This paper introduces a multi-user uplink system, utilizing the orthogonal time frequency space (OTFS) modulation scheme. The proposed design employs the generalized singular value decomposition (GSVD) of the channels connecting the base station and the two users. Precoding and detection matrices are constructed using the right and left singular vectors, respectively. Analytical expressions are derived for a practical antenna configuration, and corresponding simulation results are presented. Simulation results for uncoded, coded, and coded scenarios with channel estimation errors reveal that the proposed GSVD-based approach outperforms MMSE equalization in certain antenna configurations. Notably, GSVD shows advantages under channel estimation error in the coded scenario.
Ensuring reliable communication for high-speed users is a key requirement in sixth-generation (6G)
wireless networks. OTFS modulation, tailored for such environments, operates in the delay-Doppler
domain, enabling it to handle the efects of both delay and Doppler spread in the channel [
Various studies have explored orthogonal multiple access (OMA) techniques in orthogonal time
frequency space (OTFS) modulation; however, these methods are not based on precoding approaches
and do not employ the spatial multiplexing capabilities of MIMO systems for transmission. In [
In contrast to methods that do not utilize precoding or spatial multiplexing, several studies have
investigated precoding schemes for OTFS systems to enhance performance. In [
In the context of OTFS modulation, non-orthogonal multiple access (NOMA) has also been
investigated. For instance, a deep learning-based approach for signal detection in a two-user downlink
SISO-NOMA system was introduced in [
Although extensive research has been conducted, equalization remains a significant challenge in
OTFS modulation. The transceiver’s input-output relationship in the DD domain is characterized by a 2
dimensional (2D) phase-rotated convolution between the transmitted signal and the channel impulse
response in this domain [
The concept of generalized singular value decomposition (GSVD) decomposition was first presented
in [
Our contribution in this paper is the development of a precoding and detection scheme for a MU MIMO-OTFS system based on GSVD decomposition. This decomposition allows us to reduce the channel matrix to a diagonal form, thereby avoiding the need to handle a 2D circular convolution at the receiver. We then evaluate the performance of the proposed system and compare it with minimum mean square error (MMSE) equalization. Our proposed method demonstrates superiority over the mentioned methods.
Notation: In the remainder of this paper, boldface uppercase letters represent matrices, boldface lowercase letters denote vectors, and standard lowercase letters correspond to scalars. The symbols (.) , (.) , and vec(.) are used to signify Hermitian, transpose, and column-wise vectorization operations, respectively. Additionally, A{:,1:} denotes the selection of the first to the ℎ columns of the matrix A. The terms I and 0 refer to the identity matrix and the zero matrix, respectively.
Consider a DD grid of size × representing a discrete delay-Doppler (DD) domain. The delay
axis is divided into bins indexed by = 0, 1, . . . , − 1, where each bin corresponds to a delay of
= 1 seconds, while the Doppler axis is divided into bins indexed by = 0, 1, . . . , − 1, where
each bin corresponds to a Doppler shift of = 1 Hz. Here, denotes the available bandwidth, and
represents the frame length. Consequently, the corresponding sub-carrier spacing and subsymbol
duration in the time-frequency (TF) domain are given by Δ = and = , respectively. The
relationship between the DD domain and the TF domain is established through the symplectic finite
Fourier transform (SFFT) [
y = ∑︁ ∑︁ H,,x, + n
=1 =1 where y ∈ C× 1 for = 1, . . . , − 1 and = 2. The matrix H,, ∈ C× represents the channel matrix between the ℎ antenna of user and the ℎ antenna of the base station.2 For the remainder of this paper, it is assumed that the channel matrices H,, are full rank, indicating that the physical environment is rich in terms of the number of multipath components. The vector x, ∈ C× 1 is the vectorized form of the matrix X,. The vector n represents the additive white Gaussian noise (AWGN) where its elements are independently distributed n(.) ∼ (0, 2).
By considering all antennas together, the input-output relationship of the overall system can be written as
y = ∑︁ Hx + n
=1 where ⎡ H1,,1 ⎢ H2,,1 H = ⎢⎢⎣ ...
H,,1
H1,,2 · · · H2,,2 · · ·
. . . . . .
H,,2 · · ·
H,, H1,, ⎤ H2,., ⎥⎥⎥ ∈ C× . .
⎦ is the channel matrix between the ℎ user and the receiver. The vectors x ∈ C× 1, y ∈ C× 1, and n ∈ C× 1 represent the transmitted vector of the ℎ user, the received vector, and the noise vector from all antennas of the receiver, respectively. These vectors are given by ⎡ x,1 ⎤ ⎢ x,2 ⎥ x = ⎢⎣⎢ ... ⎥⎥⎦ , x, ⎡ y1 ⎤ ⎢ y2 ⎥ y = ⎢⎢⎣ ... ⎥⎥⎦ , y ⎡ n1 ⎤ ⎢ n2 ⎥ n = ⎢⎢⎣ ... ⎥⎥⎦ .
n
1This work can be extended to support multiple users by applying HO-GSVD [
0︀] Q = UΣV .
Here: • U ∈ C× are unitary matrices • Σ ∈ C× are block diagonal matrices • W ∈ C× , Q ∈ C× are unitary matrices • R ∈ C× is an invertible matrix • 0 ∈ R× (− ) • V ∈ C×
In this case, and with the assumption that the channel matrices are full rank, the matrices Σ1 and Σ2 can be expressed as block diagonal matrices with the following specifications:
⎡I1 Σ1 = ⎣
S1 01 ⎤ ⎡02 ⎦ , Σ2 = ⎣
S2
I2 ⎤ ⎦ where: • I1 ∈ R(− 2)× (− 2), I2 ∈ R(− 1)× (− 1) • 01 ∈ R(1− 1)× (− 1),02 ∈ R(2− 2)× (− 2)
the channel gains, where √︁ 2,1 + 2,2 = 1 for = 1, . . . , 1 + 2 − • S = diag ( 1,, 2,, . . . , 1+2− ,) ∈ C(1+2− )× (1+2− ) are diagonal matrices containing Based on the GSVD in (5), the precoding and detection matrices are respectively defined as 3 P = U,
D = WR− 1(Q{:,1:}) ∈ C× Considering data streams for each user, the precoded vector in the transmitter of each user is x = Ps where the vector s ∈ C× 1 contains the information quadrature amplitude modulation (QAM) symbols for each user. 3In simulation, we used the equivalent detection matrix considering the noise reduction as D = (V V + 2I)− 1VH (5) (6) (7) (8) .
T.v . matrix is applied to the received signal y in (2), resulting in
By substituting (5), (7), and (8) into (9), the following is obtained r = Dy = DH1x1 + DH2x2 + Dn
r = Σ1 s1 + Σ2 s2 + Dn
In (10), the term Dn leads to noise enhancement since the matrix D is not unitary. However, in (7), the precoding matrix U is unitary.
In this paper, we consider a realistic scenario where 1 + 2 ≤ . Given that the matrix H has full rank and 1 + 2 < , the matrices Σ1 and Σ2 reduce to the following forms because, in this case, = (1 + 2), 1 = 1, and 2 = 2: Comparing (11) and (10) yields Σ1 = ︂[
I1 0(− 1)× (− 2) , Σ2 = ︂[ 0(− 2)× (− 1)︂]
I2 ︂] ︂[ s1]︂
s2 r = + Dn Now, the information from each user can be easily extracted in the presence of noise. Fig. 2 illustrates the uplink setup and user configuration in a high-speed environment.
This section focuses on evaluating the performance of the proposed GSVD approach. The simulation setup parameters are detailed in Table 1. The channel model considered is the Extended Vehicular A (EVA) channel, as defined in the 3rd Generation Partnership Project (3gpp) TS 36.104 [ 20]. The maximum environmental speed is set to max = 500 km/h, resulting in a maximum Doppler frequency shift and normalized Doppler shift of max = ligmhatxs×peed = 1853 Hz and max = max = 0.9883, respectively. The modulation scheme employed is 4-QAM, which maps the information bits into corresponding symbols. We compared the performance of our proposed GSVD method with MMSE equalization in terms of bit error rate (BER), using two MU-MIMO antenna configurations: (3 + 2) × 6 and (1 + 1) × 6. The evaluation was conducted across three scenarios: one without channel coding, one using a 3/4 low-density parity check (LDPC) code as specified in the IEEE 802.11 standard with a block length of 648, and another incorporating channel coding while accounting for channel estimation errors.
Fig. 3 illustrates the BER curve as a function of signal-to-noise ratio (SNR). For the (1 + 1) × 6 configuration, the simulation results show a slight gap between the GSVD and MMSE methods, with MMSE exhibiting marginally better performance. Notably, both users achieve identical performance. 10-1 10-2 ER10-3 B 10-4 10-5-4
GSVD User 1, User 2 - (1+1) 6 MMSE User 1, User 2 - (1+1) 6 GSVD User 1 - (3+2) 6 MMSE User 1, User 2 - (3+2) 6 GSVD User 2 - (3+2) 6
4
SNR(dB) -2 0 2 6 8 10 12 As we will demonstrate later, this gap can be mitigated through the use of channel coding. In the (3 + 2) × 6 configuration, the MMSE method achieves consistent performance for both users, while the GSVD approach shows improved performance for the first user compared to MMSE. However, for the second user, the GSVD approach performs similarly to MMSE, with a slight degradation at higher SNR values.
Fig. 4 presents the results for the same configurations but with LDPC coding applied. For the (1 + 1) × 6 setup, the performance of GSVD matches that of MMSE, as predicted. In the (3 + 2) × 6 configuration, the gap between the GSVD method for the second user and the MMSE method for both users is reduced. Additionally, the GSVD approach for the first user demonstrates better performance compared to MMSE.
In Fig. 5, the BER versus SNR for the coded case is plotted, considering an antenna configuration of (3 + 2) × 6 in the presence of an estimation error. The channel coeficient in the time domain, ℎ, is considered as a reference, while the estimated channel coeficient is expressed as ℎ,est = ℎ + where is a Gaussian random variable of zero mean. In this formulation, satisfies 0 ≤ ≤ 1, and the error term follows {| |2} = (1 − 2){|ℎ|2}. Here, {·} represents the expected value. The analysis is conducted specifically for User 1. This figure demonstrates that the performance of GSVD is comparable to MMSE for the cases where = 0.99 and = 0.995, while exhibiting a slight improvement when = 0.999.
In this study, we introduced the application of the GSVD technique within an uplink two-user MIMOOTFS framework and provided analytical formulations for practical antenna configurations. The 10-1 10-2 R E B 10-3 10-4 10-5-6 10-1 10-2 R E B 10-3 10-4-4
GSVD User 1, User 2 - (1+1) 6 MMSE User 1, User 2 - (1+1) 6 GSVD User 2 - (3+2) 6 MMSE User 1, User 2 - (3+2) 6
GSVD User 1 - (3+2) 6 0
SNR(dB) -4 -2 2 4
6 GSVD = 0.99 GGMMSSMMVVSSDDEE ====0000....99999999595 MMSE = 0.999
2
SNR(dB) -2 0 4 6 8 methodology employs GSVD-based channel decomposition for each user, alongside the implementation of precoding and detection matrices. Simulation outcomes demonstrate that the proposed GSVD approach outperforms the conventional MMSE scheme under certain antenna configurations, while showing comparable results under others.
Exploring the extension of the GSVD approach to multi-user scenarios via HO-GSVD, for both uplink and downlink communication, represents a direction for future research.
This work was conducted within the MiFuture project, which has received funding from the European Union’s Horizon Europe (HE) Marie Skłodowska-Curie Actions MiFuture HORIZON-MSCA2022-DN-01, under Grant Agreement number 101119643 and YAHYA/6G HORIZON-MSCA-2022-PF-01, under Grant Agreement number 101109435. It was partially supported by FCT/MECI through national funds and when applicable co-funded EU funds under UID/50008: Instituto de Telecomunicações.
During the preparation of this work, the authors used CoPilot provided by Microsoft in order for editing and spell-checking the text.