With the above $\mathsf{q}_{k}^{*}$, the achievable rate of user $k$ can be
calculated as

$\displaystyle R_{k}$ $\displaystyle=$ $\displaystyle
I(X_{k};\hat{Y}_{k}|X_{1},\cdots,X_{k-1})$ (11) $\displaystyle=$
$\displaystyle
h(\hat{Y}_{k}|X_{1},\cdots,X_{k-1})-h(\hat{Y}_{k}|X_{1},\cdots,X_{k})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\frac{\mathsf{q}_{k}^{*}+N_{0}+\sum_{j\geq
k}h_{jk}^{2}P_{j}}{\mathsf{q}_{k}^{*}+N_{0}+\sum_{j>k}h_{jk}^{2}P_{j}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\frac{N_{0}+\sum_{j>k}h_{jk}^{2}P_{j}+h_{kk}^{2}P_{k}}{N_{0}+\sum_{j>k}h_{jk}^{2}P_{j}+2^{-2C_{k}}h_{kk}^{2}P_{k}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\frac{1+\mathsf{\overline{SINR}}_{k}}{1+2^{-2C_{k}}\mathsf{\overline{SINR}}_{k}}.$

Finally, note that
$\mathsf{\overline{SINR}}_{k}=\frac{h_{kk}^{2}P_{k}}{N_{0}+\sum_{j>k}h_{jk}^{2}P_{j}},$which
is equivalent to (4). ∎

The rate expression (3) shows how the achievable rates are affected by the
limited capacities of the digital backhaul links under the proposed per-base-
station SIC decoding framework. Fig. 3 plots the achievable rate of $R_{k}$ as
a function of the backhaul link capacity $C_{k}$ with
$\mathsf{\overline{SINR}}_{k}$ equal to $20$dB. When $C_{k}$ is small, $R_{k}$
grows almost linearly with $C_{k}$, which means that each bit of the backhaul
link provides approximately one bit increase in the achievable rate for user
$k$. The digital backhaul is efficiently exploited in this regime. However, as
$C_{k}$ grows larger, each bit of the backhaul link gives increasingly less
achievable rate improvement. In the limit of unbounded backhaul capacity,
i.e., $C_{k}=\infty$, $R_{k}$ saturates and approaches
$\frac{1}{2}\log(1+\mathsf{\overline{SINR}}_{k})\stackrel{{\scriptstyle\triangle}}{{=}}\bar{R}_{k}$,
which is the upper limit for the rate of user $k$ when the SIC decoder is
employed.

Since backhaul link capacity can be costly in practical implementations, it is
natural to ask how large $C_{k}$ needs to be in order to achieve a rate
$R_{k}$ that is close to the maximum SIC rate with unlimited backhaul. It is
easy to see that when $C_{k}=\frac{1}{2}\log(1+\mathsf{\overline{SINR}}_{k})$,
$\bar{R}_{k}-R_{k}$ is upper bounded by one-half bit, i.e.,

$\displaystyle\bar{R}_{k}-R_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log(1+\mathsf{\overline{SINR}}_{k})$ (12)
$\displaystyle-\left.\frac{1}{2}\log\frac{1+\mathsf{\overline{SINR}}_{k}}{1+2^{-2C_{k}}\mathsf{\overline{SINR}}_{k}}\right|_{C_{k}=\frac{1}{2}\log(1+\mathsf{\overline{SINR}}_{k})}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(1+\frac{\mathsf{\overline{SINR}}_{k}}{1+\mathsf{\overline{SINR}}_{k}}\right)$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}.$

Therefore, when the backhaul link has capacity
$C_{k}=\frac{1}{2}\log(1+\mathsf{\overline{SINR}}_{k})$, the achievable rate
is half a bit away from the SIC upper limit. This is also the point under
which the utilization of $C_{k}$ is the most efficient, as shown in Fig. 3.

\begin{overpic}[width=231.26378pt]{./fig/Rk_vs_Ck}
\centering\put(75.0,77.0){\tiny$\frac{1}{2}\log(1+\mathsf{\overline{SINR}}_{k})$}
\put(43.0,64.0){\tiny$\left(\frac{1}{2}\log(1+\mathsf{\overline{SINR}}_{k}),\frac{1}{2}\log\displaystyle\frac{(1+\mathsf{\overline{SINR}}_{k})^{2}}{1+2\mathsf{\overline{SINR}}_{k}}\right)$}
\put(26.0,69.0){$\leq\frac{1}{2}$} \@add@centering\end{overpic} Figure 3:
Achievable rate of user $k$ versus the backhaul capacity $C_{k}$

It is worth noting that the above result, which suggests that $C_{k}$ should
scale as $\log(\mathsf{\overline{SINR}}_{k})$, is analogous to the conclusion
of [11, Corollary IV.6], which states that the backhaul rate should scale as
$\log(P)$ in order to approach the capacity of the infinite backhaul case for
the symmetric Wyner model. The main difference is that this paper uses SIC, so
interference not yet cancelled still appears in the SINR expression.

####  III-B2 Per-Base-Station SIC Without Wyner-Ziv

Implementing the Wyner-Ziv compression may not be easy in practice. The main
difficulty lies in the binning operation that is necessary in order to take
advantage of the side information. To make the proposed per-base-station SIC
scheme more amenable to practical implementation, the Wyner-Ziv quantizer can
be replaced by a simple vector quantizer that does not take side information
(i.e., previously decoded messages in the SIC framework) into account. In this
way, the following rate can be achieved for user $k$:

$R_{k}=I(X_{k};\hat{Y}_{k}|X_{1},\cdots,X_{k-1})$ (13)

subject to the constraint

$I(Y_{k};\hat{Y}_{k})\leq C_{k},$ (14)

for $k=1,2,\cdots,L$. Following the same lines of the proof of Theorem 1, it
is easy to see that by replacing (9) with
$C_{k}=\frac{1}{2}\log\left(1+\frac{N_{0}+\sum_{j}h_{jk}^{2}P_{j}}{\mathsf{q}_{k}}\right)$,
which gives
$\mathsf{q}_{k}^{*}=\frac{N_{0}+\sum_{j}h_{jk}^{2}P_{j}}{2^{2C_{k}}-1}$, we
obtain

$\displaystyle R_{k}$
$\displaystyle=\frac{1}{2}\log\frac{\mathsf{q}_{k}^{*}+N_{0}+\sum_{j\geq
k}h_{jk}^{2}P_{j}}{\mathsf{q}_{k}^{*}+N_{0}+\sum_{j>k}h_{jk}^{2}P_{j}}$
$\displaystyle=\frac{1}{2}\log\frac{N_{0}+\sum_{j\geq
k}h_{jk}^{2}P_{j}+2^{-2C_{k}}\sum_{j<k}h_{jk}^{2}P_{j}}{N_{0}+\sum_{j>k}h_{jk}^{2}P_{j}+2^{-2C_{k}}\sum_{j\geq
k}h_{jk}^{2}P_{j}}.$ (15)

Define

$\mathsf{\overline{SINR}^{\prime}}_{k}=\frac{h_{kk}^{2}P_{k}}{N_{0}+\sum_{j>k}h_{jk}^{2}P_{j}+2^{-2C_{k}}\sum_{j<k}h_{jk}^{2}P_{j}}.$
(16)

We have just proved that the following rate is achievable using per-base-
station SIC without Wyner-Ziv compress-and-forward.

######  Theorem 2.

For the uplink multicell joint processing channel depicted in Fig. 1, the
following rate is achievable using vector quantization at the base-stations
followed by SIC at the centralized processor with a fixed decoding order:

$R_{k}=\frac{1}{2}\log\frac{1+\mathsf{\overline{SINR}^{\prime}}_{k}}{1+2^{-2C_{k}}\mathsf{\overline{SINR}^{\prime}}_{k}},$
(17)

where

$\mathsf{\overline{SINR}^{\prime}}_{k}=\frac{\mathsf{SNR}_{k}}{1+\sum_{j>k}\mathsf{SIR}_{j,k}+2^{-2C_{k}}\sum_{j<k}\mathsf{SIR}_{j,k}}.$
(18)

Comparing Theorem 2 with Theorem 1, it is easy to see that achievable rates in
(3) and (17) share the same structure. However, without Wyner-Ziv compression,
the effective SINR (18) becomes smaller due to the extra terms
$2^{-2C_{k}}\sum_{j<k}\mathsf{SIR}_{j,k}$. This means that the impact of
previously decoded signals $X_{1},\cdots,X_{k-1}$ is not fully cancelled at
stage $k$. We also note that as $C_{k}\rightarrow\infty$, we have
$\mathsf{\overline{SINR}^{\prime}}_{k}\rightarrow\mathsf{\overline{SINR}}_{k}$.
Thus, Wyner-Ziv coding is most useful only when the backhaul link capacity is
small. For reasonably large $C_{k}$, the benefit of Wyner-Ziv coding is
expected to be marginal.

###  III-C Joint Base-Station SIC Schemes

The per-base-station SIC scheme takes advantage of multicell processing only
in so far as to the use of the decoded messages as side information. It is
possible to further improve upon these schemes by joint decoding across the
base-stations.

Note that in the per-base-station SIC scheme, the decoding of source messages
and quantized messages follow the order $\hat{Y}_{1}\rightarrow
X_{1}\rightarrow\hat{Y}_{2}\rightarrow
X_{2},\rightarrow\cdots,\rightarrow\hat{Y}_{L}\rightarrow X_{L}$, where in the
$k$th compression and decoding stage, previously decoded source signals
$X_{1},\cdots,X_{k-1}$ are used as side information. However, note that in the
decoding process the quantization codewords $\hat{Y}_{1},\cdots,\hat{Y}_{k-1}$
are also decoded along the way, and thus are also available for possible joint
processing at stage $k$, which can lead to better performance. In particular,
we can incorporate $\hat{Y}_{1},\cdots,\hat{Y}_{k-1}$ in the expressions for
$C_{k}$ and $R_{k}$, leading to the following achievable rate:

$R_{k}=I(X_{k};\hat{Y}_{1},\cdots,\hat{Y}_{k}|X_{1},\cdots,X_{k-1}),$ (19)

subject to

$C_{k}\geq
I(Y_{k}:\hat{Y}_{k}|X_{1},\cdots,X_{k-1},\hat{Y}_{1},\cdots,\hat{Y}_{k-1}),$
(20)

for $k=1,2,\cdots,L$.

Alternatively, we can also proceed in a two-stage successive process of
decoding all of $\\{\hat{Y}_{k}\\}_{k=1}^{L}$ first, then
$\\{X_{k}\\}_{k=1}^{L}$ [15, 14]. Further, each of these two stages can be
accomplished in an SIC fashion, resulting in rate expressions

$R_{k}=I(X_{k};\hat{Y}_{1},\cdots,\hat{Y}_{L}|X_{1},\cdots,X_{k-1}),$ (21)

subject to

$I(Y_{k};\hat{Y}_{k}|\hat{Y}_{1},\cdots,\hat{Y}_{k-1})\leq C_{k},$ (22)

for $k=1,2,\cdots,L$. We note here that the above rate expressions (19) and
(21) can outperform the rates (3) in Theorem 1 and (17) in Theorem 2, because
in the above expressions each $X_{k}$ is decoded based on the quantized
observations of all base-stations (or that of all previous decoded base-
stations), rather than just the $k$th base-station in the per-base-station SIC
scheme. However, for this same reason, the implementations of the above joint-
base-station SIC schemes are also expected to be somewhat more complicated.
For the rest of this paper, we only focus on the per-base-station SIC schemes
of Theorem 1 and Theorem 2, and leave the joint-base-station SIC schemes as
subject for future studies.

##  IV Approximate Optimality of Per-Base-Station SIC in a Wyner Model

The proposed per-base-station SIC scheme is in general sub-optimal. However,
in certain asymmetric settings, the per-base-station SIC scheme can be shown
to be approximately optimal in the sense of achieving the sum capacity to
within a constant gap. This section studies a class of such asymmetric
channels in which each transmitter-receiver pair interferes only with one
neighbor and gets interfered by only one other neighbor as shown in Fig. 4.
This model is called the soft-handoff Wyner model in the literature [23]. The
Wyner model is an abstraction of the cellular network to a one-dimensional
setting. It can, for example, model a communication scenario where base-
stations are placed along a highway and mobile terminals travel between the
base-stations. This section proves that the per-base-station SIC scheme
achieves to within a constant gap to capacity for the Wyner model. This result
holds either with or without Wyner-Ziv coding.

\begin{overpic}[width=231.26378pt]{./fig/Z-channel}
\centering\put(50.0,47.0){\small$Y_{1}:\hat{Y}_{1}$}
\put(50.0,33.0){\small$Y_{2}:\hat{Y}_{2}$}
\put(49.3,4.3){\small$Y_{L}:\hat{Y}_{L}$} \put(-5.0,47.5){\small$X_{1}$}
\put(-5.0,33.5){\small$X_{2}$} \put(-5.0,4.5){\small$X_{L}$}
\put(36.0,53.0){\small$Z_{1}$} \put(36.0,39.0){\small$Z_{2}$}
\put(35.0,-1.0){\small$Z_{L}$} \put(13.0,50.0){\small$h_{11}$}
\put(21.5,44.0){\small$h_{21}$} \put(21.5,30.0){\small$h_{32}$}
\put(15.0,35.5){\small$h_{22}$} \put(13.0,2.0){\small$h_{LL}$}
\put(69.0,44.0){\small$C_{1}$} \put(68.0,34.0){\small$C_{2}$}
\put(68.0,7.0){\small$C_{L}$} \put(80.0,34.0){\small$\hat{Y}_{1}$}
\put(74.0,25.7){\small$\hat{Y}_{2}$} \put(79.0,16.0){\small$\hat{Y}_{L}$}
\@add@centering\end{overpic} Figure 4: Wyner model where each transmitter-
receiver only interferes with one neighbor.

Consider first the per-base-station SIC with Wyner-Ziv coding. Intuitively,
decoding order plays an important role in this asymmetric setting. Since
receiver $L$ sees a non-interfered version of the source signal $X_{L}$,
decoding should take place starting from user $L$. After $X_{L}$ is decoded,
it can serve as side information, which facilitates the decoding of user
$L-1$, then $L-2$, etc. With this decoding order and applying the result of
Theorem 1 to the channel setting of Fig. 4, we arrive at the following
achievable rate region:

$\left\\{\begin{array}[]{l}R_{i}=I(X_{i};\hat{Y}_{i}|X_{i+1}),\;\;i=1,2,\cdots,L-1\\\
R_{L}=I(X_{L};\hat{Y}_{L})\end{array}\right.$ (23)

subject to

$\left\\{\begin{array}[]{l}I(Y_{i};\hat{Y}_{i}|X_{i+1})\leq
C_{i},\;\;i=1,2,\cdots,L-1\\\ I(Y_{L};\hat{Y}_{L})\leq
C_{L}\end{array}\right.$ (24)

which, when specialized to Gaussian inputs $X_{i}\sim\mathcal{N}(0,P_{i})$ and
the Gaussian quantization scheme: $\hat{Y}_{i}=Y_{i}+e_{i}$, where $e_{i}$ is
the quantization noise independent of everything else and follows Gaussian
distributions $e_{i}\sim\mathcal{N}(0,\mathsf{q}_{i})$, gives the following
achievable rate

$R_{i}=\displaystyle\frac{1}{2}\log\left(\displaystyle\frac{1+\mathsf{SNR}_{i}}{1+2^{-2C_{i}}\mathsf{SNR}_{i}}\right),\;\;i=1,2,\cdots,L.$
(25)

The achievable sum rate for the Wyner model with decoding order from user $L$
to user $1$ is then

$R_{\mathrm{sum}}=\sum_{i=1}^{L}\frac{1}{2}\log\left(\displaystyle\frac{1+\mathsf{SNR}_{i}}{1+2^{-2C_{i}}\mathsf{SNR}_{i}}\right).$
(26)

The above sum rate can be slightly improved by noticing that the $L$th base-
station can use decode-and-forward instead of compress-and-forward, which
leads to

$R_{L}=\min\left\\{\frac{1}{2}\log(1+\mathsf{SNR}_{L}),C_{L}\right\\}.$ (27)

The main result of this section is that under a mild weak interference
condition, the sum rate achieved by the per-base-station SIC scheme is at most
a constant number of bits away from the sum capacity for the Wyner model, so
it is approximately optimal in the high SNR regime.

######  Theorem 3.

For a multicell processing Wyner model shown in Fig. 4, in the weak-
interference regime of
$\mathsf{INR}_{i+1,i}\leq\mathsf{SNR}_{i},i=1,2,\cdots,L-1$, the per-base-
station SIC scheme with Wyner-Ziv coding achieves a sum rate that is within
$L-\frac{1}{2}$ bits of the sum capacity.

###### Proof:

To show that the difference between the achievable sum rate in (26) and the
sum capacity is bounded by a constant gap, we first write down the cut-set
bound [25, Theorem 14.10.1] for the Wyner channel model. Let
$\mathcal{L}=\\{1,2,\cdots,L\\}$ represent the index set for the base-
stations. We partition $\mathcal{L}$ into two sets, $\mathcal{S}$ and
$\mathcal{S}^{c}$, and only consider cuts for which all the $X_{i}$’s and a
selected subset of $Y_{i}$’s, with $i\in\mathcal{S}$, are on one side, and the
rest of the $Y_{i}$’s, with $i\in\mathcal{S}^{c}$, and the central processor
are on the other side. Denote $\mathbf{X}(\mathcal{L})=\\{X_{i}\ |\
i\in\mathcal{L}\\}$ and $\mathbf{Y}(\mathcal{S}^{c})=\\{Y_{i}\ |\
i\in\mathcal{S}^{c}\\}$. We have an upper bound to the cut-set bound as
follows:

$\displaystyle R_{\mathrm{sum}}^{\mathrm{cut-set}}$ $\displaystyle\leq$
$\displaystyle\max_{p(\mathbf{X}(\mathcal{L}))}\min_{\mathcal{S}\subseteq\mathcal{L}}\left\\{I\left(\mathbf{X}(\mathcal{L});\mathbf{Y}(\mathcal{S}^{c})\right)+\sum_{i\in\mathcal{S}}C_{i}\right\\}$
$\displaystyle\leq$
$\displaystyle\min_{\mathcal{S}\subseteq\mathcal{L}}\left\\{\max_{p(\mathbf{X}(\mathcal{L}))}I\left(\mathbf{X}(\mathcal{L});\mathbf{Y}(\mathcal{S}^{c})\right)+\sum_{i\in\mathcal{S}}C_{i}\right\\},$

where the maximization is taken over all the possible joint distributions of
$\mathbf{X}(\mathcal{L})$, and the minimization is taken over $2^{L}$
different choices of cuts that separate the $L$ transmitters and the
centralized processor.

Evaluate the mutual-information term in (LABEL:cutset_bound) as follows:

$\displaystyle
I\left(\mathbf{X}(\mathcal{L});\mathbf{Y}(\mathcal{S}^{c})\right)$ (31)
$\displaystyle=$ $\displaystyle
h\left(\mathbf{Y}(\mathcal{S}^{c})\right)-\sum_{i\in\mathcal{S}^{c}}h(Z_{i})$
$\displaystyle\leq$
$\displaystyle\sum_{i\in\mathcal{S}^{c}}\left(h(Y_{i})-h(Z_{i})\right)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$
$\displaystyle\sum_{i\in\mathcal{S}^{c}}\frac{1}{2}\log(1+\mathsf{SNR}_{i}+\mathsf{INR}_{i+1,i})$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$
$\displaystyle\left\\{\begin{array}[]{ll}\sum_{i\in\mathcal{S}^{c}}\frac{1}{2}\log(1+\mathsf{SNR}_{i})+\frac{|\mathcal{S}^{c}|}{2},&{\rm
if\ }L\in S\\\
\sum_{i\in\mathcal{S}^{c}}\frac{1}{2}\log(1+\mathsf{SNR}_{i})+\frac{|\mathcal{S}^{c}|-1}{2},&{\rm
if\ }L\in S^{c}\\\ \end{array}\right.$

where (a) holds because Gaussian distribution maximizes the entropy $h(Y_{i})$
and (b) is due to the fact that $\mathsf{INR}_{i+1,i}\leq\mathsf{SNR}_{i}$
thus
$\frac{1}{2}\log(1+\mathsf{SNR}_{i}+\mathsf{INR}_{i+1,i})\leq\frac{1}{2}\log(1+\mathsf{SNR}_{i})+\frac{1}{2}$
for $i=1,\cdots,L-1$. For $i=L$,
$h(Y_{L})-h(Z_{L})=\frac{1}{2}\log(1+\mathsf{SNR}_{L})$.

Now, we are going to choose a particular cut to further upper bound the cut-
set bound. We compare $\frac{1}{2}\log(\mathsf{SNR}_{i})$ at each base-station
with the backhaul capacity $C_{i}$, and let the cut go through either the
user-base-station link or the backhaul link, whichever corresponds to the
smaller of the two quantities above, as shown in Fig. 5. More precisely,
define

$\mathcal{S}=\left\\{i\ \left|\frac{1}{2}\log\mathsf{SNR}_{i}\geq
C_{i}\right.\right\\}.$ (32)

With this particular cut, we can now bound the difference between the cut-set
bound and the achievable sum rate. First, consider the case that $L\in S^{c}$,
we use the achievable sum rate as given in (26):

$\displaystyle R_{\mathrm{sum}}^{\mathrm{cut-set}}-R_{\mathrm{sum}}$ (33)
$\displaystyle\leq$
$\displaystyle\sum_{i\in\mathcal{S}^{c}}\left\\{\frac{1}{2}\log(1+\mathsf{SNR}_{i})-\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{i}}{1+2^{-2C_{i}}\mathsf{SNR}_{i}}\right)\right\\}$
$\displaystyle+\sum_{i\in\mathcal{S}}\left\\{C_{i}-\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{i}}{1+2^{-2C_{i}}\mathsf{SNR}_{i}}\right)\right\\}+\frac{|\mathcal{S}^{c}|-1}{2}$
$\displaystyle=$
$\displaystyle\sum_{i\in\mathcal{S}^{c}}\frac{1}{2}\log\left(1+2^{-2C_{i}}\mathsf{SNR}_{i}\right)$
$\displaystyle+\sum_{i\in\mathcal{S}}\frac{1}{2}\log\left(\frac{2^{2C_{i}}+\mathsf{SNR}_{i}}{1+\mathsf{SNR}_{i}}\right)+\frac{|\mathcal{S}^{c}|-1}{2}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$
$\displaystyle\frac{|\mathcal{S}^{c}|}{2}+\frac{|\mathcal{S}|}{2}+\frac{|\mathcal{S}^{c}|-1}{2}$
$\displaystyle\leq$ $\displaystyle L-\frac{1}{2}$

where in (a) we used the definition of the cut (32), i.e.,
$\mathsf{SNR}_{i}\leq 2^{2C_{i}}$ for $i\in\mathcal{S}^{c}$, and
$\mathsf{SNR}_{i}\geq 2^{2C_{i}}$ for $i\in\mathcal{S}$, and the last
inequality is due to the fact that $|\mathcal{S}^{c}|+|\mathcal{S}|=L$ and
$|\mathcal{S}^{c}|\leq L$.

Now, consider the case that $L\in\mathcal{S}$. We tighten the sum rate
expression (26) by noticing that since by the definition of $\mathcal{S}$ we
have $\frac{1}{2}\log\mathsf{SNR}_{L}\geq C_{L}$, so by (27) we have
$R_{L}=C_{L}$. In this case,

$\displaystyle R_{\mathrm{sum}}^{\mathrm{cut-set}}-R_{\mathrm{sum}}$ (34)
$\displaystyle\leq$
$\displaystyle\sum_{i\in\mathcal{S}^{c}}\left\\{\frac{1}{2}\log(1+\mathsf{SNR}_{i})-\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{i}}{1+2^{-2C_{i}}\mathsf{SNR}_{i}}\right)\right\\}$
$\displaystyle+\sum_{i\in\mathcal{S}\setminus\\{L\\}}\left\\{C_{i}-\frac{1}{2}\log\left(\frac{1+\mathsf{SNR}_{i}}{1+2^{-2C_{i}}\mathsf{SNR}_{i}}\right)\right\\}+\frac{|\mathcal{S}^{c}|}{2}$
$\displaystyle\leq$
$\displaystyle\frac{|\mathcal{S}^{c}|}{2}+\frac{|\mathcal{S}|-1}{2}+\frac{|\mathcal{S}^{c}|}{2}\leq
L-\frac{1}{2}$

This completes the proof. ∎

\begin{overpic}[width=231.26378pt]{./fig/cutset}
\centering\put(50.0,55.0){\small$Y_{1}:\hat{Y}_{1}$}
\put(50.0,42.5){\small$Y_{2}:\hat{Y}_{2}$}
\put(49.3,14.5){\small$Y_{L}:\hat{Y}_{L}$} \put(-4.0,56.0){\small$X_{1}$}
\put(-4.0,42.0){\small$X_{2}$} \put(-4.0,14.0){\small$X_{L}$}
\put(36.0,62.0){\small$Z_{1}$} \put(36.0,47.0){\small$Z_{2}$}
\put(35.0,9.0){\small$Z_{L}$} \put(10.0,58.0){\small$h_{11}$}
\put(11.5,49.0){\small$h_{21}$} \put(17.5,38.0){\small$h_{32}$}
\put(25.0,44.5){\small$h_{22}$} \put(25.0,13.0){\small$h_{LL}$}
\put(69.0,52.0){\small$C_{1}$} \put(68.0,43.0){\small$C_{2}$}
\put(68.0,18.0){\small$C_{L}$} \put(80.0,41.0){\small$\hat{Y}_{1}$}
\put(73.0,34.0){\small$\hat{Y}_{2}$} \put(77.0,25.0){\small$\hat{Y}_{L}$}
\put(13.0,2.0){$\mathcal{S}$} \put(22.0,2.0){$\mathcal{S}^{c}$}
\@add@centering\end{overpic} Figure 5: Choice of cut-set for the Wyner model.
The cut goes through the access link or the backhaul link depending on whether
$\frac{1}{2}\log(\mathsf{SNR}_{i})$ is greater or smaller than $C_{i}$.

It turns out that even without Wyner-Ziv compress-and-forward, it is still
possible to achieve the sum capacity to within a constant, albeit larger, gap
under the same weak-interference condition.

######  Theorem 4.

For a multicell processing Wyner model shown in Fig. 4, in the weak-
interference regime of
$\mathsf{INR}_{i+1,i}\leq\mathsf{SNR}_{i},i=1,2,\cdots,L-1$, the per-base-
station SIC scheme without Wyner-Ziv coding achieves a sum rate that is within
$\frac{1}{2}(1+\log(3))L-\frac{1}{2}$ bits of the sum capacity.

###### Proof:

Applying Theorem 2 to the Wyner model of Fig. 4 with the decoding order from
user $L$ to user $1$, it is easy to see that the achievable rate of user $L$
remains the same while the rates of other users become

$R_{i}=\frac{1}{2}\log\frac{1+2^{-2C_{i}}\mathsf{INR}_{i+1,i}+\mathsf{SNR}_{i}}{1+2^{-2C_{i}}\mathsf{INR}_{i+1,i}+2^{-2C_{i}}\mathsf{SNR}_{i}}.$
(35)

Under the weak-interference condition
$\mathsf{INR}_{i+1,i}\leq\mathsf{SNR}_{i}$, when $i\in{\cal S}^{c}$, we have
$0\leq 2^{-2C_{i}}\mathsf{INR}_{i+1,i}\leq 1$, and when when $i\in{\cal S}$,
we have $0\leq 2^{-2C_{i}}\mathsf{INR}_{i+1,i}\leq
2^{-2C_{i}}\mathsf{SNR}_{i}$. Replace the $R_{i}$ expressions in (33) and (34)
by (35) while noting the inequalities above, we obtain the
$\frac{1}{2}(1+\log(3))L-\frac{1}{2}$ bound. ∎

Note that a looser bound can be obtained by noting that under the weak-
interference condition $\mathsf{INR}_{i+1,i}\leq\mathsf{SNR}_{i}$, the
achievable rate without Wyner-Ziv coding (35) is already within $\frac{1}{2}$
bits of the rate with Wyner-Ziv coding as in (25). As a result, a looser sum-
rate gap to the cut-set upper bound can immediately be obtained as
$L-\frac{1}{2}+\frac{L-1}{2}=\frac{3}{2}L-1$ bits222The authors wish to thank
Yuhan Zhou to point out the tighter bound in Theorem 4. .

##  V Optimal Rate Allocation with a Total Backhaul Capacity Constraint

A practical system may have a constraint on the sum capacity of all digital
backhaul links, for example, when the backhaul links are implemented in a
wireless medium with shared bandwidth. So, it may be of interest to optimize
the allocation of backhaul capabilities among the base-stations in order to
achieve an overall maximum sum rate under a total backhaul capacity
constraint. The structure of the solution of such an optimization can also
yield useful insight. The optimization problem can be formulated as the
following:

$\displaystyle\mathrm{maximize}\quad$ $\displaystyle\sum_{k=1}^{L}R_{k}$
$\displaystyle\mathrm{subject\;to}\quad$ $\displaystyle C_{k}\geq
0,\;\;k=1,2,\cdots,L$ $\displaystyle\sum_{k=1}^{L}C_{k}\leq C$

where $R_{k},k=1,2,\cdots,L$ are functions of $C_{k}$ as derived in Theorem 3,
and $C>0$ is the total available backhaul capacity. The following theorem
provides an optimal solution to the above optimization problem for the per-
base-station SIC scheme with Wyner-Ziv coding.

######  Theorem 5.

For the uplink multicell joint processing model shown in Fig. 1, with the per-
base-station SIC at the central processor with Wyner-Ziv coding, the optimal
allocation of backhaul link capacities subject to a total backhaul capacity
constraint $C$ is given by

$C_{k}^{*}=\max\left\\{\frac{1}{2}\log(\mathsf{\overline{SINR}}_{k})-\alpha,0\right\\},$
(37)

where $\alpha$ is chosen such that $\sum_{k=1}^{L}C_{k}^{*}=C$.

###### Proof:

Substituting the rate expression (19) for $R_{k}$ into the optimization
problem (V), we obtain the following equivalent minimization problem:

$\displaystyle\mathrm{minimize}\quad$
$\displaystyle\sum_{k=1}^{L}\frac{1}{2}\log\left(1+2^{-2C_{k}}\mathsf{\overline{SINR}}_{k}\right)$
$\displaystyle\mathrm{subject\;to}\quad$ $\displaystyle C_{k}\geq
0,\;\;k=1,2,\cdots,L.$ $\displaystyle\sum_{k=1}^{L}C_{k}\leq C$

It can be easily seen that (V) is a convex optimization problem, as the
constraints are linear and the objective function is the sum of convex
functions, as can be verified by taking their second derivatives.

Now introducing Lagrange multipliers $\nu$ $\in\mathbb{R}_{+}^{L}$ for the
positivity constraints $C_{k}\geq 0,k=1,2,\cdots,L$, and
$\lambda\in\mathbb{R}_{+}$ for the backhaul sum-capacity constraint
$\sum_{k=1}^{L}C_{k}\leq C$, we form the Lagrangian

$L(C_{k},{\boldmath\boldsymbol{\nu}},\lambda)=\sum_{k=1}^{L}\frac{1}{2}\log\left(1+2^{-2C_{k}}\mathsf{\overline{SINR}}_{k}\right)\\\
-\sum_{k=1}^{L}\nu_{k}C_{k}+\lambda\left(\sum_{k=1}^{L}C_{k}-C\right)$ (39)

Taking the derivative of the above with respect to $C_{k}$, we obtain the
following Karush-Kuhn-Tucker (KKT) condition

$-\frac{2^{-2C_{k}^{*}}\mathsf{\overline{SINR}}_{k}}{1+2^{-2C_{k}^{*}}\mathsf{\overline{SINR}}_{k}}-\nu_{k}+\lambda=0,$
(40)

for the optimal $C_{k}^{*}$, where $k=1,2,\cdots,L$. Note that $\nu_{k}=0$
whenever $C_{k}>0$. Now, the optimal $C_{k}^{*}$ must satisfy the backhaul
sum-capacity constraint $\sum_{k=1}^{L}C_{k}^{*}\leq C$ with equality, because
the objective of the minimization $R_{k}$ monotonically increases with
$C_{k}$. Solving the condition (40) together with the fact that
$\sum_{k=1}^{L}C_{k}^{*}=C$ gives the following optimal $C_{k}^{*}$:

$C_{k}^{*}=\max\left\\{\frac{1}{2}\log\frac{\mathsf{\overline{SINR}}_{k}}{\beta},0\right\\},$
(41)

where $\beta$ is chosen such that $\sum_{k=1}^{L}C_{k}^{*}=C$. This is
equivalent to (37). ∎

An interpretation of (41) is that whenever the SINR of user $k$ is above a
threshold $\beta$, $\frac{1}{2}\log\frac{\mathsf{\overline{SINR}}_{k}}{\beta}$
bits of the backhaul link should be allocated to user $k$. Otherwise, this
user should not be active as far as maximizing the uplink sum rate is
concerned. This optimal rate allocation is quite similar to the classic water-
filling solution for the sum-capacity maximization problem for a parallel set
of Gaussian channels, in which more power (backhaul capacity in this case) is
assigned to users with a better channel.

When written as (37), the optimal backhaul capacity allocation can be
interpreted as follows: $C_{k}=\frac{1}{2}\log(\mathsf{\overline{SINR}}_{k})$
can be thought of as the nominal optimal backhaul link capacity. If the total
backhaul capacity is above (or below) the nominal
$\sum_{k}\frac{1}{2}\log(\mathsf{\overline{SINR}}_{k})$, then the extra
capacity must be distributed (or taken away) from each base-station equally.
In other words, all base-station should nominally operate at the point 1/2
bits away from the SIC limit (as shown in Fig. 3). If more (or less) backhaul
capacity is available than the nominal value, all base-stations should move
above (or below) that operating point in the same manner.

Finally, we remark that the decoding order at the centralized processor plays
an important role in the optimal rate allocation. Different decoding orders
result in different rate expressions in Theorem 1 and thus different rate
allocations in Theorem 5, and as a consequence different achievable sum rates.
In order to determine the decoding order that results in the largest sum rate
(or the maximum network utility), we need to exhaustively search over $K!$
different decoding orders. This is a nontrivial problem that is also
encountered in other contexts involving successive decoding such as in V-BLAST
[26].

##  VI Numerical Examples

###  VI-A Two-User Symmetric Scenario

(a) $\mathsf{SNR}=30$dB, $\mathsf{INR}=20$dB, $C_{1}=C_{2}=5$ bits

(b) $\mathsf{SNR}=30$dB, $\mathsf{INR}=5$dB, $C=5$ bits

(c) $\mathsf{SNR}=30$dB, $\mathsf{INR}=20$dB, $C=10$ bits

(d) $\mathsf{SNR}=30$dB, $\mathsf{INR}=20$dB, $C=2$ bits

Figure 6: Comparison of the proposed achievability scheme and another two
schemes

To obtain numerical insight on the per-base-station SIC-based schemes proposed
in this paper, the achievable rate regions of Theorem 1 and Theorem 2 are
compared with that obtained by two other schemes: 1) single-user decoding
without joint processing; 2) joint base-station processing as defined in (21)
in Section III-C, for a two-user symmetric scenario where $L=2$,
$P_{1}=P_{2}=N_{0}=1$, $\mathsf{SNR}=h_{11}^{2}=h_{22}^{2}$,
$\mathsf{INR}=h_{12}^{2}=h_{21}^{2}$, and $C=C_{1}=C_{2}$. Under the symmetric
setting, both Theorem 1 and Theorem 2 give two symmetric achievable rate pairs
depending on the decoding order. Time-sharing of the two achievable rate pairs
gives a pentagon shaped achievable rate region.

Single-user decoding without joint processing is considered as a baseline, in
which each receiver decodes its own signal while treating the other user’s
signal as noise. This gives the following achievable rate pair

$R_{1}=R_{2}=\min\left\\{\displaystyle\frac{1}{2}\log\displaystyle\left(1+\frac{\mathsf{SNR}}{1+\mathsf{INR}}\right),C\right\\}$
(42)

which in the symmetric setting results in a square shaped achievable rate
region with ($R_{1},R_{2}$) as the top-right corner.

We also compare with the joint-base-station processing SIC scheme of (21). We
restrict ourselves to symmetric quantization noise levels here. The
quantization noise level $\mathsf{q}$ is chosen such that the resulting
average backhaul link capacity is $C$, i.e.,

$I(Y_{1};\hat{Y}_{1})+I(Y_{2};\hat{Y}_{1}|\hat{Y}_{1})=2C.$ (43)

This condition gives the following analytic expression for the quantization
noise level:

$\mathsf{q}=\frac{a+\sqrt{4b+2^{4C}\left(a^{2}-4b\right)}}{2^{4C}-1},$ (44)

where $a=1+\mathsf{SNR}+\mathsf{INR}$, and $b=\mathsf{SNR}\cdot\mathsf{INR}$.
Plots of the joint-base-station SIC are then obtained using (21) with this
quantization noise level.

The achievable rate regions obtained above are compared for the following
channel settings:

  * •

$\mathsf{SNR}=30$dB, $\mathsf{INR}=20$dB, $C=5$ bits;

  * •

$\mathsf{SNR}=30$dB, $\mathsf{INR}=5$dB, $C=5$ bits;

  * •

$\mathsf{SNR}=30$dB, $\mathsf{INR}=20$dB, $C=10$ bits;

  * •

$\mathsf{SNR}=30$dB, $\mathsf{INR}=20$dB, $C=2$ bits.

Fig. 6(a) shows that at relatively strong interference level
$\mathsf{INR}=20$dB, the proposed per-base-station SIC schemes (both with and
without Wyner-Ziv compression) expand the baseline achievable rate region by
about $2.8$ bits on either of the individual rates and on the sum rate. The
joint base-station SIC regions further outperforms the proposed scheme in sum
rate by about $2.5$ bits due to the benefits of joint decoding.

However, when the interfering links are weak, as shown in Fig. 6(b) where
$\mathsf{INR}=5$dB, all four achievable rate regions are close to each other.
This is the regime where treating interference as noise is close to optimal,
so multicell processing does not provide significant benefits.

In the above two examples, the capacities of the backhaul links are already
quite abundant, since they are set to be the rate supported by the direct
links: $\frac{1}{2}\log(1+\mathsf{SNR})\approx 5$ bits. In Fig. 6(c), we
further increase the backhaul capacity to 10 bits, and show that doing so does
not significantly improve the achievable rate region for either proposed SIC-
based schemes or the joint base-station processing scheme.

Lastly, we decrease the backhaul capacity from 5 bits to 2 bits.
Interestingly, this is a situation in which the base-line scheme can
outperform per-base-station SIC as shown in Fig. 6(d). Therefore, the proposed
per-base-station schemes can be inefficient in term of sum rate, when the
backhaul rates are very limited. Observe that the largest sum rate is still
obtained with joint base-station processing.

###  VI-B Multicell OFDMA Network

To further understand the performance of the proposed per-base-station SIC
scheme in practical systems, in this section, the achievable rates of the two
variations of the per-base-station SIC, i.e., with and without Wyner-Ziv
coding, are evaluated for a wireless cellular network setup with $19$ cells,
$3$ sectors per cell, and $10$ users per sector, where an orthogonal
frequency-division multiple-access (OFDMA) scheme with $64$ tones over a fixed
$10$Mbps bandwidth is employed. The cellular topology is shown in Fig. 7. A
19-cell wrap-around layout is used to ensure uniform interference statistics
throughout the network. The assignments of frequency tones to users within
each cell are non-overlapping. As a result, users experience only intercell
interference and no intracell interference. Both the base-stations and the
mobile users are equipped with a single antenna each. Each of the $19$ base-
stations is connected to the centralized processor via a rate-limited backhaul
link. Perfect channel estimation is assumed, and the CSI is made available to
all base-stations and to the centralized processor. In the simulation, uniform
power allocation of $-27$dBm/Hz is assumed at all the mobile users. For
convenience, a round-robin scheduler is used for user assignment. The base-
station-to-base-station distance is set to $600$m corresponding to a typical
urban deployment. Detailed system parameters are outlined in Table I.

TABLE I: Wireless Cellular Model Parameters Cellular Layout | Hexagonal, 19 cells, 3 sectors/cell
---|---
BS-to-BS Distance | 600 m
Frequency Reuse | 1
Number of User per Sector | 10
Channel Bandwidth | 10 MHz
MS Transmit PSD |  $-27$ dBm/Hz
Antenna Gain | 15 dBi
Background Noise |  $-169$ dBm/Hz
Noise Figure |  $7$ dB
Multipath Time Delay Profile | ITU-R M.1225 PedA
Distance-dependent Path Loss | $128.1+37.6\log_{10}(d)$
Number of Tones | 64

In the first part of the simulation, the capacities of the backhaul links are
fixed per base-station and uniformly distributed across the frequency tones,
e.g., if the capacity of a backhaul link is $64$Mbps, each frequency tone is
assumed to have a backhaul of $64\mathrm{Mbps}/64=1\mathrm{Mbps}$. Cumulative
distribution function (CDF) of the user rates is plotted in order to visualize
the performance gain of the proposed schemes over a baseline system, in which
base-stations decode the user messages without joint processing at the
centralized processor. For the proposed per-base-station SIC scheme, to
account for the fairness among users, the decoding order across the cells is
chosen to be in decreasing order of the user SINRs (prior to SIC). This
decoding order is adopted on each of the OFDM tones independently.

Figure 7: A cellular topology with $19$ cells, $3$ sectors per cell, and $10$
users per sector placed randomly. The distance between the neighboring base-
stations is set to $600$m.

Fig. 8 shows the CDF plots of user rates with a backhaul capacity at each
base-station of $180$Mbps (i.e., $60$Mbps per sector). It is seen that the two
per-base-station SIC schemes (with or without Wyner-Ziv coding) both
significantly outperform the baseline system. The user rate at $50$th-
percentile is around $0.8$Mbps for the baseline, $2.1$Mbps for the per-base-
station SIC scheme without Wyner-Ziv coding, and $2.8$Mbps for the per-base-
station SIC scheme with Wyner-Ziv coding. Compared with the baseline curve,
the per-base-station SIC curves also have a better distribution of user rates
in terms of fairness. There is a noticeable performance gap between the SIC
curve with Wyner-Ziv compression and without Wyner-Ziv-compression. This gap
is due to the compression gain brought by side information.

When the capacity of the backhaul per base-station increases to $270$Mbps, the
proposed per-base-station SIC schemes produce a further performance gain as
compared to the $180$Mbps case. As can be seen in Fig. 9, the $50$-percentile
rate becomes $2.6$Mbps for the per-base-station SIC scheme without Wyner-Ziv
compression, and $3.05$Mbps for the per-base-station SIC scheme with Wyner-Ziv
compression. However, the gap between the two curves becomes smaller. As the
capacity of the backhaul per base-station further increases to $360$Mbps, Fig.
10 shows that the per-base-station SIC schemes now only perform marginally
better than the $270$Mbps case. This is when the benefit of multicell SIC
starts to saturate.

Further, as shown in Fig. 10, the CDF curves for the SIC schemes with and
without Wyner-Ziv compression are very close to each other for the $360$Mbps
backhaul case. Thus, the benefit of performing Wyner-Ziv compress-and-forward
relaying at the base-stations becomes negligible when the capacities of
backhaul links are high, thus confirming our earlier theoretical analysis.

In order to quantitatively evaluate the performance gain brought by the
centralized processor, Table II shows the average per-cell sum rate obtained
by different schemes. The baseline scheme gives an average per-cell sum rate
of $55.5$Mbps. By utilizing a high-capacity backhaul with Wyner-Ziv
compression, up to 94% sum rate improvement can be obtained. Note that
although the rate improvement without Wyner-Ziv is lower than that with Wyner-
Ziv, considerable performance gains in the range of 37% to 84% can still be
obtained without Wyner-Ziv coding. As noted before, the gain due to the
backhaul saturates at around $270$Mbps.

The above simulation is performed assuming that the backhaul capacity is the
same for each base-station and is uniformly allocated across the frequencies.
It is possible to further optimize the backhaul capacity allocation using (37)
of Theorem 5. In the next set of simulations, we choose $\alpha$ in (37) to
satisfy an average backhaul constraint across the cells, and present the
resulting performance with optimized backhaul allocation in Table II and Fig.
11. It can be seen that $120-150$Mbps optimized backhaul already achieves
about the same performance as that of $180$Mbps uniform backhaul. Likewise,
$180$Mbps optimized backhaul already achieves about the same performance as
that of $270$Mbps uniform backhaul. Thus, the optimization of the backhaul is
quite beneficial.

Further, it can be seen from Fig. 11 that under infinite backhaul, the
achieved per-cell sum rate is about $110$Mbps for this cellular setting. But
when optimized, a finite backhaul capacity at about 1.5 times of the user sum
rate (i.e., at about $150$Mbps) is already sufficient to achieve about
$100$Mbps user sum rate, which is 90% of the full benefit of uplink network
MIMO. Note that the gain in per-cell sum rate due to the optimization of the
backhaul becomes smaller as the backhaul capacity increases, due to the fact
that increasing the backhaul capacity eventually offers diminishing return.

Figure 8: CDF of user rates with 180Mbps backhaul per base-station uniformly allocated across the frequencies Figure 9: CDF of user rates with 270Mbps backhaul per base-station uniformly allocated across the frequencies Figure 10: CDF of user rates with 360Mbps backhaul per base-station uniformly allocated across the frequencies Figure 11: Per-cell sum rate vs. average per-cell backhaul capacity using per-base-station SIC with uniform and optimized backhaul capacity allocation TABLE II: Improvement in per-cell sum rate over 19 cells, 3 sectors per cell, 10 users per sector. Cell diameter is 600m. Baseline per-cell sum rate is $55.5$Mbps Per-cell Average | Backhaul Capacity | Per-base-station SIC | Improvement | Per-base-station SIC | Improvement
---|---|---|---|---|---
Backhaul (Mbps) | Allocation | without WZ (Mbps) | over Baseline (%) | with WZ (Mbps) | over Baseline (%)
$180$ | uniform | $75.3$ | $37\%$ | $92.7$ | $67\%$
$270$ | uniform | $93.2$ | $68\%$ | $104.2$ | $88\%$
$360$ | uniform | $102.3$ | $84\%$ | $107.6$ | $94\%$
$120$ | optimized | $75.1$ | $35\%$ | $89.1$ | $60\%$
$180$ | optimized | $83.3$ | $50\%$ | $102.9$ | $85\%$

Finally, we mention that the performance gain presented here is idealistic
because the achievable rates are computed using information theoretical
expressions assuming ideal coding, modulation, and perfect CSI. In addition,
all users in the 19-cell cluster are assumed to participate in cooperative
multicell processing, and no out-of-cluster interference is accounted for. The
results in this paper nevertheless serve as upper bound to what is achievable
in an uplink network MIMO system.

##  VII Conclusion

This paper presents an information-theoretical study of a novel uplink
multicell processing scheme employing the compress-and-forward technique with
the per-base-station SIC receiver structure for the uplink of a network MIMO
system, in which the base-stations are connected to a centralized processor
with finite capacity backhaul links. The main advantage of the proposed
schemes is that it achieves significant performance gain over the conventional
scenario where no centralized processor is deployed, while having an
achievable rate region which is easily computable, and that it leads to an
architecture that is more amendable to practical implementations than the
joint decoding scheme. Furthermore, theoretical analysis shows that the
proposed per-base-station SIC scheme is within a constant gap to the sum
capacity for a class of Wyner channel models.

The results of this paper also show that when employing the proposed per-base-
station SIC scheme, the capacities of the backhaul links should scale with the
logarithm of the SINR at each base-station, both from a point of view of
approaching the theoretical maximum SIC rate with unlimited backhaul, as well
as for maximizing the overall sum rate subject to a total backhaul rate
constraint. Numerical simulations reveal that significant sum-rate gain can be
obtained by the proposed SIC-based schemes with modest backhaul capacity
requirement.

## Acknowledgements

The authors would like to thank Dimitris Toumpakaris for helpful discussions
and valuable comments.

## References

  * [1] S. Venkatesan, A. Lozano, and R. Valenzuela, “Network MIMO: Overcoming intercell interference in indoor wireless systems,” in _Conf. Record Forty-First Asilomar Conf. Signals, Systems and Computers_ , Nov. 2007, pp. 83–87.
  * [2] D. Gesbert, S. Hanly, H. Huang, S. Shamai, O. Simeone, and W. Yu, “Multi-cell MIMO cooperative networks: A new look at interference,” _IEEE J. Sel. Areas Commun._ , vol. 28, no. 9, pp. 1380–1408, Dec. 2010.
  * [3] I.-H. Wang and D. N. C. Tse, “Interference mitigation through limited receiver cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2913–2940, May 2011.
  * [4] A. Høst-Madsen, “Capacity bounds for cooperative diversity,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 4, pp. 1522–1544, Apr. 2006.
  * [5] C. T. K. Ng, N. Jindal, A. J. Goldsmith, and U. Mitra, “Capacity gain from two-transmitter and two-receiver cooperation,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 10, pp. 3822–3827, Oct. 2007.
  * [6] V. M. Prabhakaran and P. Viswanath, “Interference channels with destination cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 1, pp. 187–209, Jan. 2011.
  * [7] H. Do, T. Oechtering, and M. Skoglund, “Noisy network coding approach to the interference channel with receiver cooperation,” in _Proc. Forty-Ninth Annual Allerton Conf. Commun, Control and Computing_ , Sep. 2011, pp. 839–846.
  * [8] P. Marsch and G. Fettweis, “Uplink CoMP under a constrained backhaul and imperfect channel knowledge,” _IEEE Trans. Wireless Commun._ , vol. 10, no. 6, pp. 1730–1742, Jun. 2011.
  * [9] O. Simeone, O. Somekh, H. Poor, and S. Shamai, “Local base station cooperation via finite-capacity links for the uplink of linear cellular networks,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 1, pp. 190–204, Jan. 2009.
  * [10] A. Chowdhery, W. Yu, and J. Cioffi, “Cooperative wireless multicell OFDMA network with backhaul capacity constraints,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , Jun. 2011.
  * [11] A. Sanderovich, O. Somekh, H. V. Poor, and S. Shamai, “Uplink macro diversity of limited backhaul cellular network,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 8, pp. 3457–3478, Aug. 2009.
  * [12] A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer, “Communication via decentralized processing,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 7, pp. 3008–3023, Jul. 2008.
  * [13] O. Somekh, B. Zaidel, and S. Shamai, “Sum rate characterization of joint multiple cell-site processing,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 12, pp. 4473–4497, Dec. 2007.
  * [14] A. del Coso and S. Simoens, “Distributed compression for MIMO coordinated networks with a backhaul constraint,” _IEEE Trans. Wireless Commun._ , vol. 8, no. 9, pp. 4698–4709, Sep. 2009.
  * [15] S.-H. Park, O. Simeone, O. Sahin, and S. Shamai, “Robust and efficient distributed compression for cloud radio access network,” _IEEE Trans. Veh. Technol._ , vol. 62, no. 2, pp. 692–703, Feb. 2013.
  * [16] ——, “Joint decompression and decoding for cloud radio access networks,” _IEEE Signal Process. Lett._ , vol. 20, no. 5, pp. 503–506, May 2013.
  * [17] A. Sanderovich, M. Peleg, and S. Shamai, “Scaling laws and techniques in decentralised processing of interfered Gaussian channels.” _European Trans. Telecommun._ , vol. 22, no. 5, pp. 240–253, 2011.
  * [18] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai, “Downlink multicell processing with limited-backhaul capacity,” _EURASIP J. Advances Signal Process._ , 2009.
  * [19] S. Avestimehr, S. Diggavi, and D. N. C. Tse, “Wireless network information flow: A deterministic approach,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 4, pp. 1872–1905, Apr. 2011.
  * [20] S. H. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisy network coding,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 3132–3152, May 2011.
  * [21] O. Somekh, B. M. Zaidel, and S. Shamai, “Sum rate characterization of joint multiple cell-site processing,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 7, pp. 4473–4497, Dec. 2007.
  * [22] Y. Zhou, W. Yu, and D. Toumpakaris, “Uplink multi-cell processing: Approximate sum capacity under a sum backhaul constraint,” Apr. 2013, [Online] Available: http://arxiv.org/abs/1304.7509.
  * [23] A. D. Wyner, “Shannon-theoretic approach to a Gaussian cellular multiple-access channel,” _IEEE Trans. Inf. Theory_ , vol. 40, no. 6, pp. 1713–1727, Nov. 1994.
  * [24] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,” _IEEE Trans. Inf. Theory_ , vol. 25, no. 5, pp. 572–584, Sep. 1979.
  * [25] T. M. Cover and J. A. Thomas, _Elements of Information Theory_ , 1st ed. Wiley, 1991.
  * [26] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in _Proc. URSI Int. Symp. Signals, Syst. and Electron. (ISSSE)_ , Sep. 1998, pp. 295–300.

## References

  * [1] S. Venkatesan, A. Lozano, and R. Valenzuela, “Network MIMO: Overcoming intercell interference in indoor wireless systems,” in _Conf. Record Forty-First Asilomar Conf. Signals, Systems and Computers_ , Nov. 2007, pp. 83–87.
  * [2] D. Gesbert, S. Hanly, H. Huang, S. Shamai, O. Simeone, and W. Yu, “Multi-cell MIMO cooperative networks: A new look at interference,” _IEEE J. Sel. Areas Commun._ , vol. 28, no. 9, pp. 1380–1408, Dec. 2010.
  * [3] I.-H. Wang and D. N. C. Tse, “Interference mitigation through limited receiver cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 2913–2940, May 2011.
  * [4] A. Høst-Madsen, “Capacity bounds for cooperative diversity,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 4, pp. 1522–1544, Apr. 2006.
  * [5] C. T. K. Ng, N. Jindal, A. J. Goldsmith, and U. Mitra, “Capacity gain from two-transmitter and two-receiver cooperation,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 10, pp. 3822–3827, Oct. 2007.
  * [6] V. M. Prabhakaran and P. Viswanath, “Interference channels with destination cooperation,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 1, pp. 187–209, Jan. 2011.
  * [7] H. Do, T. Oechtering, and M. Skoglund, “Noisy network coding approach to the interference channel with receiver cooperation,” in _Proc. Forty-Ninth Annual Allerton Conf. Commun, Control and Computing_ , Sep. 2011, pp. 839–846.
  * [8] P. Marsch and G. Fettweis, “Uplink CoMP under a constrained backhaul and imperfect channel knowledge,” _IEEE Trans. Wireless Commun._ , vol. 10, no. 6, pp. 1730–1742, Jun. 2011.
  * [9] O. Simeone, O. Somekh, H. Poor, and S. Shamai, “Local base station cooperation via finite-capacity links for the uplink of linear cellular networks,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 1, pp. 190–204, Jan. 2009.
  * [10] A. Chowdhery, W. Yu, and J. Cioffi, “Cooperative wireless multicell OFDMA network with backhaul capacity constraints,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , Jun. 2011.
  * [11] A. Sanderovich, O. Somekh, H. V. Poor, and S. Shamai, “Uplink macro diversity of limited backhaul cellular network,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 8, pp. 3457–3478, Aug. 2009.
  * [12] A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer, “Communication via decentralized processing,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 7, pp. 3008–3023, Jul. 2008.
  * [13] O. Somekh, B. Zaidel, and S. Shamai, “Sum rate characterization of joint multiple cell-site processing,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 12, pp. 4473–4497, Dec. 2007.
  * [14] A. del Coso and S. Simoens, “Distributed compression for MIMO coordinated networks with a backhaul constraint,” _IEEE Trans. Wireless Commun._ , vol. 8, no. 9, pp. 4698–4709, Sep. 2009.
  * [15] S.-H. Park, O. Simeone, O. Sahin, and S. Shamai, “Robust and efficient distributed compression for cloud radio access network,” _IEEE Trans. Veh. Technol._ , vol. 62, no. 2, pp. 692–703, Feb. 2013.
  * [16] ——, “Joint decompression and decoding for cloud radio access networks,” _IEEE Signal Process. Lett._ , vol. 20, no. 5, pp. 503–506, May 2013.
  * [17] A. Sanderovich, M. Peleg, and S. Shamai, “Scaling laws and techniques in decentralised processing of interfered Gaussian channels.” _European Trans. Telecommun._ , vol. 22, no. 5, pp. 240–253, 2011.
  * [18] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai, “Downlink multicell processing with limited-backhaul capacity,” _EURASIP J. Advances Signal Process._ , 2009.
  * [19] S. Avestimehr, S. Diggavi, and D. N. C. Tse, “Wireless network information flow: A deterministic approach,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 4, pp. 1872–1905, Apr. 2011.
  * [20] S. H. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisy network coding,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 5, pp. 3132–3152, May 2011.
  * [21] O. Somekh, B. M. Zaidel, and S. Shamai, “Sum rate characterization of joint multiple cell-site processing,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 7, pp. 4473–4497, Dec. 2007.
  * [22] Y. Zhou, W. Yu, and D. Toumpakaris, “Uplink multi-cell processing: Approximate sum capacity under a sum backhaul constraint,” Apr. 2013, [Online] Available: http://arxiv.org/abs/1304.7509.
  * [23] A. D. Wyner, “Shannon-theoretic approach to a Gaussian cellular multiple-access channel,” _IEEE Trans. Inf. Theory_ , vol. 40, no. 6, pp. 1713–1727, Nov. 1994.
  * [24] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,” _IEEE Trans. Inf. Theory_ , vol. 25, no. 5, pp. 572–584, Sep. 1979.
  * [25] T. M. Cover and J. A. Thomas, _Elements of Information Theory_ , 1st ed. Wiley, 1991.
  * [26] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in _Proc. URSI Int. Symp. Signals, Syst. and Electron. (ISSSE)_ , Sep. 1998, pp. 295–300.

|  Lei Zhou (S’05-M’12) received the B.E. degree in Electronics Engineering
from Tsinghua University, Beijing, China, in 2003 and M.A.Sc. and Ph.D.
degrees in Electrical and Computer Engineering from the University of Toronto,
ON, Canada, in 2008 and 2012 respectively. During 2008-2009, he was with
Nortel Networks, Ottawa, ON, Canada. Since 2012, he has been with Qualcomm
Technologies Inc., Santa Clara, CA. His research interests include
multiterminal information theory, wireless communications, and signal
processing. He is a recipient of the Shahid U.H. Qureshi Memorial Scholarship
in 2011, the Alexander Graham Bell Canada Graduate Scholarship in 2011, and
the Chinese government award for outstanding self-financed students abroad in
2012.
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|  Wei Yu (S’97-M’02-SM’08) received the B.A.Sc. degree in Computer
Engineering and Mathematics from the University of Waterloo, Waterloo,
Ontario, Canada in 1997 and M.S. and Ph.D. degrees in Electrical Engineering
from Stanford University, Stanford, CA, in 1998 and 2002, respectively. Since
2002, he has been with the Electrical and Computer Engineering Department at
the University of Toronto, Toronto, Ontario, Canada, where he is now Professor
and holds a Canada Research Chair in Information Theory and Wireless
Communications. His main research interests include information theory,
optimization, wireless communications and broadband access networks. Prof. Wei
Yu currently serves as an Associate Editor for IEEE Transactions on
Information Theory. He was an Editor for IEEE Transactions on Communications
(2009-2011), an Editor for IEEE Transactions on Wireless Communications
(2004-2007), and a Guest Editor for a number of special issues for the IEEE
Journal on Selected Areas in Communications and the EURASIP Journal on Applied
Signal Processing. He is member of the Signal Processing for Communications
and Networking Technical Committee of the IEEE Signal Processing Society. He
received the IEEE Signal Processing Society Best Paper Award in 2008, the
McCharles Prize for Early Career Research Distinction in 2008, the Early
Career Teaching Award from the Faculty of Applied Science and Engineering,
University of Toronto in 2007, and the Early Researcher Award from Ontario in
2006.
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