1. Introduction

Visible light communication (VLC) has been a promising candidate for future wireless systems in specific vertical industries due to its high security and electromagnetic silence [1–3]. Intensity modulation and direct detection (IM/DD) are typically adopted, where light-emitting diodes (LEDs) are employed as transmitters for low cost and easy deployment [4,5]. To fully utilize the bandwidth, optical orthogonal frequency division multiplexing (O-OFDM) is proposed [6,7]. To satisfy real-value and non-negativity requirements, O-OFDM schemes like direct-current-biased optical OFDM (DCO-OFDM) [8], asymmetrically clipped optical OFDM (ACO-OFDM) [9], multi-band OFDM [10], and flip OFDM [11] have been developed.

In [12], a total rate of 15.73 Gb/s is achieved using DCO-OFDM with adaptive bit-loading over a $1.6$ m link. An aggregate data rate of $4.5$ Gb/s is successfully achieved for $2.0$ m indoor free space transmission [13]. Work [14] realized an aggregate rate of $6.36$ Gb/s using a DCO-OFDM-based multi-color VLC system. All these mentioned above and many other instances show the great prospect of O-OFDM in VLC.

Owing to high efficiency and implementation simplicity, DCO-OFDM is adopted in this work, and similar approaches can be extended to other O-OFDM systems. Besides, the advantages of avalanche photodiode (APD) over photon diode are obvious, since APDs can provide greater power gain and sensitivity, enabling high-speed transmission or large coverage, thus an APD is adopted as the receiver [15,16]. Under a line-of-sight (LOS) link between LED and APD, considering the intrinsic lowpass frequency response characteristics due to the LED and APD, we assume a constant normalized channel gain vector, which will be verified experimentally [17,18]. VLC systems generally require accurate channel state information (CSI) for adaptive transmission and coherent detection. OFDM systems typically adopt comb-type or block-type pilot arrangements, which are suitable for fast and frequency-selective fading channels, respectively [19–21]. With the constant normalized channel gain vector, we prove that a single pilot is sufficient for accurate channel estimation without clipping, which significantly reduces the pilot overhead. We propose to optimize the pilot position, power allocation, and coding threshold according to different channel characteristics. The LEDs and power amplifiers at the transmitter usually have limited linear dynamic ranges, so nonlinear distortion often appears due to O-OFDM’s high peak-to-average power ratio (PAPR) [22,23]. The pilot optimization under transmitter-side clipping is further considered, and the channel estimation convergence is also verified experimentally under the clipping case. The impact of link gain and clipping level on the optimal pilot position and the achievable rate is investigated. Furthermore, we adopt a resource-efficient blind estimation scheme under constant normalized link gain assumption in a LOS link. Fast convergence and low estimation error are also observed experimentally.

Typical indoor VLC can ensure that there is always a direct path between the LED and the user by deploying multiple LEDs on the ceiling. This work mainly considers pilot optimization to obtain the optimal average throughput in the case of varying channel strength due to user mobility. We mainly focus on the low-rank channel characteristics superimposed single pilot scheme, which reduces the complexity of channel estimation and achieves the maximum achievable rate under limited feedback.

The remainder of this paper is organized as follows. In Section 2, we propose a channel model of indoor VLC with a constant normalized channel gain vector under a LOS link and verify its correctness experimentally and theoretically. In Section 3, we cope with the pilot optimization problem for VLC without clipping and investigate the impact of power budget and channel characteristics on the optimal pilot position and achievable rate. In Section 4, we consider the nonlinearity triggered by LED or power amplifier (PA) and investigate the impact of LED upper saturation voltage on the optimal pilot position and achievable rate. In Section 5, we further propose a blind channel estimation approach based on constant normalized link gain vector assumption and experimentally verify its convergence. Finally, Section 6 concludes this paper.

2. System model

In Sections 2.1 and 2.2, we present the channel model and prove its correctness experimentally and theoretically. In Section 2.3, we further consider the inconsistencies of the gain vectors for different clipping levels.

2.1 Channel characteristic assumptions

2.1.1 Consistent normalized channel gain vector under user mobility

Consider the VLC system with a dominant LOS link, which can be characterized via a frequency-selective channel due to the inherent lowpass mechanism of LED and optical detectors. The channel fluctuation caused by the variations in the distance and angle is typically modeled as a time-varying attenuation scalar. Thus, we assume a constant normalized channel gain vector and time-varying channel strength scalar due to the receiver mobility [24]. In an indoor VLC scenario, the typical LOS link distance between the transmitter and the receiver is $3.0$ m to $5.0$ m. In this work, DCO-OFDM is adopted for its high efficiency and implementation simplicity, and similar approaches can be extended to other O-OFDM systems.

2.1.2 Ideal channel model without transmitter clipping and receiver SDN

We first consider the ideal channel model without the transmitter clipping and receiver signal-dependent noise (SDN). Assume that there are $2K$ subcarriers in total. Let $x_k$ denote the transmitted symbol on the $k$-th subcarrier, $0\le k\le 2K-1$. To obtain real-valued signals in the time domain, the input symbols should satisfy the Hermitian symmetry constraint, i.e., $x_k=x^*_{2K-k}$ and $x_0=x_K=0$. As a result, only subcarriers $1$ to $K-1$ can be used for data transmission. We focus on these $K-1$ subcarriers, among which a set of $r$ subcarriers are chosen as pilot subcarriers for estimating the channel strength. Let $\boldsymbol {h}=[h_1,h_2,\ldots,h_{K-1}]^T$ denote the normalized channel gain vector, where $h_k$ denotes the frequency response on the $k$-th subcarrier. The overall channel gain vector can be expressed as $\alpha \boldsymbol {h}$, where $\alpha$ denotes the varying channel strength coefficient due to user mobility. With a known normalized channel gain vector $\boldsymbol {h}$, the channel strength coefficient $\alpha$ can characterize the overall channel gain vector. In the ideal case without transmitter clipping and receiver SDN, the receiver output on the $k$-th subcarrier is given by

2.1.3 Channel model with transmitter clipping and receiver SDN

According to the central limit theorem (CLT), approximately the time-domain signals follow a Gaussian distribution after inverse discrete Fourier transform (IDFT). Considering the nonlinearity caused by the LED or PA, we adopt the double-sided clipper model mentioned in [25], where the LED upper saturation voltage is $A$, and the signals beyond $A$ or below zero will be truncated. Based on the Bussgang theorem and linear invariance of IFFT, nonlinear signal distortion induced by LED or PA in the time domain can be modeled as an attenuation effect plus additive noise in the frequency domain [26–29]. Specifically, the frequency-domain signal on the $k$-th subcarrier after transmitter clipping is expressed as

Considering that the shot noise critically depends on the received optical intensity, we first investigate the transmitted optical intensity by the LED, which is expressed as

Let $\boldsymbol {w}_i$ denote the $i$-th column of $\boldsymbol {W}^H$, and $\tilde {x}_n$ denote the $n$-th element of $\tilde {\boldsymbol {x}}$. The $(i, k)$-th element of $\boldsymbol {W}{\rm diag}\{\tilde {\boldsymbol {x}}\}\boldsymbol {W}^H$ is given by

Since $\{\tilde {x}_n\}_{n=0}^{2K-1}$ are i.i.d. random variables, Eq. (7) can be rewritten as

2.2 Experimental validation on channel model

We measure the link gains at different locations and calculate the correlation coefficient matrix to verify the strong correlation among the channel gain vectors under receiver mobility. The experimental VLC system is shown in Fig. 1, where an LED and an APD are adopted at the transmitter and receiver sides, respectively. Keeping the same position and angle for the LED, the APD is placed in front of the LED with the transceiver distance varying from $3.0$ m to $5.0$ m at a step of $0.2$ m. At the transmitter, the alternating current (AC) signals produced by the AWG are amplified by a commercial PA, and then coupled with the DC bias generated by the DC power supply through a Bias-Tee to drive the LED. At the receiver, the APD converts the received optical signals to electrical signals. An oscilloscope is adopted to record the output waveform of APD for further offline processing. The transmitter and receiver sides are shown in Fig. 1. In the tests, the peak-to-peak voltage (Vpp) of the AC signals generated by the AWG is set to be $1.0$ V, and then amplified to $15.8$ V by the PA. The DC offset added on the LED is set to be $13.6$ V.

 

Fig. 1. The diagram of the established VLC system.

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Fig. 2. The graphical representation of the correlation coefficient matrix for the link gains at eleven positions. (a) The real part. (b) The imaginary part.

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Fig. 3. The grayscale of the correlation coefficient matrix. (a) Among the equivalent clipping noises on $(K-1)$ subcarriers. (b) Between the signals and equivalent clipping noises.

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We calculate the correlation coefficient matrix of link gain vectors estimated at eleven positions, whose real part and imaginary part are shown in Fig. 2(a) and Fig. 2(b), respectively. It is seen that the correlation coefficients of these estimated link gain vectors are close to one, indicating an approximately fixed ratio among the channel gain vectors. Figure 3(a) presents the grayscale of the correlation coefficient matrix. It is seen that the equivalent noises on $K-1$ subcarriers are not correlated. Thus, they can be modeled as i.i.d. Gaussian noises, as stated in Section 2.1. The correlation coefficient matrix between the original signals and equivalent clipping noise is presented in Fig. 3(b), demonstrating that the clipping noises on the $K-1$ subcarriers are independent of the original signals.

2.3 Inconsistent normalized gain vector for different clipping levels

In Sections 2.1 and 2.2, we have theoretically and experimentally verified that the normalized channel gain vectors remain constant at different locations. In this subsection, we conduct experiments to investigate the impacts of clipping levels on the normalized channel gain vector. In the tests, the Vpp of the AC signals remains the same as in Section 2.2, and the DC offsets are set to be $13.0$ V, $13.3$ V, and $13.6$ V. Note that the adopted commercial LED has a large upper saturation voltage, such that the downside clipping dominates over saturation truncation. Keeping the AC power at a constant relatively low level, the DC offset value of $13.6$ V means little clipping effect at the transmitter, while a severe clipping effect exists at $13.0$ V DC offset.

The normalized channel gain vectors are estimated for different DC offsets, and the correlation coefficient matrix is shown in Table 1. It is seen that the normalized channel gain vectors are not precisely consistent with each other for different clipping levels. Although the normalized channel gain vector varies with clipping levels, it will be verified experimentally in Section 5.3 that a specific normalized channel gain vector is sufficient to obtain good performance in blind estimation.

Table 1. Correlation Coefficient Matrix of Channel Gain Vectors

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3. Pilot optimization for VLC without transmitter clipping and receiver SDN

3.1 LMMSE channel estimation and pilot optimization

We first consider the pilot optimization problem under an ideal VLC scenario without transmitter clipping and receiver SDN. Assume that the normalized channel gain vector $\boldsymbol {h}=[h_1,h_2,\ldots,h_{K-1}]^T$ is known. The AC power budget is $P$, i.e., $\sum _{k=1}^{K-1} P_k \le P$ with $P_k$ being the electrical AC power on the $k$-th subcarrier. Let ${\cal L}=\{i_1, i_2,\ldots, i_r\}$ denote the set of subcarrier indices for pilot transmission with $1\leq i_1 i_2 \cdot \cdot \cdot i_r\leq K-1$, and ${\cal L}^c$ denote the set of subcarrier indices for data transmission. In the following, we aim to maximize the achievable rate by optimizing the pilot subcarrier set and power allocation among the $K-1$ subcarriers. Let $\boldsymbol {h}_{\cal L}=[h_{i_1},h_{i_2},\ldots,h_{i_r}]^T$ denote the pilot subcarrier vector. The received signal vector on the $r$ pilot subcarriers is expressed as

In the following, we will prove that a single pilot is sufficient to maximize the achievable outage rate. Consider the case with a single pilot for channel estimation, i.e., $|{\cal L}| = 1$. Under the same optimization objective and constraints as Problem (12) together with $|{\cal L}| =1$, the optimization problem is formulated as follows,

The following results show that additional constraint $|{\cal L}| = 1$ also yields an optimal solution to the optimization problem (12).

Proposition 1: The same optimization target value is obtained for Problem (12) and Problem (13), i.e., $C=C_1$.

Proof: Since an additional constraint $|{\cal L}|=1$ is imposed in Problem (13) compared with Problem (12), we have $C\geq C_1$.

On the other hand, let $S^*=\left \{{\cal L}^*,f^*,g^*,\{P_k^*\}_{k=1}^{K-1}\right \}$ be the optimal solution to optimization problem (12), achieving objective value $C$. We can construct another feasible solution $S'=\left \{{\cal L}',f',g',\{P_k'\}_{k=1}^{K-1}\right \}$ to problem (13) from $S^*$ as follows,

Equation (14) means that the strongest subcarrier in ${{\cal L}^*}$ is adopted as the single pilot subcarrier, while the other $(r-1)$ subcarriers are left free. Equations (15) and (16) represent that the same estimator and threshold mapping functions in $S^*$ are adopted in $S'$. Equation (17) means that among the subcarriers in ${{\cal L}^*}$, we only assign power to the strongest one, while the power allocated to the subcarriers in ${\cal L}^{* c}$ remains unchanged. Since $i_0={\rm arg} \mathop {\rm max}_{k \in {\cal L}^*} |h_k|$, we have $P_{i_0}'\le \sum _{k\in {\cal L}^*}P_k^*$ and further $\sum _{k=1}^{K-1}P_k' \le \sum _{k=1}^{K-1}P_k^* \le P$. With $|h_{i_0}|^2P_{i_0}' = \sum _{k\in {\cal L}^*}|h_k|^2P_k^*$, $P_{i_0}'$ yields the same minimal sufficient statistic ${\rm Re}\{\hat {y}\}$ as $\{P_k^*\}_{k \in {\cal L}^*}$, while some idle subcarriers and power budgets remain, leading to $C_1\geq C$.

Based on the above arguments, we have $C=C_1$. In other words, a single subcarrier is necessary and sufficient to achieve the maximum achievable rate.

We adopt the linear minimum mean squared error (LMMSE) criterion to estimate $\alpha$ from ${\rm Re}\{\hat {y}\}$. Denoting $\mu _{\alpha } = {\mathbb {E}}[\alpha ]$ and $C_\alpha ={\mathbb {E}}\big [(\alpha -\mu _\alpha )^2\big ]$, then the LMMSE estimate of $\alpha$ is expressed as follows,

The corresponding squared error (SE) is given by

Given channel output $y_{i_0}$ on the pilot subcarrier, the posterior probability density function of $\alpha$ is obtained as

We use $\alpha$’s posterior distribution $p(\alpha |y_{i_0})$ to calculate the connection probability since $\alpha '=g(\hat {\alpha })$ denotes that the coding threshold is adjusted to the output $y_{i_0}$ on the pilot subcarrier. The exhaustive search (ES) method is adopted to obtain the optimal pilot design and corresponding achievable rate. The results can be stored in a table for further lookup operations. The detailed procedures are given in Algorithm  1, where $\Delta _1$ and $\Delta _2$ are the ES step sizes. We traverse the three-dimensional space of the pilot index, pilot power allocation, and coding threshold to find the optimal solution. Specifically, the searches of pilot index, pilot power allocation, and coding threshold require $(K-1)$, $P/\Delta _1$, and $\mu _\alpha /\Delta _2$ times, respectively. Consequently, the total complexity of Algorithm 1 is ${\mathcal {O}}(KP\mu _\alpha /(\Delta _1\Delta _2))$.

Algorithm 1. The proposed ES algorithm for finding the maximum rate and corresponding pilot scheme.

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3.2 Experimental verification of LMMSE estimator with high DC

As shown in Eq. (18), the values of $\mu _\alpha$, $C_\alpha$, and $\sigma ^2$ need to be determined as priori information. In the experiment, the LED is placed at one end of the platform, and the APD’s position is changed randomly to simulate receiver mobility. The LED DC offset is set to be $13.6$ V, a relatively high value to avoid the downside clipping effect. The Vpp of the AC signals generated by AWG is set to be $1.0$ V, then amplified by a PA to around $15.8$ V. Afterwards, the AC signals are coupled with DC offset through a Bias-Tee and loaded onto the commercial LED. According to the experimental conditions and equipment parameters, the values of $\mu _\alpha$, $C_\alpha$ and the noise level $\sigma ^2$ are set to be $6.60$, $3.00$ and $0.08$, respectively. Multiple OFDM data symbols are generated into a block for coding, interleaving, and accurate channel estimation. We change the positions and receiving angles of the APD receiver to generate different channel strengths.

Considering the quasi-static characteristics of VLC channel, we investigate the convergence characteristics of the LMMSE estimator for given $\alpha$ in the case without transmitter clipping, as shown in Fig. 4. It is seen that as the number of data symbols increases, the estimation error converges to a constant value, which is closer to $0$ as the actual channel strength coefficient becomes closer to the priori mean $\mu _\alpha$. When the actual channel strength coefficient $\alpha$ is larger than the priori mean $\mu _\alpha$, the estimate tends to be lower than the true $\alpha$, and vice versa. It is seen that less than $20$ data symbols are enough for the convergence to a constant value within $\pm 5{\%}$.

 

Fig. 4. The normalized estimation error versus the number of data symbols without transmitter clipping (DC = 13.6 V).

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3.3 Simulation results and analysis

We conduct simulations to evaluate the pilot optimization results. The exponential attenuation model is assumed on the link gain for subcarriers, expressed as

Figure 5(a) shows the impact of power budget on the achievable rate with different values of $C_\alpha$. It is seen that the achievable rate increases with the power budget, where the optimal pilot scheme is obtained through ES in Algorithm 1. As the link power budget $P$ increases from 5 W to 30 W at a step size of 5 W, the corresponding optimal pilot subcarrier indices are $[15, 18, 22, 24, 26, 28]$. It implies that as the power budget increases, we should select a weaker subcarrier as pilot. Meanwhile, the corresponding optimal pilot power allocations are [0.21, 0.30, 0.41, 0.49, 0.58, 0.65] W, meaning that more power should be allocated to the pilot to compensate for the performance degradation caused by channel estimation inaccuracy. We investigate the impact of expected channel strength on the optimal coding threshold and achievable rate, as shown in Fig. 5(b). The achievable rate increases as the expected channel strength coefficient $\mu _\alpha$ increases. As $\mu _\alpha$ increases from 4 to 14 at a step interval of 2, the corresponding optimal pilot subcarrier indices are $[11, 15, 18, 22, 25, 28]$, implying that a weaker pilot subcarrier is sufficient to obtain good channel estimation. It is reasonable since, given the priori variance of the channel strength coefficient, there is relatively less uncertainty as $\mu _\alpha$ increases, where a weaker pilot is preferred. Meanwhile, the corresponding optimal pilot power allocations are [0.70, 0.57, 0.46, 0.41, 0.37, 0.34] W, meaning that it is preferred to distribute more power for data transmission rather than pilot under larger $\mu _\alpha$. Finally, the influence of channel fluctuation degrees is explored, as shown in Fig. 5(c). As the priori channel variance $C_\alpha$ increases from 0.5 to 3.0 at a step size of 0.5, the corresponding optimal coding threshold gradually decreases as [8.69, 8.36, 8.16, 8.03, 7.93, 7.85] to obtain a higher transmission success probability. It is demonstrated that the pilot subcarrier selection and power allocation are robust to the variation in priori channel variance $C_\alpha$. Given $P = 10$ W and $\mu _\alpha = 10$, as $C_\alpha$ increases from $0.5$ to $3.0$, the optimal coding threshold decreases, while the optimal pilot subcarrier index $k = 18$ and pilot power $P_k = 0.31$ W remain constant.

 

Fig. 5. The achievable rate. (a) Versus the power budget with different values of $C_\alpha$. (b) Versus the expected channel strength $\mu _\alpha$ with different values of $C_\alpha$. (c) Versus the priori channel variance $C_\alpha$ with different power budgets.

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4. Pilot optimization for VLC with transmitter clipping and receiver SDN

In the following, we investigate the pilot optimization problem in the case with transmitter clipping and receiver SDN.

4.1 Pilot design for VLC with varying strength coefficient

Consider the transmission with transmitter clipping. On the $k$-th subcarrier, the signal-to-noise-plus-distortion ratio (SNDR) is given as follows,

Adopting single pilot subcarrier, we have

Algorithm 1 is adopted here to solve the pilot optimization problem (24), and the only difference is that $p(\alpha |y_{i_0})$ and $R(y_{i_0})$ are expressed as Eq. (27) and Eq. (28), respectively.

4.2 Experimental results of LMMSE estimator with transmitter clipping

Typically, when the LED works at a low DC or high AC power point, the influence of the clipping effect cannot be neglected. To investigate the clipping effect, the DC offset in the experiment is changed to $13.0$ V, while other settings remain the same as in Section 3.2. Similar to Section 3.2, the LMMSE estimator is adopted, while noise level $N_0$ is set to be $0.20$ rather than $0.08$ according to actual channel statistics. We change the positions and receiving angles of the APD receiver to simulate receiver mobility and investigate the influence of channel strength on convergence.

Figure 6 shows the normalized estimation error versus the number of data symbols in the case with transmitter clipping. It is seen that $60$ data symbols are necessary for the estimation error to converge to a constant value within $\pm 5{\%}$. When the actual channel strength coefficient $\alpha$ is larger than the priori mean value $\mu _\alpha$, the LMMSE outputs a smaller estimate, leading to a negative estimation error. As the actual channel strength coefficient $\alpha$ becomes closer to the priori mean $\mu _\alpha$, the estimation error gets smaller. Even though the estimation error converges at a lower speed compared under high DC, the normalized estimation error in the steady state is also within $\pm 5{\%}$.

 

Fig. 6. The normalized estimation error versus the number of data symbols with transmitter clipping (DC = 13.0 V).

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4.3 Simulation results and analysis

The impact of LED upper saturation voltage and the channel characteristics on the pilot design and achievable rate is shown in Fig. 7. It is seen that for small upper saturation voltages such as $4$ V, adopting the weakest subcarrier as pilot is preferred. In contrast, the strongest subcarrier becomes the best choice as the upper saturation voltage increases. It is also observed that the optimal pilot choice lies either in the strongest or the weakest.

 

Fig. 7. The achievable rate versus pilot power under different LED upper saturation voltages. (a) $A$ = 4. (b) $A$ = 10. (c) $A$ = 15.

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The relationship between the LED upper saturation voltage and the achievable rate is shown in Fig. 8(a). As the LED upper saturation voltage increases, weaker clipping noise leads to higher optimal AC power. Using the strongest pilot for channel estimation results in a substantial gain in achievable rate. It is also seen that there is an intersection above which using the strongest subcarrier shows a higher achievable rate. Under the same priori variance of channel strength coefficient $\alpha$, the uncertainty decreases as the channel gets stronger, leading to less demand for channel estimation. Thus, it is preferred to adopt the weakest subcarrier as pilot. Such property is reflected by the fact that, as the channel becomes stronger, the intersection of the pairwise curves moves to the right. Moreover, the impact of expected channel strength $\mu _\alpha$ on the achievable rate is shown in Fig. 8(b). It is seen that increasing both LED upper saturation voltage and expected channel strength improves the achievable rate. Larger expected channel strength leads to the weakest subcarrier being adopted as pilot. Meanwhile, a larger linear dynamic range results in a higher AC component, which is combined with the strongest pilot subcarrier for more accurate channel estimation. It is also seen that the intersection of the two capacity curves gradually moves to the right as the LED upper saturation voltage increases, implying that the strongest pilot tends to be employed under large LED linear dynamic ranges. The impact of channel priori variance on the capacity is shown in Fig. 8(c). As the priori variance increases, the channel becomes more uncertain, decreasing the achievable rate. For constant channel priori variance, higher expected strength leads to a higher achievable rate. Given expected channel strength, higher variance leads to the strongest subcarrier being adopted as pilot. Such two properties are reflected by the fact that, as the priori variance increases, the achievable rate using the weakest subcarrier as pilot goes down faster. A certain threshold exists on the variance, above which adopting the strongest subcarrier as pilot is preferred. Furthermore, as the expected channel strength increases, the intersection of two curves moves to the right, which implies that the weakest subcarrier tends to be employed as pilot.

 

Fig. 8. The achievable rate. (a) Versus upper saturation voltage under different channel strengths. (b) Versus expected channel strength with different LED upper saturation voltages. (c) Versus priori channel variance with different channel strengths.

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The relationship between the pilot power and achievable rate is shown in Fig. 9(a). A smaller LED dynamic range results in lower optimal AC power, then the achievable rate decreases monotonically as the pilot power increases. A larger LED dynamic range leads to much higher optimal AC power. A stronger subcarrier is adopted as pilot, and it is preferred to allocate more power to the pilot for more accurate channel estimation. As the linear dynamic range of LED increases, higher AC power results in a stronger pilot for channel estimation. Furthermore, the capacity variation with pilot power allocation is shown in Fig. 9(b). It is seen that under slight channel priori variance, lower demand for channel estimation leads to adopting the weakest subcarrier for channel estimation. As the priori variance of channel strength increases, the channel uncertainty increases, thus decreasing the achievable rate. As the channel variance increases, it is preferred to adopt a stronger subcarrier as pilot, and the optimal pilot power increases with the channel variance.

 

Fig. 9. The achievable rate versus pilot power allocation. (a) Different upper saturation voltages. (b) Different priori variance $C_\alpha$.

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5. Blind channel estimation

In Sections 3 and 4, We have designed a single pilot scheme, which is resource-efficient for frequency-domain channel estimation of DCO-OFDM. To further enhance resource efficiency, we investigate blind estimation for DCO-OFDM without resource overhead for pilots. We focus on the channel model described in Eq. (9), as Eq. (1) can be regarded as a special case of Eq. (9) with $\gamma =1$ and $z_{k,c}=z_{k,s}=0$ for $1\le k\le K-1$.

5.1 Principle and experimental validation

Based on the channel model described in Eq. (9), the second order statistics of observed channel output $\boldsymbol {y}$ is obtained as

To investigate the linear relationship between $\{|h_k^2|\}_{k=1}^{K-1}$ and corresponding noise variances on these $K-1$ subcarriers, we adopt the same experimental equipment and devices as described in Section 2.2. The LED transmitter and APD receiver are placed $3.0$ m apart. The channel gain and noise variance vectors are calculated using received data. Two thousand data symbols are transmitted to investigate whether the linear fitting error can approach zero as the number of data symbols increases. As shown in Fig. 10(a), the relationship between the noise variances and squared tap coefficients can be well linearly fitted. Moreover, the root mean squared error (RMSE) of the linear fitting is plotted in Fig. 10(b), and the RMSE decreases to $0.010$ with only $50$ data symbols.

 

Fig. 10. Linear fitting of the noise variance on the $k$-th subcarrier and {$|h_k^2|$}. (a) RMSE of linear fitting. (b) Linear fitting result.

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5.2 Blind estimation approach

Based on the statistical characteristics described in Eq. (29), an effective blind estimation algorithm can be designed. Without loss of generality, we assume equal power allocation on subcarriers $1$ to $K-1$, i.e., $P_k = P_1$ for $1 \le k \le K-1$. It is seen from Eq. (29) that for a specific receiver location with a constant $\alpha$, there exists a linear relationship between the diagonal elements of $\boldsymbol {C}_y$ and $\boldsymbol {\mathcal {H}}$, i.e., ${\boldsymbol {\rm vdiag}}\{\boldsymbol {C}_y\}=c{\boldsymbol {\rm vdiag}}\{\boldsymbol {\mathcal {H}}\} + d{\boldsymbol {1}}$, where $\boldsymbol {\rm vdiag}\{\cdot \}$ is the vectorization operation on the diagonal elements in $(\cdot )$, and $c=\alpha ^2(\gamma ^2P_1+\sigma _c^2)$, $d=2K\alpha B\sigma _0^2+\sigma ^2$. The covariance matrix $\boldsymbol {C}_y$ can be estimated using the testing samples. Specifically, we collect all the received signal vectors as

Using the least squares method, the fitting parameters are obtained as

When the receiver moves around, the channel strength coefficient $\alpha$ varies randomly, and there exists a proportional relationship between the slope coefficient $c$ and channel strength square $\alpha ^2$, i.e., $\alpha ^2=\beta c$. The scaling factor is calculated in the training process as $\beta =\alpha ^2/\hat {c}$. When the VLC receiver moves to a new position, the covariance matrix is again estimated using the received data. We calculate the new slope coefficient $\hat {c}_1$ by (32), and then estimate the channel strength as $\hat {\alpha }_1=\sqrt {\beta \hat {c}_1}$. For the blind channel estimation method, the major complexity comes from the estimation of the covariance matrix using Eq. (31). The dimension of matrix $\boldsymbol {Y}$ is $(K-1)\times L$, such that the blind channel estimation method involving the calculation of $\hat {\boldsymbol {C}}_y$ requires a complexity of ${\mathcal {O}}(K^2L)$.

5.3 Experimental results of blind estimation

Three experiments are carried out at DC offsets of $13.0$ V, $13.3$ V, and $13.6$ V, respectively, corresponding to different clipping levels. The Vpp of the AC signals generated by AWG remains $1.0$ V and is amplified by PA to be around $15.8$ V. In each experiment, $1500$ OFDM symbols are transmitted as the test data, where the Vpp of APD response signals is around $200$ mV. The position and receiving angle of the APD are changed three times to obtain different channel strengths, and $500$ OFDM symbols are transmitted each time as the training data to obtain $\beta$. As the number of test data symbols increases, the SE of estimation is plotted in Fig. 11. Considering that the LED used in the experiment has a large dynamic range, the lower truncation rather than the upper saturation effect mainly occurs. $13.0$ V has a large degree of lower truncation. There is basically no clipping effect when it is increased to $13.6$ V, and it can be seen that the convergence error of the final estimate is getting smaller as the DC bias gets larger. Meanwhile, we use the slope of the linear fitting of the second-order statistics to obtain the estimate of the link strength coefficient. Sending different data causes random fluctuations in the positions of the sample points used for linear fitting, which will lead to large fluctuations in the slope of linear fitting, and thus the estimation error. It is seen that around $50$ DCO-OFDM data symbols are enough to make the squared error converge below $0.05$. It is seen that larger Vpp at the APD receiver leads to larger converged SE.

Since according to the experimental results in Section 2.3, the normalized channel gain vector varies for different clipping levels, we investigate whether a particular normalized channel gain vector can adapt to all these different clipping levels. We investigate the adaptation of $\beta$ and $\boldsymbol {h}$ to other DC/AC configurations. In the experiment, $\beta$ and $\boldsymbol {h}$ are obtained under DC offset $13.6$ V and then applied to DC offsets $13.0$ V as well as $13.3$ V. The SE versus the number of data symbols is plotted in Fig. 12. It is seen that a particular normalized channel gain vector can adapt well to different clipping levels.

 

Fig. 11. Squared error of the blind estimation scheme versus the number of data symbols.(a) DC = 13.0 V. (b) DC = 13.3 V. (c) DC = 13.6 V.

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Fig. 12. Squared error of the blind estimation scheme under a LOS link.

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6. Conclusion and future prospects

In this work, we experimentally verified the constant normalized channel gain under a LOS link between the transmitter and receiver. Based on this, we proved that a single pilot can maximize the achievable rate without transmitter-side clipping. We explored the impact of channel conditions and power budget on pilot design and achievable rate. Additionally, we considered transmitter-side clipping for DCO-OFDM and experimentally verified the convergence of LMMSE estimator. Furthermore, we conducted numerical simulations to investigate the impact of link characteristics and LED linear dynamic ranges on pilot design. We also adopted blind estimation based on second-order statistics and experimentally verified its convergence under the assumption of a consistent normalized channel gain vector. The constant normalized subcarrier gain assumption simplifies the CSI for feedback, which can be utilized for future adaptive modulation and coding design.