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On BER analysis of nonlinear VLC systems under ambient light and imperfect/outdated CSI

Sandesh Jain, Rangeet Mitra, and Vimal Bhatia

Open Access Open Access

Abstract

Visible light communication (VLC) has emerged as a promising supplement to the existing radio frequency (RF) technology, which facilitates high speed communication for next generation communication systems. Although promising, the performance in a practical VLC system is severely impaired by ambient light interference. Furthermore, nonlinearity of light emitting diodes (LEDs) and imperfect/oudated channel state information (CSI) are other factors that degrade the achievable bit error rate (BER) performance. In this work, an analytical framework for analyzing the expression for BER of a wide class of modulation techniques is derived for an impaired VLC system considering: (a) ambient light interference, (b) LED nonlinearity, and (c) imperfect and outdated CSI. The derived analytical expressions for BER are validated through extensive computer simulations performed over typical VLC channels. Lastly, the utility of the derived analytical framework is demonstrated in the context of high data-rate VLC systems, wherein the performance of DC biased optical orthogonal frequency division multiplexing (DCO-OFDM) is quantified for VLC links impaired by the aforementioned degradations.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Visible light communication (VLC) has evolved as a promising supplement to existing RF technology in indoor scenarios [1–3]. The transmission of signals in a VLC system is achieved by modulating the intensity of LED luminaries at a speed invisible to the human eye, and photodetectors are used at the receiver for conversion of optical signal to electrical current. The various desirable features of VLC over RF based systems are as follows [4,5]: (1) unlicensed and large bandwidth of the order of terahertz (THz), which alleviates the spectrum-crunch faced by RF communication systems, (2) cost-effective since LED’s are already present in the existing infrastructure, which facilitates the two-fold goal of illumination and data transmission, (3) high security since VLC signals could not penetrate through walls, (4) free from electromagnetic interference, which makes VLC to be used safely in aircrafts, hospitals, and war sites, (5) high energy efficiency, and (6) high signal to noise ratio due to the illuminance of several hundred lux. Due to the aforementioned advantages, VLC’s are used in wide variety of applications such as light fidelity (Li-Fi) systems [4], vehicle to vehicle (V2V) communication, underwater communication [6], and internet of things (IoTs) based systems.

However, from recent studies, it has been found that the performance of VLC based systems severely degrades in the presence of interference due to ambient light [7,8], which reduces the overall signal to noise ratio (SNR) at the receiver. The work in [9] have designed a VLC receiver for mitigating ambient light interference. Authors in [7] have derived a closed form expression for mutual information for VLC systems in the presence of ambient light. However, the aforementioned works in [7,9] have not done bit error rate (BER) performance analysis for VLC systems under ambient light interference.

LED nonlinearity [10,11] is another factor which limits the throughput of the overall VLC system. Initial works on LED nonlinearity were based on characterization of LED nonlinearity models; prime among them are Rapp’s model [10], Volterra model, Wiener model, and Hammerstein model [12]. Further studies on VLC were focused towards design of various pre-distortion [13,14] and post-distortion algorithms [11,12,15] for mitigating LED nonlinearity [16]. Furthermore, there are various works in the literature based on BER performance analysis for VLC systems [11,17,18]. Authors in [17] perform a BER analysis for optical wireless systems impaired by LED nonlinearity. Next, the authors in [11] have proposed post-distortion algorithm for VLC, and also perform BER analysis for VLC systems impaired by LED nonlinearity. The authors in [18] have derived a closed form BER expression for VLC systems impaired by LED nonlinearity and DC-bias error.

Apart from ambient light and LED nonlinearity, imperfect channel state information (CSI) (attributed to estimation-error), and outdated CSI (attributed to user-mobility/shadowing effects) [19,20] are the other performance limiting factors, which results in degradation of the overall BER performance. Various models have been proposed in the literature for modeling user mobility in VLC systems. Authors in [21] have proposed a mobile VLC channel model using non-sequential ray tracing whilst considering different trajectories of a user talking on a cell phone in a room. Next, the authors in [22] have proposed a random waypoint model for VLC channel with user mobility, and the work in [23] have derived the statistical distribution for VLC channel with user mobility whilst considering the random orientation of the receiver. However, exact CSI or statistics of channel due to user mobility is not known at the receiver. Hence, CSI becomes outdated due to variations in channel gain due to user mobility. Furthermore, authors in [24] derive a closed form BER expression for VLC systems under imperfect CSI.

The VLC links are further degraded by intersymbol interference, which arises due to limited modulation bandwidth of LED (i.e., low pass response of LED) [25] particularly in the high data-rate regime. The resulting effect of finite modulation bandwidth can be mitigated by using DC biased optical orthogonal frequency division multiplexing (DCO-OFDM) scheme [26,27]. The other forms of OFDM for optical systems are: (a) asymmetrically clipped optical OFDM (ACO-OFDM), (b) ADO-OFDM (combination of DCO-OFDM, and ACO-OFDM) [27]. DCO-OFDM scheme is preferred over ACO-OFDM since it gives better spectral efficiency as compared to ACO-OFDM. There are several works in the literature based on BER performance of optical OFDM systems [26,28]. Authors in [26] have evaluated the BER performance for DCO-OFDM, and ACO-OFDM scheme for various clipping levels. Next, authors in [28] have derived the BER experssion for DCO-OFDM scheme over precoded massive MIMO channel.

There are various other works in the literature based on BER performance analysis for VLC systems. The work in [29] have done BER analysis for non orthogonal multiple access (NOMA) VLC systems using conventional modulation techniques like M-ary pulse amplitude modulation (M-PAM) and M-ary quadrature amplitude modulation (M-QAM) while the authors in [30,31] perform BER analysis for VLC systems using color shift keying (CSK) modulation scheme. Next, the authors in [32] have derived a closed form BER expression for CSK-VLC systems impaired by LED nonlinearity, and the work in [31] performs a BER analysis for VLC systems using circular CSK modulation scheme. Lastly, the authors in [33] have done BER performance analysis for hybrid power line communication (PLC) and VLC (PLC-VLC) systems, and the researchers in [34] have derived a closed form BER expression for VLC systems impaired by signal dependent noise.

However, to the best of author’s belief, none of the prior works reported so far have done the BER analysis of VLC systems impaired by ambient light, LED nonlinearity, and imperfect/outdated CSI together. Hence, in this work, for the first time, we attempt to derive the analytical BER for VLC systems under the aforementioned artifacts.

Contributions: Main contributions of this letter are summarized as follows: (1) The probability density function (PDF) of the overall additive distortion for a nonlinear impaired VLC link in the presence of ambient light interference under imperfect and outdated CSI is characterized, (2) An analytical expression for BER is quantified for VLC systems impaired by ambient light, and LED nonlinearity for perfect CSI scenario, (3) Using the derived PDF, an analytical expression for BER is quantified for VLC systems impaired by ambient light, LED nonlinearity, and imperfect CSI, (4) An analytical expression for BER is derived for VLC systems impaired by ambient light, LED nonlinearity, and outdated CSI, and (5) In the context of high data-rate VLC links under a finite modulation bandwidth, the BER performance of DCO-OFDM impaired by ambient light, LED nonlinearity, and imperfect/outdated CSI is also analyzed in this work.

The derived analytical expressions for BER are validated through Monte-Carlo simulations performed over realistic VLC channels impaired by imperfect and outdated CSI. A close overlap is observed between the analytical and simulated BER for various simulation parameters, which verifies the derived BER analysis in this paper.

Rest of the paper is structured as follows: The considered VLC system model is described in Section II. The analytical expressions for PDF for the overall additive distortion for a VLC link impaired by ambient light, LED nonlinearity, and imperfect/outdated CSI are derived in Section III. The proposed BER analysis for imperfect CSI and outdated CSI is given in Section IV. Simulation results are given in Section V for validating the presented BER analysis. Finally, conclusions are drawn in Section VI.

Notations: Following notations are used throughout the paper: Scalars are represented by simple lowercase characters. $\mathbb {R}$ denotes field of real numbers. $\mathbb {E}\{\cdot \}$ denotes the statistical expectation, and $\textrm {Pr}(\cdot )$ denotes probability of an event. A Gaussian distribution with zero mean, and variance $\sigma ^2$ is denoted as $\mathcal {N}(0, \sigma ^2)$ while a uniform random variable in the range $(a, b)$ is denoted by $U(a, b)$. $\tilde {\gamma }(\cdot , \cdot )$ denotes regularized incomplete gamma function, and $\tilde {\Gamma }(\cdot , \cdot )$ denotes regularized upper incomplete gamma function. An exponential integral function is denoted as $E_1(\cdot )$.

2. System model

In this section, the system model is described for a VLC system impaired by ambient light, LED nonlinearity, and imperfect/outdated CSI. In the first subsection, we describe the system model for single carrier scheme, and in the next subsection, we describe the system model for multiple carrier based DCO-OFDM scheme, which is suited for high data rate applications.

2.1 System model for VLC for single carrier scheme

Let $x \in \mathbb {R}$ be the transmitted symbol drawn from pulse amplitude modulation (PAM) constellation. Next, an appropriate DC bias is added to the transmitted signal for driving the LEDs in the forward biased operating region [35]. Biased symbols are then transmitted by the LED, and passed through the VLC channel. The VLC channel gain $h$ is given as [20,36]

$$h = \begin{cases} \frac{A_e(k+1)}{d^2 \sin^2(\Psi_c)} \cos^k(\phi) \cos(\psi), \quad 0 < \phi < \Psi_c \\ ~0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ~~~~ \textrm{otherwise}, \end{cases}$$
where $A_e$ denotes the effective area of the photodetector (PD), $d$ is the distance between the LED and the PD, $\phi$, and $\psi$ denote the angle of irradiance, angle of incidence, respectively, $\Psi _c$ denotes the field-of-view (FOV) for each photodetector, $k = -\frac {\ln (\cos (\phi _{\frac {1}{2}}))}{\ln (2)}$ is the order of Lambert emission, where $\phi _{\frac {1}{2}}$ is the LED semi angle at half illuminance. Furthermore, nonlinear characteristics of LED is modeled by Rapp’s model as follows [10,37]:
$$f(x) = \begin{cases} \frac{(x-V_{\textrm{th}})}{\Big(1+\big(\frac{(x-V_{\textrm{th}})}{V_{\textrm{max}}}\big)^{2p}\Big)^{\frac{1}{2p}}} & \quad x \geq V_{\textrm{th}}\\ 0 & \quad x < V_{\textrm{th}}, \end{cases}$$
where $V_{\textrm {max}}$, $V_{\textrm {th}}$, and $p$ is saturation voltage, threshold voltage of LED, and knee factor, respectively [10]. The received signal at the photodiode (PD) (denoted by $y$) can be written as
$$y = (h+\Delta)f(x) + \underbrace{\alpha n + \beta(w-k)}_{\textrm{overall noise}},$$
where $\hat {h} = h +\Delta$ is the estimate of $h$ known at the receiver, and $\Delta$ is the channel estimation error which can be modeled as follows [19,20]: (1) For imperfect CSI, $\Delta$ can be assumed as Gaussian distribution $\mathcal {N}(0, \sigma _{\Delta }^2)$ [19], and (2) For outdated CSI (which is caused by mobility/shadowing), $\Delta$ can be modeled as a uniform distribution $U(-\epsilon , \epsilon )$ [20], where $\epsilon$ is the variation range. The overall noise comprises of two components [38,39]: (1) The thermal noise $\alpha n \sim \mathcal {N}\{0, \alpha ^2\}$ is modeled as zero mean Gaussian random variable with variance $\alpha ^2$, and (2) $w$ is a Chi-squared distribution with $k$ degrees of freedom [7,40], which models the interference due to ambient light (which can be considered as polarized thermal light [40]), and $\beta$ denotes the attenuation in ambient light after blue filtering [25]. From Bussgang’s theorem (for decomposition of nonlinear functions $f(x) = \zeta x + v$) [41,42], and after DC-bias cancellation, (3) can be rewritten as
$$y = h \zeta x + \Delta v + \zeta \Delta x + hv + \alpha n + \beta(w-k),$$
where $\zeta = \frac {\mathbb {E}\{f(x)x\}}{\mathbb {E}\{x^2\}}$ is the scaling correlation coefficient between $x$ and $f(x)$ (where $\zeta \in [0, 1]$), and $v$ is the nonlinear distortion sequence uncorrelated with $x$ (such that $\mathbb {E}\{xv\} = 0$), which is modeled as Gaussian distribution $\mathcal {N}\{0, \sigma _v^2\}$ [18], where $\sigma _v^2 = \mathbb {E}\{f^2(x)\} - \zeta ^2\mathbb {E}\{x^2\}$ is the variance of $v$. Hence, the overall additive distortion $\tilde {n}$ can be written as
$$\tilde{n} = \Delta v + \zeta \Delta x + h v + \alpha n + \beta(w-k).$$

2.2 System model for multiple carrier DCO-OFDM scheme

Block diagram of the considered VLC system model for DCO-OFDM scheme is shown in Fig. 1. First, Hermitian symmetry is imposed on the data symbols. Let

$$ \mathbf{x} = [x_0~~x_1~~\ldots~x_{\frac{K}{2}-1}~~x_{\frac{K}{2}}~~ x^*_{\frac{K}{2}-1}~\ldots~x^*_1] $$
be the input of IFFT block, where $K$ is the number of subcarriers, and symbols $x_0$, and $x_{\frac {K}{2}}$ is set to $0$. Let $\tilde {\mathbf {x}}$ be the output of IFFT block. Next, cyclic prefix (CP) is added at the beginning of OFDM symbol to mitigate ISI (length of CP must be greater than maximum delay spread of the channel). ISI arises from low pass frequency response of the LED [43], which is modeled by a Lorentzian transfer function. The Lorentzian transfer function is given as follows [44]:
$$H(f) = \frac{1}{1+j2 \pi f \tau}$$
where $\tau$ is the carrier life time, and $\frac {1}{\tau }$ is the $3$ dB bandwidth of the LED. In this work, the Lorentzian frequency response of LED is modeled by a Butterworth low pass filter (LPF) with bandwidth of 20 MHz [25]. After adding the CP, symbols are transmitted through LED via VLC channel. The received symbol at the photodiode $\tilde {\mathbf {y}}$ can be written as:
$$\tilde{\mathbf{y}} = (\tilde{\mathbf{h}}+\mathbf{\delta}) \ast f(\tilde{\mathbf{x}}) + \alpha \mathbf{n} + \beta(\mathbf{w}-\mathbf{k})$$
where $\tilde {\mathbf {h}}$ is the convolution of Lorentzian LED low pass response, and VLC channel. $\alpha \mathbf {n} \sim \mathcal {N}\{0, \alpha ^2\mathbf {I}\}$ is the thermal noise, $\mathbf {w}$ is the ambient light noise, which is modeled as generalized Chi-squared distribution with $k$ degrees of freedom. The final output $\mathbf {y}$ (after removing CP, and performing FFT at the receiver, and using Bussgang’s theorem) can be written as [A4]:
$$\mathbf{y} = \tilde{\mathbf{H}} \zeta \hat{\mathbf{x}} + \mathbf{\Delta} \mathbf{v} + \zeta \mathbf{\Delta} \hat{x} + \tilde{\mathbf{H}}\mathbf{v} + \alpha \tilde{\mathbf{n}} + \beta(\tilde{\mathbf{w}}-\mathbf{\tilde{k}})$$
where $\tilde {\mathbf {H}}$ is a diagonal matrix whose diagonal elements are $N$ point FFT of $\tilde {\mathbf {h}}$, $\hat {\mathbf {x}}$, $\Delta$, $\tilde {\mathbf {n}}$, and $\tilde {\mathbf {w}}$ denotes FFT of $\tilde {\mathbf {x}}$, $\mathbf {\delta }$, $\mathbf {n}$, and $\mathbf {w}$, respectively.

 figure: Fig. 1.

Fig. 1. Block diagram of the system model for VLC for DCO-OFDM scheme

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In the next section, the PDF of the overall additive interference $\tilde {n}$ is characterized, and the derived expressions for PDF are utilized for BER analysis in further sections.

3. Analytical expressions for PDF of additive distortion

To derive the analytical BER, first the PDF of the overall additive distortion for a VLC system impaired by ambient light, LED nonlinearity, and imperfect/outdated CSI is characterized in this section.

3.1 PDF under imperfect CSI

Proposition 1: The overall additive distortion for a VLC link impaired by ambient light, LED nonlinearity, and imperfect CSI is characterized by the following distribution

$$p^{\textrm{ICSI}}_{\tilde{N}}(\tilde{n}) \approx \sum_{i = 1}^{m_1} \frac{\Theta_i}{\sqrt{\pi}} p\Bigg(\frac{\tilde{n}+k\beta - u_i \sqrt{2(\alpha^2 + \zeta^2 \sigma^2_{\Delta} + h^2\sigma^2_v + \sigma_1^2)}}{\beta}\Bigg),$$
where $p(x) = \frac {1}{2^{\frac {k}{2}}\Gamma (\frac {k}{2})} x^{\frac {k}{2}-1} e^{-\frac {x}{2}},~\forall x \in (0, \infty )$, $u_i$ is the root of $i^{\textrm {th}}$ physicists Hermite polynomial $H_i(x)$, $\Theta _i = \frac {i!2^{(i-1)}\sqrt {\pi }}{i^2 [H_{i-1}(u_i)]^2}$ is the weight of Hermite polynomial, and $\sigma _1^2 = \Big ( \frac {1}{\frac {1}{\sigma ^2_{\Delta }} + \frac {1}{\sigma ^2_{v}}}\Big )$ is the overall variance of $\Delta v$.

Proof: Please refer Appendix A.

Corollary of Proposition 1: The PDF of the the overall additive distortion for a VLC link for perfect CSI scenario under ambient light interference, and LED nonlinearity (denoted by $p^{\textrm {PCSI}}_{\tilde {N}}(\tilde {n})$) can be found out by substituting variance of channel estimation error $\sigma _{\Delta }^2 = 0$ and $\sigma _1^2 = 0$ in (9), and is given as:

$$p^{\textrm{PCSI}}_{\tilde{N}}(\tilde{n}) \approx \sum_{i = 1}^{m_1} \frac{\Theta_i}{\sqrt{\pi}} p\Bigg(\frac{\tilde{n}+k\beta - u_i \sqrt{2(\alpha^2 + h^2\sigma^2_v }}{\beta}\Bigg).$$

3.2 PDF under outdated CSI

Proposition 2: The PDF of the overall additive distortion for a VLC link impaired by ambient light, LED nonlinearity, and outdated CSI is given as

$$\begin{aligned}& p^{\textrm{OCSI}}_{\tilde{N}}(\tilde{n}) \approx \frac{1}{\sqrt{2}\pi \zeta A \epsilon} \sum_{i =1}^{m_1} \sum_{j=1}^{m_1} \sum_{n=1}^{N} \sum_{m =1}^{M} \Theta_i \Theta_j a_n a_m b_m^{-\frac{1}{2}}\\ & \times \Bigg[\tilde{\Gamma}\Bigg(\frac{k}{2}, \frac{\tilde{n} - \frac{\epsilon \sigma_v}{\sqrt{2 b_n b_m}}v_j - \zeta A \epsilon + k\beta - u_i \sqrt{2(\alpha^2 + h^2\sigma^2_v )}}{2 \beta}\Bigg)\\ & - \tilde{\Gamma}\Bigg(\frac{k}{2}, \frac{\tilde{n} -\frac{\epsilon \sigma_v}{\sqrt{2 b_n b_m}}v_j + \zeta A \epsilon + k\beta - u_i \sqrt{2(\alpha^2 + h^2\sigma^2_v )}}{2 \beta}\Bigg)\Bigg], \end{aligned}$$
where $\tilde {\Gamma }(a, z) = \frac {1}{\Gamma (a)} \int _{z}^{\infty } t^{(a-1)} e^{-t} dt$ denotes the regularized upper incomplete Gamma function, $\Theta _j$, and $v_j$ are weights, and roots of the Hermite polynomial, respectively, and $\{a_n, b_n\}_{n=1}^{N}$ and $\{a_m, b_m\}_{m=1}^{M}$ are constants used for approximation of an exponential integral function as in [45].

Proof: Please refer Appendix B.

4. Analytical expression for BER

Using the aforementioned derived PDF of the overall additive distortion for imperfect and outdated CSI scenarios, we proceed to derive an analytical expression of BER for generalized M-ary PAM constellation, i.e. $x \in \{\pm (2l+1)A\}_{l=0}^{l = (\frac {M}{2}-1)}$. Notably, the presented BER analysis can be readily generalized for multilevel modulations like M-ary carrieless amplitude and phase quadrature amplitude modulation (M-CAP-QAM) [46].

4.1 BER for imperfect CSI

Theorem 1: The analytical expression of BER for a VLC system under ambient light interference, LED nonlinearity, and imperfect CSI for M-PAM scheme is given as (denoted by $P_e^{\textrm {ICSI}}$)

$$\begin{aligned}P_e^{\textrm{ICSI}} & \approx \Big(1-\frac{1}{M}\Big)\sum_{i = 1}^{m_1} \frac{\Theta_i}{\sqrt{\pi}} \Bigg[ \tilde{\gamma}\Bigg(\frac{k}{2}, \frac{-A + k\beta - u_i \sqrt{2(\alpha^2 + \zeta^2 \sigma^2_{\Delta} + h^2\sigma^2_v + \sigma_1^2)}}{2 \beta}\Bigg)\\ & + \beta \tilde{\Gamma}\Bigg(\frac{k}{2}, \frac{A + k\beta - u_i \sqrt{2(\alpha^2 + \zeta^2 \sigma^2_{\Delta} + h^2\sigma^2_v + \sigma_1^2)}}{2 \beta}\Bigg)\Bigg], \end{aligned}$$
where $\tilde {\gamma }(a, z) = \frac {1}{\Gamma (a)} \int _{0}^{x} t^{a-1} e^{-t} dt$ is the regularized lower incomplete gamma function, and $A = \sqrt {\frac {6E_s}{M^2-1}}$ (where $E_s = h^2 \zeta ^2 \mathbb {E}\{x^2\}$ is the overall energy of signal).

Proof: The overall BER for M-PAM scheme can be written as [46]

$$P_e^{\textrm{ICSI}} \approx \Bigg(1-\frac{1}{M}\Bigg)\big( \underbrace{\textrm{Pr}(\tilde{n} < -A)}_{\mathscr{I}_1} + \underbrace{\textrm{Pr}(\tilde{n} > A)}_{\mathscr{I}_2}\big)$$
The integrals $\mathscr {I}_1$, and $\mathscr {I}_2$ need to be evaluated separately since the PDF of the overall additive distortion $p^{\textrm {ICSI}}_{\tilde {N}}(\tilde {n})$ is asymmetric (as opposed to classical BER analysis over AWGN channels wherein symmetry of the Gaussian distribution is exploited). Using (9), the integrals $\mathscr {I}_1$, and $\mathscr {I}_2$ can be computed as
$$\mathscr{I}_1 \approx \textrm{Pr}(\tilde{n} < -A) \approx \sum_{i = 1}^{m_1} \frac{\Theta_i}{\sqrt{\pi}} \frac{1}{2^{\frac{k}{2}}\Gamma(\frac{k}{2})} \Bigg[\int_{-\infty}^{-A} \Big(\frac{\tilde{n}+ z(u_i)}{\beta}\Big)^{\frac{k}{2}-1} \exp \Bigg(-\frac{\tilde{n}+ z(u_i)}{2\beta}\Bigg)\Bigg] d \tilde{n},$$
$$\mathscr{I}_2 \approx \textrm{Pr}(\tilde{n} > A) \approx \sum_{i = 1}^{m_1} \frac{\Theta_i}{\sqrt{\pi}} \frac{1}{2^{\frac{k}{2}}\Gamma(\frac{k}{2})} \Bigg[\int_{A}^{\infty} \Big(\frac{\tilde{n}+ z(u_i)}{\beta}\Big)^{\frac{k}{2}-1} \exp \Bigg(-\frac{\tilde{n}+ z(u_i)}{2\beta}\Bigg)\Bigg] d \tilde{n}$$
where $z(u_i) = k\beta - u_i \sqrt {2(\alpha ^2 + \zeta ^2 \sigma ^2_{\Delta } + h^2\sigma ^2_v + \sigma _1^2)}$. After substituting $t = \frac {\tilde {n}+ z(u_i)}{2\beta }$ in (14) and (15), and using the mathematical definition of lower and upper incomplete gamma function [43], we get the final expression of BER for imperfect CSI as given in (12).

Corollary of Theorem 1: The analytical expression of BER for a VLC system impaired by ambient light interference, and LED nonlinearity for perfect CSI scenario (denoted by $P_e^{\textrm {PCSI}}$) can be determined by substituting variance of channel estimation error $\sigma _{\Delta }^2 = 0$ and $\sigma _1^2 = 0$ in (12), which can be written as

$$P_e^{\textrm{PCSI}} \approx \Big(1-\frac{1}{M}\Big) \sum_{i = 1}^{m_1} \frac{\Theta_i}{\sqrt{\pi}} \Bigg[ \tilde{\gamma}\Bigg(\frac{k}{2}, \frac{-A + k\beta - u_i \sqrt{2(\alpha^2 + h^2 \sigma^2_v)}}{2 \beta}\Bigg) + \beta \tilde{\Gamma}\Bigg(\frac{k}{2}, \frac{A + k\beta - u_i \sqrt{2(\alpha^2 + h^2 \sigma^2_v )}}{2 \beta}\Bigg)\Bigg]$$

4.2 BER for outdated CSI

Theorem 2: The analytical expression of BER for a VLC system impaired by ambient light, LED nonlinearity, and outdated CSI for M-PAM scheme is given as (denoted by $P_e^{\textrm {OCSI}}$)

$$\begin{aligned}& P_e^{\textrm{OCSI}} \approx \Bigg(1-\frac{1}{M}\Bigg) \Bigg\{ \frac{1}{2 \sqrt{2}\pi \zeta A \epsilon} \sum_{i =1}^{m_1} \sum_{j=1}^{m_1} \sum_{n=1}^{N} \sum_{m =1}^{M} 2 \beta \Theta_i \Theta_j a_n a_m b_m^{-\frac{1}{2}} \times\\ & \Bigg[\tilde{\Gamma}_1\Bigg(\frac{k}{2}, \frac{-A - \zeta A \epsilon + c(u_i, v_j)}{2 \beta}\Bigg) - \tilde{\Gamma}_1\Bigg(\frac{k}{2}, \frac{-A + \zeta A \epsilon + c(u_i, v_j)}{2 \beta}\Bigg) +1\\ & -\tilde{\Gamma}_1\Bigg(\frac{k}{2}, \frac{A - \zeta A \epsilon + c(u_i, v_j)}{2 \beta}\Bigg) + \tilde{\Gamma}_1\Bigg(\frac{k}{2}, \frac{A + \zeta A \epsilon + c(u_i, v_j)}{2 \beta}\Bigg)\Bigg]\Bigg\}, \end{aligned}$$
where $c(u_i, v_j) = - \frac {\epsilon \sigma _v}{\sqrt {2 b_n b_m}}v_j + k\beta - u_i \sqrt {2(\alpha ^2 + h^2\sigma ^2_v )}$, and $\tilde {\Gamma }_1(\frac {k}{2}, x) = x \tilde {\Gamma }(\frac {k}{2}, x) - \tilde {\Gamma }(\frac {k}{2}+1, x)$.

Proof: The integration of regularized upper incomplete gamma function can be written as [47]

$$\tilde{\Gamma}_1(\frac{k}{2}, x) = \frac{x \Gamma(\frac{k}{2}, x) - \Gamma(\frac{k}{2}+1, x)}{\Gamma(\frac{k}{2})}.$$
Using (18), the overall CDF of $\tilde {n}$ (denoted by $P^{\textrm {OCSI}}_{\tilde {N}}(\tilde {n}))$) for outdated CSI can be written as
$$\begin{aligned}& P^{\textrm{OCSI}}_{\tilde{N}}(\tilde{n})) \approx \frac{1}{\sqrt{2}\pi \zeta A \epsilon} \sum_{i =1}^{m_1} \sum_{j=1}^{m_1} \sum_{n=1}^{N+1} \sum_{m =1}^{M+1} 2 \beta \Theta_i \Theta_j a_n a_m b_m^{-\frac{1}{2}}\\ & \times \Bigg[\tilde{\Gamma}_1\Bigg(\frac{k}{2}, \frac{\tilde{n} - \zeta A \epsilon + c(u_i, v_j)}{2 \beta}\Bigg) - \tilde{\Gamma}_1\Bigg(\frac{k}{2}, \frac{\tilde{n} + \zeta A \epsilon + c_{i,j}}{2 \beta}\Bigg) \Bigg]. \end{aligned}$$
Hence, using $P_e^{\textrm {OCSI}} \approx \Big (1-\frac {1}{M}\Big )\big ( \textrm {Pr}(\tilde {n} < -A) + \textrm {Pr}(\tilde {n} > A)\big ) \approx \Big (1-\frac {1}{M}\Big ) [P^{\textrm {OCSI}}_{\tilde {N}}(-A) + 1 - P^{\textrm {OCSI}}_{\tilde {N}}(A)]$ [46], we get the final expression of BER for outdated CSI scenario as given in (17). The presented BER analysis for M-PAM scheme can be readily generalized for any arbitrary digital modulations scheme since the integrals involved in computation of BER ($\mathscr {I}_1$, and $\mathscr {I}_2$) are generic independent of the modulation schemes.

4.3 BER for DCO-OFDM

In this section, an analytical expression of BER for DCO-OFDM scheme is derived. After removal of CP and performing FFT at the receiver, the overall channel consists of parallel flat-fading channels in the frequency domain (i.e., the channel coefficient is channel frequency response (CFR) over each subcarrier). Hence, analytical expression of BER for multicarrier DCO-OFDM scheme for imperfect, and outdated CSI can be easily deduced from the BER expressions derived for single-carrier ones in (12), and (17), respectively. Let us denote BER for imperfect CSI, and outdated CSI for single carrier by $P_e^{\textrm {ICSI}}(A^{(i)}, E^{(i)}_s)$, and $P_e^{\textrm {OCSI}}(A^{(i)}, E^{(i)}_s)$. The overall average BER for multicarrier DCO OFDM scheme for imperfect, and outdated CSI can be written as

$$P_e^{\textrm{ICSI, DCO-OFDM}} = \frac{1}{K} \sum_{i=1}^{i=K} P_e^{\textrm{ICSI}}(A^{(i)}, E^{(i)}_s)$$
$$P_e^{\textrm{OCSI, DCO-OFDM}} = \frac{1}{K} \sum_{i=1}^{i=K} P_e^{\textrm{OCSI}}(A^{(i)}, E^{(i)}_s)$$
where $K$ is the total number of subcarriers, $A^{(i)} = \sqrt {\frac {6 E^{i}_s}{M^2-1}}$, where $E^{i}_s = |\tilde {\mathbf {H}}^{(i)}|^2 \zeta ^2 \mathbb {E}\{|\hat {\mathbf {x}_i}|^2\}$ denotes the signal energy for $i^{\textrm {th}}$ subcarrier.

5. Simulations

In this section, we present computer simulations for validating the presented BER analysis for outdated/imperfect CSI in the presence of interference due to ambient light, and LED nonlinearity. Without loss of generality, the number of terms $m_1$ in the Gauss-Hermite approximation is taken as $30$, number of terms $N$ and $M$ in the approximation of an exponential integral function in (30) is considered as $2$, and the modulation is assumed to be 2-PAM/OOK. $10^{5}$ symbols are considered over an ensemble of $200$ independent Monte-Carlo simulations. Simulation parameters for VLC channel, and Rapp’s LED nonlinearity are listed in Table 1 [10,36]. In all the plots, single legend ("black circular markers without lines") are used to denote analytical BER.

Tables Icon

Table 1. Simulation parameters

View Table

Signal to thermal noise ratio ($\frac {A^2}{\alpha ^2}$) vs BER plots are shown in Fig. 2, and Fig. 3 for perfect CSI for $\phi _{\frac {1}{2}} = 40^{\circ }$, and $\phi _{\frac {1}{2}} = 60^{\circ }$, respectively, for different values of LED nonlinearity parameter $p$, and ambient light parameters $\{\beta , k\}$. It can be observed from Fig. 2 and Fig. 3 that the derived analytical BER for perfect CSI closely overlaps with the simulated BER plots for various ambient light parameter values $\{\beta , k\}$. It can be inferred from Fig. 2 and Fig. 3 that the BER performance degrades with increase in severity of LED nonlinearity parameter (i.e., $p = 1$ is more severe than $p = 2$ [10]), which is intuitive from the fact that the signal power reduces by a factor of $\zeta ^2$, and noise power increases by a factor of $\sigma _v^2$ due to LED nonlinearity. Furthermore, it can be observed from Fig. 2 and Fig. 3 that the BER performance degrades with the increase in LED semi-angle $\phi _{\frac {1}{2}}$ which is intuitive from the fact that channel gain $h$ decreases with the increase in LED semi-angle $\phi _{\frac {1}{2}}$, thereby reducing the overall SNR at the receiver.

 figure: Fig. 2.

Fig. 2. BER performance for perfect CSI with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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 figure: Fig. 3.

Fig. 3. BER performance for perfect CSI with $\phi _{\frac {1}{2}} = 60^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Next, signal to thermal noise ratio ($\frac {A^2}{\alpha ^2}$) vs BER plots are shown in Fig. 4, and Fig. 5 for imperfect CSI for $\phi _{\frac {1}{2}} = 40^{\circ }$, and $\phi _{\frac {1}{2}} = 60^{\circ }$, respectively, for different values of LED nonlinearity parameter $p$, and ambient light parameters $\{\beta , k\}$. It can be observed from Fig. 3, and Fig. 4 that the derived approximate expression for BER well-agrees with the simulated BER plots for various parameter values $\{\sigma ^2_{\Delta }, \beta \}$ for imperfect CSI, which establishes the accuracy of the presented analysis. Furthermore, it can be observed from Fig. 4 (left) and Fig. 5 (left) that the BER increases with the increase in channel estimation error $\sigma ^2_{\Delta }$, and ambient light parameter $\beta$, which is intuitive. However, it can be observed from Fig. 4 (right), and Fig. 5 (right) that the BER performance improves with the increase in channel estimation error $\sigma ^2_{\Delta }$, which is counter intuitive. This can be explained from the fact that the BER plot with the highest $\sigma ^2_{\Delta }$ is also the plot with the lowest $\beta$ (i.e. the BER is majorly controlled by ambient light parameter $\beta$ in this scenario).

 figure: Fig. 4.

Fig. 4. BER performance for imperfect CSI with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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 figure: Fig. 5.

Fig. 5. BER performance for imperfect CSI with $\phi _{\frac {1}{2}} = 60^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Next, signal to thermal noise ratio ($\frac {A^2}{\alpha ^2}$) vs BER plots are shown in Fig. 6, and Fig. 7 for outdated CSI for $\phi _{\frac {1}{2}} = 40^{\circ }$, and $\phi _{\frac {1}{2}} = 60^{\circ }$, respectively, for different simulation parameters $\{p, \epsilon , \beta , k\}$. It can be observed from Fig. 5, and Fig. 6 that the analytical BER plots closely overlaps with the simulated BER plots for various parameter values $\{\epsilon , \beta \}$ for outdated CSI, which justifies the derived BER expression for outdated CSI. Furthermore, it can be observed from Fig. 6 (left) and Fig. 7 (left) that the BER performance degrades with the increase in parameter values $\{\epsilon , \beta \}$, which is intuitive. However, similar to imperfect CSI scenario, some counter intuitive artifacts can be observed from Fig. 6 (right), and Fig. 7 (right) for outdated CSI that the BER decreases with the increase in outdated CSI parameter $\epsilon$. This can be justified from the fact that the BER plot with the highest $\epsilon$ is also the plot with the lowest $\beta$ (i.e. the BER is majorly controlled by ambient light parameter $\beta$ in this scenario). It is noteworthy that, in this case, we consider a low data rate scenario (of the order of few tens of MHz) for OOK wherein the effects of finite modulation bandwidth are negligible.

 figure: Fig. 6.

Fig. 6. BER performance for outdated CSI with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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 figure: Fig. 7.

Fig. 7. BER performance for outdated CSI with $\phi _{\frac {1}{2}} = 60^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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However, in the high data-rate regime, the effects of the Lorentzian transfer function are more pronounced. For achieving high data rates, the performance of DCO-OFDM in these scenarios is a viable solution. Therefore, to corroborate the analytical results obtained in (20) and (21), simulations studies are presented next for DCO-OFDM over VLC channels impaired by ambient interference, LED nonlinearity, and imperfect/outdated CSI. Simulation parameters for DCO-OFDM are chosen as follows: Number of data subcarriers $K = 720$, FFT size of $1024$, number of symbols per OFDM frame are chosen as $200$, total number of frames are chosen as $50$, and cyclic prefix of length $64$ (FFT size/16) is chosen. Signal to thermal noise ratio ($\frac {A^2}{\alpha ^2}$) vs BER performance for DCO-OFDM scheme are shown in Fig. 8 for imperfect CSI and outdated CSI scenarios. It can be observed from Fig. 8 that BER degrades with the increase in ambient light parameter $\beta$, and channel estimation error ($\sigma ^2_{\Delta }$ for imperfect CSI, and $\epsilon$ for outdated CSI). Furthermore, it can also be inferred from Fig. 8 that the simulated BER matches with the corresponding analytical BER, which validates the accuracy of the derived expressions for BER for DCO-OFDM scheme. For $K=720$, $M=2$ (for OOK), and $T_s = 1 \mu s$, the achieved data rate for VLC systems using DCO-OFDM scheme is $360~\textrm {Mbps}$.

 figure: Fig. 8.

Fig. 8. SNR vs BER performance for DCO-OFDM for imperfect (left), and outdated CSI (right) with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =1$, and $k = 1$.

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Lastly, BER performance for the considered impaired VLC link is monitored by varying the transmit data rate. Data rate vs BER plots are shown in Fig. 9 at an SNR of $30$ dB for imperfect, and outdated CSI for different values of ambient light parameter $\beta$, and $\sigma _{\Delta }^2$ (for imperfect CSI), and $\epsilon$ (for outdated CSI). It can be observed from Fig. 9 that a date rate of $1.08$ Gbps is achieved using DCO-OFDM scheme with BER of approx $10^{-4}$ for lower ambient light variance $\beta = 0.015$. As inferred from Fig. 9, data rate and BER degrades with the increase in ambient light variance.

 figure: Fig. 9.

Fig. 9. Data rate vs BER performance for DCO-OFDM for imperfect (left), and outdated CSI (right) with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =1$, and $k = 1$.

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6. Conclusion

In this paper, an analytical expression for BER is quantified for a VLC system in the presence of ambient light interference, LED nonlinearity, and imperfect/outdated CSI. The derived expressions for BER are validated through computer simulations performed over practical VLC channels, which indicate that the analytical BER closely overlaps with the simulated BER, which ratifies the presented analysis, and makes the proposed analysis viable for practical VLC system design.

Appendix A: proof of proposition 1

From (5), the PDF of $\tilde {n}$ is a convolution of scaled and shifted chi-squared distribution with $k$ degrees of freedom, and a zero mean Gaussian distribution with variance $\alpha ^2 + \zeta ^2 \sigma ^2_{\Delta } + h^2\sigma ^2_v + \sigma _1^2$. Hence, the PDF of $\tilde {n}$ (denoted by $p^{\textrm {ICSI}}_{\tilde {N}}(\tilde {n})$) for imperfect CSI can be written as

$$p^{\textrm{ICSI}}_{\tilde{N}}(\tilde{n}) = \frac{1}{\sqrt{2 \pi (\alpha^2 + \zeta^2 \sigma^2_{\Delta} + h^2\sigma^2_v + \sigma_1^2)}} \int_{-\infty}^{\infty} p\Big(\frac{\tilde{n} + k\beta-x}{\beta}\Big) \exp\Bigg(-\frac{x^2}{2(\alpha^2 + \zeta^2 \sigma^2_{\Delta} + h^2\sigma^2_v + \sigma_1^2)}\Bigg) dx.$$
After substituting $t = \frac {x}{\sqrt {2(\alpha ^2 + \zeta ^2 \sigma ^2_{\Delta } + h^2\sigma ^2_v + \sigma _1^2)}}$, the integral in (23) reduces to
$$p^{\textrm{ICSI}}_{\tilde{N}}(\tilde{n}) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} p\Bigg(\frac{\tilde{n} + k\beta-t\sqrt{2(\alpha^2 + \zeta^2 \sigma^2_{\Delta} + h^2\sigma^2_v + \sigma_1^2)}}{\beta}\Bigg) \exp(-t^2) dt.$$
Using Gauss-Hermite rule $\int _{-\infty }^{\infty } f(x) e^{-x^2} = \sum _{i =1}^{m_1} \Theta _1 f(u_i)$ [47], the above integral in (23) can be solved, and we get the final expression for PDF under imperfect CSI as given in (9).

Appendix B: proof of proposition 2

For outdated CSI, we can write (5) as: $\tilde {n} = \psi _1 + \zeta \Delta x + \Delta v$. Hence, PDF of the overall additive distortion $\tilde {n}$ for outdated CSI is the convolution of following three components: (a) $\Delta _1 = \zeta \Delta x \in U(-\zeta A\epsilon , \zeta A\epsilon )$, which is a uniform random variable, (b) $\psi _1 = hv + \alpha n + \beta (w-k)$, which is the sum of Gaussian random variable $\mathcal {N}(0, \alpha ^2 + h^2 \sigma _v^2)$, and scaled and shifted chi-square distribution, and (c) $\Delta v$, which is the product of uniform and Gaussian random variable. From (9), the PDF of $\psi _1$ (denoted by $p_1(\psi _1)$) can be written as:

$$p_1(\psi_1) \approx \frac{1}{\sqrt{\pi}} \sum_{i = 1}^{m_1} \Theta_i p\Bigg(\frac{\psi_1 + k\beta - u_i \sqrt{2(\alpha^2 + h^2\sigma^2_v)}}{\beta}\Bigg).$$
Hence, the PDF of $\psi _2 = \psi _1 + \Delta _1$, denoted by $p_2(\psi _2)$ can be written as
$$p(\psi_2) = \frac{1}{2\zeta A \epsilon} \int_{-\zeta A \epsilon}^{\zeta A \epsilon} p_2(\psi_2 -\Delta_1) d \Delta_1.$$
After solving the above integral, (25) reduces to
$$\begin{aligned}p_2(\psi_2) & \approx \frac{1}{2\zeta A \epsilon} \sum_{i = 1}^{m_1} \frac{\Theta_i}{\sqrt{\pi}} \Bigg[ \tilde{\Gamma}\Bigg(\frac{k}{2}, \frac{\psi_2 - \zeta A \epsilon + k\beta - u_i \sqrt{2(\alpha^2 + h^2\sigma^2_v )}}{2 \beta}\Bigg)\\ & - \tilde{\Gamma}\Bigg(\frac{k}{2}, \frac{\psi_2 + \zeta A \epsilon + k\beta - u_i \sqrt{2(\alpha^2 + h^2\sigma^2_v )}}{2 \beta}\Bigg)\Bigg]. \end{aligned}$$

Next, the overall PDF of the product distribution $z = \Delta v$, denoted as $g(z)$ (i.e., the product of uniform and Gaussian random variable) can be written as [48]

$$g(z) = \frac{1}{2 \epsilon \sqrt{2\pi \sigma^2_v}}\int_{-\epsilon}^{\epsilon} \frac{1}{|\Delta|} \exp\Bigg(-\frac{(\frac{z}{\Delta})^2}{2 \sigma^2_v}\Bigg) d \Delta.$$
After substituting $t = \frac {z^2}{2 \sigma _v^2} \Delta ^{-2}$, and using $\int _{-a}^{a} f(x) dx = 2\int _{0}^{a} f(x) dx$ for even function $f(x)$, (27) reduces to
$$g(z) = \frac{1}{2 \epsilon \sqrt{2\pi \sigma^2_v}}\int_{\frac{z^2}{2 \epsilon^2 \sigma_v^2}}^{\infty} t^{-1} e^{-t} dt = \frac{1}{2 \epsilon \sqrt{2\pi \sigma^2_v}} E_1\Bigg(\frac{z^2}{2 \epsilon^2 \sigma_v^2}\Bigg),$$
where $E_1(x) = \int _{x}^{\infty } \frac {e^{-t}}{t} dt$ denotes the exponential integral function. Therefore, the PDF of the overall additive distortion $\tilde {n} = \psi _2 + z$ (denoted by $p^{\textrm {OCSI}}_{\tilde {N}}(\tilde {n})$) can be written as convolution of $p_2(\psi _2)$ and $g(z)$ as
$$p^{\textrm{OCSI}}_{\tilde{N}}(\tilde{n}) = \int_{-\infty}^{\infty} p_2(\tilde{n}-\tau) g(\tau) d \tau.$$
From [45, eq. (7)], an exponential integral function can be approximated using a sum of exponentials as
$$E_1(x) \approx 4\sqrt{2} \pi \sum_{n=1}^{N} \sum_{m =1}^{M} a_n a_m \sqrt{b_n} e^{-4 b_n b_m x},$$
where $\{a_n, b_n\}_{n=1}^{N}$ and $\{a_m, b_m\}_{m=1}^{M}$ are constants as given in [45]. Using (30), the integral in (29) can be written as
$$p^{\textrm{OCSI}}_{\tilde{N}}(\tilde{n}) \approx \frac{1}{2 \epsilon \sqrt{2\pi \sigma^2_v}} 4\sqrt{2} \pi \sum_{n=1}^{N} \sum_{m =1}^{M} a_n a_m \sqrt{b_n} \times \int_{-\infty}^{\infty} p_2(\tilde{n}-\tau) \exp\Bigg(-\frac{2 b_n b_m}{ \epsilon^2 \sigma^2_v} \tau^2 \Bigg) d \tau.$$
After substituting $t = \frac {\sqrt {2 b_n b_m }}{\epsilon \sigma _v}\tau$ in (31), and using Gauss-Hermite integration rule, we get the final expression for $p^{\textrm {OCSI}}_{\tilde {N}}(\tilde {n})$ as given in (11).

Acknowledgement

This publication is an outcome of the R&D work undertaken project under the Visvesvaraya PhD Scheme of Ministry of Electronics & Information Technology, Government of India, being implemented by Digital India Corporation.

Disclosures

The authors declare no conflicts of interest.

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Figures (9)
Fig. 1.
Fig. 1. Block diagram of the system model for VLC for DCO-OFDM scheme

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Fig. 2.
Fig. 2. BER performance for perfect CSI with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Fig. 3.
Fig. 3. BER performance for perfect CSI with $\phi _{\frac {1}{2}} = 60^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Fig. 4.
Fig. 4. BER performance for imperfect CSI with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Fig. 5.
Fig. 5. BER performance for imperfect CSI with $\phi _{\frac {1}{2}} = 60^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Fig. 6.
Fig. 6. BER performance for outdated CSI with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Fig. 7.
Fig. 7. BER performance for outdated CSI with $\phi _{\frac {1}{2}} = 60^{\circ }$, $p =2$, $k = 1$ (left), and $p =1$, $k = 3$ (right). (S) denotes the simulated BER.

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Fig. 8.
Fig. 8. SNR vs BER performance for DCO-OFDM for imperfect (left), and outdated CSI (right) with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =1$, and $k = 1$.

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Fig. 9.
Fig. 9. Data rate vs BER performance for DCO-OFDM for imperfect (left), and outdated CSI (right) with $\phi _{\frac {1}{2}} = 40^{\circ }$, $p =1$, and $k = 1$.

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Tables (1)

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Table 1. Simulation parameters

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Equations (32)

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h = { A e ( k + 1 ) d 2 sin 2 ⁡ ( Ψ c ) cos k ⁡ ( ϕ ) cos ⁡ ( ψ ) , 0 < ϕ < Ψ c   0         otherwise ,
f ( x ) = { ( x − V th ) ( 1 + ( ( x − V th ) V max ) 2 p ) 1 2 p x ≥ V th 0 x < V th ,
y = ( h + Δ ) f ( x ) + α n + β ( w − k ) ⏟ overall noise ,
y = h ζ x + Δ v + ζ Δ x + h v + α n + β ( w − k ) ,
n ~ = Δ v + ζ Δ x + h v + α n + β ( w − k ) .
x = [ x 0     x 1     …   x K 2 − 1     x K 2     x K 2 − 1 ∗   …   x 1 ∗ ]
H ( f ) = 1 1 + j 2 π f τ
y ~ = ( h ~ + δ ) ∗ f ( x ~ ) + α n + β ( w − k )
y = H ~ ζ x ^ + Δ v + ζ Δ x ^ + H ~ v + α n ~ + β ( w ~ − k ~ )
p N ~ ICSI ( n ~ ) ≈ ∑ i = 1 m 1 Θ i π p ( n ~ + k β − u i 2 ( α 2 + ζ 2 σ Δ 2 + h 2 σ v 2 + σ 1 2 ) β ) ,
p N ~ PCSI ( n ~ ) ≈ ∑ i = 1 m 1 Θ i π p ( n ~ + k β − u i 2 ( α 2 + h 2 σ v 2 β ) .
p N ~ OCSI ( n ~ ) ≈ 1 2 π ζ A ϵ ∑ i = 1 m 1 ∑ j = 1 m 1 ∑ n = 1 N ∑ m = 1 M Θ i Θ j a n a m b m − 1 2 × [ Γ ~ ( k 2 , n ~ − ϵ σ v 2 b n b m v j − ζ A ϵ + k β − u i 2 ( α 2 + h 2 σ v 2 ) 2 β ) − Γ ~ ( k 2 , n ~ − ϵ σ v 2 b n b m v j + ζ A ϵ + k β − u i 2 ( α 2 + h 2 σ v 2 ) 2 β ) ] ,
P e ICSI ≈ ( 1 − 1 M ) ∑ i = 1 m 1 Θ i π [ γ ~ ( k 2 , − A + k β − u i 2 ( α 2 + ζ 2 σ Δ 2 + h 2 σ v 2 + σ 1 2 ) 2 β ) + β Γ ~ ( k 2 , A + k β − u i 2 ( α 2 + ζ 2 σ Δ 2 + h 2 σ v 2 + σ 1 2 ) 2 β ) ] ,
P e ICSI ≈ ( 1 − 1 M ) ( Pr ( n ~ < − A ) ⏟ I 1 + Pr ( n ~ > A ) ⏟ I 2 )
I 1 ≈ Pr ( n ~ < − A ) ≈ ∑ i = 1 m 1 Θ i π 1 2 k 2 Γ ( k 2 ) [ ∫ − ∞ − A ( n ~ + z ( u i ) β ) k 2 − 1 exp ⁡ ( − n ~ + z ( u i ) 2 β ) ] d n ~ ,
I 2 ≈ Pr ( n ~ > A ) ≈ ∑ i = 1 m 1 Θ i π 1 2 k 2 Γ ( k 2 ) [ ∫ A ∞ ( n ~ + z ( u i ) β ) k 2 − 1 exp ⁡ ( − n ~ + z ( u i ) 2 β ) ] d n ~
P e PCSI ≈ ( 1 − 1 M ) ∑ i = 1 m 1 Θ i π [ γ ~ ( k 2 , − A + k β − u i 2 ( α 2 + h 2 σ v 2 ) 2 β ) + β Γ ~ ( k 2 , A + k β − u i 2 ( α 2 + h 2 σ v 2 ) 2 β ) ]
P e OCSI ≈ ( 1 − 1 M ) { 1 2 2 π ζ A ϵ ∑ i = 1 m 1 ∑ j = 1 m 1 ∑ n = 1 N ∑ m = 1 M 2 β Θ i Θ j a n a m b m − 1 2 × [ Γ ~ 1 ( k 2 , − A − ζ A ϵ + c ( u i , v j ) 2 β ) − Γ ~ 1 ( k 2 , − A + ζ A ϵ + c ( u i , v j ) 2 β ) + 1 − Γ ~ 1 ( k 2 , A − ζ A ϵ + c ( u i , v j ) 2 β ) + Γ ~ 1 ( k 2 , A + ζ A ϵ + c ( u i , v j ) 2 β ) ] } ,
Γ ~ 1 ( k 2 , x ) = x Γ ( k 2 , x ) − Γ ( k 2 + 1 , x ) Γ ( k 2 ) .
P N ~ OCSI ( n ~ ) ) ≈ 1 2 π ζ A ϵ ∑ i = 1 m 1 ∑ j = 1 m 1 ∑ n = 1 N + 1 ∑ m = 1 M + 1 2 β Θ i Θ j a n a m b m − 1 2 × [ Γ ~ 1 ( k 2 , n ~ − ζ A ϵ + c ( u i , v j ) 2 β ) − Γ ~ 1 ( k 2 , n ~ + ζ A ϵ + c i , j 2 β ) ] .
P e ICSI, DCO-OFDM = 1 K ∑ i = 1 i = K P e ICSI ( A ( i ) , E s ( i ) )
P e OCSI, DCO-OFDM = 1 K ∑ i = 1 i = K P e OCSI ( A ( i ) , E s ( i ) )
p N ~ ICSI ( n ~ ) = 1 2 π ( α 2 + ζ 2 σ Δ 2 + h 2 σ v 2 + σ 1 2 ) ∫ − ∞ ∞ p ( n ~ + k β − x β ) exp ⁡ ( − x 2 2 ( α 2 + ζ 2 σ Δ 2 + h 2 σ v 2 + σ 1 2 ) ) d x .
p N ~ ICSI ( n ~ ) = 1 π ∫ − ∞ ∞ p ( n ~ + k β − t 2 ( α 2 + ζ 2 σ Δ 2 + h 2 σ v 2 + σ 1 2 ) β ) exp ⁡ ( − t 2 ) d t .
p 1 ( ψ 1 ) ≈ 1 π ∑ i = 1 m 1 Θ i p ( ψ 1 + k β − u i 2 ( α 2 + h 2 σ v 2 ) β ) .
p ( ψ 2 ) = 1 2 ζ A ϵ ∫ − ζ A ϵ ζ A ϵ p 2 ( ψ 2 − Δ 1 ) d Δ 1 .
p 2 ( ψ 2 ) ≈ 1 2 ζ A ϵ ∑ i = 1 m 1 Θ i π [ Γ ~ ( k 2 , ψ 2 − ζ A ϵ + k β − u i 2 ( α 2 + h 2 σ v 2 ) 2 β ) − Γ ~ ( k 2 , ψ 2 + ζ A ϵ + k β − u i 2 ( α 2 + h 2 σ v 2 ) 2 β ) ] .
g ( z ) = 1 2 ϵ 2 π σ v 2 ∫ − ϵ ϵ 1 | Δ | exp ⁡ ( − ( z Δ ) 2 2 σ v 2 ) d Δ .
g ( z ) = 1 2 ϵ 2 π σ v 2 ∫ z 2 2 ϵ 2 σ v 2 ∞ t − 1 e − t d t = 1 2 ϵ 2 π σ v 2 E 1 ( z 2 2 ϵ 2 σ v 2 ) ,
p N ~ OCSI ( n ~ ) = ∫ − ∞ ∞ p 2 ( n ~ − τ ) g ( τ ) d τ .
E 1 ( x ) ≈ 4 2 π ∑ n = 1 N ∑ m = 1 M a n a m b n e − 4 b n b m x ,
p N ~ OCSI ( n ~ ) ≈ 1 2 ϵ 2 π σ v 2 4 2 π ∑ n = 1 N ∑ m = 1 M a n a m b n × ∫ − ∞ ∞ p 2 ( n ~ − τ ) exp ⁡ ( − 2 b n b m ϵ 2 σ v 2 τ 2 ) d τ .
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