This article has been accepted for publication in IEEE Transactions on Vehicular Technology. 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/TVT.2022.3217140
1
IRS-Assisted OTFS System: Design and Analysis
Anna Thomas, Kuntal Deka, Sanjeev Sharma, and Neelakandan Rajamohan
Abstract—Intelligent reflecting surface (IRS) is a recent technology garnering lots of attention amongst the wireless communication research community due to its ability to improve
the channel conditions extravagantly. The main feature of IRS
technology is its capability to tune the phases of the reflecting
elements to trigger a cohesive amalgamation of various multipath.
IRS can easily co-exist with almost all technologies, thereby offering flexibility to the designers to come up with attractive systems.
Further, orthogonal time frequency space (OTFS) is a recent 2D
modulation technique that can mitigate the drawbacks of the
traditional orthogonal frequency division multiplexing (OFDM)
for fifth generation (5G) and beyond networks, especially in high
Doppler wireless environments like vehicle-to-everything (V2X)
and millimeter wave (mmWave) communications. This paper
presents the system architecture of the IRS-assisted OTFS scheme
in the delay-Doppler (DD) domain, including the modulation, demodulation, and detection methods. The selection of the reflection
coefficients play the most vital role in the IRS-OTFS system, like
any IRS system. IRS reflection optimization cannot perfectly line
up all multipath with the same phase in the frequency-selective
channels. We adopt an IRS phase optimization method where
only the strongest delay-Doppler path is phase-aligned, while
the paths with lesser strengths will be out of phase with the
direct path. This reconfigurability of IRS elements enhances the
OTFS modulation scheme, thereby significantly mitigating the
multipath fading effects in high Doppler channels. Additionally,
we propose a channel estimation for the IRS-OTFS system where
both data and pilots are embedded in the same OTFS frame
with minimal guard-band overhead. The paper presents extensive
simulation results on the error-rate performance of the proposed
system over Rayleigh and Rician fading channels. The effects
of the number of IRS elements, the phase selections, and the
channel estimation method on the system’s performance are
investigated. The comprehensive simulation study demonstrates
that by exploiting the features of both OTFS and IRS, the
proposed approach can provide impressive performance in high
Doppler environments.
Index Terms—OTFS, IRS, delay-Doppler domain, reflection
optimization, coherent phase.
I. I NTRODUCTION
A. Motivation:
The introduction of intelligent reflecting surface (IRS) technology is foreseen as a potential solution for several challenges
faced by 5G and beyond wireless communication networks.
Copyright (c) 2015 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
Anna Thomas is with the School of Electrical Sciences, Indian Institute of
Technology Goa, India. e-mail: (anna183422001@iitgoa.ac.in).
Kuntal Deka is with the Department of EEE, Indian Institute of Technology
Guwahati, India. e-mail: (kuntaldeka@iitg.ac.in).
Sanjeev Sharma is with the Department of Electronics Engineering,
Indian Institute of Technology (BHU), Varanasi, India, e-mail: (sanjeev.ece@iitbhu.ac.in)
Neelakandan Rajamohan is with the School of Electrical Sciences, Indian
Institute of Technology Goa, India. e-mail: (neelakandan@iitgoa.ac.in).
Manuscript received May 22, 2022; Accepted October 13, 2022.
The acceptance of IRS is mainly attributed to the high receive
power gain, supported by relatively simple infrastructural
requirements for its implementation. Although IRS-aided systems demand additional signal processing requirements, the
compatibility of IRS with most of the existing techniques
makes it an attractive approach to enhance the channel quality.
To achieve the maximum rate, we can independently tune
the reflection properties of each of the IRS elements. The
most motivating feature of IRS is that the IRS elements can
be controlled in real-time depending on the varying channel
conditions [1, 2].
Orthogonal time-frequency space (OTFS), a recently proposed 2D modulation technique, can outperform the existing
orthogonal frequency division multiplexing (OFDM) modulation in many scenarios like V2X communications and
mmWave communications [3]. OTFS operates over the delayDoppler (DD) domain, meaning that the data symbols are
inserted over the DD domain, and the channel response is specified in the DD domain. This shift from the conventional timefrequency domain to the DD domain ensures that all symbols
transmitted in a frame experience nearly the same channel.
Hence, the time-variant channel can be represented as a timeinvariant channel that provides consistent performance in high
Doppler environments. At the same time, the 2D-localisation
in the DD domain makes OTFS capable of attaining the full
diversity of the channel, thereby remarkably improving the
bit-error-rate (BER) performance.
IRS-assisted systems have been designed and analyzed
from the perspective of many existing technologies, including
OFDM [1]. Since IRS is to be implemented cohesively with
the existing techniques, every IRS-integrated system model
should be separately developed considering its specific properties. IRS and OTFS offer two distinct advantages: the enhanced
received power gain and the increase in the effective diversity
gain, respectively. IRS and OTFS can complement each other
in non-line of sight (NLoS) urban and high Doppler wireless
communication scenarios. Hence, it is highly relevant to design
and analyze an IRS-assisted OTFS system, which can explore
the features of both IRS and OTFS.
B. Related Prior Works:
This paper presents a detailed analysis of an IRS-assisted
OTFS system. This subsection briefly describes the prior
works relevant to both IRS and OTFS. First, we mention the
results related to IRS-aided systems. The tutorial paper [1]
presented a detailed study regarding diverse aspects of IRS like
active and passive elements, analysis of different IRS-aided
systems including IRS-OFDM, IRS-phase optimization techniques, and the channel estimation methods for these systems.
The performance of IRS assisted systems based on joint active
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2
and passive beamforming was analyzed in [4]. A detailed study
of the concepts of IRS related to the propagation and path loss
modeling was presented in [2]. Focusing on the significance
of the IRS phase factor, [5] dealt with the practical phase
shifting techniques and the beamforming optimization features
of IRS. The works of [6] and [7] specifically focused on IRSOFDM systems, presenting two channel estimation techniques
and IRS-phase optimization techniques. The ergodic capacity
of IRS systems and the optimal location of IRS concerning
the base station and the user were studied in [8]. The work
presented in [9] gives a detailed analysis of bridging the
gap between the theoretical and practical aspects of IRSempowered systems. Also, in [10], the authors have proposed a
new channel model for IRS-assisted communication scenarios.
Next, we discuss the key milestones of OTFS. The technique
of OTFS modulation was first introduced in [3]. A detailed
analysis of the OTFS modulation principle was presented in
[3, 11], elaborating on the transformations involved. The OTFS
input-output relation for ideal and rectangular pulse shaping
was derived in [12]. A message-passing (MP) detection algorithm was presented based on these relations. The impact
of fractional Doppler and its associated interference for both
ideal and rectangular pulse shaping were extensively studied in
[12]. The analysis of any practical pulse-shaping waveform for
reduced cyclic prefix OTFS was presented in [13]. The authors
in [14–16] presented diversity analyses of OTFS, which helped
to analyze the performance of OTFS detectors. In [17], the
principles of OTFS modulation were derived from the basic
time-domain modulation, while in [18], the OTFS modulation
was interpreted as an interleaved OFDM with a cyclic prefix. A
detailed analysis of Peak-to-Average Power Ratio (PAPR) for
OTFS modulation was presented in [19] highlighting the dependency of PAPR on OTFS grid parameters. Various channel
estimation techniques of OTFS were proposed in [20, 21]. A
low-complexity linear minimum mean square error (LMMSE)
detector for OTFS is designed in [22]. The performance of
OTFS has been analysed in multi-user scenario also: [23],
[24] and [25] deal with OTFS-OMA, while different OTFSNOMA schemes are analysed in [26], [27] and [28]. The
results presented in [12] and [13] show the advantages of
the OTFS modulation scheme over OFDM modulation, even
at a user velocity of 500 Kmph. The better performance of
OTFS over OFDM in mmWave applications is highlighted in
the relevant works [29], [30], and [31]. Other than the OTFS
modulation technique, for high Doppler channel environments,
a frequency-domain multiplexing technique called frequencydomain cyclic prefix (FDM-FDCP) is introduced in [32]. A
time-frequency signaling approach is used in [33] and [34]
for dealing with the high Doppler scenarios. In this paper, we
focus on the IRS-assisted OTFS system.
C. Contributions:
This paper proposes the model of the IRS-assisted OTFS
system and presents its detailed analysis. The major contributions of this paper are highlighted below.
1) To the best of the authors’ knowledge, this work is the
first attempt to study IRS-assisted OTFS system. In this
2)
3)
4)
5)
paper, we focus on the formulation of the system model
from the perspective of the OTFS modulation technique
and analyze the results for various scenarios.
Initially, we develop the input-output relation of the
proposed system in the time domain, considering a timevariant channel for both the base station (BS)-IRS and
IRS-user links. Later on, we apply the pre-processing
and post-processing steps to the time-domain relation to
formulate the input-output relation in the DD domain. The
analysis of the system model and the resulting achievable
rate demonstrates that the IRS-OTFS system can attain
both the high Doppler resilience of OTFS and the high
received power gain of the IRS system.
The significance of phase factors of IRS elements is studied in the context of random phase and coherent phase.
In addition, the in-feasibility of coherent phase shifting
in the case of OTFS is highlighted, and an IRS phase
optimization method is proposed. This phase optimization
technique maximizes the achievable rate, which is called
the strongest delay-Doppler channel response (DDCR)
method.
A basic transmission protocol is developed, including the
channel estimation and phase optimization techniques.
We consider the channel estimation of OTFS modulation
and the features of the IRS system for the transmission protocol. The channel estimation method is dataembedded, meaning that both data and pilot symbols
are inserted in the same OTFS frame, avoiding the
requirement of dedicated frames for pilots.
The simulation results regarding the BER performance
and the achievable rate of the proposed system for the
Rayleigh channel, Rician channel, and the standard Extended Vehicular A (EVA) wireless channel model are
presented. Also, the strongest DDCR phase optimization
technique is validated by comparing it with the random
phase. The system’s performance in the practical scenario
is analyzed for different numbers of IRS elements, locations of the IRS, and various types of channels.
D. Outline:
The rest of this paper is organized as follows. Section II
presents the preliminaries of OTFS and IRS. In Section III,
we formulate an IRS-assisted OTFS system, initially from
the time-domain analysis and extending to the DD domain.
Section IV presents the reflection optimization using the
strongest DDCR method in OTFS. The channel estimation
technique is explained in Section V. Simulation results are
presented in Section VI, and Section VII concludes the paper.
Notations: In OTFS modulation, the wireless channel is
represented in the DD domain. IRS-assisted systems involve
three independent channels: (1) BS-user direct channel,
(2) BS-IRS channel, and (3) IRS-user channel. Hence, for
the IRS-assisted OTFS system, we need to represent all these
three channels in the DD domain using their corresponding
delay and Doppler parameters, as given in TABLE I.
Throughout this paper, any lowercase letter a denotes a
scalar, bold-lowercase letter a denotes a vector, and bolduppercase letter A denotes a matrix of the specified dimension.
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3
TABLE I: List of main variables and their meaning.
Notation
M ×N
x, y ∈ CN M×1
s, r ∈ CN M×1
Q
βq ejθq
Ld
Lg
Lf
h0 ∈ CLd ×1
Lg ×1
{g q }Q
q=1 ∈ C
Lf ×1
{f q }Q
∈
C
q=1
hq,ij
hq,p
τ0,u , ν0,u / l0,u , k0,u
g
g
g
g
τq,i
, νq,i
/ lq,i
, kq,i
f
f
f
f
τq,j
, νq,j
/ lq,j
, kq,j
dg
df
ddir
d1
d2
d3
d
Description
Dimension of DD grid
Tx and Rx signal in DD domain
Tx and Rx signal in time domain
Number of IRS elements
Reflection coefficient of q th IRS element
Number of DD taps in BS-user direct channel
Number of DD taps in BS-IRS channel
Number of DD taps in IRS-user channel
DDCR of BS-user direct channel
DDCR of BS-IRS channel for q th IRS element
DDCR of IRS-user channel for q th IRS element
DD channel coefficient of the cascaded path,
gq,i fq,j
DD channel coefficient of the pth cascaded path,
gq,i fq,j with DD tap as (li + lj , ki + kj )
DD value / tap of uth path in h0
DD value / tap of ith path in g q
DD value / tap of j th path in f q
distance between BS and IRS
distance between IRS and user
distance between BS and user in direct channel
horizontal distance between BS and IRS (x-axis)
horizontal distance between BS and IRS (y-axis)
height of the BS and IRS (z-axis)
horizontal distance between BS and user
AH denotes the conjugate transpose. For any complex number
x, phase of x is denoted using ∠x. FN represents the N point FFT matrix. Any parameter x denotes the DD domian,
x̃ denotes the corresponding time domain and x̄ denotes the
corresponding frequency domain representations. C denotes
the set of all complex numbers. IN , 0N and 1N represent identity matrix, matrix of zeros, and matrix of ones, respectively,
of dimensions N × N . The convolution operation is denoted
using ∗, the notation ⊙ denotes the Hadamard product, and ⊗
denotes the Kronecker product. For any integers l and M , the
notation [l]M denotes (l mod M ). The Gaussian distribution
and the uniform distribution are represented by N and U,
respectively.
II. P RELIMINARIES
A. IRS
IRS comprises a collection of tunable reflecting surfaces,
which can improve the quality of the channel by aligning all
the paths with the same phase. IRS assists the current systems
as an intermediate beamforming structure between the base
station and the user. We briefly present the IRS system model
in flat-fading channels and IRS-assisted OFDM systems.
a) IRS in flat-fading channels: In general, for a flatfading channel model, the input-output relation in the timedomain for an IRS-assisted system is given by [1],
!
Q
X
jθq ˜
ỹ(t) = h̃0 +
(1)
βq e fq g̃q x̃(t)
q=1
where h̃0 , f˜q and g̃q represents the channel impulse response
(CIR) of BS-user direct path, BS-IRS path and IRS-user
path of q th IRS element, respectively. The cascaded BSIRS-user channel gain of the q th IRS element is given by
f˜q g̃q . The key parameter of the IRS system is the reflection
coefficient, βq ejθq , which has to be tuned by the IRS controller based on the available channel state information (CSI).
⋆
The optimal
reflection coefficient iis given by βq = 1 and
h
⋆
θ = mod ∠h̃0 − (∠f˜q + ∠g̃q ), 2π , for q = 1, 2, . . . , Q [4].
q
IRS tuning based on this phase factor selection criteria is called
coherent phase condition. Note that the optimal magnitude
of reflection coefficient is independent of the CSI, while the
optimal phase factor depends on the CSI.
b) IRS in frequency-selective channels with OFDM systems: For an IRS-OFDM system, we consider a frequencyselective fading channel. Considering the cascaded BS-IRSuser channel, let L′g and L′f be the maximum number of
channel taps in BS-IRS and IRS-user channels, respectively.
Also, let L′d be the number of delay taps in the BS-user direct
channel. For N subcarriers and sufficient cyclic prefix (CP),
the input-output relation in IRS-OFDM is given as [6, 7]
i
h
(2)
ȳ = FN h̃0 + H̃irs θ x̄
where FN is the N - point DFT matrix, h̃0 is the zeropadded N × 1 CIR vector of BS-user direct channel, H̃irs =
[h̃1 h̃2 . . . h̃Q ] with h̃q being the N × 1 zero-padded form of
f̃ q ∗g̃ q , θ = [β1 ejθ1 β2 ejθ2 . . . βQ ejθQ ]T , and x̄ and ȳ are the
input and output symbols in the frequency domain. Since, the
Fourier transform of the channel coefficients is taken, (2) is the
frequency domain input-output relation of IRS-OFDM system.
As observed, the cascaded BS-IRS-user CIR of this system is
given as f̃q ∗ g̃q . Analysing the frequency domain relation of
(2), we can observe that the phase factor of each IRS element
is the same for the cascaded channel frequency response of the
corresponding IRS element. The reflection phase factor in the
case of IRS-OFDM system is not frequency selective. Hence,
instead of coherent phase, an optimal phase selection strategy
has to be adopted. For the phase optimization of IRS-OFDM
systems, different algorithms are applicable as proposed in [7]
and [6].
B. OTFS
a) OTFS Parameters: In this section, we explain the
basic principle of OTFS modulation technique, highlighting
the transformations involved in each stage. OTFS is a 2D
modulation technique in which we have a DD grid ΓM×N
with M − delay bins and N − Doppler bins. For a given
sub-carrier frequency of △f and with T △f = 1, the total
bandwidth is B = M △f and the total duration of an OTFS
frame transmission is given by Tf = N T . The parameters M
and N are decided such that we get sufficient delay resolution,
1
and Doppler resolution △ν = N1T . Thus, we
△τ = M△f
have the maximum Doppler νmax < T1 and the maximum delay
1
τmax < △f
.
b) OTFS Modulation Principle: The specific feature of
OTFS is that the data symbols are placed in the DD grid rather
than the traditional time-frequency (TF) grid. Therefore, at the
transmitter and at the receiver, additional pre-processing and
post-processing operations are performed in the DD domain,
which makes OTFS compatible to OFDM. Let x[l, k] be the
data symbol placed in the lth delay bin and k th Doppler bin, for
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l = 0, 1, . . . , M −1 and k = 0, 1, . . . , N −1. At first, the input
symbols in DD domain is transformed to TF domain, X[m, n]
using inverse symplectic
finite Fourier transform (ISFFT):
PM−1 P
nk
N −1
−j2π( ml
M − N ) . Then,
X[m, n] = √N1 M l=0
k=0 x[l, k]e
similar to OFDM we perform Heisenberg’s Transform
to obtain
the corresponding time-domain signal, s(t) =
P
M−1 PN −1
j2πm△f (t−nT )
gtx (t − nT ), where
m=0
n=0 X[m, n]e
gtx (t) is the pulse-shaping signal at the transmitter. Considering a channel with the DD channel
h(τ, ν), the
RR response
received signal is given by r(t) =
h(τ, ν)ej2πν(t−τ ) s(t −
τ ) dτ dν + z(t), where z(t) is the additive white Gaussian noise (AWGN). The received signal has to be converted to TF domain using Wigner transform, Y [n, m].
For
any receive basis pulse grx (t), we have Y [m, n] =
R −j2πν(t−τ
) ∗
e
grx (t − τ )r(t) dt|τ =nT, ν=m△f , which is similar
to that in OFDM. As the post-processing step, finally the TF
signal P
is converted
to the DD domain using SFFT: y[l, k] =
ml
nk
M−1 PN −1
√1
Y [m, n]ej2π( M − N ) . To achieve the ulm=0
n=0
NM
timate benefit of OTFS, the pulse-shaping waveform has to
be ideal, satisfying the biorthogonal property. In practice,
biorthogonal ideal pulses do not exist. In the case of practical
√
rectangular pulse-shaping, we have gtx (t) = grx (t) = 1/ T ,
for 0 ≤ t ≤ T . The entire modulation process of OTFS can be
summarized as a linear input-output relation in DD domain
y[l, k] =
P
X
i=1
hi x[[l − li ]M , [k − ki ]N ] + z[l, k]
(3)
where z[l, k] ∼ CN (0, σ 2 ) is the AWGN noise.
c) OTFS Detection: The overall input-output relation
of OTFS given in (3) can be expressed in matrix form as
y = Hx + z, where x, y and z are the N M × 1 vectors
representing the input data vector, output data vector and
the noise vector respectively. The channel coefficient matrix
H ∈ CN M×N M is a circulant block matrix. Linear detectors
like LMMSE detectors are implemented in low-complexity
versions by exploring the properties of circulant block matrix,
[22, 35]. At the same time, H matrix is highly sparse and,
therefore, non-linear algorithms based on message-passing
(MPA) [12, 23], and iterative Rake detector [36] are also
successfully applied in OTFS detection.
III. P ROPOSED IRS-OTFS S YSTEM M ODEL
This section first describes the IRS-assisted OTFS system
model. For the OTFS modulation technique, both the data and
the channel are represented in the DD domain. Hence, the path
parameters are represented using the corresponding DDCR
h(τ, ν). Fig. 1 shows the basic model of the IRS-assisted OTFS
system in the downlink. Fig. 1(a) shows the positioning of BS,
IRS and user in (x, y, z) coordinate plane, highlighting the
3D distance and the horizontal distances. Fig. 1(b) gives the
channel representation showing the effective paths involved
in an IRS-OTFS system. The DDCR can have a sparse
representation, [3]. Following this, the DDCR of the BSuser direct channel, BS-IRSP
channel and IRS-user channel are
d
represented as, h(τ0 , ν0 ) = L
u=1 h0,u δ(τ0 −τ0,u )δ(ν0 −ν0,u ),
PLg
g
g
g
g
g
) and h(τqf , νqf ) =
h(τ , ν ) = i=1 gq,i δ(τq −τq,i )δ(νqg −νq,i
PLqf q
f
f
f
f
j=1 gq,j δ(τq − τq,j )δ(νq − νq,j ), respectively. For better
clarity, Fig. 1 shows only 2 DD taps in all the three channels:
BS-user direct channel (h0 ), BS-IRS channel (g q ), and IRSuser channel (f q ). TABLE I describes the notations used in the
model. For the q th IRS element shown in Fig. 1, it is observed
that the cascaded BS-IRS-user channel will be the Cartesian
product of g q and f q . The horizontal distance of BS↔IRS
and IRS↔user are denoted by dg and df , respectively. Thus,
the total horizontal distance between the BS and the user is
dg + df .
We develop the system model in two stages: (A) Inputoutput relation in the time domain and (B) Input-output
relation in the DD domain by applying the pre-processing
and post-processing steps. The system model considers all IRS
elements and path parameters independent of each other.
A. Formulation of the input-output relation in time domain
The input-output relation is derived by analyzing the following signals sequentially: (1) Signal impinging on the IRS
element from BS, (2) Reflected signal by the IRS element
and (3) Signal received at the user via reflection from the IRS
element.
(1) Signal impinging on the IRS element: Consider any
q
single IRS element q. Let rin
(t) be the signal incident on the
q th IRS element corresponding to the transmitted signal s(t)
from BS. Considering the delay and Doppler of each of the
q
paths, rin
(t) is given by
q
rin
(t)
=
Lg −1
X
i=0
g
g
g
).
gq,i ej2πνq,i (t−τq,i ) s(t − τq,i
(4)
g
g
where gq,i , τq,i
and νq,i
are the channel coefficient, the delay
value and the Doppler value of the ith path of BS to the
q th IRS element channel. Note that in the IRS-assisted OTFS
system, the input symbol vector x is in the DD domain, and
the transmitted symbol vector s is obtained through OTFS
modulation of x as explained in the following Section III-B.
(2) Reflected signal by the IRS element: Suppose the reflection coefficient of the q th IRS element is given by βq ejθq ,
where the amplitude attenuation is βq ∈ [0, 1] and the phase
of reflection is θq ∈ [0, 2π).
Remark 1. In the development of theoretical analysis for the
proposed system, we assume the continuous phase shifts for
the IRS elements. However, in practice only a finite number of
phase shifts are feasible. The impact of discrete phase shifts
in the proposed system is analyzed in detail with numerical
simulations.
The signal reflected by the q th IRS element is given by
q
q
(t)
(t) = βq ejθq rin
rout
= βq ejθq
Lg −1
X
i=0
g
g
g
).
gq,i ej2πνq,i (t−τq,i ) s(t − τq,i
(5)
(3) Signal received by the user via reflection from the IRS
element: Let rq (t) be the signal received at the user reflected
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5
Q - IRS elements, {βq ejθq }Q
q=1
IRS controller
DD channel taps
q
z
ddir = d2 + d23
df = (d − d1 )2 + d22 + d23
dg = d21 + d22
(d1 , d2 , d3 )
y
q
gq
q
q th IRS element
Lg = 2
fq
OTFS Modulator
Lf = 2
L d + Lg L f = 6
dg
df
(0,0,d3 )
h0
OTFS Demodu
lator
Ld = 2
ddir
Base Station (BS)
d − d1
d1
d
(d, 0, 0)
User
d1
x
d − d1
d
(a) 3D Schematic.
(b) Channel with independent DD taps in downlink.
Fig. 1: Basic model of IRS-assisted OTFS system.
from the q th IRS element. This signal can be written as [13]
q
r (t) =
Lf −1
X
fq,j βq e
jθq
Lg −1
X
e
f
g
f
)
−τq,j
(t−τq,i
j2πνq,j
r(t) =
gq,i e
f
g
g
)
−τq,j
(t−τq,i
j2πνq,i
g
f
s(t − τq,i
− τq,j
) (6)
f
f
where fq,j , τq,j
and νq,j
are the channel coefficient, the delay
value and the Doppler value of the j th path of the q th IRS
element to user channel. The complete input-output relation
can be obtained as given in (7)
r (t) =βq e
jθq
Lf −1 Lg −1
X X
j=0
g
f
g
f
fq,j gq,i ej2π(νq,i +νq,j )(t−(τq,i +τq,j ))
i=0
g
f
)) (7)
+ τq,j
s(t − (τq,i
Each cascaded BS-IRS-user path has a Doppler shift of νq,ij =
f
g
g
f
(νq,j
+νq,i
), a delay of τq,ij = (τq,i
+τq,j
), and an effective DD
channel gain of hq,ij = fq,j gq,i . Now, (7) can be simplified
as
q
r (t) =βq e
jθq
Lf −1 Lg −1
X X
j=0
i=0
Q
X
rq (t)
q=1
i=0
j=0
q
is given by
hq,ij ej2πνq,ij (t−τq,ij ) s(t − τq,ij ).
(8)
Observe that the input-output relation in (8) is essentially the
OTFS relation in the continuous time-domain scaled by the q th
IRS reflection coefficient βq ejθq . The resultant channel has
Lg Lf paths having the delay and Doppler parameters τq,ij
and νq,ij respectively, and a channel gain of hq,ij with i ∈
{0, 1, . . . , Lg − 1} and j ∈ {0, 1, . . . , Lf − 1}.
=
Q
X
βq e
jθq
q=1
Lf −1 Lg −1
X X
j=0
i=0
hq,ij ej2πνq,ij (t−τq,ij ) s(t − τq,ij )
(9)
The relation in (9) reveals that the individual channel coefficients of the BS-IRS and the IRS-user path are not relevant
in the effective channel state information (CSI). Instead, only
the channel gain of each cascaded BS-IRS-user path will
contribute to the CSI. Similarly, in place of the individual
delay and Doppler values of the BS-IRS and IRS-user paths,
the effective delay and Doppler of the cascaded paths matter.
Along with the channel from the BS to the user via the
IRS, a direct channel also exists from the BS to the user.
Considering this direct channel, the basic input-output relation
of the IRS-assisted OTFS system in the continuous time
domain is given by (10)
r(t) =
LX
d −1
u=0
Q
X
h0,u ej2πν0,u (t−τ0,u ) s(t − τ0,u ) +
βq ejθq
q=1
Lf −1 Lg −1
X X
j=0
i=0
hq,ij ej2πνq,ij (t−τq,ij ) s(t − τq,ij ).
(10)
where h0,u , τ0,u and ν0,u are the channel coefficient, the delay
value and the Doppler value of the uth path of the direct BS
to user channel.
B. System relation in DD domain
Now, we extend the single IRS case to the overall system
with Q IRS elements. As the reflection coefficients of all
the IRS elements are independent, the cascaded paths of the
BS-IRS-user channel are independent. The aggregate received
signal at the user via the reflection from all the IRS elements
First, we express the exact delay and Doppler values in the
continuous-time domain in terms of the delay and Doppler
taps for the given values of M and N of the OTFS grid.
The OTFS parameters (M, N, and △f ) are selected such
that the delay resolution (△τ ) and the Doppler resolution
1
(△ν) are M△f
and N1T respectively. For any ith path with
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the delay and Doppler values as (τi , νi ), the corresponding
delay and Doppler taps are obtained as li = ⌈τi M △f ⌉ and
ki = ⌈νi N T ⌉, respectively. In any practical channel models
like EVA, the values of M and N are selected such that they
are sufficiently larger than the maximum delay tap (lmax ≪ M )
and the maximum Doppler tap (kmax ≪ N ), respectively. For
the critical sampling condition of T △f = 1, and the sampling
frequency of fs = M
T , we can obtain N M samples from r(t),
after discarding the CP [13]. This discrete-time formulation of
(10) is obtained as shown in (11)
r[n] =
LX
d −1
h0,u ej2π
k0,u (n−l0,u )
NM
u=0
Q
X
q=1
βq ejθq
Lf −1 Lg −1
X X
j=0
hq,ij ej2π
s[[n − l0,u ]N M ]+
kq,ij (n−lq,ij )
NM
i=0
s[[n − lq,ij ]N M ].
DD channel, where Ld ≤ P ≤ Ld + QLg Lf ≪ N M . After
expanding y component-wise in (14), and considering only
these P channel coefficients, y[n] can be expressed as
!
Q
P
−1
X
X
βq ejθq hq,p x[n − p + p′ ]N M ,
h0,p +
y[n] =
p=0
q=1
(16)
for n = 0, 1, . . . , N M − 1. Also, n = M k + l, p = M k ′ + l′ ,
and p′ = M : if l < l′ , else 0.
Remark 2. The overall effective channel matrix of the system
Heff maintains the block-circulant structure of the OTFS
system. Hence the detection algorithms for conventional OTFS
systems that exploits the block circulant nature of the effective
channel matrix can be directly inherited to the proposed IRSOTFS system.
(11)
where n = 0, 1, . . . , N M − 1. Expressing (11) in the matrix
form, we get
!
Q
X
jθq
r = H̃0 +
βq e H̃q s + z = H̃s + z
(12)
q=1
where s, r ∈ CN M×1 are respectively the transmitted and the
received data vector in time domain, H̃ ∈ CN M×N M is the
superimposed channel matrix of direct paths and cascaded
IRS paths in time domain, and z ∈ CN M×1 is the AWGN
vector with its ith component zi ∼ CN (0, σ 2 ). The structures
of H̃0 and H̃q follow the matrix structure of OTFS in time
domain as described in [16]. For convenience, we neglect the
noise vector hereafter. After applying the ISFFT/SFFT and
Heisenberg/Wigner transforms for the ideal-pulse shaping case
of OTFS, (12) becomes
!
Q
X
jθq
βq e H q x
y = H0 +
q=1
= (H0 + [H1 H2 . . . HQ ]Θ) x
(13)
T
jθ
where Θ = β1 e 1 IN M , β2 ejθ2 IN M , . . . , βQ ejθQ IN M ,
H0 and Hq are N M × N M channel coefficient matrices of
the direct channel and the cascaded IRS channel respectively.
Writing (13) in compact form, we get
y = Heff x
(14)
where, Heff = (H0 + [H1 H2 . . . HQ ] Θ) .
(15)
Following the basic OTFS relation in the DD domain, the
element in nth row and mth column of Hq is defined as

′

hq,p for p = [n − m]N M , l ≥ l
Hq [n, m] = hq,p for p = [n − m + M ]N M , l < l′


0
otherwise
for q = 0, 1, . . . , Q ; n = 0, 1, . . . , N M − 1 ; m =
n
m
0, 1, . . . , N M − 1 ; l = n − M ⌊ M
⌋ ; l′ = m − M ⌊ M
⌋,
′
′
′
hq,p = hq,ij for l = lq,ij and k = kq,ij for 0 ≤ l ≤
M − 1 , 0 ≤ k ′ ≤ N − 1.
Let P be the total non-zero coefficients in the superimposed
IV. R EFLECTION O PTIMIZATION : S TRONGEST
D ELAY-D OPPLER C HANNEL R ESPONSE
The selection of the reflection coefficients of the IRS
elements plays the most significant role in determining the
performance of an IRS-assisted system. For a system with
Q IRS elements, the reflection coefficients are given by
Q
{βq ejθq }Q
q=1 , where βq ∈ [0, 1] and {θq }q=1 ∈ [0, 2π). For
the analysis purpose, without losing any generality, we choose
{βq }Q
q=1 = 1, which gives the maximum reflection amplitude.
The design of the IRS reflection coefficient generally focuses
on the optimization of {θq }Q
q=1 . The key feature of an IRS
system is its ability to re-tune the reflection coefficients
depending on the channel conditions. In OTFS modulation,
the channel is represented in the DD domain, which changes
slower than the time-frequency channel representation. Hence,
the IRS phase needs to be tuned less frequently compared to
an IRS-OFDM system. The impact of the IRS-phase factors
is studied through four different cases: (A) Random phase,
(B) Coherent phase, (C) Optimal phase, and (D) Strongest
DDCR phase.
A. Random phase
First we consider the random phase factor scenario, where
θq ∼ U[0, 2π], q = 1, 2, . . . , Q. Note that the channel
conditions are not considered in this case. As each IRS phase
factor is randomly set, the reflected beams will not superpose
constructively with the same phase. Hence, the random phase
selection does not exhibit the beamforming feature of IRS. In
this particular scenario of the random phase, the input-output
relation of IRS-OTFS is the same as given in (16) with random
θq .
B. Coherent phase (Ideal/Hypothetical case)
The desirable effect of IRS is to attain perfect phase
cancellation between all the IRS-reflected paths so that the
resulting reflected beam aligns with the direct beam. From
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7
(16), the input-output relation in the coherent phase can be
expressed as
!
Q
P
−1
X
X
j∠h0,p
|h0,p | +
e
y[n] =
|hq,p | x[n − p + p′ ] (17)
p=0
q=1
For coherent phase, the following condition must be satisfied
for every q = 1, 2, . . . , Q:
(θq + ∠hq,1 ) = ∠h0,1 ; . . . ; (θq + ∠hq,P ) = ∠h0,P . (18)
It can be observed from (18) that a single phase shift value of
each of the IRS elements cannot simultaneously accomplish
the coherent combining of all the scattered paths. Though
coherent phase selection is the ideal case to achieve maximum
rate, it is not practically feasible in multi-tap channels. Hence,
we have to look for a phase shifting technique that can achieve
a rate close to that of the ideal case.
C. Optimal phase (Practically feasible)
Pt HH
1
eff Heff
log2 IN M +
C=
NM
N M σ2
where Pt
is
the transmit power, Heff
=
(H0 + [H1 H2 . . . HQ ] Θ) and σ 2 is the noise variance.
For a given Pt , the phase factors can be selected based on
the following optimization problem
Pt HH
eff Heff
N M σ2
Θ
log2 IN M +
s.t.
βq = 1, q = 1, 2, . . . , Q ;
p∈{0,...,P −1}
0 ≤ θq < 2π
Since this optimization problem is very complex, we propose
a sub-optimal phase selection strategy with significantly lower
complexity than the selection based on (P1) above.
D. Strongest DDCR phase
One of the practically effective phase-tuning methods for
IRS-assisted OFDM system is the strongest CIR path coherent
combining method [6]. In this method, the IRS reflection is
selected so that the strongest time-domain CIR gets aligned
with the direct beam. In typical OFDM-based representation,
the channel power is significantly more concentrated in the
time domain than in the frequency domain. On the other hand,
in the case of OTFS, the channel power is highly spread in
both the time and frequency domain [13, 20], but localized and
sparse in the DD domain. The reflection optimization based
on the time-domain strongest-CIR method will be highly complex. Hence, for the case of OTFS modulation, we consider
the DDCR of the system. We propose to find the strongest-
q=1
Using Cauchy-Schwarz inequality in (19) for the upper bound,
p⋆ is obtained as
#2
"
Q
X
⋆
|hq,p | .
|h0,p | +
(20)
p = arg
max
p∈{0,...,P −1}
q=1
Now to align the phase of IRS-reflected beam with the direct
beam, we tune the IRS-phase factors as
⋆
θq = ∠h0,p⋆ − ∠hq,p⋆ .
⋆
⋆
Once θq the IRS elements are tuned with {θq }Q
q=1 , then (16)
becomes
y[n] =
The optimal phase of the IRS elements can be selected such
that the capacity is maximum [6], considering the imperfect
phase cancellation of all multipaths. Similar to a conventional
OTFS system [37], the capacity for an IRS-OTFS system can
be expressed as
(P1) : max
DDCR tap p⋆ as the path that maximizes the upper bound on
the effective channel gain of a path:
!2
Q
X
⋆
jθq
max h0,p +
. (19)
p = arg
max
e hq,p
P
−1
X
p=0
|h0,p |e
j∠h0,p
+
Q
X
e
q=1
jθq⋆
|hq,p |e
j∠hq,p
!
x[n − p + p′ ]N M
=ej∠h0,p⋆ (|h0,p⋆ | + |hq,p⋆ |) x[n − p⋆ + p′ ]N M +
P
−1
X
p=0
p6=p⋆
|h0,p |e
j∠h0,p
+
Q
X
q=1
|hq,p |e
j(∠hq,p +∠h0,p⋆ −∠hq,p⋆ )
x[n − p + p′ ]N M
!
(21)
Hence, the strongest path will be phase aligned with the
direct path and the remaining paths will have a phase difference with the corresponding direct path. The simulation results
presented in Section VI-G show that the strongest DDCR
method can be adopted as the less complex phase selection
method without compromising the performance.
V. C HANNEL E STIMATION
The knowledge of real-time CSI is crucial as it drives the
tuning of the IRS elements apart from the detection process.
In this section, we present a channel estimation method for
the IRS-OTFS system. All IRS elements are ON during the
channel estimation phase. The channel estimation is done in
two sequential stages: (a) estimation of effective channel coefficients based on the pilot-data embedded frame with guard
band and (b) estimation of direct channel coefficients and
cascaded channel coefficients from the estimated coefficients
of the effective channel.
A. Estimation of the effective channel coefficients
Consider the DD relation of IRS-OTFS given in (16). For
the purpose of channel estimation, we need to express (16)
in 2D relation, to highlight the delay and Doppler taps of the
effective channel. Let y[l, k] be the received symbol in the
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8
channel estimation
and
strongest DDCR tuning
DD1
DD2
DDQ+1
B. Estimation of direct and cascaded IRS channel coefficients
DDdata DDdata
training phase (data+pilot)
IRS phase: Φ
DDdata
data transmission with
⋆
IRS phase: θq
M −1
lp + lτ
xp
lp
xp pilot
guard band
kp
kp + 2kν
0
kp − 2kν
lp − lτ
data
N −1
Observation region
{DDq }Q+1
q=1 : M × N OTFS frame transmitted in training phase
Fig. 2: Transmission Protocol for N = M = 8 and kν = lτ = 1.
[l, k] DD bin. Expressing (16) in the M × N grid structure,
y[l, k] can be expressed as
!
Q
N
−1 M−1
X
X
X
′ ′
jθq
′ ′
βq e hq [l , k ]
y[l, k] =
h0 [l , k ] +
k′ =0 l′ =0
q=1
x[[l − l′ ]M , [k − k ′ ]N ] (22)
where hq [l′ , k ′ ] = hq,ij for l′ = lq,ij and k ′ = kq,ij , 0 ≤
l′ ≤ M − 1 , 0 ≤ k ′ ≤ N − 1. Limiting (22) to the maximum
delay and Doppler tap present in the channel, we can obtain
the relation as
y[l, k] =
lτ
kν
X
X
k′ =−kν l′ =0
heff [l′ , k ′ ]x[[l − l′ ]M , [k − k ′ ]N ] + z[l, k]
(23)
where lτ and kν denotes the maximum delay and Doppler
tap present in the cascaded DD channel by considering all
the IRS elements. The Doppler tap is taken from −kν to kν ,
considering the both positive and negative Doppler frequency.
heff [l′ , k ′ ] is the element at the [l′ , k ′ ] tap of the M × N grid
of the effective channel. For this stage of channel estimation,
we adopt the pilot-data embedded technique in DD domain,
[38], as shown in Fig. 2. A sufficient guard band is allotted to
accommodate the maximum delay and Doppler spread of the
effective channel. Hence the interference of pilot and data is
totally avoided in the received DD frame and the observation
region has only the pilot symbols or the noise depending on the
DD taps of heff . By applying the thresholding in the observation region, we can estimate all the DD taps. If |y[l, k]| > cσp ,
then the particular DD tap [l′ , k ′ ] = [l−lp , k−kp ], is present in
E[|x |2 ]
1
, where SNRp = σ2p . The constant
heff . σp2 is given by SNR
p
c for the thresholding has to be fixed from the simulation
analysis, specific to the parameters used such as the number
of IRS elements and the pilot power. And the estimates of
the corresponding channel coefficients can be obtained as
ĥeff [l′ , k ′ ] = y[l, k]/xp . Let P̂ (P̂ ≪ N M ) be the total number
of DD paths estimated. Then the estimated channel vector is
denoted as ĥeff = [ĥeff (0) ĥeff (1) . . . ĥeff (P̂ − 1)]T .
Once the effective channel coefficients are estimated, we
need to estimate the direct channel coefficients and cascaded
IRS channel coefficients separately. This stage of channel
estimation is specific to the IRS system. For the purpose of
preliminary analysis of the IRS-OTFS system, in this stage,
we consider the IRS channel estimation technique specified
in [6]. Considering only the P̂ non-zero channel coefficients,
we define ĥq = [hq,0 hq,1 . . . hq,P −1 ]T , for q = 0, 1, . . . , Q.
Hence, ĥeff can be expressed as


1
 ejθ1 


ĥeff = [h0 h1 . . . hQ ]  .  = Hφ
 .. 
ejθQ
where H ∈ CP̂ ×(Q+1) is the concatenation of all the direct
and IRS channel coefficients to be estimated. To obtain φ as
a full-rank matrix, we should have (Q + 1) independent phase
Q+1
factor vectors {φ(i) }i=1
. Using the corresponding (Q + 1)
independent estimates of heff , we can estimate the required
H as, Ĥ = [ĥ0 ĥ1 . . . ĥQ ]
(1)
(2)
(Q+1)
=[ĥeff ĥeff . . . ĥeff
][φ(1) φ(2) . . . φ(Q+1) ]−1
=Ĥeff Φ−1
(24)
C. MSE and overhead analysis
To analyse the performance of the proposed channel estimation method, we compute the MSE of the final channel
estimate obtained in (24) as, E[||Ĥ − H||2F ], where || · ||F
is the Frobenius norm. We have ĥeff = heff + ze , where
ze = (xp )−1 z is the estimation error. We use normalized
Quadrature Amplitude Modulation (QAM) symbol as pilot
such that |xp | = 1, and hence ze ∼ CN (0, σp2 IP̂ ). The
estimate of the complete channel coefficients is expressed as
(1) (2)
(Q+1)
Ĥ = H + ZΦ−1 , where Z = [ze ze . . . ze
]. Hence
the MSE is obtained as
−1
E[||Ĥ − H||2F ] =E[||ZΦ−1 ||2F ] = E[ZH Z]tr ΦH Φ
−1
=P̂ σp2 tr ΦH Φ
.
(25)
The minimum MSE is obtained when ΦH Φ = (Q + 1)I(Q+1) .
Observe that the DFT matrix Φ = F(Q+1) satisfies this
condition. Hence, during the channel estimation phase of the
first (Q + 1) OTFS frames, we select the IRS phase factors
the same as the columns of the DFT matrix. For the special
case where all the IRS elements follow the same DD paths,
we have P̂ = P = Lg Lf .
Observe from Fig. 2, the overhead for channel estimation
using this technique depends on channel conditions of both
BS-IRS and IRS-user links. In general the overhead is given
by Ng = (Q + 1)(4kν + 1)(2lτ + 1). For any given channel
condition, the overhead increases linearly with the number Q
of IRS elements. However, the achievable rate increases with
Q. Hence, in future works, we plan to focus on devising a
more efficient channel estimation technique specifically for
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9
d g = d1
d3
IRS
ddir
d1
ddir = d2 + d23
df = (d − d1 )2 + d23
100
10−1
df
10−2
user
d − d1
d
BER
BS
10−3
23 dB
Fig. 3: Schematic for simulation analysis.
10−5
IRS-OTFS system, minimizing the overhead and attaining
maximum rate.
10
VI. S IMULATION R ESULTS
10
This section validates the theoretical analysis of the IRSassisted OTFS system presented in this paper, based on
the simulation results. The results are analyzed, highlighting
the impact of different IRS-OTFS system parameters over
Rayleigh and Rician fading channels. For the simulation
analysis, in the generalised IRS-OTFS structure shown in
Fig. 1a we take d2 = 0. The positioning of BS, IRS, and user
used for the simulations is as shown in Fig. 3. Throughout
the simulations, we take QAM data symbols. SNR is defined
2
)
as E(|x[l,k]|
, where σz2 is the noise variance. The results are
σz2
presented assuming that there is no direct BS-user channel, and
the IRS is located at an equal distance from the BS and the user
unless otherwise mentioned. For the simulations, we have used
the LMMSE detector. The simulation results are explained
by mentioning the specific parameters used for each of the
analysis. Considering the practical implementation of IRS,
discrete phase factors are selected for the simulation purpose.
Allocating 3 bits for the phase factor, the possible discrete
phase factors are given as θ = 2π
8 n, for n = 0, 1, . . . , 7.
16 dB
10−4
−6
Q = 4
Q =4
Q = 8
Q =8
Q = 16
Q = 16
Q = 32
Q = 32
Q = 64
Q = 64
−7
−40
−30
−20
−10
0
10
SNR (dB)
Fig. 4: BER with reflection optimization using strongest DDCR, the random
phase curves are shown in solid lines and the strongest DDCR curves are
shown in dashed lines.
f
maximum delay tap is lq,max
= 20, the maximum delay of the
g
BS-IRS channel will be very small. Hence, we limit lq,max
=2
resulting in the number of multipaths as Lg = 3 and Lf = 9.
The Doppler taps of all the channels are generated as per
q
Jakes’ formula νiq = νmax
cos(θi ) where θi ∼ U(−π, π).
Considering the maximum user velocity of v = 500 Km/h,
the maximum Doppler shift in the channel is obtained as
v(m/s)
q
In Fig. 5, the BER performance using
νmax
= fc 3×10
8 .
TABLE II: EVA channel model.
Parameter
Carrier frequency, fc
Subcarrier spacing, △f
Number of Doppler bins, N
Number of delay bins, M
q
Maximum delay tap, lh,max
Value
4 GHz
15 KHz
16
512
20
A. Reflection optimization by the strongest DDCR method
In Section IV-D, we have described the IRS reflection optimization by the strongest DDCR method.
We select this discrete phase factor such that θq =
⋆
arg
min 14π |ejθ − ejθq |. Here we present the simula2π
8
,...,
8
10−2
}
tion analysis of this optimization method. We consider an
OTFS grid with M = N = 32, △f = 15 KHz, fc = 4
GHz, and the total number of cascaded paths associated with
each IRS element is Lg Lf = 2 × 2 = 4. Each path’s delay
and Doppler taps are randomly chosen, and the BS-IRS and
IRS-user channels are modeled as Rayleigh fading.
B. Performance of IRS-OTFS for EVA propagation model
Here, we analyze the BER performance against the number
of IRS elements present in the system for EVA channel
model. TABLE II shows the OTFS parameters used for this
channel model. The IRS reflection coefficients are selected
as per the strongest DDCR method. The BER results are
shown in Fig. 5 with the IRS-user channel following the
EVA channel model. Rayleigh fading model is assumed for all
the channels, where the channel coefficients are i.i.d. random
variables, hi ∼ CN (0, σh2 ) and σh2 is as specified in the powerdelay profile. Note that although as per the EVA model, the
BER
θ∈{0,
100
10−4
No IRS
10−6
Q = 8 (SDDCR)
Q = 16
Q = 8 (GA)
Q = 16 (GA)
10
−8
−30
−20
−10
0
10
SNR (dB)
Fig. 5: BER in the case of EVA propagation model for various numbers of
IRS elements.
SDDCR method and GA-based optimization method [39] is
presented. The SDDCR method gives better performance than
the GA-based optimization.
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10
C. Performance in the presence of direct channel
Here we analyze the BER performance of the IRS-OTFS
system when the direct link of the BS-user is present. Since,
the number of multipaths is a feature specific to the channel,
the parameters Ld , Lg and Lf are not controllable. The OTFS
parameters are taken as M = N = 32, △f = 15 KHz
and fc = 4 GHz. Fig. 6 shows the BER results for different
numbers of multipath at a fixed value of Q = 4. Note that the
100
1
. The number of IRS elements is taken
sampling time T = △f
as Q = 4. The number of multipaths are taken as Lg = 2
and Lf = 4. For the Rician channel, we take the first path
in both channels as the LoS path and the remaining paths as
non-LoS ones. The distances are taken as dg = 150 m and
df = 150 m and the corresponding path-loss coefficients as
αg = 2 and αf = 2. The path-loss at 1 m reference distance is
given by −30 dB. The standard noise power spectrum density
of N0 = −174 dBm/Hz is taken and the noise power at the
receiver is M △f N0 . Being the OTFS symbol duration as N T
2
]
seconds, the transmit power is given by E[|x[l,k]|
[37]. In
NT
100
10−2
BER
10−1
10−4
Q=4
10−6
BER
10−2
Lg Lf = 4
Ld + Lg Lf = 2 + 4
Q=4
10−3
Lg Lf = 8
Ld + Lg Lf = 2 + 8
K = 0 (Rayleigh)
10
−8
−20
−15
−10
−5
0
5
10
10−4
K =5
SNR (dB)
Fig. 6: BER for different channel conditions considering the direct channel.
total number of multipath in the system is L = Lg Lf + Ld
where Lg Lf is the total number of cascaded paths in the BSIRS-user channel, and Ld is the number of paths in the BS-user
direct channel. Observe from Fig. 6 that for a fixed Lg Lf , the
BER performance of the IRS-OTFS system with the direct
channel is better than that devoid of the direct link. From the
results obtained, we can conclude that even if the number of
IRS elements remains the same, better performance can be
achieved if more paths are present in the cascaded channel
and when the direct BS-user channel is present.
D. Performance with Rician and Rayleigh channel models
Next, we analyze the system’s performance in both Rician
and Rayleigh channel models. The BS-user direct channel is
not considered for this analysis. We consider two different
cases: (1) The BS-IRS and IRS-user channels are Rician (2)
The BS-IRS and IRS-user channels are Rayleigh. The Rician
channel model is generated as
!
r
r
q
K
1
−αγ
s̃ +
s̄
s = dγ
K +1
K +1
where ds and αs for γ ∈ {g, f } be the distance and path
loss exponents, respectively of the BS-IRS (γ = g) and IRSuser (γ = f ) links, s̃ and s̄ represents the deterministic LoS
and non-LoS components in the respective links, and K is the
Rician factor denoting the ratio of power in the LoS path to
that of the non-LoS paths. The Rayleigh channel is modelled
from the Rician channel by taking K = 0. For the simulation
purpose, we have taken the OTFS paramaters as M = 32 and
N = 32, also the subcarrier frequency △f = 15 KHz and
K =1
K = 10
10−5
10
15
20
25
30
35
Transmit Power (dBm)
40
45
Fig. 7: BER in Rician and Rayleigh channel.
Fig. 7, we analyze the performance of the system in the Rician
and Rayleigh channel models. The results are presented for
different values of K = 0, 1, 5, and 10, where the result
for K = 0 corresponds to the Rayleigh channel. As expected,
the BER performance improves as the power of LoS (K factor) increases. Observe that the performance in the Rayleigh
channel is poorer than in the Rician channel.
E. Impact of the location of the IRS
Further, we analyze the system’s performance concerning
the IRS position, assuming that BS, IRS, and user are deployed
as shown in Fig. 3. We fix the BS-IRS-user cascaded link’s
total horizontal distance as d = 300 m and vary the BS-IRS
distance, d1 = dg . The BER is analyzed for different distances
of d1 from 50 m and 250 m. All the other parameters used are
the same as given in Section VI-D. The results are shown with
K = 1 and K = 5 for a different number of IRS elements
and transmit power of 25 dBm.
From Fig. 8, we can observe that the performance degrades
when the IRS is placed precisely at the center of the BSuser distance. For better performance, IRS has to be placed
either in the vicinity of BS or the user. The influence of
IRS is symmetrical around the central location of the BS-IRS
cascaded link.
F. Comparison with IRS-assisted OFDM system
This section compares the BER performance of the IRSassisted OTFS system with the IRS-assisted OFDM system.
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11
channel matrix, Cirs-otfs = N1M E[log2 det(I + ρHH
eff Heff )]. The
achievable rate of IRS-OFDM
system
hP
i as given
is obtained
M
1
H
E
h̃
, where ρ
log
1
+
ρf
in [6], Cirs-ofdm = M
2
m
m=1
is the SNR. Observe that IRS phase optimization gives
100
10−1
6
10−3
Coherent phase
10
−4
Pt = 25 dBm
Q = 4
Q = 4
Q = 8
Q = 8
Q = 16
Q = 16
GA
10
−5
10
−6
10−7
50
90
130
170
210
BS-IRS distance dg (m)
250
Fig. 8: BER performance based on IRS position, solid lines for K = 1 and
dashed lines for K = 5.
We choose Rayleigh channel model with Lg = 2 and Lf = 4,
where all the path coefficients hi ∼ CN (0, 1). The OTFS
parameters are set to M = N = 32. The results are
presented in Fig. 9 for different numbers of IRS elements,
Q = 4, 8, 16, 32 and 64. The results of IRS-OFDM are
obtained by applying the strongest CIR phase optimization
technique proposed in [6]. The simulation results clearly show
that the proposed IRS-OTFS system can significantly improve
the BER performance over IRS-OFDM.
100
10−1
BER
10−2
10−3
10−4
10−5
10−6
10−7
−40
Q =4
Q = 4
Q =8
Q = 8
Q = 16
Q = 16
Q = 32
Q = 32
Q = 64
Q = 64
−30
−20
−10
0
10
Achievable Rate (bits/s/Hz)
BER
10−2
Strongest DDCR
Random phase
4
without IRS
IRS-OFDM (Strongest CIR)
N = 32, M = 32
2
0
10
Q = 32
15
20
25
30
35
Transmit power (dBm)
40
45
Fig. 10: Achievable rate for different IRS phase factor selections
notably higher rate than the random phase selection. The upper
bound is obtained assuming the ideal and inapplicable1 case
of the coherent phase shifting as explained in Section IV-B.
Fig. 10 also shows the achievable rate for the optimal phase
selection described in Section IV-C. We obtain these optimal
phases by solving the problem (P1) with the genetic algorithm
(GA), [39]. Note that the strongest DDCR method can achieve
comparably equal performance with that of the optimal phase
selection. However, for smaller values of Q, it can be checked
with simulations that the achievable rate of IRS-OTFS system
remains almost the same as that of the IRS-OFDM system.
Fig. 11 shows the achievable rate for different phase optimization methods with respect to the number Q of IRS
elements, for a fixed transmit power of Pt = 45 dBm. As
expected, the advantage of the IRS-assisted OTFS system
increases notably for higher values of Q. The various IRS
phase factor methods follow the same features as seen in
Fig. 10. Observe that, apart from the random phase, the IRSOTFS system can achieve a higher rate than the IRS-OFDM
system as the number of IRS elements increases.
SNR (dB)
Fig. 9: BER performance of IRS-OTFS (dashed lines) and IRS-OFDM (solid
lines) system
G. Achievable rate of IRS-OTFS system
The performance of IRS systems can also be analyzed
based on the achievable rate. We consider a Rayleigh fading
channel model with Lg = Lf = Ld = 2 and OTFS DD
grid of size N = M = 32. Fig. 10 shows the impact
of different IRS phase selection strategies on the achievable
rate of IRS-OTFS for a fixed number of IRS elements,
Q = 32. The advantages of IRS-assisted OTFS compared
to the conventional OTFS (‘without IRS’) is observed from
the results. The achievable rate of IRS-OTFS system can be
obtained as, [37] by considering the Heff matrix as effective
H. Channel estimation
Next, the simulation results for the channel estimation
method given in Section V are presented. This proposed channel estimation method is referred to as Sch-1 hereafter. In addition, another method Sch-2, is considered using a compressivesensing-based OTFS channel estimation. The subspace pursuit
algorithm given in [21] is used for the effective channel
coefficient estimation process. Considering the overhead for
ce
channel estimation, the achievable rate is given by Cirs-otfs
=
(1 − η)Cirs-otfs where, η is the overhead for channel estimation
ν +1)(2lτ +1)
[40]. For Sch-1, ηSch-1 = (Q+1)(4k
and for Sch-2,
(Q+2)N M
1 It is not feasible to attain a coherent phase in the case of IRS-OTFS. For
finding the upper bound, we have manually enforced the condition of the
coherent phase.
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12
Coherent phase
Achievable Rate (bits/s/Hz)
GA
Strongest DDCR
6
Random phase
without IRS
IRS-OFDM (Strongest CIR)
4
N = 32, M = 32
Pt = 45 dBm
2
0
4
16
28
40
52
Number of IRS elements (Q)
64
Fig. 11: Achievable rate with respect to number of IRS elements.
M
ηSch-2 = (Q+1)N
(Q+2)N M . Consider OTFS parameters N = M = 32
and lτ = kν = 2. Then the values of ηSch-1 = 0.036 and 0.041
for Q = 4 and Q = 16 respectively. And ηSch-2 = 0.83 and
0.94 for Q = 4 and Q = 16 respectively. For the analysis
purpose, we have considered small values for lτ and kν , and
the overhead requirement will be more for Sch-1 for higher
values of these parameters.
10−1
kν = 2 (Theor.)
kν = 2 (Sim. Sch-1)
kν = 4 (Theor.)
NMSE
10−2
Q=4
N = 32, M = 32
SNR= 20 dB
kν = 4 (Sim. Sch-1)
kν = 2 (Sim. Sch-2)
kν = 4 (Sim. Sch-2)
stage of the estimation of the channel coefficients of the direct
link and the cascaded link will be independent of the user
velocity. Also, observe from Fig. 12 that the MSE decreases
linearly as the pilot SNR SNRp increases. This observation
matches with (25) as σp2 = 1/SNRp . Observe that the QAMpilot based channel estimation method gives a better NMSE
performance than the compressive sensing based method. Sch1 requires guardband such that the interference among pilot
and data can be avoided, while for Sch-2 pilot vectors are
separately transmitted over a DD frame.
Fig. 13 shows the achievable rate in the case of random
phase selection and the strongest DDCR (SDDCR) with perfect CSI and with channel estimation. The SNRp value is fixed
at 20 dB, and the performance is studied for different numbers
of IRS elements. As expected, the random phase performance
cannot achieve the rate that the strongest DDCR phase will
provide. Also, the achievable rate of the SDDCR method in
the practical scenario (with channel estimation) is very close
to the case where perfect CSI is available.
5
Random phase (Q = 4)
SDDCR - perfect CSI (Q = 4)
Achievable rate (bits/s/Hz)
8
SDDCR - ch. est Sch-1 (Q = 4)
4
Random phase (Q = 16)
SDDCR - perfect CSI (Q = 16)
SDDCR - ch. est Sch-1 (Q = 16)
3
SDDCR - ch. est Sch-2 (Q = 4)
SDDCR - ch. est Sch-2 (Q = 16)
N = 32, M = 32
2
SNRp = 20 dB
1
10−3
0
0
10
−4
5
10
SNR (dB)
15
20
Fig. 13: Achievable rate with channel estimation.
10−5
20
25
30
SNRp (dB)
35
40
Fig. 12: NMSE with different maximum Doppler taps. (solid lines show
theoretical values and dashed lines show the simulated values.)
In Fig. 12, we validate the NMSE of the channel estimates
based on the theoretical analysis given in (25) of Section V-C.
E[||Ĥ−H||2 ]
Here the NMSE is given by E[||H||2 ]F . The results in Fig. 12
F
show the estimation performance for different user velocities
(kν ) as SNRp varies. The number of IRS elements is fixed
at Q = 4, and the SNR of data is fixed at 20 dB. Note that
the simulation results agree with the theoretical analysis. The
estimation performance is independent of the user velocity.
This feature of the proposed method is attributed to the retainment of a sufficient guard band considering the maximum
Doppler present in the channel while estimating the effective
channel coefficients. Hence, an increase in the user velocity
cannot cause pilot-data interference. The thresholding method
is not affected by the user velocity. Once the effective channel
coefficients are estimated correctly in the first stage, the second
The channel estimation performance is also analyzed based
on BER. In Fig. 14, the BER results with channel estimation
are given for an OTFS frame with size M = N = 32. For
a fixed value of Q = 4, the BER performance is studied
using different SNRp values using the strongest DDCR phase
optimization. For SNRp = 35 dB, we can achieve the BER
close to the case with perfect CSI. However, the channel
estimation provides better BER than tuning the IRS elements
with random phase factors.
VII. C ONCLUSION
In this paper, we have proposed a system model for IRSassisted OTFS modulation. IRS aided systems have gained
popularity of late because of their low power consumption
and the ability to reconfigure the reflection properties intelligently in real-time. Despite the significant power gain of
the IRS-assisted OFDM system, it still faces challenges in
high Doppler environments. The recently introduced OTFS
modulation technique is a capable solution that can outperform
OFDM in such scenarios. Therefore, an IRS-assisted OTFS
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13
100
10−1
[11]
BER
10−2
[12]
10−3
[13]
10−4
Random phase
SNRp = 30 dB (SDDCR) Sch-1
[14]
SNRp = 35 dB (SDDCR) Sch-1
10−5
SNRp = 35 dB (SDDCR) Sch-2
perfect CSI (SDDCR)
[15]
10
−6
−20
−15
−10
−5
0
5
10
SNR (dB)
[16]
Fig. 14: BER performance with channel estimation.
system is designed in this work to overcome the significant
drawback of IRS-OFDM systems. For the reflection optimization of the IRS-OTFS system, we have proposed the
method of the strongest DDCR cancellation. The simulation
results show that this phase optimization technique offers
significant performance gain over random phase selections
for diverse delay-Doppler channel conditions. Furthermore,
we have proposed a channel estimation method suitable for
the IRS-OTFS system where data and pilots are transmitted
over a single frame, with minimal guard-band overhead. By
using a minimum number of IRS elements, IRS-OTFS system
achieves a BER of 10−4 , at an SNR gain of around 5 dB and
10 dB over the conventional OTFS system (EVA channel) and
IRS-OFDM system respectively. The proposed IRS-assisted
OTFS system can exhibit the features of both IRS and OTFS,
which makes it a promising technology in high Doppler
environments for future wireless communication networks.
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14
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