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Digital Object Identifier 10.1109/OJCOMS.2024.011100
Tensor Signal Modelling and Channel
Estimation for Reconfigurable
Intelligent Surface-Assisted
Full-Duplex MIMO
Alexander James Fernandes1(Graduate Student Member, IEEE), Ioannis Psaromiligkos 1
(Member, IEEE)
1Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada. Email: alexander.fernandes@mail.mcgill.ca;
ioannis.psaromiligkos@mcgill.ca
CORRESPONDING AUTHOR: Alexander James Fernandes (alexander.fernandes@mail.mcgill.ca)
This work was supported in part by the Natural Science and Engineering Research Council of Canada under the Discovery Grant Program
and in part by the Vadasz Scholar McGill Engineering Doctoral Award.
ABSTRACT Channel estimation is one of the main challenges for reconfigurable intelligent surface
(RIS) assisted communication systems with passive reflective elements due to the high number of
parameters to estimate. In this paper, we consider channel estimation for a MIMO FD RIS-assisted
wireless communication system and use tensor (multidimensional array) signal modelling techniques to
estimate all channel state information (CSI) involving the self-interference, direct-path, and the RIS assisted
channel links. We model the received signal as a tensor composed of two CANDECOMP/PARAFAC (CP)
decomposition terms for the non-RIS and the RIS assisted links. Based on this model we extend the
alternating least squares algorithm to jointly estimate all channels, then derive the corresponding Cram´
er-
Rao Bounds (CRB). Numerical results show that compared to recent previous works which estimate the
non-RIS and RIS links during separate training stages, our method provides a more accurate estimate by
efficiently using all pilots transmitted throughout the full training duration without turning the RIS off
when comparing the same number of total pilots transmitted. For a sufficient number of transmitted pilots,
the proposed method’s accuracy comes close to the CRB for the RIS channels and attains the CRB for
the direct-path and self-interference channels.
INDEX TERMS channel estimation, reconfigurable intelligent surface, MIMO, CANDECOMP/PARAFAC,
tensor modelling, full-duplex.
I. Introduction
The reconfigurable intelligent surface (RIS), a 2D surface
composed of several passive reflecting elements with the
ability to electronically control the path of reflection, has
been gaining interest as an effort to improve the future of
wireless communication systems [1]–[3]. The RIS assists
in the quality and reliability of wireless transmission by
establishing a dominant line of sight communication link
between end points which mitigates undesirable multi-path
fading. The RIS also allows for in-band full-duplex (FD)
operation without requiring extra antenna processing on
the radio frequency chain, achieving higher energy effi-
ciency than FD relays at sub-6 GHz [4]. For FD RIS-
assisted communication systems, obtaining the channel state
information (CSI) is required to solve problems based on
controlling the RIS elements involving beamforming, self-
interference cancellation, and sum rate maximization [5]–
[8]. In recent years there have been significant advances
in obtaining the CSI for RIS-assisted communications as
outlined in the survey paper [9], however, only a few studies
have been conducted on channel estimation for RIS assisted
FD systems [10]–[16]. Most of these studies only consider
special cases of FD transmission where [10] only uses one
FD user, [11]–[14] neglect estimating the direct-path or self-
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Fernandes et al.: Tensor Signal Modelling and Channel Estimation for Reconfigurable Intelligent Surface-Assisted Full-Duplex MIMO
interference channels, and [15] assumes channel reciprocity,
with [16] being the only study to consider estimating all
communication channels part of the MIMO FD RIS-assisted
model without assuming channel reciprocity.
Multidimensional arrays, commonly known as tensors
have been proven useful in solving model identification
and channel estimation problems. A list of notable papers
and tutorials on signal processing with tensors include
[17]–[20]. Not surprisingly, tensor signal modelling has
been used in a variety of wireless communication system
models including multicarrier multiple-input-multiple-output
(MIMO), MIMO radar, relays, and RIS [21]–[26]. For
channel estimation in RIS-aided systems, in particular, the
CANDECOMP/PARAFAC (CP) decomposition of a tensor
is a popular method to model the receive signal for the RIS-
assisted communication link and estimate the transmitter-
RIS and RIS-receiver channels individually. Use of tensors
to obtain the RIS CSI was proposed by Wei et. al. in [27]
and de Araujo and de Almeida in [28]. In both studies,
by implementing a block transmission training scheme and
ignoring the direct-path, the HD RIS MIMO receive signal
can be reconstructed into a third order tensor. Through CP
decomposition of the tensor receive signal, the RIS channels
can then be estimated with the alternating least squares
(ALS) algorithm. Further in-depth analysis of this research
was performed in [29] and [30], deriving the corresponding
Cram´
er-Rao Bounds (CRBs) of the access point (AP) to
RIS to user equipment (UE) cascaded channel (excluding the
direct-path channel), and providing feasibility conditions on
the training scheme to obtain the CSI based on the structure
of the communication model. Afterwards, studies on tensor
signal modelling for RIS-assisted communication extended
beyond narrowband channel estimation into wideband chan-
nel estimation [31], joint channel and signal estimation with
CP decomposition [32], then with PARATUCK decompo-
sition [33], channel tracking [34], RIS in UAV-aided com-
munications [35], time-varying phase noise estimation [36],
reducing control overhead for RIS phase shift optimization
with CP and Tucker decomposition [37].
Among the papers surveyed on tensor signal modelling
for RIS channel estimation, only the HD pilot transmission
(uplink/downlink transmission) through the RIS is studied,
assuming the direct-path is blocked or estimated from a pre-
vious training stage. This is an important issue to highlight as
improving communication through FD transmission requires
knowledge of estimating both the RIS assisted channels and
the non-RIS assisted channels: the direct-path between the
AP and UE or the self-interference at the AP and UE.
However, having separate channel estimation stages where
the RIS is turned “off” then “on” leads to extra pilot overhead
compared to jointly estimating all channels with what the
least squares solution provides in [38], [39] for HD and [16]
for FD scenarios.
This work extends the results from our conference paper
[40] which described an ALS algorithm to jointly estimate
the direct-path and RIS path channels using two CP decom-
position terms for RIS MIMO HD systems. Specifically, we
propose a channel estimation method for RIS MIMO FD
systems using tensor signal modelling to estimate all com-
munication channels involving the self-interference, direct-
path, and RIS channels. Our main contributions are:
•We build upon the works in [5]–[8], [16], [18], [29],
[30] to derive a new third-order tensor signal model
for a narrowband FD RIS MIMO system allowing for
simultaneous transmission from the AP and all UEs.
•We formulate our FD tensor signal model in terms of
two CP decompositions based on the communication
link: the non-RIS assisted channels (self-interference
and direct-path channels) and the RIS assisted channels.
•We extend the ALS algorithm to jointly estimate the
direct-path, self-interference, and RIS channels of our
FD tensor signal model without requiring two separate
training stages or turning the RIS “off”.
•We derive the CRB of all channel parameters for our FD
tensor signal model. In contrast to previously derived
CRBs [29], [30], we take advantage of the tensor
signal model being composed as the summation of two
CP decomposition terms with additive white Gaussian
noise (AWGN) to obtain the individual CRBs of the
direct-path, self-interference, and RIS channels.
•We provide design recommendations on the pilots and
RIS phase shifts for our training scheme based on our
tensor composed of two CP decompositions compared
to [29], [30] for a tensor composed of one CP decompo-
sition. Based on the design recommendations, we show
that it is possible to reduce the pilot overhead training
duration depending on the structure of the channel
matrices.
To maximize the system throughput between the AP and
all UEs, it is important to have accurate estimates of all
communication channels received at the AP [8]. We note
that our tensor signal model represents the FD RIS-assisted
communication model better than other channel estimation
papers and that we are able to estimate the channels more
accurately. Other channel estimation papers do not use
an appropriate communication model when estimating the
channels by assuming channel reciprocity [15], neglect the
reflecting self-interference signal from the AP-RIS-AP [12],
the direct-path is blocked [14], or the self-interference is
part of the AWGN term [14]. In our tensor model, the non-
RIS assisted links are always present in the communication
link unless these channel paths themselves are zero which is
unlikely to happen due to at least the self-interference always
being present in FD communication [41].
The rest of the paper is organized as follows. In Section
II we provide the notation used in this paper and a review
on CP decomposition. Section III outlines the RIS assisted
FD MIMO communication system, the block transmission
scheme, and the proposed tensor signal model. In Section IV
2 VOLUME ,
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we propose channel estimation methods based on an iterative
ALS algorithm [18] and derive the corresponding Cram´
er-
Rao Bound for the tensor signal model. We discuss the
training scheme in Section V, where we provide the system
model design recommendations for the tensor signal model,
pilot transmission schemes and an RIS phase shift scheme.
The accuracy and training overhead of all channel estimation
methods are evaluated through numerical simulations in
Section VI, and we conclude this paper in Section VII.
II. Notation and Background
A. Notation
Column vectors are denoted as boldface lowercase (a),
matrices as boldface uppercase (A), and scalars as uppercase
(A) or lowercase (a). Tensors are symbolized by calligraphic
letters (A). The notation [A]i,j represents the element of the
i-th row and j-th column of the matrix A. A colon :is
used to denote all elements in a given dimension, e.g., [A]:,j
represents a column vector with all row elements from the
j-th column of A. The identity matrix of dimensions N×N
is IN,0M×Nis an M×Nmatrix of zeros, and 1M×Nan
M×Nmatrix of ones.
Operations on a matrix Aare denoted as: conjugate A∗,
transpose AT, conjugate transpose AH, inverse A−1, right
pseudoinverse A†
right =AH(AAH)−1and left pseudoinverse
A†
left = (AHA)−1AH. A diagonal square matrix with the
elements of a vector don its diagonal is expressed as
diag(d). The operator Db(A)forms a diagonal matrix with
the diagonal being the b-th row vector of A. The function
vec(A)creates a vector by stacking the columns of A.
The floor of xis ⌊x⌋, and mod is the modulo operator.
The L2norm of a vector is denoted as ∥a∥2and the
Frobenius norm of a matrix as ∥A∥F. The trace of Ais
denoted as Tr(A), and its rank as rank(A); the Kruskal rank
kA≤rank(A)of Ais the largest number ksuch that every
set of kcolumns drawn from Ais linearly independent.1
The outer product of two vectors is denoted as a◦b. The
matrix products are denoted as: Hadamard ⊙, Kronecker ⊗,
and Khatri-Rao ⋄. In this paper we make use of the following
identities:
(A⋄B)(A⋄B)H=AAH⊙BBH(1)
(A⋄B)H(A⋄B) =AHA⊙BHB(2)
and of the following Lemma:
Lemma 1 ([17]).For A∈CI×Rand B∈CJ×R, if kA≥1
and kB≥1, then
kA⋄B≥min(kA+kB−1, R)(3)
A circular complex multivariate Gaussian distribution with
mean µand covariance Σis denoted as CN(µ,Σ). Finally,
the expectation operator is denoted as E[·].
1Examples of cases where it is possible for kA<rank(A)is when
matrix Ahas an all zero-columns (i.e., kA= 0) or two columns are co-
linear (i.e., kA≤1).
FIGURE 1: CANDECOMP/PARAFAC Decomposition.
B. Review on CANDECOMP/PARAFAC Decomposition
For a third-order tensor X ∈ CI×J×K, CP decomposition
factorizes the (i, j, k)-th entry [18], [20] as:
[X]i,j,k =
R
X
r=1
[A]i,r[B]j,r [C]k,r (4)
where A∈CI×R,B∈CJ×R, and C∈CK×Rare called the
factor matrices. The r-th column vectors ar∈CI,br∈CJ,
and cr∈CKof the factor matrices A,B, and C, respectively,
are called the rank-one tensors that sum to Xthrough outer
products as depicted in Fig 1. CP decomposition notation is:
X=
R
X
r=1
ar◦br◦cr= [[A,B,C]] (5)
The rank of a tensor is the minimum number of Rrank-one
tensors needed to sum to X.
Given a tensor that has rank R, the CP decomposition is
unique if the factor matrices A,B, and Cas described in (5)
is the only way to sum to Xapart from permutation and scal-
ing of the factor matrices. Permutation indeterminacy means
that the rank-one tensor terms can be reordered arbitrarily
X= [[A,B,C]] = [[AΠ,BΠ,CΠ]] for any R×Rpermutation
matrix Π. Scaling indeterminacy means that we can scale the
rank-one tensors X=PR
r=1(αrar)◦(βrbr)◦(γrcr)as long
as αrβrγr= 1 for r∈ {1, . . . , R}. A sufficient condition
for the uniqueness of a CP decomposition is Kruskal’s rank
condition which states that if kA+kB+kC≥2R+ 2 then
the factor matrices A,B, and Cof Xwill be unique up to
scaling and permutations [17], [42].
Using the factor matrices, we can “unfold” a third-order
tensor Xinto the following matrix forms:
X1=A(C⋄B)T=[X]:,:,1. . . [X]:,:,K (6)
X2=B(C⋄A)T=([X]:,:,1)T. . . ([X]:,:,K )T(7)
X3=C(B⋄A)T=([X]:,1,:)T. . . ([X]:,J,:)T(8)
where X1∈CI×JK ,X2∈CJ×I K , and X3∈CK×I J .
The third-order tensor is often written in terms of the matrix
frontal slices of Xas:
[X]:,:,k =ADk(C)BT∈CI×J(9)
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Fernandes et al.: Tensor Signal Modelling and Channel Estimation for Reconfigurable Intelligent Surface-Assisted Full-Duplex MIMO
C. Double CANDECOMP/PARAFAC Decomposition
In this section we describe the double CANDE-
COMP/PARAFAC (DCP) decomposition for decomposing
a tensor into a summation of two CP decompositions. In
2023, B. Lovitz and F. Petrov showed that the uniqueness of
a tensor decomposition can be described using set theory
of a vector space instead of factor matrices, providing a
generalization of Kruskal’s theorem of tensors given in
[43, Theorem 2]. We describe that theorem for third-order
tensors, below.
Theorem 1. For vector spaces V1,V2,V3, over the field C,
let V=V1◦ V2◦ V3⊆CI×J×Kand let
T={ar◦br◦cr|r∈ {1, ..., R}} ⊆ V\{0}
be a multiset of outer products for R≥2rank-one nonzero
tensors. For each index r∈ {1, ..., R}, let the rank-one
tensor Yr=ar◦br◦cr. For each subset S⊆ {1, ..., R}
describing collections of rank-one tensors, let
dS
1=dimspan{ar|r∈S}
dS
2=dimspan{br|r∈S}
dS
3=dimspan{cr|r∈S}
where dimspan{·} is the dimension span of the set of the
vector space (which is equivalent to the standard rank of a
matrix). With |S|being the size of each subset S, if dS
1+
dS
2+dS
3≥2|S|+ 2, with 2≤ |S| ≤ Rthen X=PR
r=1 Yr
constitutes a unique tensor decomposition.
Given a unique CP decomposition X= [[A,B,C]] com-
posed of the factor matrices A∈CI×R,B∈CJ×R, and
C∈CK×Ras described in (5), we can split Xinto two
unique CP decompositions by partitioning the summation of
Rrank one tensors (column vectors of the factor matrices)
into two summation groups of Q≥2rank one tensors and
R−Q≥2rank one tensors such that 2≤Q≤R−2. This
means that the first Qcolumn vectors of the original factor
matrices can be grouped into one CP decomposition and the
remaining R−Qcolumn vectors can be grouped into the
factor matrices of the other CP decomposition:
X=
R
X
r=1
ar◦br◦cr= [[A,B,C]]
=
Q
X
r=1
ar◦br◦cr+
R
X
r=Q+1
ar◦br◦cr
=
Q
X
r=1
dr◦er◦fr+
R−Q
X
r=1
hr◦ir◦jr
= [[D,E,F]] + [[H,I,J]] (10)
where D∈CI×Q,E∈CJ×Q,F∈CK×Q,H∈CI×R−Q,
I∈CJ×R−Q,J∈CK×R−Qare the factor matrices of
two CP decompositions where they can be thought of as
partitions of the factor matrices of one CP decomposition.
Where we essentially group the column vectors from the
FIGURE 2: Full duplex MIMO RIS communication model.
original factor matrices into the new factor matrices as:
A= [D,H],B= [E,I], and C= [F,J].
This is particularly useful when obtaining the tensor
unfoldings of Xusing two CP decompositions:
X1=A(C⋄B)T=D(F⋄E)T+H(J⋄I)T∈CI×JK
(11)
X2=B(C⋄A)T=E(F⋄D)T+I(J⋄H)T∈CJ×IK
(12)
X3=C(B⋄A)T=F(E⋄D)T+J(I⋄H)T∈CK×IJ
(13)
III. Tensor Signal Model
The goal of this paper is to estimate the CSI at the AP
by transmitting pilots from both the AP and UE. In this
section, we first describe the system model for an in-
band FD communication scheme where the AP and UEs
each have separate receive and transmit antennas. Then, we
describe the pilot block transmission scheme that allows us
to reformulate the received signal into a tensor describing all
communication links, specifically, a tensor with two additive
CP decomposition terms representing the non-RIS and RIS
assisted links, respectively.
A. System Model
We consider a narrowband MIMO separate-antenna FD [41]
RIS-assisted communication system comprising an AP with
Mtransmit and Mreceive antennas, KUEs with one
transmit and one receive antenna each, and an RIS with
Nelements, as shown in Fig. 2 (same model as in [16]).
The channels between the (A)ccess Point, (U)ser Equipment,
and (R)IS pertinent to this work are defined in Fig. 2. For
example, HAR is the channel from the AP to the RIS, and
GAis the AP self-interference channel.
Pilots are transmitted using a time-slotted transmission
scheme over a training period of length Tslots. Let xA[t]
be the transmitted signal from the AP at the t-th time slot,
t∈ {1, . . . , T }, with E[xA[t]xH
A[t]] = PAIMwhere PA
is the AP transmit power per antenna. Similarly, let xU[t]
be the vector containing the transmitted symbols of the K
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FIGURE 3: Block transmission scheme.
UEs with E[xU[t]xH
U[t]] = PUIKwhere PUis the UEs’
transmit power.2We assume that all channels are block
fading where all CSI is assumed to be constant over T
time slots. The received signal at the AP y[t]due to the
simultaneous transmission of xA[t]and xU[t]is:
y[t] =(GA+HRA diag(ϕ[t])HAR)xA[t]
+ (HUA +HRA diag(ϕ[t])HU R)xU[t] + n[t](14)
where HUA = [hU A,1,...,hU A,K ],HU R =
[hUR,1,...,hUR ,K ], and n[t]∼ CN(0, σ2IM)is the
AWGN at the AP. Finally, ϕ[t] = [ejω1[t], . . . , ejωN[t]]T
with ωn[t]∈[−π, π)is the phase of the n-th RIS element,
1≤n≤N, at the t-th time slot.
The problem we consider in this work is to estimate the
CSI at the AP3comprising the channel matrices in Table 1.
TABLE 1: Channel matrices estimated at the AP
Channel Symbol Size
Self-interference (AP to AP) GAM×M
Direct-path (UE to AP) HUA M×K
Cascaded (RIS to AP) HRA M×N
Cascaded (AP to RIS) HAR N×M
Cascaded (UE to RIS) HUR N×K
B. Block Transmission
During channel estimation, similar to the pilots, the RIS
phase shifts are functions of time t∈ {1, . . . , T }. We
partition the total training period into Bnumber of blocks
as depicted in Fig. 3, where each block contains Ltime
slots (T=BL). During each block b∈ {1, . . . , B }, the
RIS phase values are constant and set to ϕ[t] = ϕ[b],
t∈ {(b−1)L+ 1, . . . , bL}. From one block to the next,
the RIS phase values are updated such that ϕT[b] = [Ψ]b,:,
with Ψ= [ϕ[1],ϕ[2],...,ϕ[B]]T. Where the b-th row vector
of the matrix Ψ∈CB×Ncorresponds to the NRIS phase
values ϕT[b]∈C1×Nduring the b-th block transmission.
Within each block, the AP and UEs transmit Lpilots cycling
2Without loss of generality, we assume that all UEs use the same transmit
power.
3Estimation at the UEs can be done in a similar manner.
Unfolding 1
Unfolding 2
Unfolding 3
FIGURE 4: Tensor unfoldings visualization.
over all column vectors of XA= [xA[1],...,xA[L]] ∈
CM×Land XU= [xU[1],...,xU[L]] ∈CK×L, respectively.
The receive signal during the b-th block transmission
Y[b]∈CM×Lis:
Y[b] =(GA+HRADb(Ψ)HAR )XA
+ (HUA +HRADb(Ψ)HU R )XU+N[b]
=GAIMXA
| {z }
self-interference
+HRADb(Ψ)HAR XA
| {z }
reflecting self-interference
+HUA IKXU
| {z }
direct-path
+HRADb(Ψ)HU R XU
|{z }
cascaded-path
+N[b]
|{z}
AWGN
=GDb(1B×M+K)X
| {z }
non-RIS assisted links
+HRADb(Ψ)HX
| {z }
RIS assisted links
+N[b]
|{z}
AWGN
(15)
where X=XT
AXT
UT∈C(M+K)×L,G=
GAHUA ∈CM×(M+K), and H=HAR HU R ∈
CN×(M+K). We note that Db(Ψ) = diag(ϕ[b]) is a diagonal
matrix of the NRIS phase values during the b-th block
where ϕ[b]can change phase values from one block to the
next. While the term Db(1B×M+K) = IM+Kis always
an identity matrix (and not a zero matrix) throughout all
Bblock transmissions, which represents the uncontrollable
non-RIS assisted wireless connection links between the AP
and UE that cannot be modified by the RIS.
C. Tensor Signal Modelling
As depicted in Fig. 4, using (6), (9), (10), and (11) we can
restructure the received signal (15) into a third-order tensor
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Fernandes et al.: Tensor Signal Modelling and Channel Estimation for Reconfigurable Intelligent Surface-Assisted Full-Duplex MIMO
Y ∈ CM×L×Bwith Y[b] = [Y]:,:,b as its b-th frontal face
matrix:
Y= [[G,XT,1B×(M+K)]]
| {z }
non-RIS assisted links
+ [[HRA ,ZT,Ψ]]
| {z }
RIS assisted links
+N
|{z}
AWGN
(16)
where Z=HX ∈CN×L.
Based on the two CP decompositions using (11), (12), and
(13), we can derive the unfoldings of (16) as:
Y1=G(1⋄XT)T+HRA(Ψ⋄(HX)T)T+N1(17)
Y2=XT(1⋄G)T+ (HX)T(Ψ⋄HRA)T+N2(18)
Y3=1(XT⋄G)T+Ψ((HX)T⋄HRA)T+N3(19)
where 1=1B×(M+K)in (17), (18), and (19).
TABLE 2: Factor matrices of the CP decompositions in (16)
CP decomposition term Factor matrix Size
Non-RIS assisted links
GM×(M+K)
XTL×(M+K)
1B×(M+K)B×(M+K)
RIS assisted links
HRA M×N
ZTL×N
ΨB×N
The tensor Yis composed of two CP decompo-
sitions: [[G,XT,1B×(M+K)]] corresponding to the self-
interference and direct-path links (non-RIS assisted links),
and [[HRA ,ZT,Ψ]] for the reflecting self-interference and
the cascaded-path (RIS assisted links). These factor matrices
are summarized in Table 2, and are described as follows:
G= [GA,HUA ]contains the self-interference and direct-
path channel matrices; XT= [XT
A,XT
U]contains the Lpilots
transmitted by both the AP and UE within each block;
1B×(M+K)is a matrix of ones representing the direct-path
between the AP and UE that is not controlled by the RIS;
HRA is the RIS to AP channel matrix; ZT= (HX)T=
([HAR,HU R ][XT
A,XT
U]T)Tcontains the AP and UE to RIS
channel matrices with the corresponding pilots transmitted
within each block; Ψcontains the phase values for all N
RIS elements throughout all Bblock transmissions where
the b-th row vector [Ψ]b,:∈C1×Nrepresents the values of
the NRIS elements during the b-th block transmission.
IV. Channel Estimation and Cram ´
er-Rao Bound Analysis
To estimate the CSI, we extend the traditional ALS algorithm
for our proposed tensor signal which is composed of two
CP decompositions instead of one. The proposed algorithm:
double CANDECOMP/PARAFAC decomposition - alternat-
ing least squares (DCPD-ALS) jointly estimates all channels
in one pilot transmission stage without turning the RIS “off”.
We then derive the CRB of all channel parameters in Table 1
from Section III.A for our proposed tensor signal model from
Section III.C.
A. Double CANDECOMP/PARAFAC Decomposition -
Alternating Least Squares Algorithm
To estimate all CSI, the goal is to minimize the following
cost function of the unknown factor matrices of our tensor
signal model:
J(G,HRA,Z) = ||Y − [[G,XT,1]] −[[HRA,ZT,Ψ]]||2
F
(20)
where Z=HX and 1=1B×(M+K)in (20).
We cannot use the traditional ALS algorithm [18] to
estimate the factor matrices for our tensor of two CP de-
compositions. This is because the traditional ALS algorithm
can only be used to estimate one of the CP decompositions
at a time, where as example, ALS solves the minimization
problem: ||Y − [[HRA,ZT,Ψ]]||2
Fto obtain the RIS-assisted
links by assuming the direct-path and self-interference links
are not present in the receive signal. Therefore we minimize
the cost function J(G,HRA,Z)by alternating between the
unfoldings of the noisy receive signal Y1,Y2, and corre-
sponding unfoldings of the two CP decomposition terms
described using (17) and (18) as:
ˆ
Gˆ
HRA= arg min
G,HRA
Y1−G HRA(1⋄XT)T
(Ψ⋄ZT)T
2
F
(21)
=Y1(1⋄XT)T
(Ψ⋄ZT)T†
right
(22)
ˆ
Z= arg min
Z
Y2−XT(1⋄G)T−ZT(Ψ⋄HRA)T
2
F
(23)
= (Ψ⋄HRA)†
left(YT
2−(1⋄G)X)(24)
where 1=1B×(M+K)in (21), (22), (23), and (24).
We obtain our estimate of HAR and HU R as
[ˆ
HAR,ˆ
HUR ] = ˆ
ZX†
right after the algorithm ends. The pro-
posed iterative DCPD-ALS algorithm is described in Al-
gorithm 1. The algorithm terminates when the normalized
squared Frobenius norm of the difference between two
consecutive iterations of ˆ
G,ˆ
Z, and ˆ
HRA are less than a
threshold δor when the maximum number of iterations is
exceeded (line 7 in Algorithm 1).
B. Cram´
er-Rao Bound Analysis
In this section we will derive the CRB of channel matrices
GA,HUA ,HAR ,HRA, and HU R . We consider the tensor
signal model we derived in (16) with the following unfold-
ings transposed:
YT
1= (1B×M⋄XT
A)GT
A+ (1B×K⋄XT
U)HT
UA
+ (Ψ⋄(XT
AXT
UHT
AR
HT
UR ))HT
RA +NT
1(25)
YT
2= (1B×M⋄GA)XA+ (1B×K⋄HUA )XU
+ (Ψ⋄HRA)HAR XA+ (Ψ⋄HRA )HURXU+NT
2
(26)
6 VOLUME ,
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Algorithm 1 Double CANDECOMP/PARAFAC Decompo-
sition - Alternating Least Squares (DCPD-ALS)
1: input: XA,XU,Ψ,Y
2: initialize: randomly generate ˆ
GA,ˆ
HUA ,ˆ
HAR,ˆ
HUR ,
ˆ
HRA, set algorithmic iteration as i= 1.
ˆ
Z(i=0) ←ˆ
HAR ˆ
HUR hXA
XUi
ˆ
G(i=0) ←ˆ
GAˆ
HUA
ˆ
HRA(i=0) ←ˆ
HRA
Y1←[[Y]:,:,1. . . [Y]:,:,B ]
Y2←([Y]:,:,1)T. . . ([Y]:,:,B )T
3: repeat
4: ˆ
G(i)ˆ
HRA(i)←Y1"(1⋄XT)T
(Ψ⋄ˆ
ZT
(i−1))T#†
right
5: ˆ
Z(i)←(Ψ⋄ˆ
HRA(i))†
left(YT
2−(1⋄ˆ
G(i))X)
6: i←i+ 1
7: until ∥ˆ
G(i)−ˆ
G(i−1)∥2
F∥ˆ
G(i)∥−2
F≤δand
∥ˆ
Z(i)−ˆ
Z(i−1)∥2
F∥ˆ
Z(i)∥−2
F≤δand
∥ˆ
HRA(i)−ˆ
HRA(i−1)∥2
F∥ˆ
HRA(i)∥−2
F≤δor i>Imax
8: ˆ
GAˆ
HUA ˆ
HRA←ˆ
G(i)ˆ
HRA(i)
9: ˆ
HAR ˆ
HUR ←ˆ
Z(i)hXA
XUi†
right
10: return ˆ
GA,ˆ
HUA ,ˆ
HAR,ˆ
HUR ,ˆ
HRA
To derive a meaningful CRB due to the inherent per-
mutation and scaling ambiguities, we assume that the first
column of the RIS channel estimators HAR,HT
RA, and
HUR are normalized as a column vector of all ones.
Similar to [44], we first construct the parameter vector
˜
θ∈C(M2+KM +2N(M−1)+N(K−1))×1that contains all the
free channel parameters to be estimated by excluding the
first normalized column vector of each of the RIS channel
matrices:
˜
θT=abcde(27)
a= [gT
A1,...,gT
AM]∈C1×(M2)(28)
b= [hT
UA1,...,hT
UAK]∈C1×(K M )(29)
c= [hT
RA2,...,hT
RAM]∈C1×(N(M−1)) (30)
d= [hT
AR2,...,hT
ARM]∈C1×(N(M−1)) (31)
e= [hT
UR2,...,hT
URK]∈C1×(N(K−1)) (32)
where gAi= [GT
A]:,i,hU Ai= [HT
UA ]:,i ,hRAi= [HT
RA]:,i ,
hARi= [HAR]:,i ,hU Ri= [HU R]:,i denote the i-th column
vector of the corresponding channel estimators. To simplify
the CRB derivation similar to [44], we introduce the fol-
lowing complex parameter vector to collect the free channel
parameters and their corresponding complex conjugate pa-
rameters as:
θ:=h˜
θT˜
θHi∈C1×(2(M2+KM +2N(M−1)+N(K−1)))
(33)
Since the error in the receive signal of (16) is due to a
tensor AWGN term N, we can write the likelihood function
f(Y;θ)for given values of a noisy received tensor signal Y
as a function of the unknown parameters θin two equivalent
ways with respect to unfoldings 1 and 2:
f(Y;θ) = 1
(πσ2)M LB exp (−1
σ2
M
X
m=1 ||[YT
1]:,m −y1m||2)
(34)
=1
(πσ2)M LB exp (−1
σ2
L
X
l=1 ||[YT
2]:,l −y2l||2)
(35)
where y1mand y2lare column vectors associated with
the corresponding unfoldings based on the m-th or l-th
dimension of the receive tensor which contains the complex
free channel parameters θas follows:
y1m= (1B×M⋄XT
A)gAm+ (1B×K⋄XT
U)hUAm
+ (Ψ⋄(XT
AXT
UHT
AR
HT
UR ))hRAm(36)
y2l= (1B×M⋄GA)[XA]:,l + (1B×K⋄HUA )[XU]:,l
+
M
X
m=1
(Ψ⋄HRA)hARl[XA]m,l
+
K
X
k=1
(Ψ⋄HRA)hU Rl[XU]k,l (37)
The log-likelihood function L(θ) = ln(f(Y;θ)) is then a
function of the complex free channel parameters θas:
L(θ) = −MLBln(πσ2)−1
σ2
M
X
m=1 ||[YT
1]:,m −y1m||2
(38)
=−MLBln(πσ2)−1
σ2
L
X
l=1 ||[YT
2]:,l −y2l||2(39)
Then, the complex Fisher information matrix (FIM) Ωis
the expected value of the second order partial derivatives
(hessian matrix) of the log-likelihood function L(θ)with
respect to all free channel parameters θas:
Ω=E"∂L(θ)
∂θH∂L(θ)
∂θ#(40)
because of constructing our complex parameter vector as the
transpose and conjugate transpose of the same free channel
parameters, we can simplify the FIM into the following block
diagonal matrix [44]:
Ω=Υ0
0Υ∗(41)
where Υis a FIM matrix of size (M2+KM + 2N(M−
1) + N(K−1)) ×(M2+KM + 2N(M−1) + N(K−1)).
Therefore we only need to find the elements of the FIM Υ
with respect to each of the channel parameters, which can be
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obtained as shown in Appendix A. The CRB on the variance
of any unbiased estimator of the free channel parameters ˜
θ
can then be obtained from the inverse of the FIM matrix as
CRB(˜
θ) = Υ−1.
V. Training Scheme: Design Recommendations,
Proposed Pilots, and RIS Phase Shifts
In this section we propose training schemes, i.e., the choice
of pilots and RIS phase shifts during channel estimation. We
first present design recommendations for the communication
system such that the proposed DCPD-ALS algorithm will be
able to obtain a unique estimate of the channel matrices.
Then, based on the design recommendations, we provide
orthogonal pilot transmission and RIS phase shift block
schemes. Finally, we discuss the computational complexity
of the DCPD-ALS algorithm and compare it to the ALS
and LS methods. We will design our training schemes using
block transmission, where Bblocks of Lpilot transmissions
are repeated while updating the RIS phase values over every
block transmission as described in Section III.B.
A. Design Recommendations
The design recommendations for the pilots and RIS phase
shifts throughout the training scheme of [29], [30] used
Kruskal’s rank condition to ensure that the traditional ALS
algorithm would yield a solution. However, due to our tensor
Ybeing composed of two CP decomposition terms in (16),
it is difficult to determine the conditions that guarantee the
factor matrices of Yto be unique based on Kruskal’s rank
condition or the generalized Kruskal’s rank condition in [43,
Theorem 2]. In fact, the CP decomposition composed of the
non-RIS assisted link: P= [[G,XT,1B×(M+K)]] does not
satisfy Kruskal’s rank condition kG+kXT+k1B×(M+K)≥
2(M+K) + 2 because kG≤M,kXT≤M+K, and
k1B×(M+K)= 1 making Kruskal’s rank condition invalid.
However, since the factor matrix Xis known, it is possible to
obtain a unique estimate of Gusing only the first unfolding
P1=G(1B×(M+K)⋄XT)Tof the tensor P, without
alternating between all three unfoldings, as long as the right
inverse ((1B×(M+K)⋄XT)T)†
right exists. Therefore, instead
of taking the traditional approach of using Kruskal’s rank
condition to design the training scheme like in [29], [30],
we focus on making the cost function in (20) of our tensor
model to be well conditioned to be able to obtain unique
estimates of the channels.
Since the pilots Xand RIS phase shifts Ψare fixed and
do not change when estimating the channels, we denote the
solutions (22) and (24) for each update in the DCPD-ALS
algorithm as the functions:
ˆ
Gˆ
HRA=f1(Z) = Y1(1B×(M+K)⋄XT)T
(Ψ⋄ZT)T†
right
(42)
ˆ
Z=f2(G,HRA) = (Ψ⋄HRA )†
left(YT
2−(1⋄G)X)(43)
Now each iteration update can be thought of as performing
the operation ˆ
Zi+1 =f2(f1(ˆ
Zi)) and [ˆ
Gi+1,ˆ
HRA,i+1] =
f1(f2(ˆ
Gi,ˆ
HRA,i)). We note that due to Z=HX, the matrix
Xcontaining the pilot transmission from both the AP and
UE needs to be a full rank matrix.
For a sequence of updates that converge to the limit
points ˆ
Z∗= limi→∞ ˆ
Zi,ˆ
G∗= limi→∞ ˆ
Gi, and ˆ
H∗
RA =
limi→∞ ˆ
HRA,i, we want every limit point to be a critical
point of the optimization problems (21) and (23) by making
them both strictly convex [45]. This will make the cost func-
tion J(G,HRA,Z)in (20) be strictly quasiconvex where the
local minimum is the global minimum [45]. If Jis quasicon-
vex but not strictly, then the least squares update functions
f1and f2will contain flat regions not guaranteeing the algo-
rithm will converge. To make (21) and (23) strictly convex,
the matrices R=(1B×(M+K)⋄XT) (Ψ⋄(HX)T)Tand
L=Ψ⋄HRA undergoing pseudo inverses of the least
squares solutions in f1and f2must be full rank. For any
given Hand HRA, if Rand Lhave very small or zero-
valued singular values, based on the designed pilots Xand
RIS phase shifts Ψ, then the limit points ˆ
Z∗,ˆ
G∗, and ˆ
H∗,
are not guaranteed to be critical points. This will result in the
algorithm to have slow convergence and remain in a swamp4,
where the algorithm updates corresponding to the directions
with small singular values will be insignificant compared to
the directions with the larger singular values. Therefore we
propose the following conditions must be met:
Proposition 1. Using identities (1) and (2), the necessary
conditions to obtain unique estimates of the channels up to
scaling and permutation ambiguities, are:
rank(X) = M+K(44)
RRH=1T1⊙XXH1TΨ⊙XXHHH
ΨT1⊙HXXH(ΨHΨ)∗⊙HXXHHH≻0
(45)
LHL=ΨHΨ⊙HH
RAHRA ≻0(46)
where 1=1B×(M+K), and both RRHand LHLare strictly
positive definite matrices.
There are many ways of choosing the pilots and RIS
phase shifts that satisfy Proposition 1. Therefore to evaluate
the stability of the training scheme of designed pilots and
RIS phase shifts, we propose functions g1(R)and g2(L)
that provides a measure that describes how positive definite
the matrices RRHand LHLare based on their eigenvalues.
Where when g1and g2are minimized, Rand Lshould be
full rank, satisfying conditions from Proposition 1. If the
conditions for Proposition 1 are not satisfied, then the cost
function will not be strictly quasiconvex, not guaranteeing
that the algorithm will converge to the global minimum.
We propose minimizing functions g1(R)and g2(L)subject
4A swamp is used as a term when a bottleneck occurs for the traditional
ALS algorithm, where it takes a large number of iterations until the error
is reduced, inducing an overall slow convergence rate [46].
8 VOLUME ,
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to the following constraints of the RIS-assisted FD MIMO
communication system model:
min g1(R) = Tr((RRH)−1), g2(L) = Tr((LHL)−1)
(47)
s.t. R=(1B×(M+K)⋄XT)T
(Ψ⋄(HX)T)T∈CBL×(M+K+N)
(48)
L=Ψ⋄HRA ∈CBM ×N(49)
X=XA
XU∈C(M+K)×L,rank(X) = M+K(50)
XA∈CM×L,Tr(XAXH
A)/(ML)≤PA(51)
XU∈CK×L,Tr(XUXH
U)/(KL)≤PU(52)
Ψ∈CB×N,[Ψ]b,n ∈ {ejω | − π≤ω < π}(53)
where g1(R)is a function of both pilots Xand RIS phase
shifts Ψ, while g2(L)is a function of only RIS phase shifts.
The maximum transmit power for the AP XAand UE XU
pilots are PAand PUrespectively, and the RIS is assumed to
be “on” throughout the full training duration with the ampli-
tude of all coefficients |[Ψ]b,n|= 1. This problem penalizes
the rank of Rand Laccording to Proposition 1, where for
any square matrix A=BBH=CHC,Tr(A−1) = Pi
1
λi,
λi=σ2
i, with λieigenvalues, σisingular values, and the
rank of the matrices A,Band C, are the number of nonzero
singular values. Therefore we can evaluate the stability of
the design choices of training schemes where if Ror Lare
rank deficient with very small singular values then g1or g2
will be extremely large in comparison to other designs.
A recommended design choice would be to make RRH
and LHLdiagonal matrices with nonzero diagonal values.
This will ensure that the objective functions g1and g2in
(47) are minimized5based on these diagonal entries and
that Rand Lwill have properly scaled singular values. This
will also reduce the computational complexity of inverting
RRHand LHLfrom O(n3)to O(n)that occurs within each
iteration. To diagonalize RRHand LHL, we propose the
following conditions on the RIS phase shifts and pilots:
ΨHΨ=BIN(54)
ΨT1B×(M+K)=0N×(M+K)(55)
XXH= diag([EA,1, . . . , EA,M , EU,1, . . . , EU,K ])
(56)
where EA,m >0, m ∈ {1, . . . , M}and EU,k >0, k ∈
{1, . . . , K}. From these conditions the M+K+Ndiagonal
entries (the eigenvalues) of the diagonal matrix RRHare:
EA,1,. . .,EA,M ,EU,1,. . .,EU,K ,B||([H]1,:)X||2
2,. . .,
B||([H]N,:)X||2
2. The Ndiagonal entries of the diagonal
matrix LHLare: B||[HRA]:,1||2
2,. . .,B||[HRA]:,N ||2
2. Where
the benefit of making RRHand LHLdiagonal matrices is
that the eigenvalues of both matrices depend on the channel
gains and not the eigenvalues of the channel matrices.
5[A−1]ii ≥1
[A]ii for any positive definite matrix A(see [47, Ex 3.12]).
TABLE 3: Proposed orthogonal pilot transmission schemes
Scheme 1 Scheme 2
Transmission simultaneous non-simultaneous
XA√PA[QM,QM]√PA[QM,0M×K]
XU√PU[P,−P]√PU[0K×M,QK]
L2M M +K
EA/L PAPAM
M+K
EU/L PUPUK
M+K
B. Full-Duplex Orthogonal Pilot Scheme
For all channel estimation methods we transmit orthog-
onal pilots from both the AP and UEs, XAXH
A=
diag([EA,1, . . . , EA,M ]),XUXH
U= diag([EU,1, . . . , EU,M ]),
and XAXH
U=0M×K, meaning that
XXH=XAXH
AXAXH
U
XUXH
AXUXH
U
= diag([EA,1, . . . , EA,M , EA,1, . . . , EA,M ]) (57)
We consider two orthogonal FD pilot transmission
schemes in Table 3, adopted by [16]. The two pilot schemes
are based on whether the AP and UEs are transmitting si-
multaneously or non-simultaneously. The simultaneous pilot
scheme 1 is designed to be power efficient such that both
the AP and UE are always transmitting non-zero pilots to
provide more transmit power on average giving a more
accurate estimate of the channels. The non-simultaneous
pilot scheme 2 is designed to use the minimum number of
pilots to obtain estimates of the channels. In this table, Pis
given by:
P=QK, . . . , QK
| {z }
qtimes
,Qr
0(K−r)×r∈CK×M(58)
with q=⌊M
K⌋,r=Mmod K, where QM,QKand Qr
are orthonormal matrices of size M,K, and r, respectively,
e.g., normalized DFT matrices. These pilot schemes have
been adopted from [16] which also satisfy conditions for a
unique solution from the LS estimator.
Scheme 1, where both the AP and UEs are always
transmitting non-zero pilots during each time slot, takes full
advantage of simultaneous pilot transmission. This allows
for scheme 1 to transmit more energy over one block
transmission than pilot scheme 2, where EA= Tr(XAXH
A)
and EU= Tr(XUXH
U)is the total energy transmitted per AP
antenna and UE respectively within one block duration of L
pilots. As a result shown in Table 3, scheme 1 on average,
transmits more power (EA/L and EU/L) than scheme 2
which only transmits a fraction of M
M+Kpower. However, to
maintain orthogonality, the duration of scheme 1 is slightly
longer than scheme 2 by a factor of 2M/(M+K)when
M > K.
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C. RIS Phase Shift Selection
For the DCPD-ALS method, a choice for the RIS phase
shifts is to adopt the DFT RIS phase shift method from [38],
by making 1N+1×1Ψ=FN+1 (i.e. B=N+ 1) such
that Ψ∈CN+1×Nis a Vandermonde matrix [48]. Since this
training scheme will have a large pilot overhead, it is possible
to reduce the total number of pilots transmitted by reducing
the number of block transmissions Bas long as Proposition
1 is satisfied. This can be achieved by reducing the number
of rows in Ψwhere 1N+1×1ΨB,N +1 = [FN+1]B ,N +1
such that B≤N+ 1 are the number of rows kept in the
original Ψ. This will effectively reduce the number of total
pilots transmitted throughout the training duration. However,
if B < N + 1,RRHand LHLmay no longer be diagonal
matrices and their smallest eigenvalues will decrease towards
zero. If Bis too small, Proposition 1 may not be satisfied as
these matrices will become singular, increasing the number
of iterations of the DCPD-ALS algorithm, obtaining non-
optimal channel estimates.
D. Computational Complexity
In Table 4, we summarize the computational complexity of
the proposed DCPD-ALS algorithm and compare it with the
ALS algorithm and LS solution; for fair comparison, the
total training duration is T= (N+ 1)Lby ensuring all
methods use the same pilot overhead. For a fair comparison
with the ALS algorithm [29], [30], we assume the direct-path
and self-interference channel ˆ
Gis estimated while the RIS is
“off” using Ltransmitted pilots using X†
right prior to using the
ALS algorithm. Then after ˆ
Gis estimated, the RIS is turned
“on”, NL pilots are transmitted, ˆ
Gis subtracted from the
receive signal, and the ALS algorithm estimates ˆ
Z(i)and
ˆ
HRA(i)over each iteration i. For the LS solution [16] the
only computation required is a matrix-vector multiplication
of the receive signal with the predetermined matrix W†
left
which has an overall lower complexity than the other algo-
rithms due to having no iterative computations. We note that
the following proposed training scheme uses pilot Scheme
1 or 2 with L= 2Mor L=M+Krespectively, if the
number of RIS elements Nis far greater than the number of
AP antennas Mor UEs K, then both the DCPD-ALS and
ALS will have a computational complexity of O(N3)while
the LS will only be O(N2). This means that the LS solution
will always be faster to execute than the DCPD-ALS or ALS
algorithm in practical scenarios (assuming the same training
scheme for comparison) where the number of RIS elements
Nwill be much larger than the number of AP antennas M
and UEs K.
VI. Simulations
In this section, we evaluate the performance of all channel
estimation methods and training schemes presented in this
paper via numerical simulations.
TABLE 4: Computational Complexity of channel estimation
methods with the same total training duration T
DCPD-ALS Computational Complexity
[ˆ
G(i),ˆ
HRA(i)]
2(N+M+K)2(N+ 1)L
+(N+M+K)(M(N+ 1)L+ 1)
+(N+ 1)L(M+K)
Z(i)2N3M+N2(2M+M+ML)
+N(M(M+K)L+ 2ML +M+ 1) + M L(M+K+ 1)
All channels O{(N+M+K)2N L +N3M}
ALS Computational Complexity
ˆ
GML(M+K)
ˆ
HRA(i)2N3L+N2L(M+ 1) + N
ˆ
Z(i)N3M+N2M(L+ 1) + N
All channels O{N3L+N3M+M L(M+K)}
LS Computational Complexity
ˆ
h(N+ 1)2(M3+M2K)L
All channels O{N2(M3+M2K)L}
A. Simulation Setup
Unless stated otherwise, we simulate a system with M= 4
transmit and receive antennas at the AP configured in a
uniform linear array (ULA), K= 3 single-antenna UEs, and
a uniform rectangular array (URA) RIS with N= 5×5 = 25
elements. The signal-to-noise ratio (SNR) is defined as P/σ2
with PA=PU=P. For Algorithm 1, we set the maximum
number of iterations, Imax, to 1000 and the convergence
threshold tolerance, δ, to 10−5. All results were obtained by
Monte-Carlo simulations over 10,000 independent channel
realizations.
The self-interference channel is modeled as Rayleigh
fading having normalized channel power [10]. All other
channels are modeled according to a 3D geometry-based
model [49], [50] with R= 2 paths, and are given by:
HAR =√ρAR
R
X
i=1
βARiaURA(ξAR,i , ψAR,i)aH
ULA (ϕAR,i )
(59)
HRA =√ρRA
R
X
i=1
βRAiaULA(ϕRA,i )aH
URA(ξRA,i , ψRA,i)
(60)
HUR =hU R,1. . . hUR,K (61)
hUR,k =√ρUR
R
X
i=1
βURi,k aU RA (ξU R,i,k , ψU R,i,k )(62)
HUA =hU A,1. . . hUA,K (63)
hUA,k =√ρU A
R
X
i=1
βUAi,k aU LA (ξU A,i,k )(64)
where βx∼ CN(0,1),x∈ {ARi, RAi, U Ri,k, U Ai,k},
describe the small scale fading. The channel gains
are normalized to be the same for all channels, i.e.,
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ρUA =ρRA =ρAR =ρUR = 1. This is to ensure
that the direct-path and the RIS links have the same
attenuation, i.e., the direct-path link is not considered
blocked with respect to the RIS link. In (59), (60), and (64),
aULA (ϕ) = 1ej π sinϕ· · · ej π(M−1) sin ϕdenotes
the steering vector for a ULA with Mantennas spaced
half-the-wavelength apart, while ϕRAi, ϕUAi,k ∈[0,π
2)
are the AoAs at the AP from the RIS and the k-th
UE, respectively. Similarly, in (59), (60), and (62),
aURA(ξ , ψ) = ay(ξ, ψ)⊗ax(ξ , ψ)is the steering vector for
a URA with Ny×Nxantennas, where the vertical (y) and
horizontal (x) steering vectors for the URA are ay(ξ, ψ) =
1ejπ sin ξsin ψ··· ej π(M−1) sin ξsin ψTand
ax(ξ, ψ) = 1ejπ sin ξcos ψ··· ej π(M−1) sin ξcos ψT,
respectively. Finally, at the RIS, the AoD (RIS to AP) and
AoA (UE to RIS) are ξRAi, ξURi∈[0,π
2)(elevation angle)
and ψRAi, ψURi∈[0, π)(azimuth angle). All angles for
the steering vectors of the AoAs and AoDs are uniformly
distributed over their described ranges [50].
The performance of estimating all CSI with the channel
h= [vec(GA)T,vec(HT
AR ⋄HRA)T,vec(HU A )T,vec(HT
UR ⋄
HRA)T]is measured by the normalized mean square er-
ror (NMSE), defined as NMSE(ˆ
h) = 1
SPS
s=1 ∥h(s)−
ˆ
h(s)∥2∥h(s)∥−2, where Sis the number of Monte-Carlo runs.
When calculating the NMSE for ˆ
hwe do not need to account
for scaling and permutation ambiguity as the cascaded RIS
channels are being multiplied together. To estimate the
CSI of the individual channels, the MSE is defined as
MSE(ˆ
H) = 1
SPS
s=1 ∥H(s)−ˆ
H(s)∥2
F, which is different than
the NMSE calculation because the first column of the chan-
nel matrices are normalized to account for scaling ambiguity.
This is because ˆ
HRA∆RA =HRA ,∆AR ˆ
HAR =HAR, and
∆UR ˆ
HUR =HU R , where ∆RA ∆AR =∆RA∆U R =IN,
therefore we normalize the first column of the channel ma-
trices when calculating the MSE between the true channels
Hand the estimates ˆ
H[29]. When comparing the MSE
with the CRB of all free channel parameters, the CRB is
evaluated as taking the trace of the inverse of the FIM
Matrix: MSE ≥CRB = Tr(Υ−1).
We will compare the proposed method to the following
benchmarks:
•The LS method from [16]. To the best of our knowl-
edge, this is the only study on FD RIS-assisted MIMO
channel estimation to consider estimation of all commu-
nication channels without assuming channel reciprocity.
•An ALS-type algorithm obtained by extending the
works in [29], [30] that estimate the factor matrices of
[[HRA ,ZT,Ψ]] by first estimating the direct and self-
interference paths using an initial training stage, and
then subtracting them from the receive signal.
In our comparisons, we ensure that the pilot transmission
scheme and the total training duration are the same across all
methods. For the DCPD-ALS and LS methods, all channels
0 5 10 15 20 25 30 35 40
SNR (dB)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
FIGURE 5: Performance of channel estimation methods vs
SNR. (M= 4,K= 3,N= 5 ×5 = 25)
are estimated in a single training stage without turning the
RIS “off”. Whereas for the ALS method, the non-RIS and
RIS channels are estimated in two training stages. In the first
stage, the RIS is turned “off” to estimate the direct-path and
self-interference channels, then in the second stage the RIS
is turned “on” and the RIS-assisted channels are estimated
with the ALS algorithm [30] after subtracting the direct-path
and self-interference links from the receive signal. Unless
stated otherwise, for the DCPD-ALS and LS methods we
use 1(N+1)×1Ψ=FN+1, i.e., B=N+ 1 block
transmissions while the RIS is “on”. For the ALS algorithm
as in [30], we use Ψ=FN, i.e., 1block transmission
during the RIS “off” stage and Nblock transmissions during
the RIS “on” stage. The total training duration for all three
channel estimation methods for this setup is T= 208 when
using pilot scheme 1 and T= 182 for pilot scheme 2.
B. Numerical Analysis Comparison between Channel
Estimation Methods
We compare the performance of our proposed DCPD-ALS
method with the ALS and LS method using the NMSE of
hin Fig. 5. Results show that our proposed DCPD-ALS
algorithm outperforms both the ALS and LS methods by
exploiting the structure of the tensor signal model. Across all
methods, pilot scheme 1 outperforms scheme 2 in estimation
accuracy partly due to a longer training duration but also due
to pilot scheme 1 transmitting at more power throughout the
full training duration.
Fig. 6 shows the MSE of the DCPD-ALS and ALS
methods and the corresponding CRBs of the individual
channel matrices as well as the full CRB over all free channel
parameters. One limitation of the LS method is that it is not
possible to obtain estimates of the decoupled RIS channels
(ˆ
HRA,ˆ
HAR,ˆ
HUR ) due to the vectorization operator when
obtaining all channel estimates with ˆ
hLS and therefore not
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Fernandes et al.: Tensor Signal Modelling and Channel Estimation for Reconfigurable Intelligent Surface-Assisted Full-Duplex MIMO
0 10 20 30 40
SNR (dB)
10-4
10-2
100
102
0 10 20 30 40
SNR (dB)
10-4
10-2
100
102
0 10 20 30 40
SNR (dB)
10-4
10-2
100
102
0 10 20 30 40
SNR (dB)
10-4
10-2
100
102
0 10 20 30 40
SNR (dB)
10-4
10-2
100
102
0 10 20 30 40
SNR (dB)
10-4
10-2
100
102
FIGURE 6: Performance of estimating the individual chan-
nels. (M= 4,K= 3,N= 5 ×5 = 25)
included in Fig. 6. For both methods, pilot scheme 1 is used
for a total training overhead duration of T= 208, where for
the ALS algorithm, L= 8 pilots are used to estimate the
direct-path (HUA ) and self-interference (GA) with the RIS
turned “off,” and the remaining N L = 200 pilots are used
to estimate the RIS channels (HRA,HAR and HU R ). These
results show that the ALS algorithm obtains a poor estimate
of HUA and GA; in contrast, the DCPD-ALS algorithm
attains the CRB and provides a better estimate of HRA,HAR ,
and HUR coming close to the CRBs. This is due to using a
small number of pilots to estimate the self-interference and
direct-path channels prior to the RIS channels whereas the
DCPD-ALS algorithm efficiently uses all transmitted pilots
to estimate all channels.
In Fig. 7, we compare the NMSE and average number
of iterations to convergence for the DCPD-ALS and ALS
channel estimation methods using pilot scheme 1 and 2 for
SNR of 0dB, 20dB, and 40dB (we note that the maximum
transmission power at each AP and UE antenna for all pilots
is Pand the SNR is defined as the peak transmit power
per antenna Pto noise ratio σ). For this experiment, the
self-interference and direct-path channel gain coefficients are
normalized to 1 while the RIS channel gain coefficients
are varied as ρRA =ρAR =ρUR =ρ. This means that
for ρ= 20dB, the links AP-RIS, RIS-UE, and RIS-AP
are 20dB stronger than the direct-path and self-interference
links. For ρ < −10dB we see the NMSE saturates and the
number of iterations to convergence (averaged over Monte-
Carlo simulations) is high, indicating slow convergence for
both algorithms making it more difficult to properly estimate
the CSI due to the attenuation of the receive signal through
the RIS assisted link. While for ρ > 10dB the NMSE is
lower, with only two iterations until convergence for both
algorithms indicating both algorithms perform best with a
strong RIS link. The third column represents the difference
between performance for pilot scheme 2 and 1, where for
example at SNR of 0dB, pilot scheme 1 has on average
three fewer iterations to convergence than scheme 2 for
the DCPD-ALS algorithm resulting in saving an average
execution runtime of 1 ms despite scheme 1 transmitting
more pilots than scheme 2. The reason why pilot scheme
1 is able to outperform scheme 2 is because scheme 1
transmits more power per pilot on average (higher EA/L
and EU/L from Table 3) providing a higher signal to noise
ratio in the receive signal at the AP, helping reduce the
number of iterations to convergence and runtime execution
despite transmitting more pilots. However if ρis too low for
certain transmit SNR then scheme 1 under performs scheme
2. The traditional ALS algorithm needs to estimate the
direct-path and self-interference channels in a training stage
while the RIS is turned “off”, then requires subtracting these
channels from the receive signal to estimate the RIS-assisted
channels. This subtraction of the estimates of the direct-path
and self-interference (and not subtracting the true values of
these channels) propagates the error in estimating the RIS
channels. Where as our DCPD-ALS algorithm estimates all
channels by efficiently using all transmitted pilots without
turning the RIS “off” or using separate training stages.
Overall, the DCPD-ALS outperforms the ALS algorithm
regardless of the strength of the RIS assisted link because it
jointly estimates all channels.
To summarize the comparison between the channel esti-
mation methods, the proposed DCPD-ALS method is able
to obtain the most accurate estimate of all channels by ex-
ploiting the structure of the proposed tensor signal model at
the cost of computational complexity. The LS method is not
an iterative algorithm and therefore the least computationally
expensive algorithm, however it does not exploit the structure
of the tensor signal model making it limited in not being able
to provide decoupled estimates of the RIS channel matrices
(ˆ
HRA,ˆ
HAR,ˆ
HUR ). For the DCPD-ALS and ALS iterative
methods, the choice of simultaneous or non-simultaneous
pilot transmission scheme can either increase or decrease the
runtime of the methods and should be a design consideration
depending on the wireless propagation environment.
C. Numerical Analysis of the Proposed Channel
Estimation Method and Training Scheme
In Fig. 8 and Fig. 9, we use a communication system
model with a larger number of AP antennas (M), UEs (K),
12 VOLUME ,
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-40 -20 0 20 40
10-10
100
-40 -20 0 20 40
10-10
100
-40 -20 0 20 40
10-10
100
-40 -20 0 20 40
0
5
10
15
20
-40 -20 0 20 40
0
5
10
15
20
-40 -20 0 20 40
-2
0
2
4
-40 -20 0 20 40
100
101
-40 -20 0 20 40
100
101
-40 -20 0 20 40
-1
-0.5
0
0.5
1
FIGURE 7: Performance of pilot transmissions schemes 1 and 2 for channel estimation methods, with Lpilots transmitted
per block and a total of B= 26 block transmissions. (M= 4,K= 3,N= 5 ×5 = 25)
and RIS elements (N) to analyze the effect of the training
scheme on the performance of the DCPD-ALS algorithm. We
compare geometric channels with R= 3 paths as described
in Section VI.A (HAR and HRA are low rank matrices
while other channels are full rank) and unstructured channels
with Rayleigh fading (all channel matrices are full rank).
The MSE, number of iterations, cost functions g1(R) =
Tr((RRH)−1) = Pi
1
λiand g2(L) = Tr((LHL)−1) =
Pj
1
λj(where λiand λjare eigenvalues), are calculated
as the average over 10,000 independent Monte-Carlo simula-
tions. Following the RIS phase shift selection we proposed in
Section V.C, we reduce the number of block transmissions B
of repeated Lpilot transmissions so that the overall training
duration T=BL ≤(N+ 1)L, where the minimum number
of pilots to obtain a unique solution for the LS solution is
T= (N+ 1)(M+K) = 910 pilots [16]. We see that
when B=N+ 1 = 65 (the recommended number of
blocks), the MSE, CRBs, and Monte Carlo averaged number
of iterations are the lowest in Fig. 8. This is due to both
g1(R)and g2(L)being minimized in comparison to lower
values of B. By observing the distribution of the eigenvalues
in Fig. 9 of RRHand LHL, we can see that compared
to other Bvalues, when B= 65 the difference between
the largest and smallest eigenvalues is the lowest. When
B < N + 1 = 65, for the model with geometric channels,
the MSE increases significantly above the CRB compared
to the scenario of unstructured channels. This difference in
performance between geometric vs unstructured channels
can be explained by the large gap in the cost function
g2(L)corresponding to a greater difference between the
largest and smallest eigenvalues of LHL. Using Lemma 1,
the rank of L∈CBM ×Nis min(BM, N )≥rank(L)≥
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Fernandes et al.: Tensor Signal Modelling and Channel Estimation for Reconfigurable Intelligent Surface-Assisted Full-Duplex MIMO
kΨ⋄HRA ≥min(kHRA +kΨ−1, N )where kΨ=B
(because Ψis a Vandermonde matrix [48, Lemma 4]). For
geometric channels kHRA = 1 and for unstructured channels
kHRA = rank(HRA) = M. This means that, for geometric
channels, the lower bound is rank(L)≥min(B, N )and for
unstructured channels rank(L)≥min(M+B−1, N ). Based
on this lower bound inequality, the matrix Lfor geometric
channels is at risk of being a low rank singular matrix for
B < N = 64, whereas for unstructured channels the lower
bound is B < N + 1 −M= 57 before being at risk
of a low rank singular matrix. However, this lower bound
based on the rank of the khatri-rao product matrices does
not explain the full picture, a better explanation comes from
observing the distribution of eigenvalues for RRHand LHL.
If we compare the distribution of eigenvalues for B= 65
with other values of B < 65, the eigenvalues for LHL
and RRHdecrease a lot more for geometric channels than
unstructured channels as we reduce the number of block
transmissions B. We can see that there is a sudden drop
in eigenvalue occurring for eigenvalue numbers λn≥Bfor
geometric channel models while this sudden drop is not
present for unstructured channel models. Since this proposed
DCPD-ALS algorithm alternates between solving [G,HRA]
and Z(line 4 and 5 of Algorithm 1), if the singular values of
Rand L(i.e. eigenvalues of RRHand LHL) are not properly
scaled, the search directions corresponding to the directions
with small singular values will increase the number of
iterations to find a solution while also increasing the error
in the estimate of the final iteration of the algorithm. This
shows that the stability of the DCPD-ALS algorithm depends
on the rank of the channels when reducing the number of
pilots in the training scheme. As we decrease the number of
blocks transmitted, the accuracy in estimating the channels
drastically decreases for low rank geometric channels but
less so for full rank Rayleigh fading channels. When the
design recommendations of Proposition 1 are satisfied, the
algorithm provides accurate estimates of the channels, with
the best results occurring when g1and g2are minimized.
This suggests that the DCPD-ALS algorithm is favourable
for conditions when the AP-RIS channel is full rank such as
Rayleigh fading (or possibly near-field propagation channels
[51]–[53]), to be able to reduce the number of required pilots
transmitted.
VII. Conclusion
In this paper, we provided a novel model for the FD RIS-
assisted MIMO communication system receive signal as
a tensor comprising two CP decompositions: one corre-
sponding to the non-RIS assisted channels (self-interference
and direct-path channels) and the other corresponding to
the RIS assisted channels. Then, we proposed the DCPD-
ALS channel estimation method based on the tensor signal
model, and provided the corresponding CRBs. In contrast to
traditional ALS methods, our proposed DCPD-ALS method
exploits the structure of the tensor signal model to efficiently
40 50 60
100
105
40 50 60
100
105
40 50 60
10-4
10-2
100
102
40 50 60
10-4
10-2
100
102
40 50 60
100
101
102
40 50 60
100
101
102
FIGURE 8: Analysis of training scheme for the proposed
DCPD-ALS algorithm vs number of blocks Bfor pilot
schemes 1 and 2. (M= 8,K= 6,N= 8 ×8 = 64,
SNR = 30dB)
use all transmitted pilots to jointly estimate the direct-path,
self-intereference, and RIS channels simultaneously during a
single training stage without turning the RIS “off”. Through
simulations we showed that the DCPD-ALS algorithm out-
performs both the ALS and LS methods used in previous
works, comes close to attaining the CRBs for the RIS paths,
and achieves the CRB for estimating the direct-path. We
also showed that it is possible to reduce the total training
duration, below the level required to obtain the LS solution,
while still maintaining a relatively accurate estimate.
Future works should aim for more practical scenarios such
as hardware-impairments (in both the transceivers and RIS
elements [16]), near-field channel models with appropriate
power scaling channel gains and higher channel rank than
far-field [51]–[53], and wideband scenarios by extending to
a fourth order tensor based on OFDM subcarrier (similar to
[31] but include the direct-path and self-interference links).
Further extensions could include improving computational
complexity and runtime by investigating low rank channel
matrix scenarios (e.g. far field geometric channels) by re-
14 VOLUME ,
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0 20 40 60 80
Eigenvalue Number
10-2
10-1
100
0 20 40 60
Eigenvalue Number
10-2
10-1
100
0 20 40 60 80
Eigenvalue Number
10-2
10-1
100
0 20 40 60
Eigenvalue Number
10-2
10-1
100
0 20 40 60 80
Eigenvalue Number
10-2
10-1
100
0 20 40 60
Eigenvalue Number
10-2
10-1
100
0 20 40 60 80
Eigenvalue Number
10-2
10-1
100
0 20 40 60
Eigenvalue Number
10-2
10-1
100
FIGURE 9: Analysis of Proposition 1 conditions for the
proposed training scheme through the eigenvalues of RRH
and LHLfor the proposed DCPD-ALS algorithm when
decreasing the number of Bblock transmissions. (M= 8,
K= 6,N= 8 ×8 = 64, SNR = 30dB)
ducing the number of iterations with regularized ALS meth-
ods [45], implementing time-varying channels extending the
proposed method for channel tracking through recursive least
squares methods [34], and adaptive meta-learning methods
such as a Dynamic Conjugate Gradient Network DyCoGNet
[54]. Future works could extend the adaptive meta-learning
methods for symbol detection with imperfect CSI estima-
tion and time-varying massive-MIMO channels discussed in
[54] but for RIS-assisted communication models using our
proposed tensor signal model.
Appendix A
Derivation of the Fisher Information Matrix
To find the elements of the FIM Υ, we need to calculate the
second order partial derivatives (hessian matrix) of the log-
likelihood function L(θ)with respect to each of the channel
parameters θas shown in (40), and take the expectation with
respect to the corresponding noise tensor unfoldings.
With (36) and (37), the partial derivatives of L(θ)in (38)
and (39) with respect to a single channel parameter are:
∂L(θ)
∂gAm,m′
=1
σ2([YT
1]:,m −y1m)H(1B×M⋄XT
A)ˆ
em′(65a)
∂L(θ)
∂hU Am,k
=1
σ2([YT
1]:,m −y1m)H(1B×K⋄XT
U)ˆ
ek(65b)
∂L(θ)
∂hRAm,n
=1
σ2([YT
1]:,m −y1m)H(Ψ⋄ZT)ˆ
en(65c)
∂L(θ)
∂hARn,m
=1
σ2
L
X
l=1
([YT
2]:,l −y2l)H(Ψ⋄HRA)ˆ
en[XA]m,l
(65d)
∂L(θ)
∂hU Rn,k
=1
σ2
L
X
l=1
([YT
2]:,l −y2l)H(Ψ⋄HRA)ˆ
en[XU]k,l
(65e)
where ˆ
em′∈RM×1,ˆ
ek∈RK×1, and ˆ
en∈RN×1, are
standard basis vectors. We note that although the noise is
independent in three dimensions:
E[[N]m1,l1,b1[N]∗
m2,l2,b2] = δm1,m2δl1,l2δb1,b2(66)
where the Kronecker delta δi,j = 1 when i=j, and δi,j = 0
when i=j, the unfoldings of a three dimensional AWGN
tensor Nare correlated as shown in [44] with covariance:
E[[NT
1]:,m[NT
2]H
:,l] =
σ2
0. . . 0. . . 0. . . 0. . . 0
...
0. . . 1. . . 0. . . 0. . . 0
...
0. . . 0. . . 1. . . 0. . . 0
...
0. . . 0. . . 0. . . 1. . . 0
...
0. . . 0. . . 0. . . 0. . . 0
(67)
The matrix E[[NT
1]:,m[NT
2]H
:,l]has a 1(indicating the noise
unfoldings are correlated) at the row and column indices of
(l, L +l, . . . , (B−1)L+l)and (m, M +m, . . . , (B−1)M+
m), respectively.
The elements of Υcan then be obtained by multiplying
all combinations of (65a)-(65e) with the complex conjugate
of each other, then taking the expectation with respect to the
noise, using (67) for the correlation between noise unfoldings
VOLUME , 15
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Fernandes et al.: Tensor Signal Modelling and Channel Estimation for Reconfigurable Intelligent Surface-Assisted Full-Duplex MIMO
1 and 2, we obtain:
E
∂L(θ)
∂g∗
Am1,m′
1
∂L(θ)
∂gAm2,m′
2
=δm1,m2
σ2ˆ
eH
m′
1(1B×M⋄XT
A)H
×(1B×M⋄XT
A)ˆ
em′
2(68a)
E"∂L(θ)
∂h∗
UAm1,k1
∂L(θ)
∂hU Am2,k2#=δm1,m2
σ2ˆ
eH
k1(1B×K⋄XT
U)H
×(1B×K⋄XT
U)ˆ
ek2(68b)
E"∂L(θ)
∂h∗
RAm1,n1
∂L(θ)
∂hRAm2,n2#=δm1,m2
σ2ˆ
eH
n1(Ψ⋄ZT)H
×(Ψ⋄ZT)ˆ
en2(68c)
E"∂L(θ)
∂h∗
ARn1,m1
∂L(θ)
∂hARn2,m2#=
L
X
l=1
[XA]∗
m1,lˆ
eH
n1(Ψ⋄HRA)H
×(Ψ⋄HRA)ˆ
en2[XA]m2,l
1
σ2
(68d)
E"∂L(θ)
∂h∗
URn1,k1
∂L(θ)
∂hU Rn2,k2#=
L
X
l=1
[XU]∗
k1,lˆ
eH
n1(Ψ⋄HRA)H
×(Ψ⋄HRA)ˆ
en2[XU]k2,l
1
σ2
(68e)
E"∂L(θ)
∂g∗
Am1,m′
∂L(θ)
∂hU Am2,k #=δm1,m2
σ2ˆ
eH
m′(1B×M⋄XT
A)H
×(1B×K⋄XT
U)ˆ
ek(68f)
E"∂L(θ)
∂g∗
Am1,m′
∂L(θ)
∂hRAm2,n #=δm1,m2
σ2ˆ
eH
m′(1B×M⋄XT
A)H
×(Ψ⋄ZT)ˆ
en(68g)
E"∂L(θ)
∂g∗
Am1,m′
∂L(θ)
∂hARn,m2#=
L
X
l=1
ˆ
eH
m′(1B×M⋄XT
A)H
×
E[[NT
1]:,m1[NT
2]H
:,l]
σ4
×(Ψ⋄HRA)ˆ
en[XA]m2,l
(68h)
E"∂L(θ)
∂g∗
Am,m′
∂L(θ)
∂hU Rn,k #=
L
X
l=1
ˆ
eH
m′(1B×M⋄XT
A)H
×
E[[NT
1]:,m[NT
2]H
:,l]
σ4
×(Ψ⋄HRA)ˆ
en[XU]k,l
(68i)
E"∂L(θ)
∂h∗
UAm1,k
∂L(θ)
∂hRAm2,n #=δm1,m2
σ2ˆ
eH
k(1B×K⋄XT
U)H
×(Ψ⋄ZT)ˆ
en(68j)
E"∂L(θ)
∂h∗
UAm1,k
∂L(θ)
∂hARn,m2#=
L
X
l=1
ˆ
eH
k(1B×M⋄XT
A)H
×
E[[NT
1]:,m1[NT
2]H
:,l]
σ4
×(Ψ⋄HRA)ˆ
en[XA]m2,l
(68k)
E"∂L(θ)
∂h∗
UAm,k1
∂L(θ)
∂hU Rn,k2#=
L
X
l=1
ˆ
eH
k1(1B×M⋄XT
A)H
×
E[[NT
1]:,m[NT
2]H
:,l]
σ4
×(Ψ⋄HRA)ˆ
en[XU]k2,l
(68l)
E"∂L(θ)
∂h∗
RAm1,n1
∂L(θ)
∂hARn2,m2#=
L
X
l=1
ˆ
eH
n1(Ψ⋄ZT)H
×
E[[NT
1]:,m1[NT
2]H
:,l]
σ4
×(Ψ⋄HRA)ˆ
en2[XA]m2,l
(68m)
E"∂L(θ)
∂h∗
RAm,n1
∂L(θ)
∂hU Rn2,k #=
L
X
l=1
ˆ
eH
n1(Ψ⋄ZT)H
×
E[[NT
1]:,m[NT
2]H
:,l]
σ4
×(Ψ⋄HRA)ˆ
en2[XU]k,l
(68n)
E"∂L(θ)
∂h∗
ARn1,m
∂L(θ)
∂hU Rn2,k #=
L
X
l=1
[XA]∗
m,lˆ
eH
n1(Ψ⋄HRA)H
×(Ψ⋄HRA)ˆ
en2[XU]k,l
1
σ2
(68o)
We can write Υwith subscripts denoting the FIM subma-
trices with respect to the individual channel parameters:
Υ=(69)
Υg∗
AgAΥg∗
AhUA Υg∗
AhRA Υg∗
AhAR Υg∗
AhUR
ΥH
g∗
AhUA Υh∗
UA hU A Υh∗
UA hRA Υh∗
UA hAR Υh∗
UA hU R
ΥH
g∗
AhRA ΥH
h∗
UA hRA Υh∗
RAhRA Υh∗
RAhAR Υh∗
RAhU R
ΥH
g∗
AhAR ΥH
h∗
UA hAR ΥH
h∗
RAhAR Υh∗
ARhAR Υh∗
ARhU R
ΥH
g∗
AhUR ΥH
h∗
UA hU R ΥH
h∗
RAhU R ΥH
h∗
ARhU R Υh∗
UR hU R
The CRB denoted with Cof the channel parameters is:
Υ−1=(70)
Cg∗
AgACg∗
AhUA Cg∗
AhRA Cg∗
AhAR Cg∗
AhUR
CH
g∗
AhUA Ch∗
UA hU A Ch∗
UA hRA Ch∗
UA hAR Ch∗
UA hU R
CH
g∗
AhRA CH
h∗
UA hRA Ch∗
RAhRA Ch∗
RAhAR Ch∗
RAhU R
CH
g∗
AhAR CH
h∗
UA hAR CH
h∗
RAhAR Ch∗
ARhAR Ch∗
ARhU R
CH
g∗
AhUR CH
h∗
UA hU R CH
h∗
RAhU R CH
h∗
ARhU R Ch∗
UR hU R
16 VOLUME ,
This article has been accepted for publication in IEEE Open Journal of the Communications Society. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/OJCOMS.2024.3506481
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
REFERENCES
[1] Q. Wu, B. Zheng, C. You, L. Zhu, K. Shen, X. Shao, W. Mei, B. Di,
H. Zhang, E. Basar, L. Song, M. D. Renzo, Z.-Q. Luo, and R. Zhang,
“Intelligent Surfaces Empowered Wireless Network: Recent Advances
and the Road to 6G,” Proceedings of the IEEE, pp. 1–40, 2024.
[2] S. Hassouna, M. A. Jamshed, J. Rains, J. u. R. Kazim, M. U. Rehman,
M. Abualhayja, L. Mohjazi, T. J. Cui, M. A. Imran, and Q. H.
Abbasi, “A survey on reconfigurable intelligent surfaces: Wireless
communication perspective,” IET Communications, vol. 17, no. 5, pp.
497–537, mar 2023.
[3] E. Bjornson, H. Wymeersch, B. Matthiesen, P. Popovski, L. San-
guinetti, and E. de Carvalho, “Reconfigurable Intelligent Surfaces: A
signal processing perspective with wireless applications,” IEEE Signal
Processing Magazine, vol. 39, no. 2, pp. 135–158, mar 2022.
[4] Q. Ding, J. Yang, Y. Luo, and C. Luo, “Intelligent Reflecting Surface
vs. Conventional Full-Duplex Relay in MmWave MIMO Networks:
A Comprehensive Performance Comparison,” IEEE Transactions on
Vehicular Technology, pp. 1–15, 2024.
[5] Z. Peng, Z. Zhang, C. Pan, L. Li, and A. L. Swindlehurst, “Multiuser
Full-Duplex Two-Way Communications via Intelligent Reflecting Sur-
face,” IEEE Transactions on Signal Processing, vol. 69, pp. 837–851,
2021.
[6] J. Zhao, M. Chen, M. Chen, Z. Yang, Y. Wang, B. Cao, and M. Shikh-
Bahaei, “Energy Efficient Full-Duplex Communication Systems with
Reconfigurable Intelligent Surface,” in IEEE Vehicular Technology
Conference, vol. 2020-Novem. Institute of Electrical and Electronics
Engineers Inc., nov 2020.
[7] B. C. Nguyen, T. M. Hoang, L. T. Dung, and T. Kim, “On Performance
of Two-Way Full-Duplex Communication System with Reconfigurable
Intelligent Surface,” IEEE Access, vol. 9, pp. 81 274–81 285, 2021.
[8] C. Qiu, Q. Wu, M. Hua, W. Chen, S. Ma, F. Hou, D. W. K. Ng, and
A. L. Swindlehurst, “Intelligent Reflecting Surface Empowered Self-
Interference Cancellation in Full-Duplex Systems,” IEEE Transactions
on Communications, vol. 72, no. 5, pp. 2945–2958, may 2024.
[9] B. Zheng, C. You, W. Mei, and R. Zhang, “A Survey on Channel
Estimation and Practical Passive Beamforming Design for Intelligent
Reflecting Surface Aided Wireless Communications,” IEEE Commu-
nications Surveys & Tutorials, vol. 24, no. 2, pp. 1035 – 1071, 2022.
[10] S. Lin, M. Wen, and F. Chen, “Cascaded Channel Estimation Using
Full Duplex for IRS-Aided Multiuser Communications,” in IEEE
Wireless Communications and Networking Conference, WCNC, vol.
2022-April. Institute of Electrical and Electronics Engineers Inc.,
2022, pp. 375–380.
[11] K. Tekbiyik, G. K. Kurt, C. Huang, A. R. Ekti, and H. Yanikomeroglu,
“Channel Estimation for Full-Duplex RIS-assisted HAPS Backhauling
with Graph Attention Networks,” in IEEE International Conference
on Communications. Institute of Electrical and Electronics Engineers
Inc., jun 2021, pp. 1–6.
[12] S. Yang, W. Lyu, Y. Xanthos, Z. Zhang, C. Assi, and C. Yuen, “Re-
configurable Intelligent Surface-Aided Full-Duplex mmWave MIMO:
Channel Estimation, Passive and Hybrid Beamforming,” IEEE Trans-
actions on Wireless Communications, vol. 23, no. 4, pp. 2575–2590,
apr 2024.
[13] G. Zhou, C. Pan, Z. Peng, and R. Schober, “Individual Channel
Estimation for RIS-Aided mm Wave Communication Systems,” in
GLOBECOM 2023 - 2023 IEEE Global Communications Conference.
IEEE, dec 2023, pp. 2912–2917.
[14] G. Zhou, Z. Peng, C. Pan, and R. Schober, “Individual Channel Estima-
tion for RIS-aided Communication Systems - A General Framework,”
IEEE Transactions on Wireless Communications, pp. 1–16, 2024.
[15] C. Hu, L. Dai, S. Han, and X. Wang, “Two-Timescale Channel
Estimation for Reconfigurable Intelligent Surface Aided Wireless
Communications,” IEEE Transactions on Communications, vol. 69,
no. 11, pp. 7736–7747, nov 2021.
[16] A. J. Fernandes and I. Psaromiligkos, “Channel Estimation for Recon-
figurable Intelligent Surface-Assisted Full-Duplex MIMO With Hard-
ware Impairments,” IEEE Wireless Communications Letters, vol. 12,
no. 10, pp. 1697–1701, oct 2023.
[17] A. Stegeman and N. D. Sidiropoulos, “On Kruskal’s uniqueness
condition for the Candecomp/Parafac decomposition,” Linear Algebra
and its Applications, vol. 420, no. 2-3, pp. 540–552, jan 2007.
[18] T. G. Kolda and B. W. Bader, “Tensor Decompositions and Applica-
tions,” SIAM Review, vol. 51, no. 3, pp. 455–500, aug 2009.
[19] N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E.
Papalexakis, and C. Faloutsos, “Tensor Decomposition for Signal
Processing and Machine Learning,” IEEE Transactions on Signal
Processing, vol. 65, no. 13, pp. 3551–3582, jul 2017.
[20] A. Cichocki, D. Mandic, L. De Lathauwer, G. Zhou, Q. Zhao, C. Ca-
iafa, and H. A. PHAN, “Tensor Decompositions for Signal Processing
Applications: From two-way to multiway component analysis,” IEEE
Signal Processing Magazine, vol. 32, no. 2, pp. 145–163, mar 2015.
[21] E. Kofidis, “A Tensor-Based Approach to Joint Channel Estima-
tion/Data Detection in Flexible Multicarrier MIMO Systems,” IEEE
Transactions on Signal Processing, vol. 68, pp. 3179–3193, 2020.
[22] H. Chen, F. Ahmad, S. Vorobyov, and F. Porikli, “Tensor Decomposi-
tions in Wireless Communications and MIMO Radar,” IEEE Journal
of Selected Topics in Signal Processing, vol. 15, no. 3, pp. 438–453,
apr 2021.
[23] G. Favier and D. S. Rocha, “Overview of Tensor-Based Cooperative
MIMO Communication Systems—Part 1: Tensor Modeling,” Entropy,
vol. 25, no. 8, p. 1181, aug 2023.
[24] J. Du, W. Yu, L. Jin, X. Li, and D. B. D. Costa, “Tensor-Based Angle
Estimation for Bistatic MIMO Radar Systems With Multi-Slot Gain-
Phase Error,” IEEE Transactions on Aerospace and Electronic Systems,
pp. 1–17, 2023.
[25] P. H. U. de Pinho, M. d. F. K. B. Couras, G. Favier, A. L. F.
de Almeida, and J. P. J. da Costa, “Semi-Blind Receivers for Two-Hop
MIMO Relay Systems with a Combined TSTF-MSMKron Coding,”
Sensors, vol. 23, no. 13, p. 5963, jun 2023.
[26] L. Mo, F. Saggese, X. Lu, Z. Wang, and P. Popovski, “Direct Tensor-
Based Estimation of Broadband mmWave Channels With RIS,” IEEE
Communications Letters, vol. 27, no. 7, pp. 1849–1853, jul 2023.
[27] L. Wei, C. Huang, G. C. Alexandropoulos, and C. Yuen, “Parallel
Factor Decomposition Channel Estimation in RIS-Assisted Multi-
User MISO Communication,” in 2020 IEEE 11th Sensor Array and
Multichannel Signal Processing Workshop (SAM), vol. 2020-June.
Hangzhou, China: IEEE, jun 2020, pp. 1–5.
[28] G. T. de Araujo and A. L. F. de Almeida, “PARAFAC-Based Channel
Estimation for Intelligent Reflective Surface Assisted MIMO System,”
in 2020 IEEE 11th Sensor Array and Multichannel Signal Processing
Workshop (SAM), vol. 2020-June. IEEE, jun 2020, pp. 1–5.
[29] L. Wei, C. Huang, G. C. Alexandropoulos, C. Yuen, Z. Zhang, and
M. Debbah, “Channel Estimation for RIS-Empowered Multi-User
MISO Wireless Communications,” IEEE Transactions on Communi-
cations, vol. 69, no. 6, pp. 4144–4157, jun 2021.
[30] G. T. de Araujo, A. L. F. de Almeida, and R. Boyer, “Channel
Estimation for Intelligent Reflecting Surface Assisted MIMO Systems:
A Tensor Modeling Approach,” IEEE Journal of Selected Topics in
Signal Processing, vol. 15, no. 3, pp. 789–802, apr 2021.
[31] Y. Lin, S. Jin, M. Matthaiou, and X. You, “Tensor-Based Algebraic
Channel Estimation for Hybrid IRS-Assisted MIMO-OFDM,” IEEE
Transactions on Wireless Communications, vol. 20, no. 6, pp. 3770–
3784, jun 2021.
[32] L. Wei, C. Huang, G. C. Alexandropoulos, Z. Yang, C. Yuen, and
Z. Zhang, “Joint Channel Estimation and Signal Recovery in RIS-
Assisted Multi-User MISO Communications,” in 2021 IEEE Wireless
Communications and Networking Conference (WCNC), vol. 2021-
March. Nanjing, China: IEEE, mar 2021, pp. 1–6.
[33] G. T. de Ara´
ujo, A. L. F. de Almeida, R. Boyer, and G. Fodor, “Semi-
Blind Joint Channel and Symbol Estimation for IRS-Assisted MIMO
Systems,” IEEE Transactions on Signal Processing, vol. 71, no. 7, pp.
1184–1199, jul 2023.
[34] J. Yuan, G. C. Alexandropoulos, E. Kofidis, T. L. Jensen, and E. De
Carvalho, “Channel Tracking for RIS-Enabled Multi-User SIMO Sys-
tems in Time-Varying Wireless Channels,” in 2022 IEEE International
Conference on Communications Workshops (ICC Workshops). Seoul,
Korea, Republic of: IEEE, may 2022, pp. 145–150.
[35] A. Hashi, S. Aldirmaz-Colak, M. Basaran, and L. Durak-Ata, “Channel
Estimation Effect on Sum-Rate Analysis of RIS-Aided UAV-Based
MISO Systems,” in 2022 30th Signal Processing and Communications
Applications Conference (SIU). Safranbolu, Turkey: IEEE, may 2022,
pp. 1–4.
[36] J. Du, Y. Cheng, L. Jin, and F. Gao, “Time-Varying Phase Noise
Estimation, Channel Estimation, and Data Detection in RIS-Assisted
MIMO Systems via Tensor Analysis,” IEEE Transactions on Signal
Processing, pp. 1–15, sep 2023.
VOLUME , 17
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content may change prior to final publication. Citation information: DOI 10.1109/OJCOMS.2024.3506481
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
Fernandes et al.: Tensor Signal Modelling and Channel Estimation for Reconfigurable Intelligent Surface-Assisted Full-Duplex MIMO
[37] B. Sokal, P. R. B. Gomes, A. L. F. de Almeida, B. Makki, and
G. Fodor, “Reducing the Control Overhead of Intelligent Recon-
figurable Surfaces via a Tensor-Based Low-Rank Factorization Ap-
proach,” IEEE Transactions on Wireless Communications, vol. 22,
no. 10, pp. 6578–6593, oct 2023.
[38] T. L. Jensen and E. De Carvalho, “An Optimal Channel Estimation
Scheme for Intelligent Reflecting Surfaces Based on a Minimum Vari-
ance Unbiased Estimator,” in ICASSP, IEEE International Conference
on Acoustics, Speech and Signal Processing - Proceedings, vol. 2020-
May. Institute of Electrical and Electronics Engineers Inc., may 2020,
pp. 5000–5004.
[39] A. L. Swindlehurst, G. Zhou, R. Liu, C. Pan, and M. Li, “Chan-
nel Estimation With Reconfigurable Intelligent Surfaces–A General
Framework,” Proceedings of the IEEE, pp. 1–27, may 2022.
[40] A. J. Fernandes and I. Psaromiligkos, “Joint Estimation of Direct
and RIS-assisted Channels with Tensor Signal Modelling,” in IEEE
Vehicular Technology Conference. Washington DC, USA: IEEE, oct
2024, pp. 1–6.
[41] A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and
R. Wichman, “In-band full-duplex wireless: Challenges and opportu-
nities,” IEEE Journal on Selected Areas in Communications, vol. 32,
no. 9, pp. 1637–1652, 2014.
[42] J. B. Kruskal, “Three-way arrays: rank and uniqueness of trilinear de-
compositions, with application to arithmetic complexity and statistics,”
Linear Algebra and its Applications, vol. 18, no. 2, pp. 95–138, jan
1977.
[43] B. Lovitz and F. Petrov, “A generalization of Kruskal’s theorem on
tensor decomposition,” Forum of Mathematics, Sigma, vol. 11, p. e27,
apr 2023.
[44] Xiangqian Liu and N. Sidiropoulos, “Cramer-Rao lower bounds for
low-rank decomposition of multidimensional arrays,” IEEE Transac-
tions on Signal Processing, vol. 49, no. 9, pp. 2074–2086, sep 2001.
[45] N. Li, S. Kindermann, and C. Navasca, “Some convergence results on
the Regularized Alternating Least-Squares method for tensor decom-
position,” Linear Algebra and its Applications, vol. 438, no. 2, pp.
796–812, jan 2013.
[46] P. Comon, X. Luciani, and A. L. F. de Almeida, “Tensor decomposi-
tions, alternating least squares and other tales,” Journal of Chemomet-
rics, vol. 23, no. 7-8, pp. 393–405, jul 2009.
[47] S. M. Kay, Fundamentals of statistical signal processing: estimation
theory. Prentice-Hall, Inc., 1993.
[48] H. Derksen, “Kruskal’s uniqueness inequality is sharp,” Linear Algebra
and its Applications, vol. 438, no. 2, pp. 708–712, jan 2013.
[49] K. Zheng, S. Ou, and X. Yin, “Massive MIMO Channel Models:
A Survey,” International Journal of Antennas and Propagation, vol.
2014, pp. 1–10, 2014.
[50] X. Chen, J. Shi, Z. Yang, and L. Wu, “Low-Complexity Channel Esti-
mation for Intelligent Reflecting Surface-Enhanced Massive MIMO,”
IEEE Wireless Communications Letters, vol. 10, no. 5, pp. 996–1000,
may 2021.
[51] E. Bjornson and L. Sanguinetti, “Power Scaling Laws and Near-Field
Behaviors of Massive MIMO and Intelligent Reflecting Surfaces,”
IEEE Open Journal of the Communications Society, vol. 1, pp. 1306–
1324, 2020.
[52] M. Cui, Z. Wu, Y. Lu, X. Wei, and L. Dai, “Near-Field MIMO
Communications for 6G: Fundamentals, Challenges, Potentials, and
Future Directions,” IEEE Communications Magazine, vol. 61, no. 1,
pp. 40–46, jan 2023.
[53] X. Mu, J. Xu, Y. Liu, and L. Hanzo, “Reconfigurable Intelligent
Surface-Aided Near-Field Communications for 6G: Opportunities and
Challenges,” IEEE Vehicular Technology Magazine, vol. 19, no. 1, pp.
65–74, mar 2024.
[54] T. Olutayo and B. Champagne, “Dynamic Conjugate Gradient Unfold-
ing for Symbol Detection in Time-Varying Massive MIMO,” IEEE
Open Journal of Vehicular Technology, vol. 5, pp. 792–806, 2024.
Alexander James Fernandes (S’19)
received the B.Eng. degree in biomed-
ical and electrical engineering and the
M.A.Sc. degree in biomedical engineer-
ing from Carleton University, Ottawa,
ON, Canada, in 2018 and 2020, re-
spectively. He is currently pursuing the
Ph.D. in Electrical Engineering with the
Department of Electrical and Computer Engineering, McGill
University, Montreal, QC, Canada. He was awarded by
the Natural Sciences and Engineering Research Council of
Canada (NSERC) the Industrial Undergraduate Student Re-
search Award (IUSRA) to work at Safran Electronics Canada
Inc. as a Software Engineering Student and from Ross Video
as a Hardware Product Verification Specialist. He served as a
reviewer for IEEE journals and conferences: IEEE Transac-
tions on Wireless Communications, IEEE Wireless Commu-
nications Letters, IEEE Vehicular Technology Conference.
His current research interests include channel estimation, sig-
nal processing, machine learning, wireless communication,
and reconfigurable intelligent surfaces.
Ioannis N. Psaromiligkos (S’96–M’03) received the
Diploma degree in computer engineering and science from
the University of Patras, Patras, Greece in 1995, and the M.S.
and Ph.D. degrees in electrical engineering from the State
University of New York at Buffalo in 1997 and 2001, respec-
tively. From 1995 to 1997, he was a Teaching Assistant and
from 1997 to 2001, a Research Assistant with the Commu-
nications and Signals Group in the Department of Electrical
Engineering, State University of New York at Buffalo. Since
2001, he has been with the Department of Electrical and
Computer Engineering, McGill University, Montreal, QC,
Canada, where he is currently an Associate Professor. His
research interests are in the areas of statistical detection and
estimation, adaptive signal processing, machine learning, and
wireless communications. Dr. Psaromiligkos has served as an
Associate Editor for IEEE COMMUNICATIONS LETTERS
and IEEE SIGNAL PROCESSING LETTERS and he is
currently an Associate Editor for the EURASIP Journal on
Advances in Signal Processing.
18 VOLUME ,
This article has been accepted for publication in IEEE Open Journal of the Communications Society. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/OJCOMS.2024.3506481
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
