Enhanced Spatial Modulation Based Orthogonal Time Frequency Space System

Anoop A, Christo K. THOMAS, Kala S

Summary :

In this paper, a novel Enhanced Spatial Modulation-based Orthogonal Time Frequency Space (ESM-OTFS) is proposed to maximize the benefits of enhanced spatial modulation (ESM) and orthogonal time frequency space (OTFS) transmission. The primary objective of this novel modulation is to enhance transmission reliability, meeting the demanding requirements of high transmission rates and rapid data transfer in future wireless communication systems. The paper initially outlines the system model and specific signal processing techniques employed in ESM-OTFS. Furthermore, a novel detector based on sparse signal estimation is presented specifically for ESM-OTFS. The sparse signal estimation is performed using a fully factorized posterior approximation using Variational Bayesian Inference that leads to a low complexity solution without any matrix inversions. Simulation results indicate that ESM-OTFS surpasses traditional spatial modulation-based OTFS, and the newly introduced detection algorithm outperforms other linear detection methods.

• Publication: IEICE TRANSACTIONS on Communications Vol.E107-B No.11 pp.785-796
• Publication Date: 2024/11/01
• Online ISSN: 1745-1345
• DOI: 10.23919/transcom.2023EBP3206
• Type of Manuscript: PAPER
• Category: Wireless Communication Technologies

1.  Introduction

The primary role of modern multicarrier modulation techniques in enabling future wireless communication with rigorous requirements on reliability and spectral efficiency has been extensively recognized. Future multicarrier modulation schemes are expected to handle highly fading wireless scenarios such as vehicle-to-vehicle communication, communications in high-speed drones and fast moving bullet trains, vehicle-to-infrastructure communication, etc. The recently proposed orthogonal time-frequency (OTFS) is a feasible option that can be used to combat high mobility conditions in multicarrier modulation schemes [1]. The information symbols are distributed in the delay-Doppler (DD) domain instead of in the time-frequency (TF) domain in the case of OFDM. The DD domain channel characteristics have made the design of equalizers and channel estimation relatively easy as proposed in [2]. These characteristics make OTFS a highly desirable candidate for future practical wireless communications.

The OTFS proposed in [1] is very effective in representing the time-fluctuating nature of the high-doppler channel. OTFS spreads information in the DD domain using a two-dimensional (2D) basis function [4]. So we can say that OTFS modulation has the capacity to transform a channel that varies over time in the TF domain into a channel with time-invariant properties in the DD domain. OTFS is very effective in doubly selective wireless channels because these basis functions span the entire time-frequency plane.

The performance evaluation of OTFS is done in the millimetre wave system, and it is estimated that the bit error rate (BER) performance of OTFS is much better than orthogonal frequency division multiplexing (OFDM) [3]. OTFS modulation can be put into practice by adding inverse sympletic finite Fourier transform (ISFFT) at the transmitter side and sympletic finite Fourier transform (SFFT) at the receiver [5]. So we can easily integrate OTFS into conventional OFDM systems. A simplified discrete input-output relationship for OTFS was proposed in [6]. In [7], authors investigated that OTFS exhibits excellent peak-to-average power ratio (PAPR) endurance in high mobility channels. An new method for channel estimation of OTFS system utilising embedded pilot signals was proposed in [8]. An iterative detector for OTFS was proposed in [9].

Multiple-input multiple-output (MIMO) technology has been part of fourth and fifth-generation wireless networks due to its ability to provide high spectral efficiency. But MIMO networks encounter many more challenges in environments characterised by a high degree of mobility compared to fixed MIMO networks. So conventional MIMO channels experience many significant drawbacks as hilighted in [10]. The DD domain OTFS channel has a sparse nature, and this property helps higher-order MIMO networks to mitigate channel equalization and channel estimation in high-doppler environments. MIMO-OTFS systems have been well investigated since their inception, and their detailed signal processing, detection methods, and channel estimation techniques are studied in [11]-[13].

Spatial modulation (SM) represents a cutting-edge technique in wireless communication, particularly enhancing Multiple-Input Multiple-Output (MIMO) systems by utilising the spatial dimension to transmit additional information [14]. This method efficiently combines antenna selection along with symbol modulation, significantly reducing inter-channel interference and system complexity. This means that only one transmit antenna is operational at the same instant in the case of SM, and other antennas remain inactive. This makes SM a modulation scheme with high spectral efficiency, low complexity, and good BER performance.

In order to enhance the spectral efficiency further, quadrature-spatial modulation (QSM) was proposed [22]. QSM, a derivative of SM, distinguishes itself by dividing the modulated symbol into in-phase and quadrature components. These components are then assigned to two different transmit antennas, each activated by its respective group of index bits to transmit the in-phase and quadrature components. Subsequently, these modulated signal components are transmitted using carriers that are mutually orthogonal. This method effectively prevents channel interference and enhances diversity gain. Generalized Spatial Modulation (GSM) is another innovative spatial modulation technique that selectively activates a subset of antennas for transmission at any given time [23]. In comparison to SM, GSM requires more complex signal processing and antenna selection algorithms, leading to increased computational complexity and potentially higher power consumption [24].

Enhanced spatial modulation (ESM) was first proposed in [16]. ESM was formulated by combining different ideas. ESM involves the use of primary and secondary constellations. In the case of ESM, when one transmit antenna is active, information symbols used for modulation are selected from a primary constellation, and when two transmit antennas are active, information symbols are modulated using the secondary constellation, and other antennas remain silent. The total count of information bits that can be sent in a single transmission of ESM depends on the size of the combinations of transmit antennas and constellation symbols. So this property increases the effective throughput of ESM compared to conventional SM. Furthermore, the secondary constellation is designed using geometric interpolation of the primary constellation, which optimize the minimum Euclidean distance among the transmitted signal vectors. This approach marks a significant divergence between ESM from traditional SM and GSM. When we select signal constellations that maintain the operational spectral efficiency of ESM on par with conventional SM, ESM exhibits superior performance. This concept of augmenting combination numbers to boost spectral efficiency is similarly observed in QSM. Nevertheless, QSM experiences a diminished minimum squared Euclidean distance and yields inferior performance compared to ESM.

SM demonstrates remarkable flexibility and adaptability, allowing it to be seamlessly integrated with various transmission technologies. When combined with OFDM to form SM-OFDM, this approach is capable of combating frequency-selective fading, obviating the need for complex equalization methods, and thereby significantly boosting the spectral efficiency of the system [25]. While OFDM performs well in static or low-mobility environments with multipath propagation, its performance can degrade significantly in high mobility scenarios due to Doppler shifts affecting the orthogonality of the subcarriers. To address this issue, spatial modulation based on OTFS (SM-OTFS) is proposed [15] to excel in environments with high Doppler shifts and significant delay spreads. The authors introduce SM-OTFS based on MIMO to enhance spectral efficiency and reduce detection complexity in [15]. It demonstrates through simulations that SM-OTFS provides significant performance gains over space-time-coded OTFS (STC-OTFS), especially in high-mobility scenarios.

The system model of the SM-OTFS scheme and its related signal processing techniques are elaborated in [26], and this paper also provides closed-form expressions for the average symbol error rate (ASER) and average bit error rate (ABER) over delay-Doppler channels. The authors also highlighted the superior performance of SM-OFDM over SM-OTFS under high Dopple environments in [26]. The system model and its associated signal processing techniques for genaralised spatial modulation on the OTFS (GSM-OTFS) system are outlined in [27], including the use of the union bound technique and moment generating function (MGF) for theoretical analysis of average BER performance. In the paper [27], the authors demonstrate through theoretical and simulation results that GSM-OTFS offers better BER performance and spectral efficiency compared to conventional SM-OTFS. Even if the GSM-OTFS system enhances spectral efficiency and BER performance compared to SM-OTFS, it introduces challenges such as increased system complexity, hardware demands, power consumption, and advanced signal processing requirements.

Quadrature spatial modulation based on OTFS (QSM-OTFS) proposed in [20] is another OTFS-based index modulation utilising QSM. In the paper [20], the authors detail the system model, signal processing steps, and performance analysis, including theoretical ABER analysis and an innovative detection technique named enhanced minimum mean square error (EMMSE) for reduced complexity in detection. Furthermore, it compares the proposed QSM-OTFS system with the traditional SM-OTFS system, highlighting the advantages in terms of ABER performance. Motivated by the characteristics of ESM and OTFS, we propose a novel ESM-based OTFS scheme called enhanced modulation-based OTFS (ESM-OTFS) that performs well in high mobility scenarios and provides high spectral efficiency compared to SM-OTFS. QSM-OTFS has the same spectral efficiency as ESM-OTFS but has inferior performance under the same channel conditions.

Enhanced spatial modulation combined with orthogonal time frequency space (ESM-OTFS) modulation presents a versatile and efficient solution for modern wireless communication challenges, particularly in environments requiring high mobility and robustness. ESM-OTFS is particularly suited for environments with high mobility, such as high-speed trains, vehicular networks (vehicle-to-vehicle and vehicle-to-infrastructure communication), and drones, due to its resilience to Doppler shifts and ability to maintain reliable communication at high speeds. This innovative approach is also well-suited for a wide range of applications, including 5G and beyond wireless systems, Internet of Things (IoT) networks, satellite and deep space communication, and underwater acoustic communication, as well as enhancing capacity and reliability in urban cellular networks. By offering improved spectral efficiency and resilience to Doppler shifts and multipath propagation, ESM-OTFS stands out as a promising technology for ensuring reliable, high-speed communication across various challenging environments and applications, driving advancements in both current and future wireless communication landscapes.

The transmit vectors of SM-OTFS and ESM-OTFS include significant numbers of zero entries rather than non-zero values, and this property makes the data transmit vector a sparse vector in most of the system configurations. So it is a good option that we can utilise sparse signal estimation methods for the detection of ESM-OTFS. Sparse Bayesian Learning (SBL) is one of the popular techniques used in sparse signal estimation. But it involves the use of matrix inversion in each iteration, so that it is computationally intensive even for datasets of intermediate size. Instead of using SBL, we can use the Variational Bayesian Inference method [17], [18] for sparse signal estimation.

Variational Bayesian Inference (VBI), also known as Variational Bayes, is a method used in Bayesian statistics and machine learning for approximating the posterior distribution of latent variables in a probabilistic model. This method is beneficial in situations involving complicated models where the use of exact inference or sampling techniques is computationally intractable. The main idea behind VBI is to approximate the true posterior distribution with a simpler, parameterized variational distribution that is easier to work with. These variational distributions are normally selected from a family of distributions, such as Gaussian distributions. The goal is to find the parameters of this simpler distribution that best approximate the true posterior. The SAVE (Space Alternating Variational Estimation for Sparse Bayesian Learning) algorithm proposed in [19] is an excellent algorithm for sparse signal estimation. Motivated by the SAVE algorithm, we are able to come up with a novel sparse signal estimation-based detector for the newly proposed ESM-OTFS.

2.  System Model

2.1  ESM Modulation

Spatial modulation (SM) is a wireless communication technique based on Mutiple Input Mutiple Output (MIMO) that uses transmit antenna index for transmitting information symbols. SM involves the activation of a single transmit antenna at any given moment in time and that selected antenna is used to transmit a signal from a chosen constellation. If the \(N_{T}\) is the number of transmit antennas and \(s_{a}=2^{n_{a}}\) is the size of size of signal constellation, the number of bits transmitted in SM is \(n_{a}+\log_{2}(N_{T})\).

Enhanced spatial modulation improves the efficiency and robustness of SM by transmitting different symbols from different constellations depending on the number of active antennas as established in [16]. This is done by first transmitting symbols from a primary constellation during single transmit antenna activation just like the conventional SM. When two transmit antennas are enabled, symbols are transmitted from a secondary constellation. The size of the secondary constellation is taken half of the primary constellation so that the same amount of information bits are transmitted during single antenna activation and double antenna activation periods. This allows for a higher diversity gain as the signal is transmitted from two different antennas, which makes it more robust to interference. In order to maximise the minimal Euclidean distance between transmitted signal vectors, secondary constellations are created via geometric interpolation. This ensures that the symbols in the secondary constellations are as far apart as possible, which makes it more difficult for the receiver to make errors.

In conventional SM, the number of information bits transmitted depend on the number of transmit antennas and the size of symbol constellation used for modulation. In ESM, it is decided by the antenna and constellation symbol combinations. An illustration of ESM system with \(2\) numbers of transmit antennas, QPSK as the primary constellation and BPSK as the secondary constellation is depicted in Table 1. Here the number of transmitted bits or bits per channel use (bpcu) is \(4\).

Table 1  ESM, \(2\) TX - \(4\) BPCU.

\(\text{QPSK}=\pm 1 \pm j\), BPSK0 and BPSK1 are respectively given by \(\text{BPSK0}=\pm 1\) \(\&\) \(\text{BPSK1}=\pm j\). The first two combinations \(C1\) and \(C2\) appear like the transmission from one of the antenna and convey one of symbols from QPSK constellation. The last combinations \(C3\) and \(C4\) corresponds to transmissions of symbols from secondary constellations BPSK0 or BPSK1 coming out of both antennas.

2.2  OTFS Signal Modulation and Demodulation

OTFS modulation is a way of transmitting information symbols that is robust to Doppler shift. It works by multiplexing the data symbols in the DD domain, which is another way of expressing the signal that is invariant to Doppler shift. This makes OTFS ideal for high-mobility applications. Let \(\{u[l,k],l=0,1,\ldots,M-1,k=0,1,\ldots,N-1\}\) be a two dimensional information signal in the DD domain where \(M\) denotes the number of sub carriers and \(N\) denotes the number of OFDM symbols or time slots. The time-frequency (TF) domain signal \(U[m,n]\) is obtained from \(u[l,k]\) through the inverse symplectic finite Fourier transform (ISFFT), i.e.,

\[\begin{equation*} U\left[m, n\right]=\frac{1}{\sqrt{MN}}\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}u\left[l, k\right]e^{j2\pi\left(\frac{nk}{N}-\frac{ml}{M}\right)} \tag{1} \end{equation*}\]

Using Heisenberg transform along with transmitter side pulse shaping signal \(p_{tx}(t)\), \(U[m,n]\) is converted to a time domain (TD) signal \(s(t)\)

\[\begin{equation*} s(t)=\displaystyle \sum_{m=0}^{M-1}\sum_{n=0}^{N-1}U[m,\ n]p_{\mathrm{t}\mathrm{x}}(t-mT)e^{j2\pi n\Delta f(t-mT)} \tag{2} \end{equation*}\]

where \(\Delta f\) and \(T\) represent the sub carrier spacing and OFDM symbol period respectively, and \(\Delta f = \frac{1}{T}\).

At the receiver, the received signal \(r(t)\) is matched with receiver side pulse shaping signal \(p_{rx}(t)\) and is converted to TF domain using Wigner transform and the resulting cross-ambiguity \(\Lambda_{p_{rx},r}(t,f)\) is given as

\[\begin{equation*} \Lambda_{p_{rx},r}(t,f) = \int r(t^{*})p_{\mathrm{r}\mathrm{x}}(t^{*}-t)e^{j2\pi f(t^{*}-t)}\mathrm {d}t^{*} \tag{3} \end{equation*}\]

The resultant signal is sampled with sub carrier spacing \(\Delta f\) and the frame duration \(T\). The received TF domain signal is given as

\[\begin{equation*} V[m, n]=\Lambda_{p_{\mathrm{rx},\mathrm{r}}(t, f)|_{t=nT,f=m\Delta f}} \tag{4} \end{equation*}\]

The DD domain signal \(v[l,k]\) is obtained from \(V[m,n]\) by applying symplectic finite Fourier transform (SFFT) and is given as

\[\begin{equation*} v\left[l, k\right]=\frac{1}{\sqrt{MN}}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}V\left[m, n\right]e^{-j2\pi\left(\frac{nk}{N}-\frac{ml}{M}\right)} \tag{5} \end{equation*}\]

2.3  OTFS System Model

Consider a DD domain channel with \(P\) channel taps, each with a delay of \(\tau_{i}\), a doppler of \(\nu_{i}\) and a fading channel gain of \(\kappa_{i}\). The channel impulse response in the DD domain as in [1] can be represented as

\[\begin{equation*} \kappa(\displaystyle \tau, \nu)=\sum_{i=1}^{P}\kappa_{i}\delta(\tau-\tau_{i})\delta(\nu-\nu_{i}) \tag{6} \end{equation*}\]

where \(\delta(.)\) denotes the dirac-delta function. The delay and doppler of the \(i^{th}\) can be formulated as

\[\begin{equation*} \displaystyle \tau_{i}=\frac{l_{i}}{M\Delta f}, \nu_{i}=\frac{k_{i}}{NT} \tag{7} \end{equation*}\]

Given the TD input signal \(s(t)\), the received signal \(r(t)\) is given as

\[\begin{equation*} r(t) = \int _{\nu}\int _{\tau }\kappa(\tau,\nu)s(t-\tau)e^{j2\pi \nu(t-\tau)} \mathrm{d}\tau \mathrm {d}\nu \tag{8} \end{equation*}\]

Substituting equation \((6)\) in equation \((8)\) yields as established in [5].

\[\begin{equation*} \begin{aligned} &r(t) = \\& \int _{\nu}\int _{\tau }\left(\sum_{i=1}^{P}\kappa_{i}\delta(\tau-\tau_{i})\delta(\nu-\nu_{i})\right)s(t-\tau) e^{j2\pi \nu(t-\tau)}\mathrm {d}\tau \mathrm {d}\nu \end{aligned} \tag{9} \end{equation*}\]

and can be simplified as

\[\begin{equation*} r(t)=\sum_{i=1}^{P}\kappa_{i}s(t-\tau_{i})e^{j2\pi \nu_{i}(t-\tau_{i})} + n(t) \tag{10} \end{equation*}\]

where \(n(t)\) is the noise signal in TD. The sampled TD signal \(r(p)\) is given as

\[\begin{equation*} r(p)=\sum_{i=1}^{P}\kappa_{i}s(p-\tau_{i})e^{j2\pi \nu_{i}(p-\tau_{i})} + n(p) \tag{11} \end{equation*}\]

where \(p=0,1,2\ldots,NM-1\).

It can be depicted in the matrix form as follows

\[\begin{equation*} r = \begin{bmatrix} r_{0} \\ r_{1} \\ \vdots \\ r_{MN-1} \end{bmatrix}=\bigg(\sum_{i=1}^{P}\kappa_{i}\Pi^{l_{i}}\Delta^{k_{i}}\bigg)s+w = \breve{H}s+n \tag{12} \end{equation*}\]

where \(\Pi=\begin{bmatrix} 0&\cdots&0&1\\ 1&\ddots&0&0\\ \vdots&\ddots&\ddots&\vdots\\ 0&\cdots&1&0 \end{bmatrix}\) is the permutation matrix with dimension \(MN\times MN\), \(\Delta\) is the \(MN\times MN\) dimensional diagonal matrix as \(\Delta=diag{\begin{Bmatrix}e^{j2\pi\frac{p}{MN}}\end{Bmatrix}}^{MN-1}_{p=0}\), \(\breve{H}\) is the TD channel matrix with dimension \(MN\times MN\), meanwhile without the loss of generality \(l_{i}\) and \(k_{i}\) are assumed to be integers.

At the receiver, the received signal vector \(r\) gets converted to DD domain and the input-output relationship in DD domain is given as follows

\[\begin{equation*} v=Hu+w \tag{13} \end{equation*}\]

where w is the noise vector in the DD domain and \(H\) is the DD domain channel matrix given as \(H=(F_{N}\otimes I_{M})\breve{H}(F^{H}_{N}\otimes I_{M})\) assuming the use of rectangular pulse shaping filter at the transmitter and receiver, \(F_{N}\) is the \(N\) point discrete Fourier transform (DFT) matrix, \(F^{H}_{N}\) is the \(N\) point inverse discrete Fourier transform(IDFT) matrix, \(\otimes\) is the operator of the Kronecker product and \(I_{M}\) is the \(M\) dimensional identity matrix.

2.4  ESM-OTFS

The system model of the proposed ESM-OTFS is shown in Fig. 1. From the Fig. 1, it is observed that the ESM-OTFS system is equipped with \(N_{T}\) transmit antennas and \(N_{R}\) receive antennas. A detail description of the signal processing used in ESM-OTFS is given as the following. A random bit sequence \(b=[b_{0} \hspace{2mm} b_{1}\cdots b_{totbits}]\) of an ESM-OTFS frame in the DD domain enter the ESM-OTFS system where \(totbits=MNlog_{2}(\mathbb{C} )\) and \(\mathbb{C}\) is the size of ESM constellation. The mapping rule of ESM modulation is given in Table 2. For each of \(\mathbb{C}\) incoming bits, an ESM transmit vector is selected as per the presented mapping rule. For each of \(MN\) time slots of an ESM-OTFS frame, a constellation symbol is assigned to each of \(N_T\) transmit antennas. These \(MN\) symbols enter the OTFS Block Creator unit of the corresponding transmit antenna and form the transmit DD domain vector for the respective antenna.

Fig. 1  System model of ESM-OTFS system.

Table 2  ESM mapping rule for \(N_{T}=2\) and QSPK as primary constellation.

The transmission rate of ESM-OTFS can be stated as \(R_{ESM-OTFS}=MNlog_{2}(\mathbb{C})\) whereas under the same setup, SM-OTFS has a transmission rate of \(R_{SM-OTFS}=MNlog_{2}(s_{a})\) where \(s_{a}\) is the order of the modulation used or the size of the modulation constellation used. Since the size of the ESM constellation formed by the combination of transmit antennas and primary and secondary constellation symbols is higher than the size of the modulation order used in SM, the transmission rate of ESM-OTFS is much better than that of SM-OTFS under the same MIMO environment. The dimension of the data transmit matrix of an ESM-OTFS frame is \(MN \times N_T\) and is given as follows

\[\begin{equation*} \mathbf{X_{ESM}}=\begin{bmatrix}\mathbf{x}_{0,0}^{1} & \ldots & \mathbf{x}_{0,0}^{i} & \ldots & \mathbf{x}_{0,0}^{N_{T}}\\[1.5mm] \mathbf{x}_{0,1}^{1} & \ldots & \mathbf{x}_{0,1}^{i} & \ldots & \mathbf{x}_{0,1}^{N_{T}}\\\vdots & \ddots & \vdots & \ddots & \vdots\\ \mathbf{x}_{0,N-1}^{1} & \ldots & \mathbf{x}_{0,N-1}^{i} & \ldots & \mathbf{x}_{0,N-1}^{N_{T}}\\\vdots & \ddots & \vdots & \ddots & \vdots\\ \mathbf{x}_{k,l}^{1} & \ldots & \mathbf{x}_{k,l}^{i} & \ldots & \mathbf{x}_{k,l}^{N_{t}}\\\vdots & \ddots & \vdots & \ddots & \vdots\\ \mathbf{x}_{M-1,N-1}^{1} & \ldots & \mathbf{x}_{M-1,N-1}^{i} & \ldots & \mathbf{x}_{M-1,N-1}^{N_{T}}\end{bmatrix} \tag{14} \end{equation*}\]

\(X_{i}\) is the DD domain information matrix of dimension \(M \times N\) transmitted from the \(i^{th}\) transmit antenna. \(X_i\) is formed by collecting all the elements of \(i^{th}\) column of \(\mathbf{X_{ESM}}\).

The TD signal that is being transmitted from the \(i^{th}\) transmit antenna \(s_{i}\) is derived from \(X_{i}\) by passing through ISSFT unit and OFDM modulator which simulates the Heisenberg transform and is given as follows

\[\begin{equation*} s_{i}=vec\left( F^{H}_{M}\left(F_{M}X_{i}F^{H}_{N}\right) \right)=\left(F^{H}_{N}\otimes I_{M}\right)x_{i} \tag{15} \end{equation*}\]

where \(s_{i}\) is the \(MN\times 1\) dimensional column vector and \(vec(.)\) is the column wise vector operation.

The TD signal \(s_{i}\) is transmitted from the \(i^{th}\) transmit antenna and travels through the multi-path wireless channel. The received signal at the \(j^{th}\) receive antenna from the \(i^{th}\) transmit antenna is given as

\[\begin{equation*} r_{j}=\breve{\mathbf{H}}_{ji}s_{i}+n_{j} \tag{16} \end{equation*}\]

In the receiver, the TD signal received at each receive antenna is converted to DD domain through SFFT and OFDM demodulator which simulates the Wigner transform. The DD domain signal received at \(j^{th}\) receive antenna is given as

\[\begin{equation*} y_{j}=\mathbf{H}_{j1}x_{1}+\mathbf{H}_{j2}x_{2}+\cdots+\mathbf{H}_{ji}x_{i}+\cdots+\mathbf{H}_{jN_{R}}x_{N_{R}}+w_{j} \tag{17} \end{equation*}\]

The received signal in the DD domain considering all the receive antennas is shown as follows

\[\begin{equation*} \mathbf{y}_{ESM}=\mathbf{H}_{eff}\mathbf{x}_{ESM}+\mathbf{w}_{eff} \tag{18} \end{equation*}\]

where \(\mathbf{y}_{ESM}=[y_{0},y_{1},\cdots,y_{N_{R}}]\), \(\mathbf{x}_{ESM}=[x_{0},x_{1},\cdots,x_{N_{T}}]\) and \(w_{eff}\) is the effective noise vector. At the same time, we have \(\mathbf{y}_{ESM},\mathbf{w}_{eff} \in {\mathbb{C}}^{N_{R}MN \times 1}\), \(\mathbf{x}_{ESM}\in {\mathbb{C}}^{N_{T}MN \times 1}\) and \(\mathbf{H}_{eff} \in {\mathbb{C}}^{N_{R}MN \times N_{T}MN }\). The effective channel matrix \(\mathbf{H}_{eff}\) of ESM-OTFS in the DD domain is given as

\[\begin{equation*} \mathbf{H}_{eff}=\begin{bmatrix}\mathbf{H}_{11}&\mathbf{H}_{12}&\ldots&\mathbf{H}_{1N_{T}}\\\mathbf{H}_{21}&\mathbf{H}_{22}&\ldots&\mathbf{H}_{2N_{T}}\\\vdots&\vdots&\ddots&\vdots\\\mathbf{H}_{N_{R}1}&\mathbf{H}_{N_{R}2}&\cdots&\mathbf{H}_{N_{R}N_{T}}\end{bmatrix} \tag{19} \end{equation*}\]

3.  Detection of ESM-OTFS

3.1  MMSE Detector

In order to bring down the complexity of detection, a detector combining MMSE equalization and minimum Euclidean distance detection is proposed for the detection of ESM-OTFS. Firstly an estimate of the transmitted signal \(\mathbf{\hat{x}}_{ESM}^{MMSE}\) is obtained using a MMSE based equalizer. The MMSE equalization is given as follows

\[\begin{equation*} \mathbf{\hat{x}}_{ESM}^{MMSE} =\left(\mathbf{H}^\mathbf{H}+\frac{I_{N_{T}MN}}{\rho_{snr}} \right)^{-1}\mathbf{H}\mathbf{y}_{ESM} \tag{20} \end{equation*}\]

where \(I_{N_{T}MN}\) is the identity matrix of order \(N_{T}MN\times N_{T}MN\) and \(\rho_{snr}\) is the average SNR in the DD domain. Now a minimum euclidean distance detector is applied on each row of \(\mathbf{\hat{X}}_{ESM}^{MMSE}\) against all the possible combination of ESM constellation vectors \(\mathcal{C}_{ESM}\). This can be formulated as follows

\[\begin{align} \Big \{\mathbf{\hat{X}}^{\eta}_{ESM}\Big \}=& \underset{\mathbf{C}_{\mathbf{ESM}} \in \mathcal{C}_{\mathbf{ESM}}}{\arg\min} \left | \mathbf{\hat{X}}_{ESM}^{MMSE}(\eta)- \mathbf{C}_{\mathbf{\mathbf{ESM}}} \right |^{2}, \tag{21} \\ &1\leq \eta \leq MN, \nonumber \end{align}\]

where \(\hat{\mathbf{X}}_{ESM}^{MMSE}(\eta)\) is the \(\eta^{th}\) row of the matrix \(\hat{\mathbf{X}}_{ESM}^{MMSE}\). Then the ESM constellation vector with minimum euclidean distance is taken as the detected ESM-OTFS transmit vector for each time slot of ESM-OTFS frame. After the equalization and detection, the demodulated ESM-OTFS frame \(\mathbf{\hat{X}}_{ESM} \in \mathbb{C}^{MN\times N_{T}}\) can be expressed in the matrix form as

\[\begin{equation*} \mathbf{\hat{X}}_{ESM}=\begin{bmatrix}\mathbf{\hat{x}}_{0,0}^{1} & \ldots & \mathbf{\hat{x}}_{0,0}^{i} & \ldots & \mathbf{\hat{x}}_{0,0}^{N_{T}} \\ \mathbf{\hat{x}}_{0,1}^{1} & \ldots & \mathbf{\hat{x}}_{0,1}^{i} & \ldots & \mathbf{\hat{x}}_{0,1}^{N_{T}}\\ \vdots & \ddots & \vdots & \ddots & \vdots\\ \mathbf{\hat{x}}_{0,N-1}^{1} & \ldots & \mathbf{\hat{x}}_{0,N-1}^{i} & \ldots & \mathbf{\hat{x}}_{0,N-1}^{N_{T}}\\\vdots & \ddots & \vdots & \ddots & \vdots\\ \mathbf{\hat{x}}_{k,l}^{1} & \ldots & \mathbf{\hat{x}}_{k,l}^{i} & \ldots & \mathbf{\hat{x}}_{k,l}^{N_{t}}\\\vdots & \ddots & \vdots & \ddots & \vdots\\ \mathbf{\hat{x}}_{M-1,N-1}^{1} & \ldots & \mathbf{\hat{x}}_{M-1,N-1}^{i} & \ldots & \mathbf{\hat{x}}_{M-1,N-1}^{N_{T}}\end{bmatrix} \tag{22} \end{equation*}\]

Each row of \(\mathbf{\hat{X}}_{ESM}\) represents the estimate of ESM-OTFS transmit vector for each of the \(MN\) time slots of ESM-OTFS frame. Now an estimate of the original transmitted bits \(\hat{b}\) can be recovered from comparing each row of \(\mathbf{\hat{X}}_{ESM}\) using a look-up table as shown in Table 2. The steps involved in detection of ESM-OTFS using MMSE detector is summarized in Algorithm 1.

3.2  Variational Bayesian Inference Based ESM-OTFS Detector

Since the computational complexity of the maximum likelihood detector increases exponentially with the increase in the number of transmit \(N_{T}\) and receive \(N_{R}\) antennas, we propose a new detection algorithm based on the sparsity of the ESM-OTFS transmit frame. In ESM-OTFS, only one or two transmit antennas radiate at the same time and others remain silent. This results in the formation of data transmit matrix with most of the entries are zero making the matrix sparse. The key idea behind the proposed detector is the sparse signal estimation of the received signal using Variational Bayesian Learning. So the problem of ESM-OTFS signal detection can be restated as

\[\begin{equation*} y=\mathbf{H}x+w, \tag{23} \end{equation*}\]

where \(x\) and \(y\) are transmit and receive signals of dimensions \(N_{T}MN\times 1\) and \(N_{R}MN\times 1\) respectively in the DD domain, \(\mathbf{H}\) is the effective DD domain channel matrix of dimension \(N_{R}MN\times N_{T}MN\) and \(w\) is the Gaussian distributed white noise signal with zero mean and variance \(\gamma\).

It is assumed to have a two-layer hierarchical prior for the transmit data signal \(x\) as proposed in [19] so that it motivates the sparsity property of \(x\). It is assumed that \(x\) follows a Gaussian distribution parameterized by \(\alpha=[\alpha_{1},\alpha_{2},\cdots,\alpha_{N_{T}MN}]\) where \(\alpha_{i}\) is the inverse variance parameter of \(x_{i}\) and \(x\) is given as

\[\begin{equation*} p(x/\alpha) = \prod_{i=1}^{N_{T}MN} p(x_i/\alpha_i) = \prod_{i=1}^{N_{T}MN} \mathcal{N}(0,\alpha_i^{-1}). \tag{24} \end{equation*}\]

Moreover a Gamma prior distribution is assumed for \(\alpha\)

\[\begin{equation*} p(\alpha)= \prod_{i=1}^{N_{T}MN} p(\alpha_i/a,b) = \prod_{i=1}^{N_{T}MN} \Gamma^{-1}(a) b^a \alpha_i^{a-1} e^{-b\alpha_i}. \tag{25} \end{equation*}\]

It is presumed that the variance \(\gamma\) of white noise signal \(w\) is already known and a full DD domain Channel State Information (CSI) is accessible at the receiver. The likelihood distribution of the received signal \(y\) is given as follows

\[\begin{equation*} p(y/x) = (2\pi)^{-N_{R}MN/2} \gamma^{N_{R}MN/2} e^{\frac{-\gamma \left||y - \mathbf{H} x \right||^2}{2}}. \tag{26} \end{equation*}\]

The estimation of the posterior distribution of the received signal \(y\) is very cumbersome. To cope with this problem, variational Bayesian technique is applied and the posterior distribution \(p(x/y,\alpha)\) is estimated by a varitional distribution \(q(x,\alpha)\). It can be represented as

\[\begin{equation*} q(x,\alpha)= \prod_{i=1}^{N_{T}MN}q_{x_i}(x_i)\prod_{i=1}^{N_{T}MN}q_{\alpha_i}(\alpha_i). \tag{27} \end{equation*}\]

Variational Bayesian technique calculates the factors of \(q(x,\alpha)\) by minimizing the Kullback-Leibler (KL) distance between the variational distribution \(q(x,\alpha)\) and the true posterior distribution \(p(x,\alpha/y)\). The KL distance between \(q(x,\alpha)\) and \(p(x,\alpha/y)\) is denoted as \(KLD_{VBI}\).

\[\begin{equation*} KLD_{VBI}= KL\left(p(x,\alpha/y) || q(x,\alpha) \right). \tag{28} \end{equation*}\]

Minimization of KL distance is equivalent to maximizing the evidence lower Bound (ELBO). To discuss this pont further, \(KLD_{VBI}\) can be expressed as

\[\begin{equation*} \begin{aligned} KLD_{VBI}&=-\int q(\theta) \ln \frac{p(\theta/y)}{q(\theta)} d\theta \\ &= -\int q(\theta) \ln \frac{p(y,\theta)}{p(y)q(\theta)} d\theta. \end{aligned} \tag{29} \end{equation*}\]

This further simplifies to

\[\begin{equation*} KLD_{VBI}= \ln p(y)- \int q(\theta) \ln \frac{p(y,\theta)}{q(\theta)} d\theta, \tag{30} \end{equation*}\]

where \(\theta=\left\{x,\alpha\right\}\) and can be rearranged as

\[\begin{equation*} \ln p(y)=KLD_{VBI}+L(q). \tag{31} \end{equation*}\]

Since \(KLD_{VBI}\) is a distance, \(KLD_{VBI}\geq 0\). It means that the ELBO \(L(q)\) is the lower bound on \(\ln p(y)\). As we know, \(\ln y\) is independent of \(q(\theta)\) and minimization of \(KLD_{VBI}\) is analogous to maximization of the lower bound \(L(q)\). The ELBO maximization results in the following expression

\[\begin{equation*} \ln (q_i({\theta}_i)) = <\ln p(y, \theta)>_{k\neq i} + c_i, \tag{32} \end{equation*}\]

where \(\theta\,=\,\left\{x,\alpha\right\}\) and \(\theta_i\) denotes each scalar in \(\theta\). Here \(<>_{k\neq i}\) symbolizes the operator of expectation across the distributions \(q_k\) for all \(k\neq i\). The joint probability distribution \(p(y,\theta)\) can be represented as

\[\begin{equation*} p(y,\theta) = p(y/x,\alpha)p(x/\alpha)p(\alpha) \tag{33} \end{equation*}\]

Now the focus is to find an iterative solution. For this purpose, \(\ln p(y,\theta)\) can be expanded as follows

\[\begin{equation*} \begin{aligned} \ln p(y,\theta) &= \frac{N_{R}MN}{2}\ln \gamma - \frac{\gamma}{2}\left\|y - \mathbf{H} x\right\|^2 \\ &\quad+ \sum_{i=1}^{N_{T}MN}\left(\frac{1}{2}\ln \alpha_i - \frac{\alpha_i}{2}x_i^2\right) \\ &\quad+ \sum_{i=1}^{N_{T}MN}\left((a-1)\ln \alpha_i+a\ln b - b\alpha_i\right) \\ &\quad+ \text{constants}. \end{aligned} \tag{34} \end{equation*}\]

Now using (32) and (34), we need to find the update expressions for \(\ln q_{x_i}(x_i)\) and \(\ln q_{\alpha_i}(\alpha_i)\,\).

\[\begin{equation*} \begin{array}{l} \ln q_{x_i}(x_i) = \\ -\frac{\gamma}{2}\Big\{ <\left||y-\mathbf{H}_{\bar{i}}x_{\bar{i}}\right||^2>\,-\,(y-\mathbf{H}_{\bar{i}}<x_{\bar{i}}>)^{H}\mathbf{H}_ix_i\,-\,\\ x_i\mathbf{H}_i^H(y-\mathbf{H}_{\bar{i}}<x_{\bar{i}}>)\,+\,\left||\mathbf{H}_i\right||^2x_i^2 \Big\},-\,\frac{<\alpha_i>}{2}x_i^2 + c_{x_i} \\ = \, -\frac{1}{2\sigma^2_i}\left(x_i\, - \,\mu_i\right)^2 + c_{x_i}', \end{array} \tag{35} \end{equation*}\]

where we represent \(\mathbf{H}x\) as \(\mathbf{H}x\,=\,\mathbf{H}_ix_i\,+\,\mathbf{H}_{\bar{i}}x_{\bar{i}}\) where where \(\mathbf{H}_i\) denotes the \(i^{th}\) column of \(\mathbf{H}\),\(\mathbf{H}_{\bar{i}}\) is formed by removing \(i^{th}\) column of \(\mathbf{H}\),\(x_i\) is the \(i^{th}\) element of \(x\) and \(x_{\bar{i}}\) is the vector formed by removing \(i^{th}\) element of \(x\). \(c_{x_i}\) and \(c_{x_i}'\) are the normalization constants. From \((35)\), we can understand that the expansion of \(\ln q_{x_i}(x_i)\) is quadratic in nature and can be expressed as a Gaussian distributed random variable. The mean and variance of the subsequent Gaussian distribution is given as follows

\[\begin{equation*} \begin{array}{@{}l@{}} \sigma^2_i \, = \, \frac{1}{\gamma \left||\mathbf{H}\right||^2 \, + \, \alpha_i}, \,\,\, \\ <x_i> = \mu_i \, = \, \sigma^2_i \mathbf{H}^H\left(y\,-\,\mathbf{H}_{\bar{i}}<x_{\bar{i}}>\right)\gamma, \end{array} \tag{36} \end{equation*}\]

where \(\mu_i\) is the point estimate of \(i^{th}\) element of the transmitted signal \(x\). In a similar way \(\ln q_{\alpha_i}(\alpha_i)\,\) can be expressed as follows

\[\begin{equation*} \begin{array}{@{}l@{}} \ln q_{\alpha_i}(\alpha_i) = (a-1+\frac{1}{2})\ln \alpha_i\,-\,\alpha_i\left(\frac{<x_i^2>}{2}\,+\,b\right)\,+\,c_{\alpha_i}, \\ q_{\alpha_i}(\alpha_i)\, \propto \, \alpha_i^{a+\frac{1}{2}-1}e^{-\alpha_i \left(\frac{<x_i^2>}{2}\,+\,b\right)}, \end{array} \tag{37} \end{equation*}\]

where \(c_{\alpha_i}\) is the normalization constant. From \((37)\), we can come to a conclusion that variational approximation of \(q_{\alpha_i}(\alpha_i)\) is following a Gamma distribution. The mean of the resulting Gamma pdf is given as follows

\[\begin{equation*} \begin{array}{@{}l@{}} <\alpha_i>\,=\, \frac{a+\frac{1}{2}}{\left(\frac{<x_i^2>}{2}\,+\,b\right)}, \,\,\, \mbox{where}\,\,\, <x_i^2> = \mu_i^2\,+\,\sigma^2_i. \end{array}\!\!\!\!\! \tag{38} \end{equation*}\]

Now the equalized version of the transmitted signal matrix \(\mathbf{\hat{X}}_{ESM}^{VBI}\) in the DD domain is formed by collecting all the point estimates \(\mu_{i}\) where \(i=1,2\cdots N_{T}MN\). Again, the minimum Euclidean distance detector is applied on each row of \(\mathbf{\hat{X}}_{ESM}^{VBI}\) using (21) to obtain \(\mathbf{\hat{X}}_{ESM}\) as depicted in (22). Now each row of \(\mathbf{\hat{X}}_{ESM}\) is decoded using the ESM mapping rule as illustrated in Table 2 to retrieve an estimated of the original transmitted bits. The detection of the entire ESM-OTFS frame is accomplished in this manner. The steps involved in detection of ESM-OTFS using VBI detector is summarized in Algorithm 2.

4.  The Computational Complexity of the Detector

The signal detection algorithm at the receiver is composed of two distinct components that contribute to its overall complexity. First part is the computational complexity of the algorithm used for equalisation and second part is the complexity of the demodulation algorithm. MMSE detector utilises (20) for equalisation which has a computational complexity of \(O(M^3N^3N_T^3)\) [21]. The minimum euclidean distance detector is used for demodulation purpose which has a computational load of \(O(MN\mathbb{C})\), where \(\mathbb{C}\) represents the size of ESM constellation as depicted in Table 2. There is an additional search complexity of \(O(\mathbb{C})\) for decoding the bits.

The VBI based detector does not require any matrix inversion operation for the equalisation operation. It has a computational complexity of \(O(M^2N^2N_T^2L)\) where \(L\) is the number of iteration required for convergence of the algorithm. Since VBI based detector also uses euclidean distance based detector for the demodulation part, it has the same computational complexity in comparison to MMSE detector for demodulation.

5.  Spectral Efficiency

Predominantly spectral efficiency (SE) has been characterised as the capacity of data transmission within a given bandwidth and can be expressed as a ratio between the information rate and the total bandwidth occupied. The total time span of an ESM-OTFS frame is \(NT\) and the total bandwidth engaged by a frame is \(M \Delta f\). The number of bits transmitted by an ESM-OTFS frame is \(MNlog_2(\mathbb{C})\). So we can express the SE of OTFS-ESM as follows

\[\begin{equation*} \begin{split} SE_{ESM-OTFS} & = \frac{MN \times log_2(\mathbb{C})}{NT \times M\Delta f} \\ & = log_2(\mathbb{C}), \end{split} \tag{39} \end{equation*}\]

where \(T\Delta f=1\). The SE of the OTFS-SM and OTFS-QSM [20] systems can be represented using the same parameters as

\[\begin{equation*} \begin{aligned} & SE_{SM-OTFS} =log_{2}(M_{mod})+log_{2}(N_T) \\ & SE_{QSM-OTFS} =log_{2}(M_{mod})+2log_{2}(N_T), \end{aligned} \tag{40} \end{equation*}\]

where \(M_{mod}\) is size of the modulation alphabet used. As an example, consider an ESM-OTFS system with QPSK as primary modulation and \(N_T=2\). The ESM constellation corresponding to this system configuration is depicted in Table 2 and from this, we can see that there are \(16\) distinct constellation vectors. So the spectral efficiency of ESM-OTFS is 4 bits/s/Hz and this is also referred to as bit per channel use (bpcu). Under the same settings, SE of SM-OTFS and QSM-OTFS are 3 and 4 bits/s/Hz respectively.

6.  Simulation Results and Discussions

In this section, we discuss the findings from the simulations of the bit error rate (BER) performance of ESM-OTFS compared to SM-OTFS and QSM-OTFS under different system configurations. It is assumed that the receiver has the perfect channel state information (CSI) and all the channels have the Rayleigh fading. The main simulation parameters are given in Table 3.

Table 3  Simulation parameters.

The BER performance comparison of ESM-OTFS, QSM-OTFS, SM-OTFS and SIMO-OTFS for 4 bpcu is given in Fig. 2. The constellation used in each scheme is indicated in the legend of the figure and in case of ESM-OTFS, primary constellation is indicated. A MIMO setup of \(2\times 4\) is used for this simulation. It can be seen from Fig. 2 that ESM-OTFS outperforms other schemes in BER performance. At the BER value of \(10^{-4}\), ESM-OTFS outperforms QSM-OTFS by 2.1 dB, SM-OTFS by 3 dB and SIMO-OTFS by 5.3 dB.

Fig. 2  BER performance of ESM-OTFS, QSM-OTFS, SM-OTFS and SIMO-OTFS systems based on MMSE detector with \(M=4\), \(N=4\) for 4 bits/s/Hz.

The BER performance comparison of of ESM-OTFS, QSM-OTFS, SM-OTFS and SIMO-OTFS for 6 bpcu is given in Fig. 3. A MIMO setup of \(4\times 4\) is used for this simulation. At the BER value of \(10^{-3}\), ESM-OTFS outperforms QSM-OTFS by 2 dB and SM-OTFS by 3.8 dB.

Fig. 3  BER performance of ESM-OTFS, QSM-OTFS, SM-OTFS and SIMO-OTFS systems based on MMSE detector with \(M=4\), \(N=4\) for 6 bits/s/Hz

Figure 4 depicts the BER performance of ESM-OTFS for 8 bpcu using various detectors. A MIMO setup of \(4\times 4\) is used for this simulation and 16-QAM is used as the primary modulation. It can be shown that VBI based detector outperforms the linear detectors. ESM-OTFS detector has a gain of 6.1 dB at the value of \(10^{-2}\) compared to detection using MMSE detector.

Fig. 4  BER performance of ESM-OTFS using ZF, MMSE and VBI detectors with \(M=4\), \(N=4\) for 8 bits/s/Hz.

The BER performance of ESM-OTFS for 10 bpcu using various detectors is shown in the Fig. 5. A MIMO setup of \(8\times 8\) is used for this simulation and 16-QAM is used as the primary modulation. VBI based detector has a gain of 7.8 dB over MMSE based detector. It is observed from Fig. 4 and Fig. 5 that the performance of VBI based detector improves as the transmit data matrix becomes more sparse.

Fig. 5  BER performance of ESM-OTFS using ZF, MMSE and VBI detectors with \(M=4\), \(N=4\) for 10 bits/s/Hz

ESM-OTFS is highly robust to Doppler shifts, which are prevalent in high mobility environments. Figures 6 and 7 demonstrates the performance of ESM-OTFS at different velocities. Figures 6 and 7 showcases the BER performance of ESM-OTFS for 4 bpcu and 6 bpcu respectively. From the figures, we can understand that the BER performance of ESM-OTFS remains almost same from normal velocity of 30 kmph to very high velocity of 500 kmph. There is only a slight performance degradation at higher velocities and we can estimate that ESM-OTFS is highly immune to doppler shift resulting from velocity variations.

Fig. 6  BER performance of ESM-OTFS at different velocities \(M=16\), \(N=16\) for 4 bits/s/Hz.

Fig. 7  BER performance of ESM-OTFS at different velocities \(M=16\), \(N=16\) for 6 bits/s/Hz.

The computational complexity of the proposed ESM-OTFS detectors was discussed in Sect. 4. To illustrate further, consider an ESM-OTFS system \(M=4\), \(N=4\) and \(N_T=2\). In order to compare the receiver complexity at different bpcu, QPSK,16-QAM,64-QAM,256-QAM are used as the primary constellation to have bpcu of 4, 6, 8 and 10 respectively. Since the computational complexity of the equalisation algorithms depend on \(M\), \(N\) and \(N_T\), the computational complexity remains same in all four cases of bpcu. Since the demodulation algorithm has a computational complexity of \(O(MN\mathbb{C})\), the computational load in terms of multiplication increases with increase in size of \(\mathbb{C}\). As we know, the bpcu of the ESM-OTFS system is \(log_2(\mathbb{C})\), and the computational complexity of the detector increases with an increase in bpcu. This is highlighted in Fig. 8.

Fig. 8  Trade-off between detector complexity and bpcu in ESM-OTFS systems.

As the modulation order increases, the receiver must perform more sophisticated signal processing to distinguish between closely spaced constellation points, thereby increasing the computational burden. The graph plotting computational complexity against bpcu (Fig. 8) effectively captures this relationship. As bpcu increases ― moving from QPSK to 256-QAM ― the computational complexity of the receiver escalates due to the need for more refined signal processing techniques to accurately decode the higher-order modulated signals. This trade-off is critical in designing and optimizing communication systems for specific application scenarios, balancing the need for high data rates (and thus higher bpcu) against the constraints of receiver complexity, power consumption, and real-time processing capabilities.

7.  Conclusion

In this paper, we propose an ESM-OTFS scheme which is suitable for high doppler shift wireless communication environments and enhances the system reliability and spectral efficiency. The system model and signal processing of ESM-OTFS have been discussed. A novel detector based Variational Bayesian Inference has been proposed for the detection of ESM-OTFS. It is demonstrated through the simulation that ESM-OTFS has a better BER performance compared to SM-OTFS and QSM-OTFS. It is also verified that VBI based ESM-OTFS detector outperforms the linear detectors.

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Authors

Anoop A
  Indian Institute of Information Technology

Kala S
  Indian Institute of Information Technology

Keyword

enhanced spatial modulation (ESM),  orthogonal time frequency space (OTFS),  variational Bayesian inference (VBI),  delay-Doppler (DD),  bit error rate (BER)