Rician MIMO Channel- and Jamming-Aware Decision Fusion
D. Ciuonzo A. Aubry V. Carotenuto





Rician MIMO Channel- and Jamming-Aware Decision Fusion
    D. Ciuonzo, Senior Member, IEEE, A. Aubry, Senior Member, IEEE,
and V. Carotenuto, Member, IEEEManuscript received 20th November 2015; revised 3rd August 2016; accepted
23th February 2017. The associate editor coordinating the review of
this manuscript and approving it for publication was Prof. Xavier
Mestre.

D. Ciuonzo was with University of Naples Federico II,
DIETI, Via Claudio 21, 80125 Naples, Italy. He is now with Networking
Measurement and Monitoring (NM-2) s.r.l., 80143 Naples, Itally. (e-mail:
domenico.ciuonzo@ieee.org)

A. Aubry and V. Carotenuto are with University of Naples Federico
II, DIETI, Via Claudio 21, 80125 Naples, Italy. (e-mail:
{augusto.aubry, vincenzo.carotenuto}@unina.it)


    
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In this manuscript we study channel-aware decision fusion (DF) in
a wireless sensor network (WSN) where: (i) the sensors transmit
their decisions simultaneously for spectral efficiency purposes and
the DF center (DFC) is equipped with multiple antennas; (ii) each
sensor-DFC channel is described via a Rician model. As opposed to
the existing literature, in order to account for stringent energy
constraints in the WSN, only statistical channel information is assumed
for the non-line-of-sight (scattered) fading terms. For such a scenario,
sub-optimal fusion rules are developed in order to deal with the exponential
complexity of the likelihood ratio test (LRT) and impractical (complete)
system knowledge. Furthermore, the considered model is extended to
the case of (partially unknown) jamming-originated interference. Then
the obtained fusion rules are modified with the use of composite hypothesis
testing framework and generalized LRT. Coincidence and statistical
equivalence among them are also investigated under some relevant simplified
scenarios. Numerical results compare the proposed rules and highlight
their jamming-suppression capability.



Decision Fusion, Distributed Detection, Virtual MIMO, Wireless Sensor
Networks.





§ INTRODUCTION




 §.§ Motivation and Related Literature


Decision Fusion (DF) in a wireless sensor network
(WSN) consists in transmitting local decisions about an observed phenomenon
from sensors to a DF center (DFC) for a global decision, with the
intent of surveillance and/or anomaly detection <cit.>.
Typically all the studies had been focused on a parallel access channels
(PACs) with instantaneous <cit.> or statistical
channel-state information (CSI) <cit.>, although some
recent works extended to the case of multiple access channels (MACs).
Adoption of a MAC in WSNs is clearly attractive because of its increased
spectral efficiency. 

Distributed detection over MACs was first studied in <cit.>,
where perfect compensation of the fading coefficients is assumed for
each sensor. Non-coherent modulation and censoring over PACs and MACs
were analyzed in <cit.> with emphasis on processing gain
and combining loss. The same scenario was studied in <cit.>,
focusing on the error exponents (obtained through the large deviation
principle) and the design of energy-efficient modulations for Rayleigh
and Rice fading. Optimality of received-energy statistic in Rayleigh
fading scenario was demonstrated for diversity MACs with non-identical
sensors in <cit.>. Efficient DF over MACs only with knowledge
of the instantaneous channel gains and with the help of power-control
and phase-shifting techniques was studied in <cit.>.
Techniques borrowed from direct-sequence spread-spectrum systems were
combined with on-off keying (OOK) modulation and censoring for DF
in scenarios with statistical CSI <cit.>.

DF over a (virtual) MIMO (this setup will be referred to as “MIMO-DF”
hereinafter) was first proposed in <cit.>, with focus on
power-allocation design based on instantaneous CSI, under the framework
of J-divergence. Distributed detection with ultra-wideband sensors
over MAC was then studied in <cit.>. The same model was adopted
to study sensor fusion over MIMO channels with amplify-and-forward
sensors in <cit.>. A recent theoretical study
on data fusion with amplify and forward sensors, Rayleigh fading channel
and a large-array at the DFC has been presented in <cit.>. 

Design of several sub-optimal fusion rules for MIMO-DF scenario was
given in <cit.> in a setup with instantaneous CSI and
Rayleigh fading, while the analysis was extended in <cit.>
to a large-array at the DFC, estimated CSI and inhomogeneous large
scale fading. In both cases binary-phase shift keying (BPSK) has been
employed. It is worth noticing that in MIMO-DF scenario the log-likelihood
ratio (LLR) is not a viable solution, since it suffers from the exponential
growth of the computational complexity with respect to (w.r.t.) the
number of sensors and a strong requirement on system knowledge.

However, frequently the final purpose of a WSN is anomaly detection
(viz. the null hypothesis is much more frequent than the alternative
hypothesis, denoting the “anomaly”). Such problem arises in many
application contexts, such as intrusion detection or monitoring of
hazardous events. In this case, a wise choice of the modulation format
is the on-off keying (OOK), which ensures a nearly-optimal censoring
policy (and thus significant energy savings) <cit.>.
Additionally, though the channel between the sensors and the DFC may
be accurately modelled as Rician, assuming instantaneous CSI (i.e.,
estimation of the scattered fading component) may be too energy costly
for an anomaly detection problem. This motivated the study of DF over
Rician MAC channels (in the single-antenna DFC case) with only statistical
CSI in <cit.>. We point out that statistical
CSI is instead a reasonable assumption for a WSN and can be obtained
through long-term training-based techniques (since statistical parameters
of Rician model have a slower variation with respect to the coherence
time of the channel), mimicking the procedures proposed in <cit.>.
The aforementioned problem may be further exacerbated by the presence
of a (possibly distributed) jamming device in the WSN deployment area
<cit.>. Such problem is clearly relevant in non-friendly environments,
such as the battlefield, where malicious devices (i.e., the jammers)
are placed to hinder the operational requirements of the WSN. Indeed,
due to jammer hostile nature, unknown interference is superimposed
to useful received signal (containing “informative” sensors contributions).
Therefore, additional relevant parameters may be unknown at the DFC
side. This precludes development of sub-optimal (simplified) fusion
rules based on the LLR, which assume complete specification of pdfs
under both hypotheses. To the best of our knowledge, the study of
such a setup for MIMO-DF has not been addressed yet in the open literature.



 §.§ Main Results and Paper Organization


The contributions of the present manuscript are summarized as follows.


  * We study decision fusion over MAC with Rician fading and multiple
antennas at the DFC (as opposed to <cit.>).
In the present study only the LOS component is assumed known at the
DFC. Also, by adopting the same general assumptions in <cit.>,
the considered model also accounts for unequal long-term received
powers from the sensors, through a common path loss and shadowing
model;

  * We derive sub-optimal fusion rules dealing with exponential complexity
and with required system knowledge in the considered scenario, namely,
we derive (i) “ideal sensors” (IS) (following the same spirit
as in <cit.>), (ii) “non-line of sight”
(NLOS), (iii) “widely-linear” (mimicking <cit.>)
and (iv) “improper Gaussian moment matching” (IGMM, based on
second order characterization of the received vector under both hypotheses)
rules; 

  * Subsequently, we consider DF in the presence of a (either distributed
or co-located) multi-antenna jamming device, whose communication channel
is described by an analogous Rician model. The problem is tackled
within a composite hypothesis testing framework and solved via the
generalized likelihood-ratio test (GLRT) <cit.> and similar
key simplifying assumptions as in the “no-jamming” scenario, thus
leading to IS-GLRT, NLOS-GLRT and IGMM-GLRT rules, respectively;

  * Simulation studies (along with a detailed complexity analysis) are
performed to compare the performance of the considered rules and verify
the asymptotical equivalences (later proved in Secs. <ref>
and <ref>) among them in some
specific instances. Also, the performance trend as a function of the
Rician parameters of the WSN and the jammer, the thermal noise and
the number of receive antennas are investigated and discussed.

The remainder of the manuscript is organized as follows: Sec. <ref>
introduces the model; in Sec. <ref> we derive and
study the fusion rules, while in Sec. <ref>
we generalize the analysis to the case of a subspace interference;
the obtained rules are compared in terms of computational complexity
in Sec. <ref>; in Sec. <ref>
we compare the presented rules through simulations; finally in Sec. <ref>
we draw some conclusions; proofs and derivations are contained in
a dedicated Appendix.

Notation - Lower-case (resp. Upper-case) bold letters denote
vectors (resp. matrices), with a_n (resp. a_n,m) being the
nth (resp. the (n,m)th) element of a (resp. A);
upper-case calligraphic letters denote finite sets, with 𝒜^K
representing the k-ary Cartesian power of 𝒜; O_N× K
(resp. I_N) denotes the N× K (resp. N× N)
null (resp. identity) matrix, with corresponding short-hand notation
O_N for a square matrix; 0_N (resp. 1_N)
denotes the null (resp. ones) vector of length N; a_n:m
(resp. A_n:m) denotes the sub-vector of a (resp.
the sub-matrix of A) obtained from selecting only nth to
mth elements of v (resp. nth to mth rows/columns
of A); 𝔼{·}, var{·}, (·)^T,
(·)^†, (·)^-, (·), (·)
and ‖·‖ denote expectation, variance,
transpose, conjugate transpose, pseudo-inverse, real part, imaginary
part and Euclidean norm operators, respectively; (·)_+ is
used to indicate max{0,·}; diag(A) (resp.
diag(a)) denotes the diagonal matrix extracted from
A (resp. the diagonal matrix with main diagonal given by a);
det(A) is used to denote the determinant of A;
λ_min(A) denotes the minimum eigenvalue
of the Hermitian matrix A; P_X^⊥ denotes
the orthogonal projector to the range space spanned by X;
a (resp. A) denotes the augmented
vector (resp. matrix) of a (resp. A), that is a≜[[ a^T a^† ]]^T (resp. A≜[[ A^T A^† ]]^T); P(·) and p(·) denote probability mass functions (pmf)
and probability density functions (pdf), while P(·|·) and
p(·|·) their corresponding conditional counterparts; Σ_x
(resp. Σ̅_x) denotes the covariance (resp.
the complementary covariance) matrix of the complex-valued random
vector x; 𝒩_ℂ(μ,Σ)
(resp. 𝒩_ℂ(μ,Σ,Σ̅)
denotes a proper (resp. an improper) complex normal distribution with
mean vector μ and covariance matrix Σ (resp.
covariance Σ and pseudo-covariance Σ̅),
while 𝒩(μ,Σ) denotes the corresponding
real-valued counterpart; finally the symbols ∝, and ∼
mean “statistically equivalent to” and “distributed
as”, respectively.



§ SYSTEM MODEL


Hereinafter we will consider a decentralized binary hypothesis test,
where K sensors are used to discern between the hypotheses in the
set ℋ≜{ℋ_0,ℋ_1} (e.g.
ℋ_0/ℋ_1 may represent the absence/presence
of a specific target of interest). The kth sensor, k∈𝒦≜{1,2,…,K},
takes a binary local decision ξ_k∈ℋ about the observed
phenomenon on the basis of its own measurements. Here we do not make
any conditional (given ℋ_i∈ℋ) mutual independence
assumption on ξ_k. Each decision ξ_k is mapped to a
symbol x_k∈ X={0,+1} representing an OOK modulation:
without loss of generality (w.l.o.g.) we assume that ξ_k=ℋ_i
maps into x_k=i, i∈{0,1}. The quality of the WSN is characterized
by the conditional joint pmfs P(x|ℋ_i). Also,
we denote P_D,k≜ P(x_k=1|ℋ_1)
and P_F,k≜ P(x_k=1|ℋ_0) the
probability of detection and false alarm of the kth sensor, respectively
(here we make the assumption P_D,k≥ P_F,k, meaning that
each sensor decision procedure leads to receive operating characteristics
above the chance line <cit.>). In some situations, aiming
at improving clarity of exposition, we will use the short-hand notation
(P_0,k,P_1,k)=(P_F,k,P_D,k) (and (P_0,P_1)=(P_F,P_D),
in the simpler case of conditionally i.i.d. decisions).

Sensors communicate with a DFC equipped with N receive antennas
over a wireless flat-fading MAC in order to exploit diversity so as
to mitigate small-scale fading; this setup determines a distributed
(or virtual) MIMO channel <cit.>.
Also, perfect synchronization[Multiple antennas at the DFC do not make these assumptions harder
to verify w.r.t. a single-antenna MAC.], as in <cit.>, is assumed
at the DFC.

We denote: y_n the signal at the nth receive antenna of the
DFC after matched filtering and sampling; (√(d_k,k) h̅_n,k)
the composite channel coefficient between the kth sensor and the
nth receive antenna of the DFC; w_n the additive white Gaussian
noise at the nth receive antenna of the DFC. The vector model at
the DFC is:

    y=     H̅D^1/2x+w

where y∈ℂ^N, x∈𝒳^K and
w∼𝒩_ℂ(0_N,σ_w^2I_N)
are the received-signal vector, the transmitted-signal vector and
the noise vector, respectively. Also, the matrices H̅∈ℂ^N× K
and D∈ℂ^K× K model independent small-scale
fading, geometric attenuation and log-normal shadowing. More specifically,
D≜diag([ β_1   ⋯ β_K ]^T)
is a (known) matrix with kth diagonal element d_k,k=β_k
(β_k>0) accounting for path loss and shadow fading experienced
by kth sensor. On the other hand, the kth column of H̅
models the (small-scale) fading vector of kth sensor as h̅_k=b_k a(θ_k)+√(1-b_k^2) h_k.
Here a(·) denotes the steering vector (which depends on
the angle-of-arrival θ_k, assumed known at the DFC[W.l.o.g. in this work we adopt a 1-D functional dependence for a(·)
(obtained from a “far-field” assumption), albeit more complicate
expressions could be considered as well.]) corresponding to the LOS component and h_k∼𝒩_ℂ(0_N,I_N)
corresponds to the normalized NLOS (scattered) component. Finally,
we denote b_k≜√(κ_k/1+κ_k),
where κ_k represents the (known) usual Rician factor between
kth sensor and DFC.

The matrix H̅ can be expressed compactly in terms of
the relevant matrices A(θ)≜[ a(θ_1)      ⋯ a(θ_K) ],
H≜[ h_1   ⋯ h_K ]
and R≜diag([ b_1   ⋯ b_K ]^T),
respectively, as

    H̅≜  A(θ) R+H (I_K-R^2)^1/2 .

Finally, we underline that the received scattered term from
kth sensor in Eq. (<ref>) is √((1-b_k^2) β_k) h_k∼𝒩_ℂ(0_N,ν_k I_N),
where ν_k≜[β_k (1-b_k^2)], while its LOS
term is μ_k≜[√(β_k) b_k a(θ_k)]
and corresponds to the kth column of the matrix 

    A(θ)≜(A(θ) R D^1/2) ,

denoting the matrix of received LOS terms from the WSN.



§ FUSION RULES




 §.§ Optimum (LLR) Rule 


The optimal test <cit.> is formulated on the basis of the
LLR Λ_opt≜ln[p(y|ℋ_1)/p(y|ℋ_0)],
and decides in favour of ℋ_1 (resp. ℋ_0)
when Λ_opt>γ (resp. Λ_opt≤γ),
with γ denoting the threshold which the LLR is compared to[ The threshold γ can be determined to ensure a fixed system
false-alarm rate (Neyman-Pearson approach), or can be chosen to minimize
the probability of error (Bayesian approach) <cit.>.]. After few manipulations, the LLR can be expressed explicitly as

    Λ_opt=ln[∑_x∈𝒳^KP(x|ℋ_1)/[σ_e^2(x)]^Nexp(-y-∑_k=1^Kμ_k x_k^2/σ_e^2(x))/∑_x∈𝒳^KP(x|ℋ_0)/[σ_e^2(x)]^Nexp(-y-∑_k=1^Kμ_k x_k^2/σ_e^2(x))]

where σ_e^2(x)≜(σ_w^2+∑_k=1^Kν_k x_k).
The above result follows from y|ℋ_i being a Gaussian
mixture random vector, since the pdf under each hypothesis can be
obtained as p(y|ℋ_i)=∑_xp(y|x) P(x|ℋ_i)
(the directed triple ℋ→x→y
satisfies the Markov property). It is apparent that implementation
of Eq. (<ref>) requires a computational complexity
which grows exponentially with K (namely 𝒪(2^K),
where 𝒪(·) stands for the usual Landau's notation).
Also, differently from <cit.> (where a BPSK
modulation is employed), the computation here is complicated by the
fact that each component of the mixture (under ℋ_i)
has both different mean vectors and covariance (actually scaled identity)
matrices. Therefore, sub-optimal fusion rules with reduced complexity
are investigated in what follows.



 §.§ Ideal sensors (IS) rule


The LLR in Eq. (<ref>) can be simplified under the
assumption of perfect sensors <cit.>,
i.e., P(x=1_K|ℋ_1)=P(x=0_K|ℋ_0)=1.
In this case x∈{0_K,1_K} and Eq. (<ref>)
reduces to <cit.>:

    ln[[σ_e^2(1_K)]^-N exp(-y-∑_k=1^Kμ_k^2/σ_e^2(1_K))/[σ_w^2]^-N exp(-y^2/σ_w^2)]∝
    
    2 (μ̅^†y)+ν̅/σ_w^2‖y‖ ^2≜Λ_IS

where μ̅≜1/K∑_k=1^Kμ_k
and ν̅≜1/K∑_k=1^Kν_k and terms
independent from y have been discarded (as they can be incorporated
in a suitably modified threshold γ).

It is worth noticing that the assumption of perfect local decisions
is used only for system design purposes, and does not
mean that the system is working under such ideal conditions, thus
the rule is suboptimal. Also, we observe that IS rule in (<ref>)
is formed by a weighted combination of a maximum ratio combiner (MRC,
which actually is the statistic resulting from IS assumption on the
known part of channel vector at the DFC <cit.>) and an
energy detector (ED, i.e., the statistic arising from the IS assumption
on the random part of the channel vector at the DFC <cit.>).
Clearly, from Eq. (<ref>) it is apparent that IS rule
does not require sensor performance (i.e., the pmf P(x|ℋ_i),
i∈0,1) for its implementation.



 §.§ Non line-of-sight (NLOS) rule


In this case we derive a sub-optimal rule arising from the simplifying
assumption κ_k=0 (i.e., no sensor has a LOS path), thus
leading to: 

    ln[∑_x∈𝒳^KP(x|ℋ_1)/[σ̅_e^2(x)]^Nexp(-y^2/σ̅_e^2(x))/∑_x∈𝒳^KP(x|ℋ_0)/[σ̅_e^2(x)]^Nexp(-y^2/σ̅_e^2(x))]

where we have denoted σ̅_e^2(x)≜(σ_w^2+∑_k=1^Kβ_kx_k).
We observe that in this case the LLR is function of the sole sufficient
statistic Λ_ NL≜y^2,
i.e., the energy of the received signal, which we retain as a simple
statistic for our test[We recall that, as in the case of IS rule, NLOS assumption is only
exploited at the design stage for development of simplified rule Λ_ NL.]. There is a twofold motivation for this choice. First, it was shown
in <cit.> that under identical β_k's and conditionally
independent decisions, the LLR in Eq. (<ref>) is
a monotone function of y^2 (thus y^2>γ
is the uniformly most powerful test <cit.>). Secondly,
by applying Gaussian moment matching to the simplified model in Eq.
(<ref>), the same test would be obtained. Therefore,
though we have no optimality claims for Λ_ NL
in this general case, we will consider NLOS rule as the decision statistic
due to its simplicity (and no requirements on sensors performance).



 §.§ Widely-linear (WL) rules


It can be shown that y|ℋ_i has the following statistical
characterization up to the first two order moments (the proof is given
in Appendix):

    𝔼{y|ℋ_i}    =A(θ) ρ_i
    Σ_y|ℋ_i    =A(θ) Σ_x|ℋ_i A(θ)^†+σ_e,i^2 I_N
    Σ̅_y|ℋ_i    =A(θ) Σ_x|ℋ_i A(θ)^T

where ρ_i≜[ P_i,1     ⋯ P_i,K ]^T
and σ_e,i^2≜[∑_k=1^Kν_k P_i,k+σ_w^2].
Therefore a convenient and effective approach consists in adopting
a WL statistic <cit.>. The WL approach (i.e., based
on the augmented vector y) is motivated by linear
complexity and y|ℋ_i being an improper (cf.
Eq. (<ref>)) complex-valued random vector,
that is Σ̅_y|ℋ_i≠O_N.
More specifically, WL statistic is generically expressed as:

    Λ_WL≜z^†y ,

 where the augmented vector z has to be designed
according to a reasonable criterion. Then, Λ_WL
is compared to a proper threshold γ to obtain the corresponding
test.

Clearly, several optimization metrics may be considered for obtaining
z. The best choice (in a Neyman-Pearson sense)
would be searching for the WL rule maximizing the global detection
probability subject to a global false-alarm rate constraint, as proposed
in <cit.> for a distributed detection problem. Unfortunately,
the optimized z presents the following drawbacks:
(i) it is not in closed-form, (ii) it requires
a non-trivial optimization and (iii) it depends on the prescribed
false-alarm constraint. Additionally, the problem under investigation
is not a multivariate Gauss-Gauss test (i.e., y|ℋ_i∼𝒩_ℂ(μ_i,Σ_i))
but one discerning between mixtures of complex GMs (cf. Eq. (<ref>)).
This would further complicate the optimization problem tackled in
<cit.>.

Differently, in this paper we choose z as the
maximizer of either the normal <cit.> or modified <cit.>
deflection measures, denoted as D_0( z )
and D_1( z ) respectively, that is:

    z_ WL,i≜max_z: ‖ z ‖ ^2=1D_i( z )
    where   D_i( z )≜(𝔼{Λ_WL|ℋ_1}-𝔼{Λ_WL|ℋ_0})^2/var{Λ_WL|ℋ_i}

Maximization of deflection measures is commonly used in the design
of (widely) linear rules for DF, since z_ WL,i
always admits a closed-form and also literature has shown acceptable
performance loss w.r.t. the LLR in analogous DF setups <cit.>.
The vector z_ WL,i,
being the optimal solution to the optimization in Eq. (<ref>),
is (a similar proof can be found in <cit.>):

    z_ WL,i=Σ_y|ℋ_i^-1 A(θ) ρ_1,0/||Σ_y|ℋ_i^-1 A(θ) ρ_1,0||

where ρ_1,0≜(ρ_1-ρ_0) and
Σ_y|ℋ_i is given by:

    Σ_y|ℋ_i=A(θ) Σ_x|ℋ_i A(θ)^†+σ_e,i^2 I_2N

The WL statistics are thus obtained employing Eq. (<ref>)
into (<ref>). It is worth pointing out that, from
inspection of Eq. (<ref>), WL
rules only require knowledge up to the second order of the vectors
x|ℋ_i.



 §.§ Improper Gaussian moment matching (IGMM) rule


Differently here we fully exploit the second order characterization
provided in Eqs. (<ref>-<ref>).
In fact, after fitting y|ℋ_i to an improper complex
Gaussian, the following quadratic test can be obtained <cit.>:

    Λ_ IGMM   ≜    -(y-𝔼{y|ℋ_1})^† Σ_y|ℋ_1^-1 (y-𝔼{y|ℋ_1})+
           (y-𝔼{y|ℋ_0})^† Σ_y|ℋ_0^-1 (y-𝔼{y|ℋ_0})

where 𝔼{y|ℋ_i}=A(θ) ρ_i
and Σ_y|ℋ_i is given in
Eq. (<ref>). IGMM rule presents the same
(reduced) requirements on knowledge of sensors performance as the
WL rules (cf. Eqs. (<ref>) and
(<ref>)). Differently, we expect it to perform nearly-optimal
(i.e., close to the LLR) at low SNR, as in such case both Gaussian
mixtures are well-approximated by a single Gaussian pdf.



 §.§ Asymptotic equivalences


In this sub-section, we will establish asymptotic equivalences among
the proposed rules in the form of the following lemmas. These will
be employed as useful tools to facilitate the understanding of the
numerical comparisons shown in Sec. <ref>.

As the sensors approach a NLOS condition (i.e.,
the Rician factor κ_k→0) IS and IGMM (recalling
P_D,k≥ P_F,k) rules are statistically equivalent to the
NLOS rule, i.e., they collapse to an energy detection test.


The proof is obtained by substituting κ_k=0 in Eqs. (<ref>)
and (<ref>), which respectively gives Λ_IS=y^2(∑_k=1^Kβ_k/(Kσ_w^2))
and 

    Λ_ IGMM    =2 y^2 ∑_k=1^Kβ_k(P_D,k-P_F,k)/(∑_k=1^Kβ_kP_D,k+σ_w^2)(∑_k=1^Kβ_kP_F,k+σ_w^2).

The latter result follows from 𝔼{y|ℋ_i}=0_2N
and Σ_y|ℋ_1=(∑_k=1^Kβ_kP_i,k+σ_w^2) I_2N
(since under NLOS assumption R=O_K implies A(θ)=O_N× K
and ν_k=β_k, respectively). Therefore it is apparent
that IS and IGMM (assuming (P_D,k-P_F.k)≥0) rules become
statistically equivalent to NLOS rule. 

The above lemma states that IS and IGMM rules are both statistically
equivalent to NLOS rule when each sensor has only a purely scattered
component. Indeed, in such a case (as also supported intuitively),
only a dependence on ‖y‖ ^2 is
relevant in the design of a fusion rule for the binary hypothesis
test under consideration (i.e., all the mentioned decision procedures
collapse into the received energy test). Accordingly IS rule, being
based on a weighted combination of a MRC-ED (see Eq. (<ref>)),
exploits only the non-coherent term in NLOS case. Similarly IGMM rule,
being based on second-order characterization of y|ℋ_i,
simplifies as the two hypotheses manifest in NLOS scenario with a
sole change of variance in the received signal (i.e., no mean or covariance
structure modification).

However, in the case of conditionally i.i.d. decisions, a stronger
result can be proved for IS and IGMM rules, as described by the following
lemma.

In the case of conditionally i.i.d.
decisions, viz P(x|ℋ_i)=∏_k=1^KP(x_k|ℋ_i)
and (P_D,k,P_F,k)=(P_D,P_F) (recalling that P_D>P_F),
and under a “weak-LOS assumption” (quantified as P_i(1-P_i)λ_min(A(θ) A(θ)^†)≪σ_e,i^2),
IS and IGMM rules are approximately statistically equivalent.


We begin by observing that, under conditionally i.i.d. assumption,
the covariance of y|ℋ_i simplifies
to (since Σ_x|ℋ_i=P_i(1-P_i) I_K):

    Σ_y|ℋ_i=P_i(1-P_i) A(θ) A(θ)^†+σ_e,i^2 I_2N .

Then, we express it in terms of the eigendecomposition (A(θ) A(θ)^†)=(U_M Λ_M U_M^†),
that is Σ_y|ℋ_i=U_M [P_i(1-P_i)Λ_M+σ_e,i^2I_2N] U_M^†.
If P_i(1-P_i)_min(A(θ) A(θ)^†)≪σ_e,i^2
holds, we can safely approximate Σ_y|ℋ_i≈(U_Mσ_e,i^2 U_M^†).
We refer to this assumption as a “weak-LOS” one since, as all
the κ_k's get low in R, all the eigenvalues in Λ_M
get small while σ_e,i^2=(P_i ∑_k=1^KD(I_K-R^2)1_K+σ_w^2)
increases. Also, we notice that IGMM rule in Eq. (<ref>)
is statistically equivalent to:

    y^† (Σ_y|ℋ_0^-1-Σ_y|ℋ_1^-1) y +
    
    2 y^†(Σ_y|ℋ_1^-1𝔼{y|ℋ_1}-Σ_y|ℋ_0^-1𝔼{y|ℋ_0})

Thus, by exploiting the aforementioned approximation in Eq. (<ref>),
Λ_ IGMM is shown to be approximately
expressed as:

    y^2 [1/σ_e,0^2-1/σ_e,1^2]+2 y^†[𝔼{y|ℋ_1}/σ_e,1^2-𝔼{y|ℋ_0}/σ_e,0^2]

Then, after few manipulations (and exploiting definition of σ_e,i^2
and 𝔼{y|ℋ_1}, respectively), Eq. (<ref>)
can be rewritten as:

    2 (P_D-P_F)/σ_e,0^2 σ_e,1^2 (∑_k=1^Kν_k y^2+σ_w^2 y^†A(θ) 1_K)=
    2K (P_D-P_F) σ_w^2/σ_e,0^2 σ_e,1^2 (ν̅/σ_w^2y^2+2 (μ̅^†y))

which is apparently the IS rule (except for an irrelevant positive
scalar, recalling P_D>P_F). This concludes the proof.

We underline that Lem. <ref> does not include
Lem. <ref>, since at a relatively low Rician
factor (_min(A(θ) A(θ)^†)
gets low, whereas σ_e,i^2 increases) for all the sensors,
data covariance matrix under ℋ_i will be approximately
diagonal, while the difference of the mean terms 𝔼{y|ℋ_i}
will not be negligible. In the latter case, IGMM will exhibit the
same linear-quadratic dependence on the data as the IS rule. In other
terms, it will reduce to a weighted combination of a MRC-ED (see (<ref>)
and (<ref>), respectively). In this region, however,
NLOS rule does not perform as well as those statistics, since its
dependence is only on ‖y‖ ^2.
Moreover, it is worth noticing that weak-LOS assumption P_i(1-P_i)eig_min(A(θ) A(θ)^†)≪σ_e,i^2
is also likely to be satisfied in a low-SNR regime (i.e., high σ_w^2,
right-hand increases) and for “good-quality” sensors (i.e., (P_D,P_F)→(1,0),
left-hand decreases).

Finally, we look at the extreme case given by IS assumption. In this
case, IS rule is statistically equivalent to the LLR (by construction,
cf. Sec. <ref>). On the other hand, we are able to prove
the following asymptotic equivalence properties among WL and IGMM
rules, reported in the following lemma.

Under “IS assumption”, IGMM rule is
statistically equivalent to IS rule (thus attaining optimum performance),
while WL rules are statistically equivalent and are given by the sole
“widely-linear” part of IS rule in Eq. (<ref>).


We start by recalling statistical equivalence of IGMM rule to Eq.
(<ref>). Then, we observe that IS assumption straightforwardly
implies ρ_1=1_K (resp. ρ_0=0_K)
and Σ_x|ℋ_i=O_K. Thus Eq. (<ref>)
specializes into:

    y^† (σ_e,1^2-σ_e,0^2/σ_e,0^2 σ_e,1^2) y+2/σ_e,1^2 y^†A(θ) 1_K=
    2K/σ_e,1^2 {‖y‖ ^2 (ν̅/σ_e,0^2)+2 {μ̅^†y}}

which is related to IS rule via an irrelevant positive constant (we
recall that, under IS assumption, σ_e,1^2=(∑_k=1^Kν_k+σ_w^2)
and σ_e,0^2=σ_w^2 hold). This proves the first
part of the lemma. By similar reasoning, it can be shown that both
WL rules in Eq. (<ref>), under
IS assumption, coincide with:

    y^†((A(θ) 1_K)/‖A(θ) 1_K‖)=√(2)/‖μ̅‖ {μ̅^†y} .

It is apparent that right-hand side of Eq. (<ref>)
is proportional to the first contribution of IS rule in Eq. (<ref>),
thus completing the proof.

Therefore, when sensors are ideal, IGMM rule will be statistically
equivalent to IS (viz. LLR) rule, as no covariance structure change
happens when one of the two hypotheses is in force. Differently WL
rules, lacking a ‖y‖ ^2 dependence,
do not reduce to a weighted MRC-ED combination. Based on this reason,
we expect that when the WSN operates with “good-quality” sensors,
WL rules will experience some performance loss with respect to IS
and IGMM rules.



§ JAMMER (SUBSPACE) INTERFERENCE ENVIRONMENT


In this section, we complicate the model in Eq. (<ref>)
and assume the presence of jamming devices operating on the WSN-DFC
communication channel. More specifically, we model the jamming signal
as an r-dimensional vector, whose experienced channel follows the
same Rician model as the WSN at the DFC, that is y_s=y+s_J,
where: 

    s_J    =(A_J(ϕ) R_J+H_J (I_r-R_J^2)^1/2) D_J^1/2ψ.

 In Eq. (<ref>) ψ∈ℂ^r
represents the (unknown deterministic) jamming signal. Similarly to
the WSN, A_J(ϕ)∈ℂ^N× r, H_J∈ℂ^N× r,
R_J∈ℝ^r× r and D_J∈ℝ^r× r
denote the (full-rank) steering matrix (whose ℓth column is
given by a(ϕ_ℓ) and depends on the angle-of-arrival
ϕ_ℓ), the normalized scattered matrix (whose ℓth
column h_J,ℓ∼𝒩_ℂ(0_N,I_N),
assumed mutually independent from the others), the diagonal matrix
of the Rician factors (whose ℓth element is denoted as b_ℓ,J)
and the large-scale diagonal fading matrix of the jammer (whose ℓth
element is denoted as β_ℓ,J), respectively. It is worth
noticing that Eq. (<ref>) accounts
for interfering systems with both distributed (viz. R_J
and D_J are both diagonal) or co-located (viz. R_J
and D_J are both scaled identity) transmitting antennas
in space <cit.>. It is apparent that the former case
includes the case of multiple jammers. The considered
interfering source can be classified as a “constant jammer”, according
to the terminology[We underline that the term “constant” may be misleading, as the
definition of <cit.> implies that the jammer continuously
emits a radio signal (changing with time), which is unknown at the
DFC.] proposed in <cit.>. Though it represents the simplest typology
of jammer, it is here considered as a first step toward the development
of fusion rules robust to “smarter” jammers.

In this case, the received signal y_s is conditionally distributed
as:

    y_s|ℋ_i∼∑_x∈𝒳^KP(x|ℋ_i) 𝒩_ℂ(μ_s(x,ζ),[σ_e^2(x)+σ_J^2] I_N)
    where    μ_s(x,ζ)≜A(θ) x+A_J(ϕ) ζ

where σ_J^2≜∑_ℓ=1^rν_ℓ,J |ψ_ℓ|^2,
ν_ℓ,J≜β_ℓ,J (1-b_ℓ,J^2) and ζ≜(R_J D_J^1/2 ψ),
respectively. Hereinafter we will make the reasonable assumption that
the DFC can only learn A_J(ϕ), i.e., the
DFC does not have knowledge of: (i) the diagonal matrix of the
Rician factors R_J, (ii) the large-scale fading diagonal
matrix D_J and (iii) the actual jamming (transmitted)
signal ψ. The following sub-sections are thus devoted to
the design of (sub-optimal) fusion rules in the presence of the aforementioned
(unknown deterministic) interference parameters.



 §.§ Clairvoyant LRT and GLRT 


In what follows, we will employ in our comparison the clairvoyant
LRT as a benchmark, which (unrealistically) assumes {ψ,D_J,R_J}
as known and thus implements the statistic:

    Λ_c-opt≜
    ln[∑_x∈𝒳^KP(x|ℋ_1)/[σ_e^2(x)+σ_J^2]^Nexp(-y-∑_k=1^Kμ_k x_k-A_J(ϕ) ζ^2/σ_e^2(x)+σ_J^2)/∑_x∈𝒳^KP(x|ℋ_0)/[σ_e^2(x)+σ_J^2]^Nexp(-y-∑_k=1^Kμ_k x_k-A_J(ϕ) ζ^2/σ_e^2(x)+σ_J^2)]

Clearly, the LRT is uniformly most powerful <cit.>
and thus no other fusion rule can expect to perform better. Unfortunately
the LRT cannot be implemented, as the jamming parameters are not known
in the practice. For this reason, hereinafter we will devise tests
which tackle the arising composite hypothesis testing problem. 

A widespread test for the considered problem would be the GLRT <cit.>,
requiring the maximization of pdf under both hypotheses w.r.t. the
(unknown) parameters set. The GLRT has been successfully applied to
different application contexts, such as spectrum sensing <cit.>,
allowing important design guidelines on system level performance (in
terms of optimized sensing time) <cit.>. In our case, it is
not difficult to show that optimization w.r.t. {ψ,D_J,R_J}
is tantamount to maximizing both pdfs w.r.t. σ_J^2 and
ζ as they were (parametrically) independent. Therefore,
this yields the statistic:

    Λ_ GLR≜ln[max_ζ,σ_J^2 p(y_s|ℋ_1)/max_ζ,σ_J^2 p(y_s|ℋ_0)] .

From inspection of Eq. (<ref>), it is apparent
that GLRT has no simple implementation for this problem, because of
its exponential complexity (p(y_s|ℋ_i) is a
GM with 2^K components) and required non-linear optimizations.
Thus, exact GLRT implementation appears as not feasible from a practical
point of view and will not be pursued in the following. Nonetheless,
we will show that “GLRT philosophy” of Eq. (<ref>)
can be exploited jointly with the simplifying assumptions that lead
to the sub-optimal statistics obtained in Sec. <ref>
in order to devise computationally efficient and jamming-robust
fusion rules.



 §.§ IS-GLRT rule


The GLRT in Eq. (<ref>) can be simplified under
the IS assumption, i.e., P(x=1_K|ℋ_1)=P(x=0_K|ℋ_0)=1.
Indeed, based on these assumptions, it holds:

    y_s|ℋ_0   ∼𝒩_ℂ(A_J(ϕ) ζ, [σ_w^2+σ_J^2] I_N)
    y_s|ℋ_1   ∼𝒩_ℂ(A(θ) 1_K+A_J(ϕ) ζ, [σ_e^2(1_K)+σ_J^2] I_N)

The ML estimates of ζ under ℋ_0 and ℋ_1
are obtained respectively as <cit.>:

    ζ̂_0≜    A_J(ϕ)^- y_s
    ζ̂_1≜    A_J(ϕ)^- (y_s-A(θ) 1_K)

Hence, the concentrated likelihoods are:

    p_is(y_s|ℋ_0,ζ̂_0,σ_J^2)=
    1/{π[σ_w^2+σ_J^2]}^N exp[-‖r_0‖ ^2/σ_w^2+σ_J^2]
    
    p_is(y_s|ℋ_1,ζ̂_1,σ_J^2)=
    1/{π[σ_e^2(1_K)+σ_J^2]}^N exp[-‖r_1‖ ^2/σ_e^2(1_K)+σ_J^2]

where r_0≜[P_A_J(ϕ)^⊥ y_s]
and r_1≜[P_A_J(ϕ)^⊥(y_s-A(θ) 1_K)],
respectively. Then the ML estimates[These are straightforwardly obtained by setting ∂ln p_is(y_s|ℋ_i,ζ̂_i,σ_J^2)/∂σ_J^2=0
and accounting for the constraint σ_J^2≥0.] of σ_J^2 under ℋ_0 and ℋ_1
are obtained as <cit.>:

    σ̂_J,0^2   ≜   [‖r_0‖ ^2/N-σ_w^2]_+
    σ̂_J,1^2   ≜   [‖r_1‖ ^2/N-σ_e^2(1_K)]_+

Then, we substitute Eqs. (<ref>)
and (<ref>) into (<ref>)
and (<ref>), respectively, thus obtaining:

    p_is(y_s|ℋ_0,ζ̂_0,σ̂_J,0^2)=
    1/{π[σ_w^2+σ̂_J,0^2]}^N exp[-‖r_0‖ ^2/σ_w^2+σ̂_J,0^2]
    
    p_is(y_s|ℋ_1,ζ̂_1,σ̂_J,1^2)=
    1/{π[σ_e^2(1_K)+σ̂_J,1^2]}^N exp[-‖r_1‖ ^2/σ_e^2(1_K)+σ̂_J,1^2]

Taking ln(·) of the concentrated likelihood ratio { p_is(y_s|ℋ_1,ζ̂_1,σ̂_J,1^2)/p_is(y_s|ℋ_0,ζ̂_0,σ̂_J,0^2)}
provides the final expression:

    Λ_IS-GLR≜   { N ln[σ_w^2+σ̂_J,0^2/σ_e^2(1_K)+σ̂_J,1^2]-‖r_1‖ ^2/σ_e^2(1_K)+σ̂_J,1^2.
       .+‖r_0‖ ^2/σ_w^2+σ̂_J,0^2}

The proposed rule, in analogy to Sec. <ref>, will be referred
to as IS-GLRT in the following.



 §.§ NLOS-GLRT rule


Differently, here we start by using the NLOS assumption (κ_k=0,
k∈𝒦) on the conditional received signal pdf, which
gives:

    y_s|ℋ_i∼∑_x∈𝒳^KP(x|ℋ_i) 𝒩_ℂ(A_J(ϕ) ζ, [σ̅_e^2(x)+σ_J^2] I_N)

where σ̅_e^2(x)≜(σ_w^2+∑_k=1^Kβ_kx_k).
Even under such a simplifying assumption, Eq. (<ref>)
still has the form of a complex Gaussian mixture with 2^K distinct
components, thus being intractable from a computational point of view.
Thus, we further resort to Gaussian moment matching to fit the pdf
of y_s|ℋ_i to a (proper) complex Gaussian pdf
as follows:

    𝔼{y_s|ℋ_i}=A_J(ϕ) ζ    Σ_y_s|ℋ_i=(σ_n,i^2+σ_J^2) I_N

where we have denoted σ_n,i^2≜(∑_k=1^KP_i,k β_k+σ_w^2).
Therefore, moment matching yields:

    y_s|ℋ_i∼𝒩_ℂ(A_J(ϕ) ζ, [σ_n,i^2+σ_J^2] I_N)

Now, in order to obtain a GLRT-like statistic, we would need to evaluate
the ML estimates of {ζ,σ_J^2} under ℋ_i
for the matched model in Eq. (<ref>). This
is the case for the ML estimates of ζ under ℋ_0
and ℋ_1, being both equal to ζ̂_0
(cf. Eq. (<ref>)). After substitution,
the concentrated matched likelihood of y_s|ℋ_i
is:

    p_nl(y_s|ℋ_i;ζ̂,σ_J^2)=1/{π[σ_n,i^2+σ_J^2]}^N exp[-‖r_0‖ ^2/σ_n,i^2+σ_J^2]

where r_0 has the same meaning as for IS-GLRT rule. After
substitution, it is not difficult to prove that the “moment-matched”
concentrated likelihood ratio

    p_nl(y_s|ℋ_1;ζ̂_0,σ_J^2)/p_nl(y_s|ℋ_0;ζ̂_0,σ_J^2)

is an increasing function of ‖r_0‖ ^2,
independently on the value of the (unknown) σ_J^2, whose
estimation can be thus avoided (the proof can be obtained by taking
the logarithm of (<ref>) and exploiting
P_D,k≥ P_F,k). Therefore, the test deciding for ℋ_1
when Λ_NL-GLR>γ, where

    Λ_NL-GLR≜‖P_A_J(ϕ)^⊥y_s‖ ^2

is uniformly most powerful under NLOS assumption and after moment
matching. For the mentioned reason, the present test, denoted here
as NLOS-GLRT (in analogy to Sec. <ref> and
with a slight abuse of terminology, since estimation of σ_J^2
is not needed for test implementation), will be employed in our comparison.



 §.§ IGMM-GLRT rule


It can be readily shown that the characterization up to the second
order in Eqs. (<ref>) and (<ref>)
generalizes to:

    𝔼{y_s|ℋ_i}=t_i+A_J(ϕ) ζ
    Σ_y_s|ℋ_i=Σ_y|ℋ_i+σ_J^2 I_N    Σ̅_y_s|ℋ_i=Σ̅_y|ℋ_i

where we have denoted t_i≜𝔼{y|ℋ_i}
(cf. Eq. (<ref>)). We first match the pdf
of the vector y_s|ℋ_i to that of an improper
complex Gaussian vector, that is:

    y_s|ℋ_i∼𝒩_ℂ(t_i+A_J(ϕ) ζ,Σ_y_s|ℋ_i,Σ̅_y_s|ℋ_i)

It is easy to verify that Eq. (<ref>)
is also equivalent to the following linear model:

    y_s|ℋ_i=t_i+A_J(ϕ) ζ+w_i

where w_i∼𝒩_ℂ(0_N,Σ_y|ℋ_i+σ_J^2 I_N,Σ̅_y|ℋ_i)
(i.e., a zero-mean non-circular complex Gaussian vector). Therefore,
when the hypothesis ℋ_i is in force, we define y_s,i≜(y_s-t_i)
and exploit the SVD of A_J(ϕ)=(U_J Λ_J V_J^†),
thus obtaining 

    y_s,i=U_J [        Λ_r; O_(N-r)× r ]_= Λ_J V_J^† ζ+w_i .

 where Λ_r∈ℂ^r× r denotes the (diagonal)
sub-matrix extracted from the matrix of the singular values Λ_J
(since the rank of the interference is equal to r). We then define
ζ^'≜(Λ_r V_J^† ζ)∈ℂ^r
and notice that ζ and ζ^' are in one-to-one
correspondence. Therefore, after a left-multiplication by U_J^†
(i.e., a unitary matrix, which does not entail loss of information),
Eq. (<ref>) is rewritten as follows:

    s_i    =[        I_r; O_(N-r)× r ]_≜Sζ^'+n_i

where s_i≜(U_J^† y_s,i)∈ℂ^N
and 

    n_i∼𝒩_ℂ(0_N, U_J^† Σ_y|ℋ_i U_J+σ_J^2 I_N, U_J^† Σ̅_y|ℋ_i U_J^*)

Then, we can define the following augmented model:

    s_i    =    S̅ ζ^'+n_i
    S̅   ≜   [[        I_r        O_r; O_(N-r)× r O_(N-r)× r;        O_r        I_r; O_(N-r)× r O_(N-r)× r ]]

where n_i∼𝒩_ℂ(0_2N,R_A,i),
and we have defined R_A,i≜(Σ_A,i+σ_J^2 I_2N),
Σ_A,i≜U̅_J^† Σ_y|ℋ_i U̅_J
and 

    U̅_J≜[   U_J   O_N;   O_N U_J^* ] .

Hence, the (matched) pdf of the augmented vector s_i
is given by <cit.>:

    p_igmm(s_i ;ζ^',σ_J^2|ℋ_i)=
    1/π^N(R_A,i)^1/2exp[-1/2(s_i-S̅ ζ^')^† R_A,i^-1 (s_i-S̅ ζ^')]

In order to obtain the IGMM-GLRT rule, we need the ML estimates of
{ζ^',σ_J^2}. First, the ML estimate
of ζ^' from Eq. (<ref>)
is readily given by ζ̂_̂î^̂'̂≜(S̅^† R_A,i^-1 S̅)^-1 S̅^† R_A,i^-1 s_i.
After substitution, the concentrated log-likelihood is:

    ln p_igmm(s_i;ζ̂_̂î^̂'̂,σ_J^2|ℋ_i)=-N lnπ-1/2ln(R_A,i)
    
    -1/2s_i^† [R_A,i^-1-R_A,i^-1 S̅ (S̅^† R_A,i^-1 S̅)^-1S̅^† R_A,i^-1 ] s_i

We now observe that S̅ is related to a conveniently defined
matrix T via a permutation matrix Γ, as shown
in Eq. (<ref>) at the top of next page. 

 Based on the aforementioned definition, Eq. (<ref>)
is rewritten as:

    ln p_igmm(s_i;ζ̂_̂î^̂'̂,σ_J^2|ℋ_i)=-N lnπ-1/2ln(R_A,i)
    
    -1/2m_i^† [R_p,i^-1-R_p,i^-1 T (T^† R_p,i^-1 T)^-1T^† R_p,i^-1 ] m_i

where m_i≜(Γ^† s_i)
and R_p,i≜(Γ^† R_A,i^-1 Γ)^-1=(Γ^† R_A,i Γ)
(since every permutation matrix is unitary, i.e., (Γ^†Γ)=(ΓΓ^†)=I_2N).
It can be recognized in second line of Eq. (<ref>)
that matrix in square brackets has the block structure (obtained by
exploiting the simplified structure of T) 

    [        O_2r O_2r×2(N-r); O_2(N-r)×2r    R_c,i^-1 ]

 where R_c,i^-1∈ℂ^2(N-r)×2(N-r) is the
Schur complement of the block (R_p,i^-1)_1:2r of matrix
R_p,i^-1 and can be identified from R_p,i as:

    R_p,i=[    _i    _i;  _i^† R_c,i ]

where _i∈ℂ^2r×2r and _i∈ℂ^2r×2(N-r),
respectively. Accordingly, the third term in Eq. (<ref>)
is equivalently written as -1/2 m_c,i^† R_c,i^-1 m_c,i,
where m_c,i≜(m_i)_2r+1:2N. Furthermore,
it is also apparent that R_c,i is in the form R_c,i=(Σ_c,i+σ_J^2 I_2(N-r)),
where Σ_c,i≜(Γ^†Σ_A,iΓ)_(2r+1:2N).
Therefore R_c,i^-1 has the eigenvalue decomposition R_c,i^-1=U_c,i [Λ_c,i+σ_J^2 I_2(N-r)]^-1 U_c,i^†.
Consequently, Eq. (<ref>)
can be expressed as

    ln[p_igmm(s̆_i;ζ̂^̂'̂,σ_J^2)]=     -N lnπ-1/2∑_n=1^2Nln[λ_A,i,n+σ_J^2]
        -1/2∑_ℓ=1^2(N-r)|v_i,ℓ|^2/λ_c,i,ℓ+σ_J^2 ,

 where λ_A,i,n and λ_c,i,ℓ are the eigenvalues
of Σ_A,i and Σ_c,i, respectively. Also,
in Eq. (<ref>) and we have denoted with v_i,ℓ
the ℓth element of v_i≜(U_c,i^† m_c,i).
We also remark that, because of Σ_A,i definition, the
eigenvalues λ_A,i,n are equal to those of Σ_y|ℋ_i.
Eq. (<ref>) can now be easily differentiated w.r.t.
σ_J^2 and set to zero in order to find the stationary
points. This is achieved via the solution of the polynomial equation:

    ∑_n=1^2N1/λ_A,i,n+σ_J^2=∑_ℓ=1^2(N-r)|v_i,ℓ|^2/(λ_c,i,ℓ+σ_J^2)^2

Clearly, given a set of stationary points (to which we must add the
boundary solution σ̂_J,i^2=0) say it σ̂_J,i^2(s),
the argument corresponding to the maximum likelihood of Eq. (<ref>)
is chosen as the actual σ̂_J,i^2, that is σ̂_J,i^2≜max_σ̂_J^2(s)≥0ln[p(s̆_i;ζ̂^̂'̂,σ̂_J^2(s))].
This is also implied by the objective function ln[p_igmm(s̆_i;ζ̂^̂'̂,σ_J^2)]→-∞
as σ_J^2 tends to +∞. Finally, IGMM-GLR statistic
is evaluated as

    Λ_IGMM-GLR≜-1/2∑_n=1^2Nln[λ_A,1,n+σ̂_J,1^2/λ_A,0,n+σ̂_J,0^2]
    
    -1/2{∑_n=1^2(N-r)|v_1,ℓ|^2/λ_c,1,ℓ+σ̂_J,1^2-∑_n=1^2(N-r)|v_0,ℓ|^2/λ_c,0,ℓ+σ̂_J,0^2} .

The procedure for evaluation of IGMM-GLR statistic is summarized in
Alg. <ref>. 






 §.§ Asymptotic equivalences in the presence of jammer


Hereinafter, we will turn our attention to asymptotic equivalence
properties of fusion rules which deal with the case of jammer presence,
specularly as in Sec. <ref>.

We first observe that, in the presence of jammer interference, it
is not difficult to show that a similar statement as that in Lem.
<ref> does not hold, since there is a different design
criterion between NLOS-GLRT and IS/IGMM-GLRT. Indeed, the former is
obtained by exploiting a monotonic concentrated LLR (under NLOS assumption,
after Gaussian moment matching and implicit estimation of ζ);
these assumptions allow avoiding the estimation of σ_J^2.
Therefore, NLOS-GLRT cannot be interpreted as a GLRT-like procedure
in a strict sense, since it implicitly estimates only ζ.
On the other hand, IGMM-GLRT and IS-GLRT rules are both constructed
on an estimate σ̂_J^2. Therefore, we cannot
expect the three rules to have identical performance in a NLOS setting,
as opposed to the “interference-free” scenario. However, an intuitive
argument on their NLOS behaviour can be drawn by analyzing the forms
of IS-GLR (cf. Eq. (<ref>)) and IGMM-GLR (cf. Eq. (<ref>))
under the aforementioned assumption. Indeed, by assuming that the
Rician factors κ_k→0, produces (after lengthy
manipulations):

    Λ=
    N lnσ_a^2/σ_b^2-‖r_0‖ ^2/σ_b^2+‖r_0‖ ^2/σ_a^2    if ‖r_0‖ ^2/N<σ_a^2
    
    N ln‖r_0‖ ^2/Nσ_b^2-‖r_0‖ ^2/σ_b^2+N     if σ_a^2≤‖r_0‖ ^2/N<σ_b^2
    
    0     if ‖r_0‖ ^2/N≥σ_b^2

where σ_a^2<σ_b^2 and their expressions are σ_a^2=σ_n,0^2=∑_k=1^Kβ_kP_F,k+σ_w^2
(resp. σ_b^2=σ_n,1^2=∑_k=1^Kβ_kP_D,k+σ_w^2)
for IGMM-GLR and σ_a^2=σ_w^2 (resp. σ_b^2=σ̅_e^2(1_K)=∑_k=1^Kβ_k+σ_w^2)
for IS-GLR, respectively. By looking at Eq. (<ref>),
it is apparent that both the statistics are increasing functions of
‖r_0‖ ^2 (i.e., the energy of the
received signal y_s after projecting out the LOS part of
the jammer interference) within [0,σ_b^2]. Therefore,
the higher σ_b^2 the more the statistic function in Eq. (<ref>)
will be safely approximated by an increasing function of ‖r_0‖ ^2.
Additionally, every statistic being an increasing function of ‖r_0‖ ^2
will experience the same performance as the NLOS-GLRT (we recall that
such test is constructed simply comparing ‖r_0‖ ^2
to a suitable threshold, cf. Eq. (<ref>)). Such test
is obtained without explicitly estimating σ_J^2 and by
claiming uniformly most powerfulness after moment matching of the
statistic ‖r_0‖ ^2. The use of this
test allows avoiding a performance loss attributed to the fact that,
under a NLOS assumption, we are testing (after moment matching)

    (σ_n,0^2+σ_J^2)      under ℋ_0
    
    (σ_n,1^2+σ_J^2)      under ℋ_1

with σ_J^2 being unknown. Clearly, if we are faced to
estimate σ_J^2 under the condition σ_J^2≥(σ_n,1^2-σ_n,0^2),
discrimination among the two hypotheses is not achievable. Indeed,
the uncertainty interval of σ_J^2 (i.e., [0,+∞))
produces overlapping intervals for the overall variance under both
hypotheses (i.e., [σ_n,0^2,+∞) and [σ_n,1^2,+∞),
respectively) and therefore, when the aforementioned condition is
satisfied, the correct hypothesis cannot be declared on the basis
of a simple variance estimation. Additionally, since σ_b^2
is higher for IS-GLR than for IGMM-GLR (as P_D,k≤1, k∈𝒦),
we can expect IS-GLRT to perform better than IGMM-GLRT in a
NLOS WSN situation, especially when σ_J^2 becomes large
(which is either the case of a jammer emitting a high power signal
or experiencing mostly a NLOS channel condition).

Finally, we show that an analogous form of Lem. <ref>
holds for IS-GLRT and IGMM-GLRT in a setup with an operating jammer,
as stated hereinafter.

Under “IS” assumption, IGMM-GLRT rule is statistically equivalent
to IS-GLRT rule (and thus attains exact GLRT performance).


Clearly, under IS assumption, IS-GLRT is statistically equivalent
to the exact GLRT in Eq. (<ref>), by construction.
Then, we need only to show that IGMM-GLRT is statistically equivalent
to IS-GLRT. Indeed, under IS assumption, 𝔼{x|ℋ_1}=1_K,
𝔼{x|ℋ_0}=0_K and Σ_x|ℋ_i=O_K
hold, respectively. Therefore, the second order characterization needed
for IGMM-GLRT in Eqs. (<ref>) and (<ref>)
reduces to:

    𝔼{y_s|ℋ_i}=μ_i+A_J(ϕ) ζ
    Σ_y_s|ℋ_i=(σ_e,i^2+σ_J^2) I_N    Σ̅_y_s|ℋ_i=O_N

where the equalities σ_e,1^2=σ_e^2(1_K),
σ_e,0^2=σ_w^2, μ_1=A(θ) 1_K
and μ_0=0_N hold, respectively. It is apparent
that the simplified characterization in Eqs. (<ref>)
coincides with that in Eq. (<ref>). Since both
rules are obtained with a GLRT-like approach, this proves their statistical
equivalence.

Then, when sensors are ideal, IGMM-GLRT rule will be statistically
equivalent to IS-GLRT (viz. GLRT) rule, as there is no covariance
structure change between the two hypotheses. On the other hand, we
expect that when the WSN operates with “good-quality” sensors,
NLOS-GLRT will experience some performance loss with respect to IS-GLRT
and IGMM-GLRT rules, since it does not exploit the LOS part of the
sensors channel vectors.



§ COMPLEXITY ANALYSIS


In Tab. <ref> we compare the computational
complexity of the proposed rules, where 𝒪(·) indicates
the usual Landau notation (i.e., the order of complexity). The results
underline the computations required whenever each new y is
transmitted (assuming static parameters pre-computed and stored in
a suitable memory). First, as previously remarked, it is apparent
that the optimum rule (i.e. the LLR) is unfeasible, especially when
K is very large. Differently, all the proposed rules have polynomial
complexity w.r.t both K and N (as well as r, when jammer-robust
rules are considered). The computational complexity of IS rule is
mainly given by the computation of the scalar product and energy needed
to evaluate Eq. (<ref>), while the dominant term in the
case of IS-GLRT is represented by the evaluation of the energy of
r_0 and r_1, respectively (recall that the orthogonal
projector of interference can be written as P_A_J(ϕ)^⊥=U_J,⊥U_J,⊥^†,
where U_J,⊥ collects the last (N-r) columns of the
eigenvector matrix U_J). Similar considerations (as IS rule)
hold for NLOS (which simply requires ‖y‖ ^2),
whereas NLOS-GLRT similarly (as IS-GLRT) requires first a projection
operation, that is, evaluation of P_A_J(ϕ)^⊥y_s.
Furthermore, a linear dependence with N, as IS and NLOS rules,
holds for WL rules (see Eq. (<ref>)). Differently,
IGMM rule is based on the computation of a quadratic form of y,
which leads to 𝒪(N^2) complexity. A higher complexity
is also required by IGMM-GLRT, whose dominant terms are given by:
(i) the computation of v_i (see definition provided in
Sec. <ref>) and the solution to a polynomial
equation of order p_ord≜2N+4(N-r)-1. The solution
is known to have a complexity 𝒪(p_ord^4τ^2)
(e.g. following Sturm approach <cit.>), where τ
is a parameter related to the bit resolution of the maximum value
among the known coefficients.






§ SIMULATION RESULTS




 §.§ Setup description and measures of performance


We consider sensors deployed in a 2-D circular area around the DFC
(placed in the origin, whose cartesian coordinates are denoted as
(x_dfc,y_dfc)) with radius r_max=1000 m.
Sensors are located uniformly at random (in Cartesian coordinates,
denoted as (p_x,k,p_y,k), k∈𝒦) and we assume
that no sensor is closer to the DFC than r_min=100 m.
The large-scale fading is modelled via β_k=ξ_k(r_min/r_k)^L,
where ξ_k is a log-normal random variable, i.e., 10log_10(ξ_k)∼𝒩(μ_P,σ_P^2),
where μ_P and σ_P are the mean
and standard deviation in dBm, respectively. Moreover,
r_k denotes the distance between the kth sensor and the DFC
and L represents the path-loss exponent (for our simulations, we
choose L=2). In the following, we assume (μ_P,σ_P)=(15,2)
for the WSN. Additionally, we suppose that the DFC is equipped with
a half-wavelength spaced uniformly linear array and that kth sensor
is seen at the DFC as a point-like source, that is 

    a(θ_k)=[                 1      e^jπcos(θ_k)                 ⋯ e^jπ(N-1)cos(θ_k) ]^T

where clearly θ_k=arccos[x_dfc-p_x,k/y_dfc-p_y,k].
A similar procedure is employed for the generation of jammer parameters,
with reference to a case of a jamming device distributed in angular
space. The sole difference is in the choice (μ_P,σ_P)=(25,2),
reflecting a non-negligible jammer power received by the DFC.

Also, the Rician factors of the sensors κ_k, k∈𝒦,
are uniformly generated within [κ_min,κ_max].
Such interval will be varied in order to generate three typical scenarios
corresponding to a WSN with “LOS”, “Intermediate” and “NLOS”
channel situations, in order to comprehensively test the proposed
fusion rules. More specifically, we will consider Rician factors generated
randomly as: (i) [κ_min,κ_max]=[10,20] (dB)
(LOS scenario), (ii) [κ_min,κ_max]=[-10,10] (dB)
(Intermediate scenario) and [κ_min,κ_max]=[-20,-10] (dB)
(NLOS scenario). Similar reasoning is applied to the generation of
Rician factors for the jammer, where two different scenarios are also
considered: (a) [κ_min,κ_max]=[10,20] (dB)
(LOS jammer) and (b) [κ_min,κ_max]=[-10,10] (dB)
(weak-LOS jammer).

The three generated WSN examples are shown in Fig. <ref>,
where the corresponding angles-of-arrival (θ_k, k∈𝒦),
the averaged total received and LOS powers per antenna ((β_k,b_k^2β_k),
k∈𝒦 ) are shown for the case of K=14 sensors. Also,
in each of the subfigures, we illustrate the corresponding DOAs (ϕ_ℓ,
ℓ=1,…,r), the averaged total received and LOS
powers per antenna ((β_ℓ,J,b_ℓ.J^2β_ℓ,J),
ℓ=1,…,r) of a jammer distributed in the angular
space with r=2, whose Rician factors are generated according to
scenarios (a) (LOS jammer scenario) and (b) (weak-LOS jammer
scenario), respectively. Finally, for simplicity we assume conditionally
i.i.d. decisions, that is P(x|ℋ_i)=∏_k=1^KP(x_k|ℋ_i)
with (P_1,P_0)=(P_D,P_F)=(0.5,0.05). In this case, ρ_i=P_i1_K
and Σ_x|ℋ_i=P_i(1-P_i) I_K
hold, respectively.

The performances of the proposed rules are analyzed in terms of system
probabilities of false alarm and detection, defined respectively as

    P_F_0≜ P(Λ>γ|H_0),   P_D_0≜ P(Λ>γ|H_1),

with Λ representing the statistic associated to the generic
fusion rule and γ the corresponding threshold.






 §.§ Fusion Rules Comparison


P_D_0 vs. noise level σ_w^2 (No-interference):
First, the scenario with no jammer is addressed. In Figs.
<ref>, <ref>
and <ref>, we show P_D_0 vs. σ_w^2,
under the constraint P_F_0=0.01 for the “LOS”, “Intermediate”
and “NLOS” setups in Figs. <ref>, <ref>
and <ref>, respectively (K=14 sensors and
N∈{2,6} antennas at the DFC). Clearly, LLR performs the best
among all the considered rules. Secondly, WL rules are very close
to the LLR in the “LOS” setup (indeed in the conditionally i.i.d.
case and at high SNR, for a LOS condition it approximately holds z_ WL,i∝(A(θ) A(θ)^†)^-1A(θ)1_k,
that is WL rules both approximate through right-pseudoinverse
operation a counting rule, being optimal in this specific scenario)
with increasing performance loss in the “Intermediate” and “NLOS”
setups, respectively. Such a trend is in agreement with Lem. <ref>,
which states that as NLOS assumption is verified, the optimum statistic
should possess a dependence on ‖y‖ ^2,
which is not the case of WL rules. Also, IGMM, IS and NLOS rules have
a performance behaviour in line with the asymptotic equivalences shown
in Sec. <ref>. Clearly, NLOS setup is
such that performance of IGMM, IS and NLOS rules (almost) coincide.
On the other hand, in the LOS scenario, IS and IGMM rules are very
close (the “weak-LOS” assumption is almost satisfied), while NLOS
rule experiences a certain performance loss. Finally, we underline
that the benefit of improved number of antennas is only experienced
by LLR, WL and IGMM rules. Differently, NLOS and IS rules do
benefit of a larger DFC array only in the case of low SNR or
NLOS setup. This can be attributed to the fact that only in these
conditions there is no significant (pseudo-)covariance structure change
between the two hypotheses (see (<ref>) and
(<ref>)). Then NLOS and IS rules, not exploiting
(at least) a second-order characterization of y|ℋ_i,
are not able to benefit from increase of N in the remaining cases.





P_D_0 vs. noise level σ_w^2 (Interference):
A similar scenario is shown in Figs. <ref>,
<ref> and <ref>,
where we show P_D_0 vs. σ_w^2, under the constraint
P_F_0=0.01 for the “LOS”, “Intermediate” and “NLOS”
setups in Figs. <ref>, <ref>
and <ref>, respectively (K=14, both jammer
scenarios considered), and N=6 antennas at the DFC. For the sake
of completeness, the performance of clairvoyant LRT are also reported
(cf. Eq. (<ref>)). We first notice that
IS-GLRT, NLOS-GLRT and IGMM-GLRT outperform IS, NLOS and IGMM rules
(whose performance are obtained by ignoring the presence of the jamming
signal), respectively, unless there is a significant receive noise
σ_w^2 (i.e., low SNR); such trend is more apparent when
moving to a WSN-DFC channel which experiences a LOS scenario (cf.
Fig. <ref>). Indeed, in such a case,
jammer interference suppression may come up at the expenses of (partial)
cancellation of some of the sensors contributions. Indeed, in a LOS
scenario and at low SNR, jammer interference suppression may not be
beneficial as the scenario is noise-dominated. On the other
hand, in a LOS scenario and at high SNR, the problem becomes interference-dominated;
therefore an effective jammer signal suppression significantly improves
performance, even at the expenses of (partial) elimination of some
sensors contributions. The sole exception to these considerations
is represented by IGMM-GLRT in a NLOS WSN scenario (cf. Fig. <ref>),
where performance are observed to be worse than its interference-unaware
counterpart (i.e., IGMM rule) over all the σ_w^2 range
considered. Such evidence can be attributed to the overlapping of
unknown parameter support under the two hypotheses, due to σ_J^2
(cf. Sec. <ref>), which does
not allow to achieve satisfactory performance.





P_D_0 vs. number of antennas N (Interference): The
benefits of increasing the number of antennas on jammer suppression
capabilities for the designed rules are illustrated in Figs. <ref>,
<ref>, and <ref>,
respectively. More specifically, it is shown P_D_0 vs. N,
under the constraint P_F_0=0.01 and σ_w^2=0 dBm
for the “LOS”, “Intermediate” and “NLOS” setups in Figs.
<ref>, <ref>
and <ref>, respectively. First of all, we
notice that P_D_0 for all the “interference-aware” fusion
rules increases with N. Furthermore, the gain with respect to their
corresponding “interference-unaware” counterparts improves as
well. This is true in the case of IS-GLRT and NLOS-GLRT for all the
scenarios considered, since the considered noise level σ_w^2
implies a moderate SNR and due to improved jamming-suppression capabilities
with higher N. Again, the only exception is given by IGMM-GLRT
in the NLOS case, given the overlapping of unknown parameter support
under both the hypotheses due to σ_J^2. By looking at
the specific example, in the LOS WSN scenario, the IGMM-GLRT has the
best trend with the number of antennas, as significant pseudo-covariance
structure change in the hypotheses is implied in such scenario (therefore
second-order characterization of y|ℋ_i is beneficial).
Differently, in the “Intermediate” and “NLOS” WSN setups,
IS-GLRT and NLOS-GLRT represent the best alternatives, with IS-GLRT
slightly outperforming NLOS-GLRT.








§ CONCLUSIONS


In this paper we studied channel-aware DF in a WSN with interfering
sensors whose channels are modelled as Rician and their NLOS components
are not known at the DFC (i.e., they are not estimated), focusing
on anomaly detection problems. We developed five sub-optimal fusion
rules (i.e., IS, NLOS, WL and IGMM rules) in order to deal with the
exponential complexity of LRT. For the present setup, the following
performance trends have been observed:


  * In a WSN with a LOS setup, WL rules represent the best (and more convenient)
alternative to the LLR, whereas the same rules suffer from severe
performance degradation in a NLOS setup. On the other hand, NLOS rule
is mainly appealing in a NLOS setup, as the IS rule, which also achieves
satisfactory performance in a weak-LOS condition. Indeed, in the latter
case they are both able to exploit an increase in the number of receive
antennas, as well as in the case of low SNR. Finally IGMM rule, exploiting
a second-order characterization of the received vector under both
hypotheses, has the most appealing performance when considering all
the three scenarios.

Successively, we considered a scenario with a (possibly distributed)
“Rician” jamming interference and tackled the resulting composite
hypothesis testing problem within the GLRT framework. More specifically,
we developed sub-optimal GLRT-like decision rules which extend IS,
NLOS and IGMM rules to the case of subspace interference. With reference
to these rules, the following trends have been observed:


  * All the considered “interference-aware” rules (IS-GLRT, NLOS-GLRT
and IGMM-GLRT) significantly outperform the “interference-unaware”
counterparts in the case of a moderate-to-high SNR level and non-negligible
LOS condition, as in such case system performance is interference-dominated
and thus interference suppression leads to a remarkable gain. Also,
it has been shown that all these rules benefit from increase of N
for enhanced interference suppression, with the sole exception of
IGMM-GLRT in a NLOS case (due to lack of identifiability). Numerical
evidence has also underlined the appeal of IGMM-GLRT and IS-GLRT in
LOS and Intermediate/NLOS setups, respectively;

Finally, asymptotic equivalences established among all these rules
in the case of either interference-free or interference-prone scenarios
were confirmed by simulations. Future research tracks will concern
theoretical performance analysis of the proposed rules and design
of advanced fusion schemes robust to smarter jammers.



In this appendix we will provide the second-order characterization
of y|ℋ_i.
The mean vector 𝔼{y|ℋ_i} is evaluated
as:

    𝔼{y|ℋ_i}=𝔼{H̅D^1/2x+w|ℋ_i}=
    𝔼{H̅} D^1/2 𝔼{x|ℋ_i}=A(θ) ρ_i

where we have exploited 𝔼{w}=0_N and statistical
independence between fading coefficients and sensors decisions, respectively.
Finally in Eq. (<ref>) we have recalled the definitions
of matrix A(θ), whose kth column
equals μ_k=b_k √(β_k) a(θ_k),
and of ρ_i=[ P_i,1     ⋯ P_i,K ]^T.

Differently, the covariance matrix is expressed as:

    Σ_y|ℋ_i=
    𝔼{(y-A(θ) ρ_i) (y-A(θ) ρ_i)^† |ℋ_i}=
    A(θ) 𝔼{(x-ρ_i)(x-ρ_i)^T |ℋ_i} A(θ)^† +
    𝔼{(H B_s x)(H B_s x)^† |ℋ_i}+𝔼{ww^†}=
    A(θ) Σ_x|ℋ_i A(θ)^†+
    𝔼{(∑_k=1^Kh_k √(ν_k) x_k)(∑_ℓ=1^Kh_ℓ^† √(ν_ℓ) x_ℓ)|ℋ_i} +σ_w^2 I_N

where B_s≜(I_K-R^2)^1/2D^1/2
and we recall ν_k=(1-b_k^2)β_k. The second term in
Eq. (<ref>) can be simplified as

    𝔼{(∑_k=1^Kh_k √(ν_k) x_k)(∑_ℓ=1^Kh_ℓ^† √(ν_ℓ) x_ℓ)|ℋ_i} =
    ∑_k=1^K𝔼{h_k h_k^†} ν_k 𝔼{x_k^2|ℋ_i}=∑_k=1^Kν_k P_i,k I_N

which follows from mutual independence of vectors h_k, k∈𝒦.
Then, substituting back Eq. (<ref>) in Eq. (<ref>)
gives:

    Σ_y|ℋ_i=A(θ) Σ_x|ℋ_i A(θ)^†+σ_e,i^2 I_N

where σ_e,i^2≜[∑_k=1^Kν_k P_i,k+σ_w^2].
Analogously, we can evaluate the pseudo-covariance of y|ℋ_i
as

    Σ̅_y|ℋ_i=𝔼{(y-A(θ) ρ_i) (y-A(θ) ρ_i)^T |ℋ_i}
    
    =A(θ) 𝔼{(x-ρ_i)(x-ρ_i)^T |ℋ_i} A(θ)^T +
    𝔼{(H B_s x)(H B_s x)^T |ℋ_i}
    
    =A(θ) Σ_x|ℋ_i A(θ)^T+
    𝔼{(∑_k=1^Kh_k √(ν_k) x_k)(∑_ℓ=1^Kh_ℓ^T √(ν_ℓ) x_ℓ)|ℋ_i}

since 𝔼{w w^T}=O_N (i.e., the noise
is assumed circular). Also, it can be shown that the second term in
Eq. (<ref>) is a null matrix, since

    𝔼{(∑_k=1^Kh_k √(ν_k) x_k)(∑_ℓ=1^Kh_ℓ^T √(ν_ℓ) x_ℓ)|ℋ_i} =
    ∑_k=1^K𝔼{h_k h_k^T} ν_k 𝔼{x_k^2|ℋ_i}=O_N

since the NLOS fading vector h_k is assumed circular. Therefore
the final expression for the pseudo-covariance is:

    Σ̅_y|ℋ_i=A(θ) Σ_x|ℋ_i A(θ)^T

Eq. (<ref>) is not a null matrix, thus
motivating augmented form processing.

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