Typical behaviour of relays in communication channels
Alamino, R.C., Saad, D.
Neural Computing Research Group, Aston University, Birmingham, United Kingdom
The typical behaviour of the relay-without-delay channel under LDPC coding and its multiple-
unit generalisation, termed the relay array, is studied using methods of statistical mechanics. A
demodulate-and-forward strategy is analytically solved using the replica symmetric ansatz which is
exact in the system studied at Nishimori’s temperature. In particular, the typical level of improve-
ment in communication performance by relaying messages is shown in the case of small and large
number of relay units.
PACS numbers: 02.50.-r, 02.70.-c, 89.20.-a
Keywords: statistical physics, replica theory, relay channel, LDPC codes
I. INTRODUCTION
Methods of statistical mechanics have recently become
increasingly more important in the study of communi-
cation channels. The development of the replica and
cavity methods for analysing disordered systems [1, 2]
and the related recent introduction of systematic rigor-
ous bounds [3, 4] made new theoretical tools available for
their analysis.
More specifically, the replica method has been applied
to a wide range of problems in information theory, from
error correcting codes [5, 6] to multi-user communica-
tion [7]. It facilitates the derivation of practical and
theoretical limits in various communication channels and
provides typical results in cases that are difficult to tackle
via traditional methods of information theory.
The growing use of information networks, both physi-
cally connected and wireless, and the increasing number
of services taking place in the Internet, have made the
study of multi-user communication highly attractive and
relevant from a practical point of view, in addition to
being a challenging and exciting field for theoretical re-
search.
Up to date, there is no generalised theory for multi-
user channels within the framework of information theory
and analytical results are only known for special cases.
The main difficulty being that multi-user networks do not
admit the source-channel separation principle, which al-
lows one to separate the information transmission process
into the two successive steps of source coding (compres-
sion) and channel coding (error-correction); this principle
plays an essential role in the information theoretic anal-
ysis of communication channels. In spite of their incom-
plete theoretical foundations, multi-user communication
networks play an important role in a variety of commu-
nication devices ranging from mobile phones to comput-
ers. We strongly believe that a statistical physics-based
analysis may offer answers where the current information
theory methodology fails, especially in the limit of a large
number of users.
With the technological demand and the possibility of
providing a principled analysis by the methods of statisti-
cal mechanics, early results for multi-user communication
are being revisited and analysed from different and com-
plementary points of view, resulting in new insights and
developments [7, 8]. One of the more interesting and rel-
evant communication channels is the relay channel [9].
The generic relay channel is characterised by an auxil-
iary user between transmitter and receiver, which assists
in the transmission of the message. Due to the increase
in the number of multi-user networks, such as mobile
phones and computer networks, the transfer of informa-
tion with the help of relays has become an attractive op-
tion. As these networks are becoming more distributed,
the assisted transmission supported by arrays of relays
has become feasible and merits further analytical explo-
ration.
This paper is organised as follows. In section II we
define the general relay array and introduce as particular
cases the classical relay channel and the recently investi-
gated relay-without-delay. In section III we outline the
statistical physics methods used to analyse the problem
which will be based on a replica approach detailed in
section IV. Section V contains our conclusions and final
comments.
II. MODEL
A. LDPC Codes
Low-Density Parity-Check (LDPC) codes [10] are
state-of-the-art error-correcting codes with performance
that is second to none, especially within the high code
rate regime. In the notation we will be using here, N -
dimensional messages s are encoded into M -dimensional
codewords t. LDPC codes are defined by a binary
parity-check matrix A = [C1|C2], concatenating side-
by-side two very sparse matrices known to both sender
and receiver: C2 that is invertible and of dimensionality
(M−N)×(M−N) andC1 of dimensionality (M−N)×N .
The matrix A can be either random or regular, charac-
terised by the number of non-zero elements per row (K)
and column (C). Irregular codes show superior perfor-
mance to regular structures [11, 12] if constructed care-
fully. In order to simplify our treatment, we focus here

2on regular constructions; the generalisation to irregular
codes is straightforward [13, 14].
Encoding refers to the linear mapping of a N -
dimensional original message s ∈ {0, 1}N to a M -
dimensional codeword t ∈ {0, 1}M (M > N)
t = Gs (mod 2) , (1)
where all operations are performed in the field {0, 1},
indicated by (mod 2), and the M ×N generator matrix
is
G =
(
I
C−12 C1
)
(mod 2) , (2)
where I is the N × N identity matrix. By construction
AG = 0 (mod 2) and the first N bits of t correspond to
the original message s.
Decoding is carried out by estimating the most proba-
ble transmitted vector from the received corrupted code-
word [6, 13]. For mathematical convenience, in the
present work we map the Boolean variable t ∈ {0, 1}M
into a spin variable t ∈ {1,−1}M by the transformation
x → (−1)x.
B. Relay Array
The relay array is a multiple-units generalisation of
the (single unit) relay channel of [9]. Since the single
relay is a special case of this general framework, we will
first explain the principles of relay-based communication
using the more general scenario.
The LDPC codeword t is transmitted to each one of L
relay units through noisy channels, corrupted by a global
Additive White Gaussian Noise (AWGN) ν0 and by local
independent AWGNs νi, both of zero mean and variances
σ20 and σ2i , respectively. Each relay processes the received
corrupted message ri and encodes the acquired informa-
tion into a vector ti which is then transmitted to a final
receiver. The final receiver receives an algebraic sum-
mation of the relay outputs plus a direct transmission
from the original sender, corrupted also by ν0, subject
to a final AWGN ν of zero mean and variance σ2. The
exact form of the channel is depicted in Fig. 1 and the
corresponding equations are
r = at +
L
∑
i=1
biti + ν + ν0, (3)
ri = cit + νi + ν0. (4)
The variables a, bi and ci (i = 1, ..., L) are the relative
gains of each transmission and can be random or set to
constant values. The power from the original source to
each relay is c2i , to the final receiver is a2 and the power
from each relay to the final receiver is b2i . For simplicity,
as well as for comparison with results reported in the
literature, we will mostly consider the case of unit relative
gain parameters.
ν1
νL
ν0
t r
t L
1t1r
Lr
...
ν
FIG. 1: The L-component relay array. The transmitter sends
a codeword t to the final receiver and to each of the L relays.
Each relay receives a message ri which is a corrupted ver-
sion of the original codeword subject to the AWGNs ν0 and
νi. It then sends to the final receiver the encoded vector ti.
The final receiver receives the original transmitted codeword
summed with all the relayed messages ti and corrupted by
the AWGN vectors ν0 and ν .
When L = 1, we refer to the channel simply as the re-
lay channel. In the classical relay channel (CRC), stud-
ied by Cover and El-Gamal [15], the messages sent by
the relays to the final receiver are only allowed to de-
pend on the set of symbols received by each of the relays
before the current time step, tµi = f(t1, ..., tµ−1), which
corresponds to the fact that it takes the relay some time
to process the information before relaying it. However,
if the time delay in the direct transmission to the final
receiver is much longer than in the transmission to the
relay units, one can allow the message sent by the relay to
depend on the present received symbol as well such that
tµi = f(t1, ..., tµ). This last case, termed relay-without-
delay (RWD), created significant interest recently and
was studied by El-Gamal and Hassanpour [16]. For the
case of a relay array where all communication is carried
out through the relays and there is no direct transmis-
sion to the final receiver, the restriction of the CRC, to
consider all but the last received symbol, is unnecessary.
The most studied strategies used by the relay units
are the Amplify-and-Forward (A&F) and the Decode-
and-Forward (D&F) strategies. In A&F, which we will
only mention briefly in this paper, the relay just retrans-
mits its received vector after possible amplification, e.g.,
ti = ri. In D&F, the relays decode the message and
transmit their estimates to the final receiver. The math-
ematical treatment of both strategies will be discussed in
the corresponding sections.
III. STATISTICAL PHYSICS OF DECODING
We transform the decoding problem of the final re-
ceiver into a statistical physics system by defining a dy-
namical variable τ ≡ (τ1, ..., τM ), which represents the
candidate variable vectors at the receiver. Each τµ plays
a role equivalent to a spin located in the µ-th site of an
M -site lattice.

3The final receiver generates an estimate tˆ of the orig-
inal codeword using the Marginal Posterior Maximiser
(MPM) estimator
tˆµ = sgn 〈τµ〉P(τ |r), (5)
which minimises the probability of bit error [13, 17].
Other estimators can be used depending on the error
measure considered. For example, minimisation of block
error is obtained using the Maximum a Posteriori (MAP)
estimator tˆ = maxτ P(τ |r).
The posterior probability density is calculated by
Bayes’ rule as
P(τ |r) = P(r|τ )P(τ )P(r) , (6)
with
P(r) =
∑
t
P(r|t)P(t)
=
∑
t,{ti},{ri}
P(t)P(r|t, {ti})
∏
i
P(ti|ri)P(ri|t).
(7)
One of the basic quantities of interest is the overlap
between the codeword and the decoded message. Our
analysis, focuses on the typical behaviour of the decod-
ing process and, accordingly, we take averages over all
possible codewords, all received messages and all allowed
encodings, which we consider as quenched disorder in the
corresponding physical system. The overlap between de-
coded and original messages takes the form
d = 1M
M
∑
µ=1
〈
tµ sgn 〈τµ〉P(τ |r)
〉
A,r,t
. (8)
This quantity can be derived from the free-energy
f = − lim
M→∞
1
βM 〈lnZ〉A,r,t, (9)
with the partition function
Z =
∑
t
e−βH(t;r), (10)
and the corresponding Hamiltonian
H(t; r) = − lnP(r|t)P(t). (11)
Usually we disregard the normalisation of the distri-
butions within the Hamiltonian as they merely add con-
stants that shift the zero energy. In the case of LDPC
codes, P(t) turns out to be a constraint on the summa-
tion variables.
In the above Hamiltonian, the parity-check matrix A
defines an interaction between the τ variables while t and
r induce local fields at the corresponding sites. The in-
verse temperature β is the ratio between the true and
the decoder’s assumed noise level. In our numerical cal-
culations, we adopt β = 1, also known as Nishimori’s
temperature, which means that the decoder assumes the
correct noise level for the channel. It can be shown that
at Nishimori’s temperature the system never enters the
glassy phase [2, 18] and the thermodynamically dominant
solution is always Replica Symmetric (RS); we therefore
restrict our analysis to the RS treatment.
One of the important properties and the novelty in
the statistical physics formulation of the problem is that
looking at the problem as a dynamical spin system, one
can interpret the results in terms of phase transitions,
which are directly related to the overlap between the orig-
inal and estimated message and the entropy function of
the obtained solutions. Combining this extra informa-
tion we can have a better understanding of the way the
system changes from a phase of perfect decoding (termed
the ferromagnetic phase) to a phase where the message
is recovered only up to a certain amount of error (the
paramagnetic phase).
As the replica treatment of A&F turns out to be the
same as for the simple Gaussian channel with a modified
power and noise level, the solution is obtained straight-
forwardly by applying the results of [13] and will not be
studied here.
Full use of LDPC decoding in the relays in the D&F
strategy is made when each relay decodes the received
vector ri by the MPM estimator using the fact that the
codeword was encoded by an LDPC code. The message
transmitted to the final receiver by each relay would then
be
tµi = sgn 〈τµi 〉P(τ i|ri). (12)
In equation (6) this is equivalent to setting
P(ti|ri) =
M
∏
µ=1
δ
(
tµi − sgn 〈τµi 〉P(τ i|ri)
)
. (13)
As tµi ∈ {±1}, we can rewrite this probability density
as
P(ti|ri) =
M
∏
µ=1
θ
(
〈tµi τµi 〉P(τ i|ri)
)
, (14)
where θ is the Heaviside step function.
The replica treatment of the LDPC D&F turns out
to be extremely involved due to the introduction of a
θ function with an average over the variables τ i inside
it, which includes a term dependant on the parity-check
matrix. Analytical studies of this rather difficult case are
under way.
In the present work we focus on a simplification of
this strategy, also known in the literature as Demodulate-
and-Forward. In it, the relays do not have the complete
information about the encoding mechanism and therefore
assume a uniform prior for the transmitted codeword. In

4this case, the posterior distribution of the bits in the
message for the relay is
P(ti|ri) =
M
∏
µ=1
1
1 + exp [−2tµi rµi /(σ2i + σ20)]
, (15)
and it is straightforward to show that the MPM estimator
is given simply by
tµi = sgn (r
µ
i ). (16)
The fact that the disorder with respect to the selected
code does not appear in the estimate of the relays makes
the replica calculations feasible in this case, as follows.
IV. REPLICA SYMMETRIC ANALYSIS
As the RS analysis of LDPC coding systems has been
introduced and carried out in a number of publications
(e.g. [13]) we will omit the detailed derivation and con-
centrate on the final expressions. The derivation follows
exactly the same steps as in [19] where quenched aver-
ages over all possible parity-check matrices are first car-
ried out, followed by the RS assumption which enables
the representation of the order parameters in the form of
field distributions (see also [20]) with each order parame-
ter containing m replicated dynamical variables τi being
written as
q(τia1 , ..., τiam) =
∫ ∞
−∞
dxpi(x)xm−∆ ≡
〈
xm−∆
〉
x, (17)
qˆ(τia1 , ..., τiam) =
∫ ∞
−∞
dxˆ pˆi(xˆ) xˆm−∆ ≡
〈
xˆm−∆
〉
xˆ, (18)
where ∆ is 1 if the zeroth replica is included and 0 oth-
erwise. The field distributions pi and pˆi act as generators
of the order parameters in the replica symmetric analysis
of diluted systems.
A set of self-consistent equations is obtained by the
saddle point method in the thermodynamic limit where
the extremisations are made with respect to the field dis-
tributions pi and pˆi resulting in
pˆi(xˆ) =
〈
δ
(
xˆ−
K−1
∏
m=1
xm
)〉
x
, (19)
pi(x) =
〈
δ
(
x−
∑
τ τ [Ψ(τ, r)]
β∏C−1
l=1
(
1 + τxˆl
)
∑
τ [Ψ(τ, r)]
β∏C−1
l=1 (1 + τxˆl)
)〉
r,xˆ
,
where
Ψ(τ, r) ≡
∫
{ L
∏
i=1
dri exp
[
− (ri − ciτ)
2
2(σ2i + σ20)
]}
× exp

− 12(σ2 + σ20)
(
r − aτ −
∑
i
bi sgn ri
)2
.
(20)
The expression 〈ti〉 is the mean of the variable ti and
P(r) ∝ Ψ(1, r). The overlap is
d = 〈sgnu〉u, with (21)
P(u) =
〈
δ
(
u−
∑
τ τ [Ψ(τ, r)]
β∏C
l=1
(
1 + τxˆl
)
∑
τ [Ψ(τ, r)]
β∏C
l=1 (1 + τxˆl)
)〉
r,xˆ
,
(22)
the free energy is given by
βf = CK ln 2 + C〈ln(1 + xxˆ)〉x,xˆ
− CK
〈
ln
(
1 +
K
∏
m=1
xm
)〉
x
−
〈
ln
{
∑
τ
[Ψ(τ, r)]β
C
∏
l=1
(
1 + τxˆl
)
}〉
xˆ,r
, (23)
and the internal energy, the derivative with respect to β
of the above equation becomes
u = −
〈
∑
τ Ψβ(lnΨ)
∏C
l=1
(
1 + τxˆl
)
∑
τ Ψβ
∏C
l=1 (1 + τxˆl)
〉
xˆ,r
. (24)
For any number L of relays, the results can be obtained
by a numerical solution of the equations (19). Note the
summation over the internal variables, i.e., the messages
received and sent by the relays. This comes from the
Bayesian formulation of the problem where the final re-
ceiver has access just to r and, therefore, one must inte-
grate over all unknown variables.
We also note that the above equations are fairly gen-
eral. Using the appropriate function Ψ one can recover all
previous results for single user channels and apply them
to more general channels when the intermediate process-
ing of the message does not involve any knowledge of the
parity-check matrices.
The ferromagnetic state, which corresponds to perfect
decoding, is given by the following solution to the saddle
point equations (19)
pˆi(xˆ) = δ(xˆ− 1), pi(x) = δ(x− 1). (25)
Substitution of these distributions in equation (21)
gives d = 1. By substituting the ferromagnetic solution
into the formulas for the free and internal energies, we
obtain (at Nishimori’s temperature)
u = f = −〈lnΨ(1, r)〉r, (26)
meaning that the entropy of this phase is zero.
The Hamiltonian of the relay array is gauge invariant
with respect to the gauge transformation
rµ → γµrµ,
tµ → γµtµ, (27)

5where the vector γ obeys the parity-check constraints.
We can verify that the transition probabilities P(rµ|tµ)
are also invariant under this gauge transformation. Note
that if a channel is symmetric, i.e., exhibits a similar
probability for cross-symbol error flips (for a detailed def-
inition see [21]), it is automatically gauge invariant under
the above transformation. For gauge invariant channels
the internal energy is
U = 〈H(τ ; r)〉τ ,r,t
=
∑
τ ,t
∫
drP(τ |r, β)P(r|t)P(t)H(τ ; r), (28)
where
P(τ |r, β) ∝ e−βH, (29)
is the thermal Gibbs probability at inverse temperature β
which obeys P(τ |r, β = 1) = P(τ |r). Since under such a
gauge transformation the Hamiltonian remains invariant,
we have H(t; r) = H(1; tr), where tr ≡ (t1r1, ..., tMrM )
and 1 is an M -dimensional vector with all entries equal
to 1. Therefore, one can write the following expression
for the internal energy
U =
∑
τ ,t
∫
dr P(r|τ , β)P(τ |β)P(r|β) P(r|t)P(t)H(1; τr).
(30)
Gauging the variables τr → r, reorganising the terms
and taking β = 1, we finally get
U =
∫
drP(r|1)H(1; r). (31)
The meaning of this is that, for a gauge invariant chan-
nel of the type described above (which includes general
symmetric channels), the internal energy is independent
of the configuration. In special cases, as can be found
in [2], the gauge symmetry allows for an analytical ex-
pression to be found. The same method can be used
to prove that the probability distribution for the mag-
netisation is equal to the probability distribution for the
two-point correlations in Nishimori’s temperature, which
indicates the absence of a spin glass phase and no replica
symmetry breaking.
A. Relay Channel
In order to compare our results with those of [16], we
analyse the RWD for the setup sketched in Fig. 2 with
σ21 = ησ2, a = b1 = 1 and c1 = (1 + σ2)−1/2. The
corresponding function Ψ is then given by
Ψ(τ, r) = e−(r−τ−1)2/2σ2 erfc
(
− τ√
2ησ2
)
+ e−(r−τ+1)2/2σ2 erfc
(
+ τ√
2ησ2
)
, (32)
ν1
t r
1t1r
c1 1b
νa
FIG. 2: Schematic drawing of the relay-without-delay (RWD)
setup to be analysed.
where η is an arbitrary positive constant and erfc (x) is
the complementary error function
erfc (x) = 2√pi
∫ ∞
x
e−y2 dy. (33)
For these values of noise and gains, the capacity of this
channel as derived in [16] is
C = 12 log2
(
1 + 1 + c
2
1
σ2
)
. (34)
The numerical results for the overlap between the re-
trieved and the original codewords, obtained by solving
recursively equations (19), are given in Fig. 3 for K = 4,
C = 3, β = 1 and η = 0.1. Shannon’s limit, marking the
noise level below which error-free communication is theo-
retically possible, is indicated by the vertical dashed line
and corresponds to a noise level σ2 ≈ 8.79. The dashed
curve shows the overlap for a simple Gaussian channel
with noise level σ2 and the continuous one shows the
overlap for the RWD. The improvement in the practi-
cal limit for error-free communication, indicated by the
highest noise level for which d = 1 is clear. However,
the distance between the dynamical transition thresh-
old σ2d ≈ 2.22, marking the point where sub-dominant
metastable states emerge, and Shannon’s limit for the
channel is greater than in the case of the simple Gaussian
channel (for numerical results for the Gaussian channel
see [21]). Numerical calculations point to the expected
result that decreasing the noise level from the source to
the relay brings the dynamical transition threshold σ2d
closer to Shannon’s limit. However, one must remem-
ber that the relay strategy examined does not use the
full potential of the relay and the additional information
embedded in the LDPC codes. We expect that a LDPC
decoding in the relay will improve the communication
performance and currently focus on the analysis of this
scenario.
We can also see in Fig. 3 that, as the noise level in-
creases, the channel becomes closer to the Gaussian chan-
nel. This is just a consequence of the fact that, for high
noise level, the additional information provided by the re-
lay becomes negligible as both relay and receiver decode
the message poorly.

7ν0
t r
1t1r
ν
FIG. 6: Schematic drawing of the classical relay channel
(CRC) setup. The relative gains are all equal to 1 and not
shown in the picture.
0.0 0.5 1.0 1.5 2.0
λ
0
2
4
6
8
10
N
oi
se
le
ve
l
Shannon Limit
Thermodynamical Transition
Dynamical Transition
FIG. 7: The continuous/dashed line shows the dynam-
ical/thermodynamical transition noise levels of the RWD
against λ in the setup of Fig. 6. The continuous line without
marked symbols is Shannon’s limit for a CRC with the same
noise levels and transmission powers.
is
C =





1
2 log2
(
1 + 1λσ2
)
, λ ≥ 1,
1
2 log2
(
1 + 4(1+λ)2σ2
)
, λ < 1.
(35)
In Fig. 7 we compare the dynamical and thermody-
namical threshold noise levels of a RWD with Shannon’s
limit for the CRC, both with the setup described above,
for different values of λ, the ratio between the noise levels
applied at the transmission and reception points.
We can see that, although the practical decoding line
(dynamical transition) falls below Shannon’s limit for all
calculated values, the thermodynamical transition goes
above it for the CRC case at higher values of λ. Figure 7
shows that the capacity of the RWD is indeed higher
than the CRC for the case studied and quantifies the
gain in allowing the message sent by the relay to depend
on the current transmitted symbol (which is excluded
in the CRC); the RWD result being calculated with the
practical LDPC coding scheme. Although allowing this
instantaneous dependence would at first sight seem just
a small modification, insignificant in the infinite block
length limit, it indeed gives relevant extra information
which facilitates more efficient retrieval at the final re-
ceiver. The insight gained is that for the RWD and large
λ, the relay transmission tµ1 is correlated with the origi-
nal codeword tµ, which is not the case in the CRC; this
allows for an improvement in the information extraction
at the receiver.
B. Large Relay Array
Now, we will use the central limit theorem to obtain the
result for large L in the relay array setup given in Fig. 1.
As the relay messages are correlated and to guarantee
that the quantities have the same order, we introduce a
1/L scaling in the summation over relay messages. The
function Ψ for this model becomes
Ψ(τ, r) ≡
∫
{ L
∏
i=1
dri exp
[
− (ri − τ)
2
2(σ2i + σ20)
]}
× exp

− 12(σ2 + σ20)
(
r − τ − 1L
∑
i
sgn ri
)2
, (36)
where we assumed, for simplicity, a = bi = ci = 1.
For L  1, the central limit theorem gives rise to a
modified distribution of the variable r given by
P(r) =
∫
[ L
∏
i=1
dri P(ri)
]
F
(
1
L
L
∑
i=1
sgn ri
)
= 〈F (ω)〉ω ,
(37)
where
F (ω) = 1√
2piσ2
exp
[
− (r − 1− ω)
2
2(σ2 + σ20)
]
, (38)
P(ri) =
1
√
2piσ2i
exp
[
− (ri − 1)
2
2(σ2i + σ20)
]
, (39)
(40)
with
P(ω) = N
(
1
L
L
∑
i=1
〈sgn ri〉ri ,
1
L2
L
∑
i=1
(
1− 〈sgn ri〉ri
)2
)
.
(41)
For simplicity, we consider the case where the noise
level is the same for all relays σ2i = σ21 and define
σ2r ≡ σ21 + σ20 , σ2f ≡ σ2 + σ20 . (42)
The corresponding distribution for ω then becomes
P(ω) = N
(
erf
(
1/
√
2σr
)
, 1L erfc
2
(
1/
√
2σr
)
)
. (43)
Consequently, the contribution for the final noise level
coming from the relay transmission decreases as L−1. In

81 2 3 4 5
L (Relay units)
1.0
1.5
2.0
2.5
N
oi
se
le
ve
l
Thermodynamical Transition
Dynamical Transition
FIG. 8: Dynamical and thermodynamical transition points
for many relays. The exact formula is used to calculate the
points L = 1, 2, 3, 4, 5. The horizontal lines represent the large
L limit.
the limit L →∞, this distribution becomes a δ function
centred at the error function value and therefore
P(r) = N
(
1 + erf
(
1/
√
2σr
)
, σ2f
)
. (44)
Accordingly, the function Ψ becomes
Ψ(τ, r) = exp
{
− 12σ2f
[
r − τ − erf
(
τ/
√
2σr
)]2
}
. (45)
Figure 8 compares the dynamical and thermodynami-
cal transition points for L = 1, 2, 3, 4, 5 calculated by the
exact formula and the result obtained by the approxima-
tion for large L. Again, we consider the case of K = 4,
C = 3 and β = 1.
It is clear from Fig. 8 that already at L = 5, both
dynamical and thermodynamical transition points ap-
proach the large L limit solution, thus making this ap-
proximation attractive already for low L values.
V. CONCLUSIONS
In this work we analysed the behaviour of relay arrays
using methods of statistical mechanics. These commu-
nication networks are of growing significance due to the
increase of multi-user, mobile and distributed communi-
cation systems.
We found an analytical solution for the relay-without-
delay (RWD) channel given by the RS ansatz, which due
to the gauge symmetry of the channel, is exact at Nishi-
mori’s temperature that correspond to a choice of the cor-
rect prior within the Bayesian framework. We showed the
level of improvement with respect to a simple Gaussian
channel without relaying which, even for the naive re-
lay strategy of Demodulate-and-Forward analysed here,
is significant.
We compared the RWD dynamical and thermodynam-
ical transition points, for different noise ratios, between
the relay and the direct channels; and found that al-
though these points are far from Shannon’s theoretical
limit, the difference between the dynamical and the ther-
modynamical transition decreases. The relevance of the
relay is clearly decreasing as its noise level increases as
the level of additional information it conveys diminishes.
We also were able to compare the RWD case to the
classical relay channel (CRC) for different noise ratios
between the relay and the direct channel. We found that
the capacity of the RWD is higher than the CRC for a
high relay noise, showing the significance of the extra
information conveyed by the relay on the current trans-
mitted symbol, which is absent in the CRC framework.
The performance of a large array of relays was anal-
ysed and compared against results obtained for a small
number of units. The results obtained are consistent and
indicate that this useful approximation provides accurate
results already for a small number of units. For a large
array, we also found that the increase in noise tolerance
levels off.
We have demonstrated the usefulness of methods
adopted from statistical physics for analysing multi-user
communication systems. While we have concentrated
on limited scenarios of relay channels, we believe that
these methods hold a promising alternative to the infor-
mation theory methodology which, in general, has not
been successful in dealing with multi-user communica-
tion systems. The study of different relay channels and
other multi-user communication networks is underway.
Acknowledgements
Support from EVERGROW, IP No. 1935 in FP6 of
the EU is gratefully acknowledged.
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