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arXiv:1710.08671v1 [cs.IT] 24 Oct 2017

Linear State Estimation via 5G C-RAN Cellular

Networks using Gaussian Belief Propagation

Mirsad Cosovic, Dejan Vukobratovic, Vladimir Stankovic

Abstract—Machine-type communications and large-scale

information processing architectures are among key

(r)evolutionary enhancements of emerging ﬁfth-generation (5G)

mobile cellular networks. Massive data acquisition and

processing will make 5G network an ideal platform for

large-scale system monitoring and control with applications in

future smart transportation, connected industry, power grids,

etc. In this work, we investigate a capability of such a 5G

network architecture to provide the state estimate of an

underlying linear system from the input obtained via

large-scale deployment of measurement devices. Assuming that

the measurements are communicated via densely deployed

cloud radio access network (C-RAN), we formulate and solve

the problem of estimating the system state from the set of

signals collected at C-RAN base stations. Our solution, based

on the Gaussian Belief-Propagation (GBP) framework, allows

for large-scale and distributed deployment within the emerging

5G information processing architectures. The presented

numerical study demonstrates the accuracy, convergence

behavior and scalability of the proposed GBP-based solution to

the large-scale state estimation problem.

I. INTRO DUC TI ON

With transition towards ﬁfth generation (5G), mobile

cellular networks are evolving into ubiquitous systems for

data acquisition and information processing suitable for

monitoring and control of large-scale systems. At the

forefront of this evolution is the transformation of radio

access network (RAN) to support massive-scale

machine-type communications (MTC) [1] and transformation

of core network (CN) to support large-scale centralized or

distributed information processing through Cloud-RAN

(C-RAN) and Fog-RAN (F-RAN) architecture [2], [3]. MTC

services in 5G will offer both massive-scale data acquisition

from various machine-type devices through massive MTC

(mMTC) service, but also, provide ultra reliable and

low-latency communication (URLLC) service for

mission-critical applications [4]. Complemented with

ultra-dense RAN deployment and ﬂexible and virtualized

signal processing architecture, novel 5G network services

that are particularly suitable for large-scale system

monitoring and control of various smart infrastructures are

emerging [5].

In this work, we focus on a generic state estimation

problem placed in the context of a future 5G-inspired

M. Cosovic is with Schneider Electric DMS NS, Novi Sad, Serbia

(e-mail: mirsad.cosovic@schneider-electric-dms.com). D. Vukobratovic

is with Department of Power, Electronic and Communications

Engineering, University of Novi Sad, Novi Sad, Serbia (e-mail:

dejanv@uns.ac.rs). V. Stankovic is with Department of Electronic and

Electrical Engineering, University of Strathclyde, Glasgow, UK (e-

mail:vladimir.stankovic@eee.strath.ac.uk).

C-RAN-based cellular network. We consider an underlying

large-scale physical system characterized by the state vector

sthat contains values of Nsystem state variables. The state

variables are observed through the set of Mmeasurements x

of physical quantities collected at the measurement devices

spread across the system. This paper considers linear system

model in which measured quantities are linear functions of

the (sub)set of state variables. Further, we assume

measurements are wirelessly communicated across

C-RAN-based cellular network. In C-RAN, large number of

spatially distributed remote radio heads (RRH) constitutes an

ultra-dense RAN infrastructure that receives signals from

densely populated MTC devices (e.g., the measurement

devices under consideration) [2]. The signal vector y

collected at RRHs is forwarded via backhaul links to a

central C-RAN location where it is fed into a collection of

base-band units (BBU) for signal detection and recovery. In

the standard C-RAN signal detection problem, the goal is to

recover the signal xtransmitted by the set of MTC devices

from the signal yreceived at RRHs and gathered centrally at

BBUs [6] [7]. However, in this paper, focusing on widely

applicable linear system state estimation, we extend this goal

and investigate the problem of recovering the system state s

directly from the signal ycollected across the C-RAN.

The problem we observe represents a concatenation of the

two well-studied problems: the linear system state estimation

problem (see, e.g., [8], for the case of power system state

estimation) and the problem of uplink signal detection in

C-RAN [6]. For the joint problem, it is straightforward to

derive (and implement at a central location) the standard

minimum mean-square error (MMSE) estimator, however,

such a solution comes with prohibitive O(N3)-complexity

that hinders its application for large-scale systems. By

exploiting inherent sparsity within both of the component

problems, an approximate MMSE solution for each problem

can be obtained using the tools from probabilistic graphical

models, as recently investigated for both (power system)

state estimation [9] and uplink signal detection in C-RANs

[7]. In particular, an instance of the Belief-Propagation (BP)

algorithm, called Gaussian BP (GBP) [10], can be applied to

produce an exact MMSE estimate with O(N)-complexity,

thus scaling the MMSE solution to large-scale system

scenarios.

In this paper, we motivate, formulate and solve the linear

system state estimation problem considered jointly with the

signal detection problem in C-RAN-based cellular networks.

We cast the problem of estimating the system state sfrom

the received vector yinto an equivalent maximum

a-posteriori (MAP) problem, and place it into the framework

of a popular class of probabilistic graphical model called

factor graphs. The state estimate ˆ

sis then derived as a

solution of the GBP algorithm applied over a speciﬁc

bi-layer structure of the factor graph. Throughout the paper,

we use state estimation in power systems with the

measurements collected via 5G-inspired C-RAN network as

a running example. Our initial numerical results demonstrate

the viability of the proposed approach, both in terms of

accuracy and convergence.

The paper is organized as follows. In Section II, we present

the joint state estimation and C-RAN uplink communication

system model. In Section III, this model is mapped into a

corresponding factor graph, and the state estimate is obtained

via GBP. Section IV provides numerical results of the proposed

GBP state estimator. The paper is concluded in Section V.

II. SY S TE M MO DE L

We consider a generic state estimation problem where a

set of state variables sis estimated from a set of observed

noisy linear measurements x. However, unlike the traditional

setup where the measurements in xare assumed available at

a central node, here we assume they are transmitted via radio

access network (RAN) of a mobile cellular system based on

a cloud-RAN (C-RAN) architecture. The received signal yis

collected at a large-number of densely deployed remote radio

heads (RRHs) and jointly processed at the C-RAN base-band

units (BBUs). The problem we consider is that of estimating

the system state sfrom the received signal y.

Linear system measurements model: We consider a

system described via the set of Nstate variables

s= (s1, s2,...,sN)T∈CN×1. The system is observed via a

set of Mmeasurements x= (x1, x2,...,xM)T∈CM×1.

Each measurement is a linear function of the state variables

additionally corrupted by the additive noise, i.e.,

xi=ai·s+ni,1≤i≤M, where

ai= (ai,1, ai,2,...,ai,N )∈C1×Nis a vector of coefﬁcients,

while ni∈Cis a complex random variable. Overall, the

system is represented via noisy linear observation model

x=A·s+n, where the matrix A∈CM×Ncontains

vectors ai,1≤i≤M, as rows, while

n= (n1, n2,...,nM)T∈CM×1is a vector of additive noise

samples. We assume noise samples niare independent

identically distributed (i.i.d.) Gaussian random variables with

variance σ2

ni. For simplicity, we assume the measurement

noise variances are equal, i.e., σ2

ni=σ2

n,1≤i≤M. In

other words, nrepresents a complex Gaussian random vector

with zero means µn=0and the covariance matrix

Σn=σ2

nI∈CM×M(Iis an identity matrix).

C-RAN uplink communication model: In the standard

state estimation models, measurements are either assumed

available, or they are communicated to the central node,

where the state estimation problem is solved.

Communication models typically involve point-to-point

communication links between the measurement devices and

the central node, affected by communication impairments

such as delays, packet losses, limited bit rates, etc. In the

cellular networks context, this assumes reservation of uplink

resources and subsequent non-orthogonal transmission,

which typically incurs signiﬁcant communication delays.

Inspired by the recent evolution of massive MTC and

ultra-dense C-RAN architectures in upcoming 5G mobile

cellular networks, in this work, we consider different

grant-less and non-orthogonal communication model, as we

detail next. Note that the following C-RAN cellular network

model could provide ultra-low latency for the state

estimation application under consideration. In other words,

such an architecture could produce the system state estimate

at the central network node with very low delay after the

measurements are acquired, which is crucial for emerging

mission-critical 5G MTC use cases [11].

In the mobile cellular system under consideration,

measurements are collected by the measurement devices that

we refer to as MTC user equipment (MTC-UE). We consider

uplink transmission of Msingle-antenna MTC UEs towards

Lsingle-antenna RRHs. We assume both MTC UEs and

RRHs are randomly and uniformly distributed across a given

geographic area (note that this placement model is somewhat

reﬁned in the numerical results section). The signal

x∈CM×1, representing the set of collected noisy

measurements, is transmitted1by MMTC UEs, while the

signal y= (y1, y2,...,yL)∈CL×1is received at LRRHs,

where y=H·x+m. The matrix H∈CL×Mrepresents

the channel matrix, where hi,j represents a complex channel

coefﬁcient between the j-th MTC UE and the i-th RRH,

while m= (m1, m2,...,mL)∈CLis a vector of additive

noise samples. As for the measurement process, for the

communication process we also assume noise samples mi

are i.i.d. zero-mean Gaussian random variables with variance

σ2

m, i.e., the mean value and the covariance matrix of mis

given as µm=0and Σm=σ2

mI∈CL×L, respectively.

Linear system state estimation problem: From the

received signal ycollected across RRHs, we are interested in

ﬁnding an estimate ˆs of the state vector s. In this paper, we

focus on the centralized C-RAN architecture, where all the

BBUs are collocated at the central C-RAN node. Thus, due

to availability of yat the central location, we consider

centralized algorithms for the state estimation problem.

However, the solution we propose in this paper is based on

GBP framework, thus it is easily adaptable to a distributed

scenario that we refer to as fog-RAN (F-RAN), where BBUs

are distributed across different geographic locations closer to

the MTC UEs. We refer the interested reader to our recent

overview of distributed algorithms for solving the state

estimation problem in the context of upcoming 5G cellular

networks [5].

A common centralized approach to solve the above state

estimation problem is to provide the corresponding minimum

mean-square error (MMSE) estimate. One can easily obtain

the MMSE estimate ˆs for the underlying linear model in the

form:

ˆs = (HA)HΣ·(HA)Σ(HA)Hy,(1)

1Note that, at this point, one can insert speciﬁc linear modulation scheme

xm=fm(x). For simplicity, we assume fm(x) = x.

C-RAN BBUs

C-RAN RRHs

Power System

Fig. 1. System model example: State estimation in Smart Grid via C-RAN-based mobile cellular network.

where (·)His the conjugate-transpose matrix operation, and

Σ=HΣnHH+Σm. However, solving (1) scales as O(N3)

which makes linear MMSE state estimation inapplicable in

large-scale systems which are of interest in this paper.

In the following, we cast the MMSE estimation into an

equivalent MAP state estimator as follows:

ˆs = arg max

s∈CNP(s|y),(2)

where P(s|y)is the posterior probability of the state safter

the signal yis observed at the RRHs. As we will demonstrate

in the sequel, if certain sparsity arguments are applicable in the

system model under consideration, the solution of the MAP

problem can be efﬁciently calculated using the framework of

factor graphs and belief-propagation (BP) algorithms.

System model example (Smart Grid): Before

continuing, it is useful to consider an example of the above

state estimation setup. We consider the state estimation

problem in an electric power system, where the goal is to

estimate the state of the power system s, containing complex

voltages of Nsystem buses, via the set of measurements x

obtained using measurement devices. Measurement devices

are geographically distributed across the power system and

we assume they are equipped with wireless cellular

interfaces, i.e., they represent MTC UEs connected to the

C-RAN based cellular network. The signal yreceived from

the set of RRHs densely deployed across the cellular

network coverage area is processed centrally within the

C-RAN system architecture. Fig. 1 illustrates the smart grid

example that we will further reﬁne in Section IV and use as

a running example throughout this paper.

III. STATE EST IM ATI ON V IA GAUS SIA N BELIEF

PROPAG ATION

In this section, we provide a solution to the combined state

estimation and uplink signal detection problem deﬁned in (2),

by applying factor graphs and GBP framework. In fact, for

both constituent scenarios: the conventional state estimation

(in case of power systems) and the uplink signal detection in

C-RAN, the GBP has already been proposed and analyzed (see

details in [9] and [7]). Thus in this work, we propose using

GBP in a joint and combined setup of extracting the system

state directly from the observed C-RAN signals.

We note that the various properties of the proposed GBP

approach (e.g., complexity, convergence, etc.) will strongly

depend on the structure of the underlying factor graph. For

example, in terms of complexity (we will come back to

convergence in the next section), for GBP to scale well to

large-scale systems, it is fundamental that both matrices A

and Hdeﬁning the two linear problems are sparse, i.e., that

for both Aand H, the number of non-zero entries scales as

O(N). In many real-world scenarios, the sparsity typically

arises from geographic constraints and reﬂect locality that is

typically present in both the measurement and the

communication part of the system model. More detailed

account on the sparsity of matrices Aand Hclearly depends

on the speciﬁc scenario under consideration, and we relegate

these details to Section IV where we will explicitly deal with

the smart grid example introduced earlier.

Factor Graph System Representation: The MAP problem

under consideration can be rewritten as follows:

ˆs = arg max

s∈CNP(s|y)∝arg max

s∈CNP(s,y) = (3)

= arg max

s∈CN,x∈CMP(s,x,y)(4)

Assuming that the system state shas a Gaussian prior, and

given that the measurement and communication noise is

assumed Gaussian, the distribution P(s,x,y)is jointly

Gaussian. In addition, due to the problem structure where s

and yare conditionally independent given x, we obtain:

ˆs = arg max

s∈CN,x∈CMP(y|s,x)P(s,x) = (5)

= arg max

s∈CN,x∈CMP(y|x)P(x|s)P(s).(6)

As noted before, in many real-world systems of interest, a

measurement xjis a linear function of a small subset of local

state variables sN(xj), where N(xj)is the index set of the

state variables that affect xj, and sN(xj)={si|i∈ N (xj)}.

In other words, the row-vector ajhas non-zero components

only on a small number of positions indexed by the set N(xj),

thus making the matrix Asparse2. Using this fact and the fact

that the measurements xjare mutually independent, we obtain:

P(x|s) =

M

Y

j=1

P(xj|sN(xj)).(7)

On the other hand, in the C-RAN communication part,

although in theory the received signal yidepends on all the

transmitted symbols in x, the channel coefﬁcients between a

RRH and a geographically distant MTC UE can be

considered negligible, thus leading to matrix Hsparsiﬁcation

[7]. Upon distance-based sparsiﬁcation proposed in [7], the

received symbol yidepends only on a small number of

symbols xN(yi), where N(yi)is the index set of symbols

transmitted by the set of MTC UEs in geographic proximity

of the i-th RRH. Taking the channel sparsiﬁcation into

account, we obtain:

P(y|x) =

L

Y

i=1

P(yi|xN(yi)).(8)

Finally, assuming that the state vector is apriori given as

a set of i.i.d Gaussian random variables, we obtain the ﬁnal

factorized form of the initial MAP problem:

ˆs = arg max

s∈CN,x∈CM

L

Y

i=1

P(yi|xN(yi))·

M

Y

j=1

P(xj|sN(xj))·

N

Y

k=1

P(sk).(9)

2More precisely, the number of state variables that affect certain

measurement is limited by a constant, independently of the size Nof the

system.

The factor graph representation of the MAP problem

follows the factorization presented in (9) and is illustrated in

Fig. 2. Factor graph G=G(V ∪ F,E)is a bipartite graph

consisting of the set of variable nodes V, the set of factor

nodes F, and the set of edges E. In our setup, the set Vcan

be further divided as V=S ∪ X ∪ Y ,where

S={s1, s2,...,sN}is the set of state nodes,

X={x1, x2,...,sM}is the set of measurement nodes,

while Y={y1, y2,...,yL}is the set of received symbol

nodes. The set of factor nodes can be divided as

F=FH∪ FA∪ Fy∪ Fx∪ Fs,where

FH={fh1, fh2,...,fhL}and FA={fa1, fa2,...,faM}

represent factor nodes that capture linear relationships

between variable nodes described by the rows of matrices H

and A, respectively. In addition, Fy={fy1, fy2,...,fyL}

and Fs={fs1, fs2,...,fsN}represent the factor nodes that

provide inputs due to observations of yand the prior

knowledge about x, respectively, while

Fx={fx1, fx2,...,fxM}serve as virtual inputs needed for

initialization of measurement nodes. Similarly, the set of

edges Ecan be divided as E=EH∪ EA∪ Ey∪ Ex∪ Es,

where EH∪ Ey∪ Exand EA∪ Es∪ Excan be considered as

the set of edges of two bipartite subgraphs

GH= (Y ∪ X ∪ FH∪ Fy∪ Fx,EH∪ Ey∪ Ex)and

GA= (X ∪ S ∪ FA∪ Fs∪ Fx,EA∪ Es∪ Ex), obtained as

the subgraphs of Ginduced from the set of factor nodes

FH∪ Fy∪ Fxand FA∪ Fs∪ Fx, respectively3.

.

.

.

y

F

Y

x

F

A

F

H

F

S

X

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Fig. 2. Factor graph representation of the system model.

As noted earlier, the state estimation problem using GBP

over factor graph GA, and the uplink C-RAN signal detection

problem using GBP over factor graph GH, have been recently

investigated in detail in [9] and [7], respectively.

Gaussian Belief-Propagation and GBP Messages: To

estimate the state variables s, we apply message-passing

GBP algorithm [10]. GBP operates on the factor graph Gby

exchanging messages between factor nodes and variable

3As noted in footnote 1, if the signal xis modulated prior to transmission,

one can easily add an additional “layer” to the factor graph in Fig. 2 containing

a set of Mmodulated signal variable nodes Xmconnected via modulation

factor nodes Fmwith the corresponding measurement nodes X.

nodes in both directions. As a general rule, at any variable or

factor node, an outgoing message on any edge is obtained as

a function of incoming messages from all other edges, using

the message calculation rules presented below. In general, the

underlying factor graph describing joint state estimation and

signal detection problem (Fig. 2) will contain cycles, thus the

resulting GBP will be iterative, which means that all nodes

will iteratively repeat message updates on all of the outgoing

edges according to a given message-passing schedule. We

provide details on message-passing schedule, correctness and

convergence of GBP on loopy graphs later in this section.

Let us consider a variable node vi∈ V incident to a factor

node fj∈ F. Let N(vi)denote the index set of factor nodes

incident to vi, and N(fj)denote the index set of variable

nodes incident to fj. We denote messages from vito fjand

from fjto vias µvi→fj(vi) = (mvi→fj, σ2

vi→fj)and

µfj→vi(vi) = (mfj→vi, σ2

fj→vi), respectively. Note that, in

the GBP scenario, all messages exchanged across the factor

graph represent Gaussian distributions deﬁned by the

corresponding mean-variance pairs (m, σ2). Thus to describe

processing rules in a variable and a factor node, it is

sufﬁcient to provide equations that map input (m, σ2)-pairs

into the output(m, σ2)-pair, as detailed below.

Message from a variable node to a factor node: the

equations below are used to calculate

µvi→fj(vi) = (mvi→fj, σ2

vi→fj):

mvi→fj= X

k∈N (vi)\j

mfk→vi

σ2

fk→vi!σ2

vi→fj(10a)

1

σ2

vi→fj

=X

k∈N (vi)\j

1

σ2

fk→vi

.(10b)

Message from a factor node to a variable node: In the

setup under consideration, factor nodes represent linear

relations between variable nodes. Thus, e.g., for a factor

node fj, we can write the corresponding linear relationship

as:

fj(vN(fj)) = Civi+X

k∈N (fj)\i

Ckvk.(11)

With this general notation, the equations below provide

µfj→vi(vi) = (mfj→vi, σ2

fj→vi):

mfj→vi=1

Ci X

k∈N (fj)\i

Ckmvk→fj!(12a)

σ2

fj→vi=1

C2

i X

k∈N (fj)\i

C2

kσ2

vk→fj!.(12b)

Calculation of marginals: Applying the above rules in

variable and factor nodes of the factor graph results in the

sequence of updates of messages exchanged across the edges

of the graph. To complete description of loopy GBP, we

need to deﬁne message initialization at the start, and

message scheduling during the course of each iteration,

which is done next. After sufﬁcient number of GBP

iterations, the ﬁnal marginal distributions of the random

variables corresponding to variable nodes is obtained as:

ˆmvi= X

k∈N (vi)

mfk→vi

σ2

fk→vi!σ2

vi(13a)

1

ˆσ2

vi

=X

k∈N (vi)

1

σ2

fk→vi

.(13b)

GBP Message-Passing Schedule, Correctness and

Convergence: We adopt standard synchronous GBP

schedule in which variable node processing is done in the

ﬁrst half-iteration, followed by the factor node processing in

the second half-iteration. The iterations are initialized by

input messages from Fygenerated from the received signal

y, and initial messages from Fxand Fsthat follow certain

prior knowledge (as detailed in the next section).

GBP performance on linear models deﬁned by loopy

factor graphs is fairly well understood. For example, if the

GBP converges, it is known that the GBP solution will

match the solution of the MMSE estimator. The convergence

criteria can also be derived in a straightforward manner, by

deriving recursive ﬁxed point linear transformations that

govern mean value and variance updates through the

iterations and investigating spectral radius of such

transformations. Due to space restrictions, we leave the

details of the convergence analysis in our scenario for the

future work.

IV. NUM ERI CA L CASE ST UDY: SMA RT GRID STATE

EST IM ATI ON IN 5G C- RAN

In this section, we specialize our state estimation setup for a

case study in which we perform power system state estimation

by collecting measurements via 5G-inspired C-RAN.

Power system state estimation - DC model: For the sake

of simplicity, in the following, we consider the linear DC

model of a power system. The DC model is an approximate

model obtained as a linear approximation of the non-linear

AC model that precisely follows the electrical physical laws

of the power system. In the DC model, the power system

containing Nbuses is described by Nstate variables

s= (s1, s2,...,sN)T, where each state variable si=θi

represents the voltage angle θi(in the DC model, the

magnitudes of all voltage phasors are assumed to have unit

values). In the DC model, the measurements include only

active power ﬂow Prk at the branch (r, k)between the bus r

and the bus k, active power injection Printo the bus r, and

the voltage angle θr. Collecting Mof such arbitrary

measurements across the power system, we obtain the

measurement vector x= (x1, x2,...,xM)T, where each

measurement xi∈ {Prk , Pr, θr}is a linear function4of the

(sub)set of state variables s, additionally corrupted by

additive Gaussian noise of ﬁxed (normalized) noise standard

deviation of σnper unit (p.u.). The noisy measurements x

are then transmitted via C-RAN network as described below.

4More precisely, we have that Prk =−brk (θr−θk)and Pr=

−Pk∈Nrbrk (θr−θk), where Nris the set of adjacent buses of the bus

rand brk is susceptance of the branch (r, k).

We illustrate the methodology using the IEEE test bus case

with 30 buses shown in Fig. 3 (N= 29, since one of the bus

voltage angles is set to the reference value zero) that we use

in the simulations. The example set of Mmeasurements is

selected in such a way that the system is observable with the

redundancy M/N. For each simulation scenario, we generate

1000 random (observable) measurement conﬁgurations.

10 21

22

24

25

26

27

28

29

30

6

20

17

9

11

15 18 19

23

13 12

14

16

8

1 3 4

5

7

2

Fig. 3. The IEEE 30 bus test case divided into disjoint sub-rectangles.

C-RAN cellular network model: The set of M

MTC-UEs simultaneously transmit their measurements to the

set of LRRHs during a given allocated time-frequency slot

shared by all MTC-UEs. We assume MTC-UEs and RRHs

are placed uniformly at random following independent

Poisson Point Process (PPP) in a unit-square area, however,

with slight reﬁnement of the PPP placement strategy.

Namely, to account for neighboring relations within logical

topology of IEEE 30 bus test case, we ﬁrst divide a

unit-square into w×qdisjoint sub-rectangles as shown in

Fig. 3, and then we assign MMTC-UEs to one of w·q

sub-rectangles. We also balance the number of RRHs per

sub-rectangle, thus allocating ∼L/(w·q)RRHs per

sub-rectangle. Finally, all RRHs and MTC-UEs allocated to

a given sub-rectangle are placed using the PPP within a

given sub-rectangle.

After the placement, we assume MMTC-UEs transmit

their signals x, where each measured signal is normalized to

its expected normalization value5. For the channel

coefﬁcients between the MTC UEs and RRHs, we assume

the following model:

hi,j =γi,j d−α

i,j (14)

where γi,j is the i.i.d. Rayleigh fading coefﬁcient with zero

mean and unit variance, di,j is distance between i-th

MTC-UE and j-th RRH and αis the path loss exponent. We

5We assume normalization constants are known in advance at MTC-UEs

and C-RAN nodes, either as a prior knowledge or by long-term averaging.

use channel sparsiﬁcation approach proposed in [7], with

threshold distance set to d0=pw2+q2(i.e., equal to the

diagonal length of each sub-rectangle). The received signal

y= (y1, y2,...,yL)collected at LRRHs is additionally

corrupted by additive Gaussian noise, whose standard

deviation is selected so as to provide ﬁxed and pre-deﬁned

signal-to-noise ratio (SNR) value. Finally, noisy received

signal yis forwarded via high-throughput backhaul links to

C-RAN BBUs.

GBP-based State Estimation: Using the approach

presented in Section III, we apply GBP across the factor

graph illustrated in Fig. 2 to recover the state estimate x

from the received signal y. More precisely, for each random

measurement conﬁguration, we generate the part of the

factor graph GAand, similarly, from known MTC-UE and

RRH random positions, we derive6the part of the factor

graph GH. Upon reception of y, the GBP runs until it

converges. We adopt a synchronous scheduling of GBP

messages where messages are synchronously ﬂooded from

the factor nodes to variable nodes and back within a single

GBP iteration. For a linear model, it is well known that if

the GBP converges, it will converge to the minimum

mean-square error (MMSE) estimate of the state x.

Simulation Results: In the ﬁrst set of experiments, we ﬁx

the relative RRH density L/M = 1,SNR = 10, and

redundancy M/N = 3. We investigate the accuracy of the

GBP solution of state estimate as a function of different

values of measurements noise σn={10−1,10−2,10−3,

10−4}.

10−110−210−310−4

0.00

2.00

4.00

6.00

8.00

Measurement noise σn(p.u.)

RMSE

Fig. 4. The root mean square error of estimate vector of power system state

variables obtained by C-RAN and without C-RAN model.

Fig. 4 shows the root mean square error

RMSE = (1/N)||ˆsc−ˆs¯c ||2, where ˆscand ˆs¯c are estimate

vectors of power system state variables obtained with and

without C-RAN model discussed in this paper, respectively,

for different values of measurement noise σn. For the case

without C-RAN model, we assume measurements xare

available at BBUs as they are, i.e., without additional noise

or errors. In practice, this could be obtained via standard

grant-based uplink procedures where each MTC UE is

6We note that, in case the small-scale fading is included in the model, one

can assume that the channel state information is available at the C-RAN.

allocated separate orthogonal resources. However, such a

strategy incurs signiﬁcant delay as the underlying system

scales, due to message exchange delay, resource allocation

delay, as well as ARQ-based error-correction strategies. Note

that the C-RAN model described in this paper admits very

low latency as all MTC UEs transmit their signals

immediately and concurrently. According to the box plot in

Fig. 4, the C-RAN approach is able to reach nearly identical

solution as the approach without C-RAN (e.g., RMSE →0),

if the value of measurements noise is sufﬁciently low. Note

that the typical value (standard deviation) of the

measurement noise for devices located across a power

system are in the range between 10−2p.u.and 10−3p.u., for

legacy measurement devices, and between 10−4p.u.and

10−5p.u., for phasor measurement units. Consequently, the

presented approach is suitable for the state estimation in

power systems.

In the next simulation experiment, we investigated the

system observability as a function of the number of RRHs L

deployed in the system, for different values of redundancy

M/N. We start with L/M = 0.2and increase the RRH

density in order to evaluate its effect on the system

observability.

0.2 0.4 0.6 0.8 1 1.2

0

50

100

Relative RRH density L/M

Unobservable Topologies (%)

M/N

1.5

2.0

2.5

3.0

3.4

Fig. 5. The fraction of unobservable system topologies for different values

of measurement redundancies M/N versus relative RRH density L/M.

Fig. 5 shows the fraction of instances GBP was not able

to converge due to insufﬁcient rank of the underlying system

as a function of the number of base stations L. Note that the

fundamental condition for the system to have full rank is

that L≥N. By slightly expanding this condition, we get

(L/M)·(M/N)≥1. For all the points in Fig. 5 for which

this condition is not satisﬁed, the system is unobservable. If

the condition is satisﬁed, then in each simulation run, a

random measurement conﬁguration is veriﬁed to provide an

observable system, thus the rank insufﬁciency may only

appear as a consequence of the C-RAN topology and the

channel matrix sparsiﬁcation. According to Fig. 5, for the

parameters used in our simulations, we can see that GBP

generally performs well, however, in the region where

(L/M)·(M/N)is slightly above 1, rank insufﬁciency may

deteriorate the performance.

Overall, for systems with large number of MTC UEs M,

the state can be estimated with relatively small number of

RRHs L. In contrast, for the scenario where a number of

MTC UEs is small, the number of RRHs must be increased

for successful reconstruction. In addition, simulation results

point to capability of the proposed scheme to provide

successful reconstruction if the underlying system is

observable (i.e., of full rank), while the accuracy of

reconstruction (i.e., the accuracy of the state estimator) will

depend on the parameters such as channel sparsiﬁcation,

SNR, measurement standard deviations and number of MTC

UEs and RRHs. We leave detailed study of these

inter-dependencies for our future work.

V. CO NC LUS IO N S

Motivated by the development of 5G massive MTC and

large-scale distributed 5G C-RAN architecture, in this paper,

we proposed a scalable and efﬁcient linear state estimation

framework. The proposed framework is based on the GBP

algorithm and jointly combines linear state estimation with

signal detection in 5G C-RANs. The advantage of GBP

solution is accuracy that matches the MMSE estimation, low

complexity due to lack of scheduling MTC-UE

transmissions, low latency due to simultaneous data transfer,

scalability to large-scale systems (due to the fact that the

underlying factor graph is usually sparse), and ease of

parallelization and distributed implementation in future

distributed F-RAN architectures. For the future work, we aim

to provide rigorous convergence analysis of GBP in the

presented framework, motivated by similar analysis in [7]

and [9], and provide extensive numerical simulation study.

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