1 Introduction

The visible light communication (VLC) has emerged as a preferred complementary technology to the existing congested traditional radio frequency (RF) communication because of its high bandwidth, wide licence-free spectrum, and immunity to electromagnetic interference from various sources [1]. In this respect, VLC has emerged to be the upcoming futuristic technology for the fifth generation (5 G) and beyond communication systems [2]. Owing to the use of light-emitting diodes (LEDs) as transmitters, VLC facilitates both illumination and data communication simultaneously.

Fig. 1
figure 1

Block diagram of the considered system model

Full size image

Although VLC has demonstrated itself to be one of the preferred communication technique, the user mobility and multipath channel between the user and the LEDs due to reflection from walls and other objects in the room produces interference due to frequency domain spreading [3]. For optical wireless communication systems, to mitigate multipath-induced distortion, conventional multi-carrier modulation schemes such as optical-orthogonal frequency division multiplexing (O-OFDM) are proposed in the literature [4, 5]. However, the receiver complexity of O-OFDM increases with mobile users and with an increase in the multipath channel between the receiver and the transmitter. Recently, a less sophisticated multi-carrier modulation approach called orthogonal time frequency space (OTFS) has been suggested in the literature to alleviate multipath-related impairments [6].

Contrary to OFDM’s time-frequency (TF) plane modulation, OTFS proposes to modulate data to be transmitted in the delay-Doppler (DD) domain. OTFS exhibits superior performance compared to OFDM, by taking advantage of the channel’s dispersive effects by considering diversity in both the time and the frequency domain. Since the first paper on OTFS [6], the OTFS has significantly outperformed OFDM in terms of performance over a variety of dynamic and static channels. In [6], the bit error rate (BER) and packet error performance (PER) of OTFS were compared with the conventional OFDM technique for RF communication. For indoor VLC channels, authors in [7, 8] have investigated quad and dual LED-OTFS and observed superior BER performance of OTFS over OFDM. Authors in [9,10,11,12] have shown that for static and mobile multipath VLC channels in an indoor environment, the BER performance of OTFS outperforms the conventional OFDM technique. Effective representation of the channel in the DD domain is inherently sparse when the number of channel paths is small compared to the number of symbols transmitted per frame [6]. Various channel estimation approaches for OTFS have been proposed in the literature. Authors in [13] have proposed time domain channel estimation and equalization method for OTFS with fractional Doppler shifts. For an RF-based communication system, authors in [14] have proposed sparse coding-based channel estimation approach for OTFS-sparse code multiple access (SCMA) in the uplink. Taking advantage of inherent sparsity, authors in [15] have presented sparse signal recovery methods such as orthogonal matching pursuit (OMP) and modified subspace pursuit (MSP) for channel estimation in uplink-OTFS. For massive-multiple input multiple output OTFS, authors in [16] have proposed a three-dimensional structured orthogonal matching pursuit (3D-SOMP) for channel estimation in the downlink with low pilot overhead. However, techniques based on greedy algorithms, like OMP and its derivatives, heavily rely on calculating the precise stopping criteria and might result in high convergence error, which reduces overall performance [17]. For static VLC systems, authors in [18] have proposed ZA-LMS-based sparse channel estimation algorithm. For mobility-impaired OTFS-VLC systems, the channel estimation problem is not yet investigated thoroughly. To estimate sparse dispersive OTFS-VLC channels and to overcome the shortcomings of the previous greedy-algorithm-based schemes, a zero-attracting least mean square (ZA-LMS)-based channel estimator is proposed with analysis in this paper.

Contrary to OFDM’s time-frequency (TF) plane modulation, OTFS proposes to modulate data to be transmitted in the delay-Doppler (DD) domain. OTFS exhibits superior performance compared to OFDM, by taking advantage of the channel’s dispersive effects by considering diversity in both the time and the frequency domain. Since the first paper on OTFS [6], the OTFS has significantly outperformed OFDM in terms of performance over a variety of dynamic and static channels. In [6], the bit error rate (BER) and packet error performance (PER) of OTFS were compared with the conventional OFDM technique for RF communication. For indoor VLC channels, authors in [7, 8] have investigated quad and dual LED-OTFS and observed superior BER performance of OTFS over OFDM. Authors in [9,10,11,12] have shown that for static and mobile multipath VLC channels in an indoor environment, the BER performance of OTFS outperforms the conventional OFDM technique. Effective representation of the channel in the DD domain is inherently sparse when the number of channel paths is small compared to the number of symbols transmitted per frame [6]. Various channel estimation approaches for OTFS have been proposed in the literature. Authors in [13] have proposed time domain channel estimation and equalization method for OTFS with fractional Doppler shifts. For an RF-based communication system, authors in [14] have proposed sparse coding-based channel estimation approach for OTFS-sparse code multiple access (SCMA) in the uplink. Taking advantage of inherent sparsity, authors in [15] have presented sparse signal recovery methods such as orthogonal matching pursuit (OMP) and modified subspace pursuit (MSP) for channel estimation in uplink-OTFS. For massive-multiple input multiple output OTFS, authors in [16] have proposed a three-dimensional structured orthogonal matching pursuit (3D-SOMP) for channel estimation in the downlink with low pilot overhead. However, techniques based on greedy algorithms, like OMP and its derivatives, heavily rely on calculating the precise stopping criteria and might result in high convergence error, which reduces overall performance [17]. For static VLC systems, authors in [18] have proposed ZA-LMS-based sparse channel estimation algorithm. For mobility-impaired OTFS-VLC systems, the channel estimation problem is not yet investigated thoroughly. To estimate sparse dispersive OTFS-VLC channels and to overcome the shortcomings of the previous greedy-algorithm-based schemes, a zero-attracting least mean square (ZA-LMS)-based channel estimator is proposed with analysis in this paper.

Recognizing the inherent sparse nature of effective channels in DD domain, in this paper, we have proposed a ZA-LMS-based channel estimation method for the VLC-OTFS system. Simulations performed over a realistic mobile VLC channel modeled by random way-point model indicate that OTFS with ZA-LMS mitigate distortions due to the user mobility and multipaths and gives better performance compared to the conventional LMS algorithm and OMP algorithm. Rest of the paper is organized as follows: The considered system model and channel model are described in Sect. 2. ZA-LMS for OTFS-VLC is described in Sect. 3. Analytical BER expression for the considered VLC-OTFS system is derived in Sect. 4. Simulation results are given in Sect. 5. Lastly, Sect. 6 concludes the paper.

Notations: The notations s, \({\textbf{s}}\), and \({\textbf{S}}\) stand for scalars, vectors, and matrices, respectively. \({\textbf{s}}[i]\) and \({\textbf{S}}[k,l]\) represent the \(k^{th}\) element of vector \({\textbf{s}}\) and \((k,l)^{th}\) element of matrix \({\textbf{S}}\). The set of matrices with dimension \(A\times B\) having each entry from the complex plane is denoted by \({\mathbb {C}}^{A\times B}\). Let \({\textbf{S}}=\text {circ}[{\textbf{S}}_{0},\ldots ,{\textbf{S}}_{B-1}] \in {\mathbb {C}}^{AB\times AB}\) represent the circulant matrix. Transpose of a vector \((\cdot )\) is denoted by \((\cdot )^{T}\). \({\mathbb {E}}\{.\}\) denotes the statistical expectation operator. \({\mathcal {N}}(\mu ,\sigma ^{2})\) denote the Gaussian distribution with mean \(\mu\) and variance \(\sigma ^{2}\). The \(l_{p}\) norm of the vector \({\textbf{s}}\) is denoted by \(\Vert {\textbf{s}}\Vert _{p}\).

2 System model

In this section, a block diagram of the considered system model of the OTFS-VLC system effected by impairments due to user mobility, multipath between the receiver and transmitter and ambient light noise and thermal noise is depicted in Fig. 1. Let \(N_{s} = KL\) represent the number of symbols transmitted in each frame, where K and L represent the number of symbols and sub-carriers, respectively. Let \({\textbf{x}}\in {\mathbb {C}}^{N_{s}\times 1}\) be transmitted binary phase shift keying (BPSK) symbols. For OTFS modulation, Zac transformation is done on the input vector to transform DD-mapped symbols to the time domain for transmission. Zac transform is computationally complex and performed in two steps. First, the input BPSK modulated vector \({\textbf{x}}\) is transformed into the TF domain using the two-dimensional (2D) inverse symplectic fast Fourier transform (ISFFT) such that

$$\begin{aligned} \mathbf {X_{t}}[v,u] = \sum _{l=0}^{K-1}\sum _{k=0}^{L-1}{\textbf{x}}_{l,k}e^{-j2\pi (\frac{ul}{K}-\frac{vk}{L})} \end{aligned}$$
(1)

In the second step, Heisenberg transform on the output of ISFFT is applied to transform it into the time domain

$$\begin{aligned} {\tilde{\textbf{x}}}(t) = \sum _{u=0}^{K-1}\sum _{v=0}^{L-1}\mathbf {X_{t}}[v,u]e^{j2\pi u\Delta f(t-vT)}g(t-vT) \end{aligned}$$
(2)

where g(t) denotes the pulse transmitted. To create a 2D lattice in the TF domain, sampling is done at intervals T and \(\Delta f\), respectively, where \(\Lambda = (vT,u\Delta f)\), and \(v=0,\ldots ,L-1\), and \(u=0,\ldots ,K-1\).

Before transmitting the time domain data, the output of Heisenberg transform \({\tilde{\textbf{x}}}\) in (23) is prefixed with cyclic prefix (CP) of length \((C_{p}-1)\), where \(C_{p}\) is the total number of channel paths. The symbols are broadcasted through LED in the time domain after OTFS modulation and adding CP. The output is transmitted over a mobile VLC channel, \({\textbf{h}}\), modeled by the random-way point channel model explained in subsection 2.1. The channel is denoted by the expression \({\textbf{h}}=[h_{0}, h_{1},\ldots ,h_{C_{p}-1}]^{T}\). After removing the CP, the received information signal in the temporal domain can be expressed as

$$\begin{aligned} {\textbf{r}}={\textbf{H}}{\tilde{\textbf{x}}}+{\tilde{\textbf{w}}} \end{aligned}$$
(3)

where \({\textbf{H}}\) is estimated as

$$\begin{aligned} {\textbf{H}}\left( \tau ,\nu \right) = \sum _{i=1}^{C_{p}} h_{i}\delta \left( \tau -\tau _{i}\right) \delta \left( \nu -\nu _{i}\right) , \end{aligned}$$
(4)

where \(\nu _{i}\), \(\tau _{i}\) are Doppler shift and delay, respectively, for the \(i^{th}\) cluster, and \(\delta (\cdot )\) denotes the Dirac delta function. In this work, both ambient light noise and thermal noise are approximated by a zero mean Gaussian distribution denoted by \({\tilde{\textbf{w}}} \in {\mathbb {C}}^{N_{s}\times 1}\) and is additive independent and identically distributed (i.i.d.) whose \(i^{th}\) entry is defined as \(w_{i}\sim \mathcal{C}\mathcal{N}(0,\sigma ^{2})\). Where \(\sigma ^{2}=\sigma ^{2}_{a}+\sigma ^{2}_{t}\) and \(\sigma ^{2}_{a}\) and \(\sigma ^{2}_{t}\) is the variance of ambient light noise and thermal noise, respectively.

Similar to the transmitter side, at the receiver side, the symbols received by photodetector \({\textbf{r}}(t)\) are in the time domain and are transformed back to the information domain using the inverse Zac transformation. Similar to Zac transformation, inverse Zac transformation can be done in two following simple steps. First, the received time domain symbols are transformed to TF domain \({\textbf{Y}}[v,u]\) by applying the Wigner transform

$$\begin{aligned} {\textbf{Y}}[v,u] = \int {\textbf{r}}(\tau )p^{*}(\tau -t)e^{-j2\pi f(t-\tau )}d\tau \end{aligned}$$
(4)

where p is the received pulse. Pulses g and p are ideal such that they satisfy bi-orthogonality and robustness. Then SFFT is applied on the output of the Wigner transform \({\textbf{Y}}_{v,u}\) [6] to transform signal mapped in TF to DD, i.e., information domain.

$$\begin{aligned} {\textbf{y}}_{l,k}= & {} \frac{1}{\sqrt{KL}}\sum _{v=0}^{L-1}\sum _{u=0}^{K-1}{\textbf{Y}}[v,u]e^{-j2\pi (\frac{ul}{K}-\frac{vk}{L})}+{\textbf{w}} \end{aligned}$$
(5)
$$\begin{aligned} {\textbf{y}}= & {} \textbf{H}^{{\mathrm{\tiny {eff}}}}{\textbf{x}}+{\textbf{w}} \end{aligned}$$
(6)

where \({\textbf{y}} \in {\mathbb {C}}^{N_{s}\times 1}\) is the symbol received at the receiver in the information domain, i.e., DD domain, \(\textbf{H}^{{\mathrm{\tiny {eff}}}} \in {\mathbb {C}}^{N_{s}\times N_{s}}\) is the effective channel matrix which is sparse in nature, \({\textbf{x}} \in {\mathbb {C}}^{N_{s}\times 1}\) is the transmitted BPSK symbols mapped in DD domain and, \({\textbf{w}}\) is the noise having the same statistical properties of \({\tilde{\textbf{w}}}\). Alternatively, the relation in (6) can be written as:

$$\begin{aligned} {\textbf{y}}= {\textbf{X}}\mathbf {h_{b}}+{\textbf{w}} \end{aligned}$$
(7)

where \(\mathbf {h_{b}} \in {\mathbb {C}}^{N_L\times 1}\) is a \(N_L \times 1\) vector with \(C_p\) non-zero elements and \({\textbf{X}}\in {\mathbb {C}}^{N_s\times N_L}\). Based on the received observations ZA-LMS-based receiver is trained, and symbols are estimated by zero-forcing (ZF) using the channel estimated after training. The estimated symbols are then detected by maximum likelihood (ML) detector [19]. The detected symbols are then passed through a BPSK demodulator to receive the transmitted bits.

2.1 Random way-point channel model for VLC

In this Subsection, the mobility-impaired channel model modeled by RWP model is described [20]. For the RWP model, the probability distribution function (pdf) of the channel \(\mathbf {h_b}\) is given as [21]

$$\begin{aligned} p(h)=\left\{ \begin{array}{ll} \sum _{i=1}^{4}Q_{i}h^{-\beta _{i}}, &{} {h_{MN}\le h\le h_{MX};}\\ 0, &{} \hbox {otherwise} \end{array} \right. \end{aligned}$$
(8)

where \(Q_{1}=Q[27+\frac{35D^{2}}{r_{max}^{2}}+\frac{8D^{4}}{r_{max}^{4}}]\), \(Q_{2}=-Q\frac{35}{r_{max}^{2}}{\mathfrak {V}}^{\frac{2}{a+3}}\), \(Q_{3}=-Q\frac{8}{r_{max}^{4}}{\mathfrak {V}}^{\frac{4}{a+3}}\), and \(Q_{4}=-Q\frac{16D^{2}}{r_{max}^{4}}{\mathfrak {V}}^{\frac{2}{a+3}}\), \(Q=\frac{12{\mathfrak {V}}^{\frac{2}{a+3}}}{73(a+3)r_{max}^{2}}\), \(\beta _{1}=\frac{2}{a+3}+1\), \(\beta _{2}=\beta _{4}=\frac{4}{a+3}+1\), and \(\beta _{3}=\frac{6}{a+3}+1\), where \({\mathfrak {V}}=b(a+1)D^{a+1}\), \(b=\frac{R}{2\pi }\). The line of sight distance of the LED from the user is denoted as D, the effective geometric area of the detector is denoted by R, and \(r_{max}\) is the radius of the maximum coverage area. \(h_{MN}=\frac{{\mathfrak {V}}}{(r_{max}^{2}+D^{2})^{\frac{a+3}{2}}}\), \(h_{MX}=\frac{{\mathfrak {V}}}{D^{(a+3)}}\) and, \(a=\frac{-1}{\log (\cos (\phi _{\frac{1}{2}}))}\) where \(\phi _{\frac{1}{2}}\) is the half-angle of the fixature of the LED transmitting.

3 ZA-LMS for OTFS-VLC system

In this Section, the ZA-LMS-based channel estimation algorithm for the OTFS-VLC system impaired by dispersive VLC channel is described. As \(C_{p}<<KL\), effective channel matrix \(\textbf{H}^{{\mathrm{\tiny {eff}}}}\) in (22) is sparse in nature. Hence, in this paper, the ZA-LMS algorithm is implemented for channel estimation as it takes advantage of inherent channel sparsity [22]. From (7), the channel estimation problem can be described as a non-convex combinatorial problem that is formulated as

$$\begin{aligned}{} & {} {\underset{{\textbf{h}}_b}{\text {min}}\ \Vert {\textbf{h}}_b \Vert _0}, \nonumber \\{} & {} {\text {s.t.} \ \Vert {\textbf{y}}_{p} - {\textbf{X}}_{p}{\textbf{h}}_{b} \Vert _2^2 \le \beta }, \end{aligned}$$
(9)

where \({\textbf{y}}_{p}\) and \({\textbf{X}}_{p}\) are the received, and the transmitted pilots, and \(\beta\) is the error tolerance parameter which always has a positive value. Various offline training methods for sparse channel estimation are proposed in the literature to solve the aforementioned problem, such as OMP [23] and sparse Bayesian learning (SBL) [24]. However, because these methods are offline, they have a significant propagation latency and high computational cost since they must calculate the matrix inversions for each iteration. The ZA-LMS algorithm is proposed to address the problem statement without having the drawbacks of offline techniques. The mean square deviation-based cost function \(J_{ZA}(n)\) for ZA-LMS [25] is therefore defined as

$$\begin{aligned} J_{ZA}(j) = {\mathbb {E}}\{\Vert {\textbf{y}}_{p}(j)- {\textbf{X}}_{p}(j) {\hat{\textbf{h}}}_b(j) \Vert ^2 \} + \gamma f({\hat{\textbf{h}}}_b(j)), \end{aligned}$$
(10)

where \({\hat{\textbf{h}}}_b\) is the estimated channel, \(\gamma\) is the regularization parameter, and \(f(\cdot )\) is the penalty term inducing sparsity. Following the use of the traditional steepest descent algorithm [26] the estimated channel \({\hat{\textbf{h}}}_b(j+1)\) can be iteratively updated as

$$\begin{aligned} {\hat{\textbf{h}}}_b(j+1) = {\hat{\textbf{h}}}_b(j)-\frac{\mu }{2}\nabla _{{\hat{\textbf{h}}}_b(j)}(J_{ZA}(j)). \end{aligned}$$
(11)

where the step-size parameter is denoted as \(\mu\). The gradient \(\nabla _{{\hat{\textbf{h}}}_b(j)}\) of the cost function considered earlier is estimated as

$$\begin{aligned} \nabla _{{\hat{\textbf{h}}}_b(j)}(J_{ZA}(j)) = 2\mathbf {R_{xx}}{\hat{\textbf{h}}}_b(j) -2\mathbf {R_{xy}} - \rho g(f({\hat{\textbf{h}}}_b(j))), \end{aligned}$$
(12)

where \(g(f({\hat{\textbf{h}}}_b(j))) = \nabla _{{\hat{\textbf{h}}}_b(j)}(f({\hat{\textbf{h}}}_b(j)))\) represents the gradient of the penalty function \(f(\cdot )\) which is inducing sparsity, \(\rho = \frac{\gamma \mu }{2}\) denotes regularization step-size, \(\mathbf {R_{xx}}\) is the auto-covariance of the transmitted pilot in DD domain \({\textbf{X}}\) computed as \({\mathbb {E}}\{{\textbf{X}}_{p}^{T}{\textbf{X}}_{p}\}\), and \(\mathbf {R_{xy}}\) is the cross-covariance between the transmitted and received pilot vectors \({\textbf{X}}_{p}\) and \({\textbf{y}}_{p}\) computed as \({\mathbb {E}}\{{\textbf{X}}_{p}^{T}{\textbf{y}}_{p}\}\). The gradient of the cost function can be substituted to simplify the weight update equation from (12) to (11) such that

$$\begin{aligned} {\hat{\textbf{h}}}_b(j+1) = {\hat{\textbf{h}}}_b(j) + \mu (\mathbf {R_{xy}}-\mathbf {R_{xx}}{\hat{\textbf{h}}}_b(j)) - \rho g(f({\hat{\textbf{h}}}_b(j))). \end{aligned}$$
(13)

Pursuing the stochastic-gradient approach, the final update expression of the estimated channel can be obtained as

$$\begin{aligned} {\hat{\textbf{h}}}_b(j+1) = {\hat{\textbf{h}}}_b(j) + \mu {\textbf{X}}_{p}^T(j){\textbf{e}}(j) - \rho g(f({\hat{\textbf{h}}}_b(j))), \end{aligned}$$
(14)

where \({\textbf{e}}(j)\) represents the instantaneous observation error estimated as

$$\begin{aligned} {\textbf{e}}(j) = {\textbf{y}}_{p}(j) - {\textbf{X}}_{p}(j) {\hat{\textbf{h}}}_b(j), \end{aligned}$$
(15)
figure a

ZA-LMS-based channel estimation

In this paper, \(l_1\)-norm-based sparsity-inducing penalty functions is considered.

3.1 ZA-LMS using \(l_1\)-norm approximation

The \(l_1\)-norm approximation represented as \(f_1(\cdot )\) can be determined as [25],

$$\begin{aligned} f_1({\hat{\textbf{h}}}_b(j)) = \Vert {\hat{\textbf{h}}}_{b}(j)\Vert _1 = \sum _{i=1}^{L^2 }{|{\hat{\textbf{h}}}_b(j)(i)|}. \end{aligned}$$
(16)

The gradient term \(g(f_1({\hat{\textbf{h}}}_{b}(j)))\) can be estimated as follows

$$\begin{aligned} g(f_1({\hat{\textbf{h}}}_b(j))) = \text {sgn}({\hat{\textbf{h}}}_b(j)). \end{aligned}$$
(17)

where \(\text {sgn}(\cdot )\) is the signum function. The update equation for ZA-LMS-\(l_1\)-norm is given as:

$$\begin{aligned} {\hat{\textbf{h}}}_b(j+1) = {\hat{\textbf{h}}}_b(j) + \mu {\textbf{X}}(j){\textbf{e}}(j) - \rho \text {sgn}({\hat{\textbf{h}}}_{b}(j)) \end{aligned}$$
(18)

Upon adaptation, the tap coefficients of the weight to be updated are attracted to zero by the third term present in these equations (also known as zero attractor), i.e., \(\rho \text {sgn}({\hat{\textbf{h}}}_{b}(j))\). The strength of the zero attractor is regulated by the regularization parameter which is represented as \(\rho\). The speed of convergence of the proposed algorithm depends on the sparsity of the channel matrix.

4 Analytical BER expression for VLC-OTFS system over mobility impaired channel

In this section, we derive the BER expression of the mobility-impaired VLC-OTFS system, assuming a transmitted constellation of BPSK. The average pairwise error probability (PEP) between symbol matrices given by (7) can be written as:

$$\begin{aligned} P({\textbf{X}}_{A}\rightarrow {\textbf{X}}_{B})={\mathbb {E}} \left[ { Q \left( {\sqrt{\frac{\gamma ~\Vert \mathbf {h_b} ({\textbf{X}}_{A}-{\textbf{X}}_{B})\Vert ^{2}}{2}}\, }\right) }\right] \!.\end{aligned}$$
(19)

where \(\gamma\) is the signal-to-noise ratio. This can be further simplified by writing:

$$\begin{aligned} \Vert \mathbf {h_b}({\textbf{X}}_{A}-{\textbf{X}}_{B})\Vert ^{2} =\mathbf {h_b}({\textbf{X}}_{A}-{\textbf{X}}_{B}) ({\textbf{X}}_{A}-{\textbf{X}}_{B})^{H}\mathbf {h_b}^{\prime H}. \end{aligned}$$
(20)

The matrix \(({\textbf{X}}_{A}-{\textbf{X}}_{B}) ({\textbf{X}}_{A}-{\textbf{X}}_{B})^{H}\) is Hermitian and can by diagonalized as:

$$\begin{aligned} ({\textbf{X}}_{A}-{\textbf{X}}_{B})({\textbf{X}}_{A}- {\textbf{X}}_{B})^{H}=\mathbf {U\Lambda U}^{H}\end{aligned}$$
(21)

where U is unitary and \(\varvec{\Lambda }=\text {diag}\lbrace \lambda _{1}^{2}, \cdots \lambda _{P}^{2} \rbrace\), \(\lambda _{i}\) is the \(i_{th}\) singular value of difference matrix \(\varvec{\Delta }_{AB} = ({\textbf{X}}_{A}-{\textbf{X}}_{B})\). Therefore, (19) can be expressed simply as:

$$\begin{aligned} P({\textbf{X}}_{A}\rightarrow {\textbf{X}}_{B})= {\mathbb {E}}\left[ Q\sqrt{\frac{\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{4}}\right] \end{aligned}$$
(22)

Using an approximation of Q-function:

$$\begin{aligned} Q(\sqrt{x}) \approx \frac{1}{12} e^{\frac{-x}{2}}+\frac{1}{4} e^{\frac{-2x}{3}} \end{aligned}$$
(23)

Therefore, (22) can be written as:

$$\begin{aligned} P({\textbf{X}}_{A}\rightarrow {\textbf{X}}_{B})&\approx {\mathbb {E}}\left[ \frac{1}{12} e^{\frac{-\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{8}}+\frac{1}{4} e^{\frac{-\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{6}}\right] \nonumber \\&\approx \!\frac{1}{12}{\mathbb {E}}\!\left[ \!e^{\frac{-\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{8}}\!\right] \!+\!\frac{1}{4} {\mathbb {E}}\left[ \!e^{\frac{-\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{6}}\!\right] \end{aligned}$$
(24)
$$\begin{aligned} {\mathbb {E}}\left[ e^{\frac{-\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{8}}\right]&= {\mathbb {E}}\left[ e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}}e^{\frac{-\gamma |h_{2}|^2\lambda _{2}^2}{8}}\ldots e^{\frac{-\gamma |h_{L}|^2\lambda _{L}^2}{8}}\right] \nonumber \\&= {\mathbb {E}}\left[ e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}}\right] {\mathbb {E}}\left[ e^{\frac{-\gamma |h_{2}|^2\lambda _{2}^2}{8}}\right] \ldots \nonumber \\&\quad \cdot {\mathbb {E}}\left[ e^{\frac{-\gamma |h_{L}|^2\lambda _{L}^2}{8}}\right] \end{aligned}$$
(25)
$$\begin{aligned} {\mathbb {E}}\left[ e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}}\right]&= \int _{h_{min}^2}^{h_{max}^2}e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}} \sum ^{4}_{i=1}\frac{Q_{i}}{2}h^{\frac{-\beta _{i}}{2}} dh \nonumber \\&= \sum ^{4}_{i=1}\frac{Q_{i}}{2}\int _{h_{min}^2}^{h_{max}^2}e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}} h^{\frac{-\beta _{i}}{2}} dh \end{aligned}$$
(26)

Let, \(\frac{\gamma \lambda _{1}^{2}}{8}=a\) and \(\frac{\beta _{i}}{2}=b\). Thus,

$$\begin{aligned} {\mathbb {E}}\left[ e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}}\right]&= \sum ^{4}_{i=1}\frac{Q_{i}}{2}\int _{h_{min}^2}^{h_{max}^2}e^{-ah} h^{-b} dh \nonumber \\&= \sum ^{4}_{i=1}\frac{Q_{i}}{2} \left[ -a^{b-1}\Gamma (1-b,ah) \right] _{h_{min}^{2}}^{h_{max}^{2}} \nonumber \\&= \sum ^{4}_{i=1}-a^{b-1}\frac{Q_{i}}{2} \nonumber \\&\quad \cdot \left[ \Gamma (1-b,ah_{max}^{2})-\Gamma (1-b,ah_{min}^{2}) \right] \end{aligned}$$
(27)

The upper incomplete gamma function at high signal-to-noise ratio can be approximated as,

$$\begin{aligned} \Gamma (\frac{-a_{i}+1}{2},\beta _{i}h^{2})\approx e^{-\beta _{i}h^{2}}(\beta _{i}h^{2})^{\frac{-a_{i}-1}{2}} \end{aligned}$$

Thus, (27) can be approximated as,

$$\begin{aligned} {\mathbb {E}}\left[ e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}}\right]&= \sum ^{4}_{i=1}-a^{b-1}\frac{Q_{i}}{2} \nonumber \\&\quad \cdot \left[ e^{-ah_{max}^2}(ah_{max}^2)^{-b}-e^{-ah_{min}^2}(ah_{min}^2)^{-b} \right] \nonumber \\&= \sum ^{4}_{i=1}\frac{-Q_{i}}{2a} \nonumber \\&\quad \cdot \left[ e^{-ah_{max}^2}(h_{max}^2)^{-b}-e^{-ah_{min}^2}(h_{min}^2)^{-b} \right] \end{aligned}$$
(28)

Substituting a and b in (28) we get,

$$\begin{aligned}&{\mathbb {E}}\left[ e^{\frac{-\gamma |h_{1}|^2\lambda _{1}^2}{8}}\right] =\sum ^{4}_{i=1}\frac{-4Q_{i}}{\gamma \lambda _{1}^2} \nonumber \\&\quad \cdot \left[ e^{\frac{-\gamma \lambda _{1}^2h_{max}^2}{8}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{1}^2h_{min}^2}{8}}(h_{min})^{-\beta _{i}} \right] \end{aligned}$$
(29)

Thus, first part of (24) can be written as,

$$\begin{aligned}&\frac{1}{12}{\mathbb {E}}\left[ e^{\frac{-\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{8}}\right] =\frac{(-1)^{L}}{12}\prod _{l=1}^{L}\sum _{i=1}^{4}\frac{4Q_{i}}{\gamma \lambda _{l}^2}\nonumber \\&\quad \cdot \left[ e^{\frac{-\gamma \lambda _{l}^2h_{max}^2}{8}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{l}^2h_{min}^2}{8}}(h_{min})^{-\beta _{i}} \right] \end{aligned}$$
(30)

Similarly, the second part of (24) can be estimated as,

$$\begin{aligned}&\frac{1}{4}{\mathbb {E}}\left[ e^{\frac{-\gamma \sum _{l=1}^{L}|h_{l}|^2\lambda _{l}^2}{6}}\right] =\frac{(-1)^{L}}{12}\prod _{l=1}^{L}\sum _{i=1}^{4}\frac{3Q_{i}}{\gamma \lambda _{l}^2}\nonumber \\&\quad \cdot \left[ e^{\frac{-\gamma \lambda _{l}^2h_{max}^2}{6}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{l}^2h_{min}^2}{6}}(h_{min})^{-\beta _{i}} \right] \end{aligned}$$
(31)

Thus, (22) is finally,

$$\begin{aligned}&P({\textbf{X}}_{A}\rightarrow {\textbf{X}}_{B}) = \frac{(-1)^{L}}{12}\prod _{l=1}^{L}\sum _{i=1}^{4}\frac{4Q_{i}}{\gamma \lambda _{l}^2}\nonumber \\&\quad \cdot \left[ e^{\frac{-\gamma \lambda _{l}^2h_{max}^2}{8}}(h_{max})^{-\alpha _{i}}-e^{\frac{-\gamma \lambda _{l}^2h_{min}^2}{8}}(h_{min})^{-\alpha _{i}} \right] \nonumber \\&\quad + \cdot \frac{(-1)^{L}}{4}\prod _{l=1}^{L}\sum _{i=1}^{4}\frac{3Q_{i}}{\gamma \lambda _{l}^2}\nonumber \\&\quad \cdot \left[ e^{\frac{-\gamma \lambda _{l}^2h_{max}^2}{6}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{l}^2h_{min}^2}{6}}(h_{min})^{-\beta _{i}} \right] \end{aligned}$$
(32)

The PEP with a minimum value of L dominates the overall BER. Thus, we can assume L=1.

$$\begin{aligned}&P({\textbf{X}}_{A}\rightarrow {\textbf{X}}_{B}) = \frac{-1}{12}\sum _{i=1}^{4}\frac{4Q_{i}}{\gamma \lambda _{1}^2}\nonumber \\&\quad \left[ e^{\frac{-\gamma \lambda _{1}^2h_{max}^2}{8}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{1}^2h_{min}^2}{8}}(h_{min})^{-\beta _{i}} \right] \nonumber \\&\quad - \frac{1}{4}\sum _{i=1}^{4}\frac{3Q_{i}}{\gamma \lambda _{1}^2}\left[ e^{\frac{-\gamma \lambda _{1}^2h_{max}^2}{6}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{1}^2h_{min}^2}{6}}(h_{min})^{-\beta _{i}} \right] \end{aligned}$$
(33)

Finally,

$$\begin{aligned}&P({\textbf{X}}_{A}\rightarrow {\textbf{X}}_{B}) = \sum _{i=1}^{4}\frac{-Q_{i}}{3\gamma \lambda _{1}^2}\nonumber \\&\quad \left[ e^{\frac{-\gamma \lambda _{1}^2h_{max}^2}{8}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{1}^2h_{min}^2}{8}}(h_{min})^{-\beta _{i}} \right] \nonumber \\&\quad -\frac{3Q_{i}}{4\gamma \lambda _{1}^2}\left[ e^{\frac{-\gamma \lambda _{1}^2h_{max}^2}{6}}(h_{max})^{-\beta _{i}}-e^{\frac{-\gamma \lambda _{1}^2h_{min}^2}{6}}(h_{min})^{-\beta _{i}} \right] \end{aligned}$$
(34)

The exact expression for the PEP using the characteristic function of the random-way point channel model is given by (32). Using the PEP expression, we have obtained an upper bound on the BER given by (34). From the simulation results, the analytical results are verified, and it is observed that the BER bound is tight at high SNRs.

4.1 Computational complexity analysis

The computational complexity of the channel estimation in each iteration for both LMS and ZA-LMS-\(l_1\) is in the order of \({\mathcal {O}} (2N_{t})\), while for the OMP is \({\mathcal {O}} (N_{t}^3)\), which is significantly higher in comparison to the proposed scheme. From Fig. 4, it can be observed that the rate of increase in the number of computations with input data size is more in OMP as compared to the traditional LMS and proposed ZA-LMS algorithm for the VLC-OTFS system. Further, it will be shown in the simulation results of Section VI that the ZA-LMS-\(l_1\) scheme provides better mean square deviation in comparison to the existing OMP technique, thus making ZA-LMS-\(l_1\) extremely well suited for the considered system.

Fig. 2
figure 2

Computational complexity of OMP and ZA-LMS for OTFS-VLC system

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5 Simulation results

In this section, we demonstrate the simulation results to illustrate the enhanced performance of the ZA-LMS-based channel estimator over the classical LMS-based channel estimator and OMP-based channel estimator for the dispersive OTFS-VLC system, with channel modeled by random way-point model. The system parameters for simulations are listed in Table 1. We have considered \(N_{s}= 512\) for simulations. The BPSK modulation scheme is used to modulate symbols mapped in DD domain. For channel-estimation, step-size \((\mu )\) is considered to be 0.005, regularization parameter \((\gamma )\) for ZA-LMS is \(5\times 10^{-8}\). After OTFS demodulation at the receiver, the ZA-LMS-based channel estimation algorithm is applied to estimate the CIR from the pilot symbols. Results are compared with the LMS and OMP estimator.

Table 1 Simulation Parameters
Full size table
Fig. 3
figure 3

mean square deviation performance for OTFS-VLC system at signal-to-noise ratio 30 dB

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In Fig. 2, the convergence performance of ZA-LMS, OMP, and LMS estimators is compared for signal-to-noise ratio of 50 dB. The convergence plot of ZA-LMS falls below both the OMP and the LMS on saturation, i.e., ZA-LMS has lower mean square deviation than OMP and LMS upon saturation. Thus, it can be inferred that for sparse OTFS- VLC systems ZA-LMS is a better alternative to the OMP and traditional LMS method.

Fig. 4
figure 4

BER performance for OTFS-VLC system

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Figure 3 presents the BER performance of OMP, LMS and ZA-LMS. OTFS with ZA-LMS and LMS-based channel estimator gives considerable gain compared to OMP-based receiver while ZA-LMS gives a gain of approximately 4 dB at BER of \(10^{-3}\). Thus, it can be concluded that the proposed ZA-LMS-based channel estimator is a better estimator as compared to the conventional techniques for exploiting the inherent sparsity of the OTFS-VLC system.

6 Conclusion

In this paper, ZA-LMS-based channel estimator is proposed for a VLC-OTFS system with the dispersive mobile multipath channel. Furthermore, it was observed from the simulations that due to the sparse nature of the VLC channel represented in the DD domain, ZA-LMS performed better than the traditional LMS and OMP algorithm. The simulated findings show that ZA-LMS is a more suitable low-complexity solution for channel estimation in the OTFS-VLC system and supports its deployment for communication systems beyond 5 G and 6 G.