An α-Rotated Fourier Transform Used as OTFS Enhancement .pdf

An α-Rotated Fourier Transform Used
as OTFS Enhancement
Marwa Rjili1(B)
, Abdelhakim Khlifi2
, Fatma Ben Salah1
, Belghacem Chibani1
,
and Said Chniguir3
1
MACS laboratory: Modeling Analysis and Control of Systems LR16ES22,
National Engineering School of Gabes, University of Gabes, Gabes, Tunisia
marwa.rjili@isimg.tn, fatmabensalah89@gmail.com, abouahmed17@gmail.com
2
Innov’COM Laboratory, Sup’COM, University of Cartahge, Tunis, Tunisia
abdelhakim.khlifi@gmail.com
3
Mathematical Department, IPEIG, University of Gabes, Gabes, Tunisia
saidchneguir@yahoo.fr
Abstract. Orthogonal-time-frequency-space (OTFS) modulation has
received significant attention in wireless communication research due to
its exceptional ability to reliable mobile communication. Network links
censure high-speed connection. OTFS represents a promising candidate
for next-generation wireless communication systems. Operating in the
delay-Doppler (DD) domain, it relies on channel invariance. Furthermore,
the OTFS system can be implemented on top of an existing Orthog-
onal Frequency Division Multiplexing (OFDM) system, thereby reduc-
ing installation expenses. This advantage is especially noteworthy, given
that the Fractional Fourier Transform (FrFT)-based OFDM scheme has
demonstrated its superiority over traditional OFDM schemes. We utilize
the FrFT-based OFDM system to enhance the performance of the OTFS
system. The Weighted Fractional Fourier Transform (WFrFT), as a vari-
ant of FT, is a generalized Fourier transform this induces the discrete
Fourier transform (DFT) as a special case. In this article, we have cho-
sen to apply FrFT-based OFDM to assess the performance of the OTFS
system. Specifically, we have employed the Weighted Fractional Fourier
Transform (WFrFT) as a variant of FrFT. We evaluate the usage of an
OTFS system that incorporates the inherent WFrFT- based OFDM pro-
cessing. We obtained various optimal alpha values for SNR values ranging
from 0 to 9 dB. Additionally, we have demonstrated that the OTFS sys-
tem based on WFrFT maintains a lower error rate compared to the con-
ventional OTFS system for SNR ranging from 0 to 9 dB.
1 Introduction
The interaction between emitter and a specific receiver, depends on the chan-
nel character. Consequently, it becomes imperative to meticulously select the
appropriate waveforms to overcome potential constraints. Addressing this chal-
lenge, numerous novel waveforms have been proposed to cater to the multifaceted
c
The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
L. Barolli (Ed.): AINA 2024, LNDECT 204, pp. 249–260, 2024.
https://doi.org/10.1007/978-3-031-57942-4_25

250 M. Rjili et al.
requirements and scenarios of next mobile network generation that extend past
the fifth generation [1].
A variety of waveforms have been proposed as an enhanced version of orthog-
onal frequency division multiplexing (OFDM) using various techniques includ-
ing, generalized frequency division multiplexing (GFDM)[2,3], Filtered OFDM
(F-OFDM), and Asymmetric OFDM (A-OFDM) [4,5]. Recently, a novel wave-
form known as OTFS has emerged, demonstrating an effective capacity to handle
time-varying channels with pronounced Doppler effects [6,7]. OFDM modulation
creates a channel that is equivalent to the Time-Frequency domain (TF), while
OTFS modulation creates a channel that is equivalent to the Delay Doppler
domain [8].
In a recent study [9], an integrated waveform framework was introduced,
wherein the Weighted Fractional Fourier Transform (WFrFT) serves as a pre-
coding technique in the OTFS system [1]. In summary, the Fractional Fourier
Transform (FrFT) emerges as a potent signal processing tool that enhances sys-
tem robustness, also finding application in OFDM systems as a replacement
for the conventional Fourier Transform (FT) to enhance performance [9]. The
DFrFT OFDM system exhibits a lower symbol error rate compared to con-
ventional OFDM [10,11]. This holds true even in frequency-selective channels,
effectively mitigating Inter-Symbol Interference (ISI) [12,13]. However, these lim-
itations can potentially be overcomemed by adopting the newly proposed OTFS
operator with the use DFrFT [9].
This paper presents the evaluation analysis of the OTFS system combined
with WFrFT- OFDM. We will evaluate the design of an OTFS system based
on WFrFT-OFDM and determine the optimal fractional orders for various SNR
values, ranging from 0 to 9 dB with a 3 dB increment, based on Monte Carlo
simulations.
2 Fractional Fourier Transform
The time-frequency plan has orthogonal axes that are formed by frequency and
time. In comparison, the conventional Fourier transform can be considered as
a transformation that redefines a signal in the time domain with respect to
frequency; this transformation can be represented as a rotation up to π/2 radians
in the trigonometric direction within the time-frequency plan.
On the other hand, the Fractional Fourier Transform is an extension of the
classic Fourier Transform, it can be rotated to an angle in the time-frequency
plane as shown in Fig. 1. The Fractional Fourier Transform (FrFT) presents the
original signal function in time for each unique value of α. As a result, the
FrFT can be considered as a one-parameter family of transformation. When
α = 0, the Fractional Fourier Transform is used as an identity transformation
which essentially restores the original signal in the time domain conversely, when
α = π/2, it gives the conventional Fourier Transform, the FrFT of a signal is
mathematically defined as [14]:

Rotated Fourier Transform Used as OTFS Enhancement 251
Fig. 1. (Ox, Oy) plane rotation
Xα =
N/2
−N/2
x(t)kα(t, v) dt (1)
where α = aπ/2 is the angle of FrFT, a is the real number defined as FrFT order
and kα(t, v) is defined as transformation Kernel [14]:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
δ(t + v) if α + π is a multiple of 2π
δ(t − v) if α is a multiple of 2π

1−j cot α
2π
exp(j
(t2
+v2
)
2
cot(α) − jvt cos(α)) if α is not a multiple of 2π)
(2)
Then, the inverse Fractional Fourier Transform can be expressed as [14]:
x(t) =
1
2π
N/2
−N/2
Xα(v)k−α(t, v) dv (3)
2.1 The Weighted Fourier Fractional Transform
A weighted Fractional Fourier Transform consists of Fourier Transforms of
four different integer orders with different weighting factors which is defined
4-WFrFT. This approach allows for the development of a new method to defin-
ing the FrFT [19], you can express it’s definition as:
Fα
[f(t)] =
3

k=0
Bα
k fk(t) (4)
with
Bα
k = cos(
(α − k)π
4
) cos(
2(α − k)π
4
) exp(
−3(α − k)π
4
). (5)
You can write:
f0(t) = f(t)
f1(t) = F[f0(t)]
f2(t) = F[f1(t)]
f3(t) = F[f2(t)]
(6)

252 M. Rjili et al.
where F represents the Fourier transform. Equation (7) presents an alternative
definition of four-WFrFT primarily utilized in signal processing [19,20].
Fα
[f(t)]) =
3

k=0
Bα
k fk(t)
= Bα
0 .f0(t) + Bα
1 .f1(t) + Bα
2 .f2(t) + Bα
3 .f3(t)
= Bα
0 .I.f(t) + Bα
3 .F.f(t) + Bα
3 .F2
.f(t) + Bα
3 .F3
.f(t)
= (Bα
0 .I + Bα
1 .F + Bα
2 .F2
+ Bα
3 .F3
)f(t)
(7)
In this context, I represents the identity matrix. while I = F4
denotes the
use of the Fourier Transform in both vector and matrix forms. This allows us to
write the equation as follows:
Fα
[f(t)] =

I, F1, F2, F3

⎡
⎢
⎢
⎣
Bα
0
Bα
1
Bα
2
Bα
3
⎤
⎥
⎥
⎦ f(t) (8)
To maintain the consistency of dimension, the correct form of Bα
k in Eq. (8)
must be Bα
l .I. To make things easier; it is also known as Bα
k (k = 0, 1, 2, 3), the
weightings coefficients Bα
k can be defined as [19]:
⎡
⎢
⎢
⎣
Cα
0
Cα
1
Cα
2
Cα
3
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
1 1 1 1
1 i −1 −i
1 −1 1 −1
1 −i −1 i
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
Bα
0
Bα
1
Bα
2
Bα
3
⎤
⎥
⎥
⎦ (9)
By using this transformation, the coupled equations are completely separated
into a new set of equation that have considerably simpler forms:
C
(α+β)
0 = Cα
0 Cβ
0 , C
(α+β)
1 = Cα
1 Cβ
1
C
(α+β)
2 = Cα
2 Cβ
2 , C
(α+β)
3 = Cα
3 Cβ
3
(10)
where Cα
n = exp(2πnα/2), α = 0, 1, 2, 3. Solutions for the original set of coeffi-
cients are obtained by inverse transformation of Eq. (10) as follows:
Bα
0 = exp(3iπα/4) cos(πα/2) cos(πα/4), Bα
1 = B
(α−1)
1 , Bα
2 = B
(α−2)
0 , Bα
3 = B
(α−3)
3
(11)
As a result, the fractional-order Fourier transform can be expressed in the fol-
lowing manner:
Fα
[f(t)]) = Bα
0 f0(t) + Bα
1 f1(t) + Bα
2 f2(t) + Bα
3 f3(t)
=
3

n=0
exp(
i3π(α − n)
4
) cos(
π(α − n)
2
) cos(
π(α − n)
4
)fn(t)
(12)

3 WFrFt-OFDM Based OTFS System
3.1 OTFS Transmitter
At the transmitter, in the ﬁrst step, the information symbols X reside in the
Delay Doppler domain (DD) of size M × N, where M represents the number of
delay intervals and N represents the doppler intervals respectively. As demon-
strated in Fig. 2, the time-frequency domain OTFS symbols are realized based on
X via the Inverse Symplectic Fast Fourier Transform (ISFFT), the transmitter
signal is given as [16]:
XT F =
1
MN
N−1

k=0
M−1

l=0
XDDe(j2π[ nk
N − ml
M ])
(13)
where the variable XT F represents values in the TF domain, while XDD repre-
sents values in the DD domain. Additionally, the index m takes integer values
from 0 to M −1, and the index n takes integers from 0 to N −1. A more concise
representation of the Inverse Symplyctic Fast Fourier Transform (ISFFT) can
be obtained using the discrete Fourier Transform (DFT) matrices TN ∈ CN×M
and TM ∈ CN×M
, let XDD ∈ CN×M
contain the symbols x[k, l] of the delay
doppler domain, and XT F ∈ CN×N
contains the symbols X[m, n] of the time
frequency domain. Then, the Eq. (14) can be written in matrix form as [9]:
XT F = TH
N XDDTM (14)
Then, Time Frequency domain signal X is processed with an IWFrFT transform
of order α [16] to generate the signal as shown in the following Fig. 2 [9]: which
is referred as WFrFT-OFDM modulation.
Fig. 2. OTFS transmitter
S = XT F T−α,M (15)
where XT F represents the time-frequency signal matrix and T−α,M represents
the DFT matrix of order α. Using (14) in (15), the equation for S can be reduced
to:
S = TH
N XDDTM T−α,M (16)
The signal matrix in the delay-doppler domain, denoted as XDD, can be trans-
formed into an equivalent vector signal x, within the same domain. This vector,

254 M. Rjili et al.
represented as x, has a size of MN × 1 and is obtained by vectorizing X. This
means that each element in X, at position (k, l), corresponds to the (kM + l)th
indexed element in the resulting x vector. As a result, the time-domain (TD)
signal can be represented as a vector as it follows:
s = vec{TM T−α,M XDDTH
N } = (TH
N T1−α,M ) (17)
where s is located within the TD domain signal vector of size MN × 1 . The
notation denote the Kronecker product operation and T(1−α,M) represents
(1 − α)th
order WFrFT matrix of size M. Then the signal is transformed to
passband signal and a cyclic prefix(cp) with length lcp is added.
3.2 DD Domain Channel
The signal s(t) is conveyed through a time-varying channel possessing a complex
baseband channel impulse response denoted as h(τ, v). This impulse response
characterizes how the channel responds to an impulse signal with a delay of τ
and a Doppler shift of v, as described in [17]. The received signal, R(t) can be
expressed as follows [18]:
R(t) =

h(τ, v)kαs(t, v)e2jπ(t−τ)
dτdv (18)
Equation (18) signifies a continuous Heisenberg transform that is parameterized
by s(t), as described in [17]. Since there are generally only a limited number of
reflectors within the channel, each associated with specific delays and Doppler
shifts, we require only a minimal number of parameters to characterize the chan-
nel within the delay-Doppler domain. The representation of the channel h(τ, v)
is presented as [18]:
h(t − v) =
q

j=1
hiδ(t − τj)δ(t − vj) (19)
In this context, where q represents the number of propagation paths, hj,
τj, and vj respectively stand for the path gain, delay, and Doppler shift (or
frequency) associated with the j − th path. Additionally, δ(.) represents the
Dirac delta function. We label the delay and Doppler taps for the i − th path as
follows:
τl =
lj
MΔf
and vl =
kj
NT
(20)
where lj and kj represents the delay and doppler taps, respectively corresponding
to the lth
path can take numbers that are either integers or fractions. At the
receiver, the received signal S is converted to baseband. The cyclic prefix of the
baseband signal is then removed to obtain a vector/signal of the form:
R = {R(n)}MN−1
n=0 (21)

The relationship between the entries of R(n), the transmitted data symbols
S(n) and the additive white Gaussian noise w(n) in Time Delay can be described
by the following equation [16].
R(n) =
q

j=1
hjej2π(kj (n−lj ))/MN)
S([n − lj]MN + w(n) (22)
The notation [.]n represents the modulo-n operation. Equation (22) can also
be expressed as a generalized linear system equation involving a channel matrix
and transmitted and received vectors, as illustrated below [9]:
R = Hs + w (23)
Let R, s, w denote the vectors that respectively represent the received signal,
and Additive white Gaussian Noise (AWGN) samples of size MN × 1. On the
order, H ∈ CMN×MN
refers to a complex matrix.
3.3 OTFS Receiver
After having prior knowledge that relevant information has been obtained, appro-
priate treatments were performed in the transmitting unit, which then transferred
them to a desired receiving destination. A The receiver, an orthogonal time fre-
quency space (OTFS) demodulator initially, the signal is received, and the cyclic
preﬁx is removed, as shown in the following Fig. 3 [9]. Subsequently, the resulting
discrete signal, as deﬁned in Eq. (15), is converted into a 2-D signal represented in
matrix form as R = vec−1
(R). Following this, the Weighted Fractional Fourier
Transform (WFrFT) is applied to this matrix to compensate for the Inverse
Weighted Fractional Fourier Transform (IWFrFT), resulting in the generation
of the Time-Frequency (TF) domain matrix denoted as Y , as depicted below:
Fig. 3. OTFS receiver
Y = RTα,M (24)
The SFFT is used to transform the symbols Y obtained in the time frequency
domain into the delay doppler domain after the Wigner transform [19]:
y =
1
MN
N−1

n=0
M−1

m=0
Y e−j2π[nk/N−ml/M]
(25)

256 M. Rjili et al.
The matrix operation equivalent to the (SFFT) can be represented as follows:
Y = TH
M Y TN (26)
where we substitute Eq. (24) for Eq. (26), we get:
Y = TH
M Fα,M RTN (27)
After TH
M = T−1,M , Eq. (27), similar to Eq. (23) can be inscribed in vector form:
y = vecT−1,M Tα,M RTN = (TN Tα−1,M )R (28)
The received signal vector y is located within the Delay Doppler domain,
and R is derived from Eq. (24). Tα−1,M represents the inverse of the WFrFT
(Weighted Fractional Fourier Transform) matrix with a fractional parameter
α − 1.
4 Simulation Results
In this section, we present an assessment of the performance of OTFS (Orthog-
onal Time Frequency Space) systems utilizing the Weighted Fractional Fourier
Transform, focusing on their bit-error-rate (BER). The outcomes are obtained
through simulations conducted on an OTFS data frame with dimensions 16×32,
where N = 16 delay bins, each further divided into M = 32 Doppler bins. These
bins are then grouped together to form an OTFS grid comprising MN cells, each
containing a single symbol. This data frame is subsequently employed in Monte
Carlo simulations. We consider the vehicular A channel for the simulations and
employ QAM modulation to generate graphical representations of error perfor-
mance.
Fig. 4. WFrFT-OTFS for SNR = 0 dB

Table 1. Various Optimal Fractional Orders
SNR(dB) 0 3 6 9
BERmin 75.10−3
72.10−3
52.10−3
33.10−3
To determine the optimal fractional value αopt that yields the lowest BER rates
under various scenarios, we conduct 1000 Monte Carlo simulations, varying the
fractional orders in the range of [0.1], with a step size of 0.01. The results reveal
diﬀerent optimal fractional order values across a range of SNR (Signal-to-Noise

258 M. Rjili et al.
Ratio) values, spanning from 0 to 9 dB, with 3 dB increments and a minimum
Bit Error Rate (BER) for each SNR value. These optimal values are summarized
in Table 1 below. In Fig. 4 can be seen that we have achieved an optimal alpha
(αopt) of 0.05 at a SNR of 0 dB, which is equivalent to a minimum Binary Error
Rate (BER) of 75.10−3
. The results of Fig. 5 indicate that we have obtained an
optimal alpha of 0.88 with a SNR of 3 dB, which is equal to a minimum Binary
Error Rate (BER) of 72.10−3
. While Fig. 6 illustrates that an optimal alpha
value of 0.95 is achieved with an SNR of 6 dB and a BER value of 52.10−3
. In
Fig. 7, we show that the optimal alpha value is 0.95 with an SNR of 9 dB and
a BER value of 33.10−3
. While Fig. 8 shows that the WFrFT-OTFS systems
outperforms the conventional OTFS one (α = 1).
Fig. 8. WFRFT-OTFS performance in terms of BER

5 Conclusion
This article presents a general OTFS system designed on the basis of the
WFRFT. Since WFRFT is considered as a generalized Fourier transform with
alpha equal to 1, inducing the conventional Fourier transform, we have also
demonstrated the relationship between the input and output signals of the OTFS
system as well as the Delay-Doppler channel. We obtained different optimal alpha
values for different SNR values, ranging from 0 to 9 dB with a step size of 3 dB.
Furthermore, we concluded that the WFRFT-based OTFS system maintains a
lower error rate than the conventional OTFS system.
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