Location-aware spectrum sensing for cognitive visible light communications over multipath channels

Zile Jiang; Zile Jiang; Xiaodi You; Gangxiang Shen; Biswanath Mukherjee

doi:10.1364/OE.445712

1. Introduction

The wide deployment of light emitting diodes (LEDs) has accelerated the development of indoor visible light communication (VLC) and visible light positioning (VLP) technologies [1–3]. Although the spectrum resource of visible light is ample, the narrow bandwidth of off-the-shelf LEDs has limited the capacity of VLC applications, especially for multi-user scenarios. To enhance the transmission quality and the capacity of a multi-user VLC system, approaches for efficient spectrum resource allocation are required.

Recently, cognitive VLC (CVLC) has received increasing attention [4–6] due to its potential for efficient spectrum utilization. In a CVLC system, users are classified into primary users (PUs) and secondary users (SUs), where the PUs can be licensed users with the privilege of transmitting data at high data rate, low latency, and reliable quality-of-service (QoS), whereas the SUs can be best-effort users without stringent requirement on data rate, latency, or QoS. In this context, the PUs own higher priorities to access the VLC spectrum and the SUs can only opportunistically use the VLC spectrum when the spectrum is not being used by the PUs. By allowing dynamic spectrum sharing between PUs and SUs, CVLC can improve spectrum utilization and enhance system transmission capacity. However, to enable CVLC, it is crucial for an SU to accurately detect PUs’ presence so as to avoid its interference with the PUs. This detection process is called spectrum sensing. This study will focus on the spectrum sensing mechanism to propose different sensing algorithms and their corresponding analytical models.

There have been extensive studies on spectrum sensing in conventional cognitive radio systems [7,8]. Different approaches have been proposed to detect the presence of PUs, e.g., energy detection (ED) [9,10], waveform detection [11,12], and cyclic-prefix-based detection (CD) [13,14]. However, for a CVLC system, although extensive efforts have been made in the aspects of handover [4] and capacity optimization [5,6], they assume the presence of PUs and ignore the process of detecting PUs. There are few studies dedicated to spectrum sensing in a CVLC system. In [15], based on a single LED transmitter, an ED scheme was proposed to detect PUs in a CVLC system and a collaborative spectrum sensing scheme was developed to improve the PU detection accuracy. However, in a realistic indoor VLC system, multiple LEDs may be deployed for adequate and uniform illumination [16–21]. This work will particularly focus on spectrum sensing in a CVLC system based on multiple LED transmitters.

Because of differences in optical paths existing between different transmitters and indoor reflections from walls and ceilings, etc., an indoor CVLC channel inherently owns a multipath characteristic, which causes inter-symbol interference (ISI) and consequently degrades signal transmission quality [22,23]. To tackle this, orthogonal frequency division multiplexing (OFDM) has been employed in a classical VLC system to combat the multipath effect [24,25], where pilot symbols are transmitted in advance to estimate channel state information (CSI) [26,27]. Unfortunately, in the current CVLC system, it is difficult for an SU to estimate its channel characteristic using pilot symbols before spectrum sensing because at this moment it knows neither exact pilot symbols used by PUs nor whether a PU is present. Moreover, in a multipath channel, PUs’ signals arrive at an SU’s receiver with different gains and time delays. Although the conventional ED scheme can optimally sense independent and identically distributed PUs’ signals [14], its performance deteriorates if the received PUs’ signals are correlated [28]. Therefore, to overcome the performance degradation of ED, modifications or new approaches are required to cater to the multipath channel conditions.

To this end, we propose a novel location-aware spectrum sensing scheme for the CVLC system, where an SU employs its location information to assist its multipath channel estimation. Here, the SU’s location can be obtained by some indoor positioning technique such as VLP [3]. Based on the location information, the SU first estimates the multipath CSI and then constructs a location-based channel matrix, which is further used to optimize a detection metric and a decision threshold for spectrum sensing. Because VLC systems usually employ intensity modulation and direct detection (IM/DD), their channels are generally much simpler and more stable than a time-varying radio frequency (RF) channel [16]. Thus, as long as the SU is fixed in its location, its multipath channel characteristic can be predicted at the receiver, which helps estimate the CSI and enables the SU to combat the multipath effect in spectrum sensing.

By considering different a priori information such as signal and noise variances, we propose three PU detection algorithms and develop their respective analytical models. The first is the location-aware likelihood-ratio-test detection (LLD) algorithm, which can achieve an optimal sensing performance over a multipath channel based on a priori knowledge of both signal and noise variances. However, LLD suffers from high computational complexity and requires full a priori information of signal and noise variances. To simplify LLD, we further propose the location-aware semi-blind detection (LSD) algorithm, which can achieve a sub-optimal sensing accuracy while not requiring the information of signal variance. Moreover, because an SU may not precisely know the noise information, to avoid performance degradation due to noise uncertainty (i.e., the difference between the noise variance used for spectrum sensing and the real noise variance), we also propose the location-aware near-blind detection (LND) algorithm, which is even simpler not requiring either signal or noise variance.

To evaluate the performance of the proposed spectrum sensing scheme, simulations are conducted in comparison with the conventional ED and CD schemes. The tolerance against location error and receiver tilt is also evaluated. Results show that the proposed PU detection algorithms are effective to achieve a high detection probability (i.e., the probability of correctly detecting a PU) based on the considered CVLC channel. Specifically, under a strong multipath effect, both LLD and LSD can improve the detection probability by up to 10% compared with the conventional ED scheme. Also, although LSD does not require a priori information of signal variance, it achieves a sensing accuracy comparable to LLD. In addition, by incorporating CD in LSD, we can further improve the detection probability by ∼20% and 8% compared with the ED and CD schemes, respectively. Finally, we evaluate how the noise uncertainty impacts the detection probability. It is found that, when the noise uncertainty is over 0.5 dB, LND performs best among all the algorithms although it requires the least a priori information. Also, the evaluations of the tolerance against location error and receiver tilt show that both LSD and LND are robust to receiver tilt, and LSD is robust to location error.

The rest of this paper is organized as follow. Section 2 analyzes the proposed location-aware spectrum sensing scheme and introduces three PU detection algorithms. Section 3 conducts simulation studies and evaluates the performance of the three proposed algorithms. Section 4 concludes this paper.

2. Location-aware spectrum sensing

2.1 Working principle

In this study, we assume a multi-cell CVLC system, in which there are multiple LEDs and PUs distributed at different indoor locations and the LEDs are divided into different groups to spatially form multiple CVLC cells. Based on this multi-cell configuration, we focus on spectrum sensing within a single cell where all the LEDs transmit the same downlink signal [16–21] and an SU tries to detect the presence of PUs. Figure 1 shows the functional diagram of the considered spectrum sensing system. Specifically, with the assumption of DC-biased optical OFDM (DCO-OFDM) signal transmission [25], a transmitter first maps random bits to bipolar OFDM symbols. Then, after digital-to-analog (D/A) conversion followed by DC-bias, multiple LEDs transmit the same DCO-OFDM signal s(t) to PUs, which can also be received by an SU. Upon receiving PUs’ signals from different directions, the SU conducts spectrum sensing to detect the presence of PUs.

Fig. 1. Functional diagram of the location-aware spectrum sensing scheme.

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There are two types of channels, i.e., line-of-sight (LOS) and non-LOS (NLOS) channels, in the indoor CVLC system. The LOS channel exists directly between the multiple LEDs and an SU’s receiver, whereas the NLOS channel exists between multiple reflective objects (e.g., walls and ceiling) and the receiver. Based on the configuration in Fig. 1, we denote all the LEDs in a cell by LED_i (i = 1, 2, …, L). Here, L is the number of LEDs deployed in the cell, which is typically small (often ≤ 4) in many VLC models [16–21]. Then, we can model the total channel impulse response (CIR) between the LEDs and the SU by:

(1)$$h(t) = \mathop \sum \limits_{i = 1}^L [{{h_{i,\textrm{ }LOS}}(t) + {h_{i,\textrm{ }NLOS}}(t)} ], $$

where h_i_{, LOS}(t) and h_i_{, NLOS}(t) denote the LOS and NLOS CIR from LED_i, respectively. The narrow modulation bandwidth of a commercial LED would result in an imperfect impulse response. With appropriate pre-equalization approaches [29], h_i_{, LOS}(t) and h_i_{, NLOS}(t) can be approximated by Dirac functions. Note that similar approximations have been made in [17] and [24]. Therefore, we can derive Eq. (1) to the following equation as:

(2)$$h(t) \approx \mathop \sum \limits_{i = 1}^L \left[{{H_{i,\textrm{ }LOS}}(0)\delta (t - {\tau_{i,\textrm{ }LOS}}) + \mathop \int \nolimits_0^{ + \infty } {A_{i,\textrm{ }NLOS}}({\tau_{i,\textrm{ }NLOS}})\delta (t - {\tau_{i,\textrm{ }NLOS}})\textrm{d}{\tau_{i,\textrm{ }NLOS}}} \right]. $$

Here, δ (·) is Dirac function. For LED_i, τ_i_{, LOS} and τ_i_{, NLOS} represent the signal time delays of the LOS and NLOS channels, respectively; H_i_{, LOS} (0) is the DC gain of the LOS channel; and A_i_{, NLOS} (τ_i_{, NLOS}) is the gain of the NLOS channel with time delay τ_i_{, NLOS}. After the signal is received by the SU’s photo-detector, its converted electrical signal x(t) is given by [16]:

(3)$$x(t) = \gamma s(t) \otimes h(t) + n(t), $$

where γ is receiver responsivity, s(t) is the optical signal transmitted by each LED with variance σ_s², and n(t) is the additive white Gaussian noise (AWGN) at the output of the SU’s photo-detector with variance σ_n².

After removing the DC component by a filter, the received signal is sampled to generate a total of N discrete data samples, based on which the SU carries out spectrum sensing. Specifically, we use a vector x to represent these data samples:

(4)$$\textbf{x}|{{\cal H}_1} = \gamma \textbf{Hs} + \textbf{n};\textbf{x}|{{\cal H}_0} = \textbf{n}. $$

Here, the length of x is N, which is the sampling size. ${\cal H}_1$ represents the event that the PU is present, ${\cal H}_0$ represents the event that no PU is present, s is a vector of discrete signal samples of s(t) with DC components removed, and n is a vector of discrete noise samples of n(t), i.e., n ∼ $\mathcal{N}$ (0, σ_n²I). H is a discrete channel matrix, given in the form of a Toeplitz matrix:

(5)$$\;\textbf{H} = {\left[{\begin{array}{ccccccc} {{h_1}}& \cdots &{{h_m}}&0& \cdots & \cdots &0\\ 0&{{h_1}}& \cdots &{{h_m}}&0& \cdots &0\\ {}&{}&{}& \vdots &{}&{}&{}\\ 0& \cdots & \cdots &0&{{h_1}}& \cdots &{{h_m}} \end{array}} \right]_{N \times (N + m - 1)}}, $$

where h₁, …, h_m are the discrete samples of multipath CIR h(t), described by h = [h₁, …, h_m], and m is the length of the discrete samples of the CLVC CIR.

Based on the discrete samples of the received signal x, we may employ the conventional spectrum sensing schemes to detect the presence of PUs. For example, for independent and identically distributed PU’s signals, the ED scheme can be employed to realize an optimal spectrum sensing [14]. However, due to the dispersiveness of multipath channel H, the same copy of the transmitted signal samples s can propagate and superimpose at the SU’s receiver with respective gains and time delays. A high correlation can occur among the received data samples x, which will influence their independence and degrade the sensing accuracy [28]. Thus, we cannot directly apply the conventional spectrum sensing schemes to the current multipath channel, but require new detection approaches. To this end, we next propose a novel location-aware spectrum sensing scheme, which includes two steps.

First, we construct a location-based channel matrix Ĥ to characterize the multipath channel. As mentioned before, in a CVLC system, it is difficult for an SU to carry out pilot-assisted channel estimation before spectrum sensing, since it does not know what PUs’ pilot symbols are or whether PUs’ signals exist. On the other hand, because indoor CVLC systems employ IM/DD, their channels are relatively stable and determinate as long as the relative locations between LEDs and an SU’s receiver are fixed. Moreover, since LEDs are usually fixed on a ceiling, the location of an SU can be easily decided using an indoor positioning technique such as VLP [3]. Thus, the CVLC channel characteristic is theoretically predictable, and by utilizing the location information of an SU, it is possible to design an efficient spectrum sensing scheme that can compensate for the predictable multipath dispersiveness. We call this location-aware spectrum sensing. We represent the discrete samples of the LOS CIR by a vector ĥ = [ĥ₁, …, ĥ_m], whose elements can be described as:

(6)$$\begin{aligned} {{\hat{h}}_j} & =\left\{ \begin{array}{l} \displaystyle\sum\limits_{i \in {I_j}} {{H_{i,\textrm{ }LOS}}(0)} ,\quad {I_j} \ne \emptyset \\ 0,\quad \quad \quad \qquad \quad \textrm{ }{I_j} = \emptyset \end{array} \right.\textrm{ },\textrm{ }j = 1, \cdots ,m,\\ {H_{i,\textrm{ }LOS}}(0) & =\left\{ \begin{array}{l} \displaystyle\frac{{(l + 1){A_r}{{\cos }^l}({\phi_i}){T_s}({\theta_i})g({\theta_i})\cos ({\theta_i})}}{{2\pi D_i^2}},\quad 0 \le {\theta_i} \le {\psi_c}\\ 0,\quad {\theta_i} > {\psi_c} \end{array} \right.,i = 1, \cdots ,L \end{aligned}, $$

where l is the Lambertian emission order of LEDs, A_r is the physical area of an SU’s photo-detector, T_s is the gain of an optical filter, g is the gain of an optical concentrator, ψ_c is the field-of-view (FOV) of an SU’s receiver, c is the speed of light, t₀ is sampling interval, I_j = {i = 1, …, L | j = [D_i / ct₀]}, and [·] denotes a rounding function. For LED_i, D_i denotes its distance from an SU, and ϕ_i and θ_i denotes the irradiance and incidence angles relative to an SU, respectively. Assuming that the coordinates of LED_i and the SU’s receiver are (x_i, y_i, z_i) and (x_r, y_r, z_r), respectively, D_i, ϕ_i, and θ_i can be derived as follows in the form of location coordinates:

(7)$$\left\{ \begin{array}{l} {D_i} = \sqrt {{{({x_i} - {x_r})}^2} + {{({y_i} - {y_r})}^2} + {{({z_i} - {z_r})}^2}} \\ {\phi_i} = \arccos ({{{({z_i} - {z_r})} / {{D_i}}}} )\\ {\theta_i} = \arccos ({{{ < {v_i},{v_O} > } / {(||{v_i}||\cdot ||{v_O}||}})} )\end{array} \right.. $$

Here, v_i = [x_i – x_r, y_i – y_r, z_i – z_r], v_O = [sinβ · cosα, sinβ · sinα, cosβ], <·, ·> denotes inner product of two vectors, and α and β are the azimuthal and polar angles of the photo-detector, respectively. The detailed descriptions of the azimuthal and polar angles can be found in [17]. Acquiring the orientation of a receiver usually requires extra sensors [30], which would be very complex for an SU. Thus, in this study, we assume that an SU is not equipped with such sensors and that the SU’s receiver is facing upward, i.e., β = 0, which results in:

(8)$${\theta _i} = {\phi _i} = \arccos ({{{({z_i} - {z_r})} / {{D_i}}}} ). $$

Based on Eqs. (6), (7), and (8), we can construct vector ĥ, which includes the discrete samples of the LOS CIR estimated from an SU’s location information. Then, by using ĥ, we can further construct the location-aware channel matrix as:

(9)$$\hat{\textbf{H}} = {\left[ {\begin{array}{ccccccc} {{{\hat{h}}_1}}&\cdots &{{{\hat{h}}_m}}&0& \cdots & \cdots &0\\ 0&{{{\hat{h}}_1}}& \cdots &{{{\hat{h}}_m}}&0& \cdots &0\\ {}&{}&{}& \vdots &{}&{}&{}\\ 0& \cdots & \cdots &0&{{{\hat{h}}_1}}& \cdots &{{{\hat{h}}_m}} \end{array}} \right]_{N \times (N + m - 1)}} \approx \textbf{H}. $$

Here, to accurately model the real indoor channel conditions, we assume that the SU can receive both LOS and NLOS signals. However, when constructing the location-based channel matrix Ĥ (from Eqs. (6) to (9)), we ignore the NLOS channel for the following two reasons. First, the NLOS components are much weaker than the LOS components [16,17]. Second, it is difficult and complex for an SU to estimate the NLOS components.

Next, based on Ĥ and x, we construct a detection metric M for spectrum sensing. By comparing the detection metric with a decision threshold K, an SU can judge whether a PU is present. Similar to [15], we define the probability of correctly detecting a PU as detection probability, denoted by P_D, and the probability of falsely detecting a PU as false alarm probability, denoted by P_F. They are mathematically expressed as:

(10)$${P_D} = \Pr (M > K|{{\cal H}_1});{P_F} = \Pr (M > K|{{\cal H}_0}). $$

Equation (10) indicates that the performance of spectrum sensing is decided by the relationship between the detection metric M and the decision threshold K. Next, we propose three different detection algorithms to optimally choose M and K based on different a priori information.

2.2 Location-aware likelihood-ratio-test detection (LLD)

The likelihood ratio test is a useful approach for binary hypothesis tests [31]. To conduct efficient spectrum sensing in a CVLC system, we propose a location-aware likelihood-ratio-test detection (LLD) algorithm, which requires full a priori knowledge of signal variance σ_s² and noise variance σ_n². Based on Central Limit Theorem [32], the transmitted OFDM signal samples s can be approximately assumed to follow a normal distribution, i.e., s ∼ $\mathcal{N}$ (0, σ_s²I). When a PU is present, we have x | ${\cal H}_1$ ∼ $\mathcal{N}$ (0, Σ), where Σ = σ_n²I + σ_s²γ ²HH^T. When the PU is absent, we have x | ${\cal H}_0$ ∼ $\mathcal{N}$ (0, σ_n²I). Thus, the log likelihood ratio function with respect to x is given by:

(11)$$\Lambda (\textbf{x}) = \ln {{[p(\textbf{x}|{{\cal H}_1})} / {p(\textbf{x}|{{\cal H}_0})]}} = \ln [{{\sigma _n^{2N}} / {\textrm{det}(\boldsymbol{\Sigma})}}] + {\textbf{x}^T}[\sigma _n^{ - 2}\textbf{I} - {\boldsymbol{\Sigma}^{ - 1}}]\textbf{x}. $$

Here, p(·) denotes probability density function, and det(·) denotes the determinant of a square matrix. By eliminating all the constants irrelevant to x and substituting the real channel matrix H with an estimated version Ĥ, we can obtain the detection metric for the LLD algorithm as:

(12)$${M_{LLD}} ={\textbf{x}^T}[\sigma _n^{ - 2}\textbf{I} - {(\sigma _n^2\textbf{I} + \sigma _s^2{\gamma ^2}\hat{\textbf{H}}{\hat{\textbf{H}}^T})^{ - 1}}]\textbf{x}.$$

Accordingly, the detection probability and the false alarm probability of the LLD algorithm with threshold K_LLD are, respectively, given by:

(13)$$\begin{aligned} {P_{D,\textrm{ }LLD}} &= 1 - {F_{{\chi ^2}}}({{{{K_{LLD}} \cdot \textrm{tr}(\textbf{T}\boldsymbol{\Sigma})} / {\textrm{tr}({{(\textbf{T}\boldsymbol{\Sigma})}^2})}};{{{{\textrm{[tr}(\textbf{T}\boldsymbol{\Sigma})]}^2}} / {\textrm{tr}({{(\textbf{T}\boldsymbol{\Sigma})}^2})}}} )\\ {P_{F,\textrm{ }LLD}} &= 1 - {F_{{\chi ^2}}}({{{\sigma_n^{ - 2}{K_{LLD}} \cdot \textrm{tr}(\textbf{T})} / {\textrm{tr}({\textbf{T}^2})}};{{{{\textrm{[tr}(\textbf{T})]}^2}} / {\textrm{tr}({\textbf{T}^2})}}} )\end{aligned}.$$

Here, T = σ_n⁻²I – (σ_n²I + σ_s²γ ²ĤĤ^T)⁻¹, tr(·) denotes the trace of a square matrix, and Fχ²(x; d) is the cumulative distribution function (CDF) of a chi-square distribution with degree of freedom d. To choose an appropriate threshold K_LLD, we adopt the Neyman-Pearson (NP) criterion [33] which is widely employed in binary hypothesis tests. In the NP criterion, the false alarm probability needs to be fixed so as to achieve an acceptable spectrum sensing performance. Thus, following the common practice in binary hypothesis tests [34], we set P_F to be 0.1. Then, based on Eq. (13), the decision threshold of the LLD algorithm is derived as:

(14)$${K_{LLD}} = \sigma _n^2 \cdot {{\textrm{tr}({\textbf{T}^2})} / {\textrm{tr}(\textbf{T})}} \cdot F_{{\chi ^2}}^{ - 1}({1 - {P_{F,\textrm{ }LLD}};{{{{\textrm{[tr}(\textbf{T})]}^2}} / {\textrm{tr}({\textbf{T}^2})}}} ). $$

Theoretically, M_LLD is an optimal detection metric for binary tests when both signal and noise variances are known. However, this metric involves matrix inverse operations, which are time-consuming when the sampling size N is large. The computational complexities of M_LLD and K_LLD are both O(N³). Moreover, calculating M_LLD requires a priori information of σ_s², which cannot be obtained at an SU since σ_s² is only correlated to PUs’ QoS. Even if the SU can estimate σ_s², it is often inaccurate under the conditions of multipath and low signal-to-noise ratio (SNR).

2.3 Location-aware semi-blind detection (LSD)

LLD has a high computational complexity and requires a priori information of signal variance σ_s². To address these limitations, we also propose a location-aware semi-blind detection (LSD) algorithm for spectrum sensing, which can avoid the matrix inverse operations and does not require a priori information of σ_s². Based on Sherman-Morrison-Woodbury Formula [35], Eq. (12) can be transformed to:

(15)$${\textbf{x}^T}[\sigma _n^{ - 2}\textbf{I} - {(\sigma _n^2\textbf{I} + \sigma _s^2{\gamma ^2}\hat{\textbf{H}}{\hat{\textbf{H}}^T})^{ - 1}}]\textbf{x} = \sigma _s^2{\gamma ^2}\sigma _n^{ - 2}{\textbf{x}^T}\hat{\textbf{H}}{(\sigma _n^2\textbf{I} + \sigma _s^2{\gamma ^2}{\hat{\textbf{H}}^T}\hat{\textbf{H}})^{ - 1}}{\hat{\textbf{H}}^T}\textbf{x}. $$

When SNR is low, we have σ_n² >> σ_s²γ ²||H^TH||≈σ_s²γ ²||Ĥ^TĤ||. Then, by using Banach’s Lemma [35], it follows that:

(16)$${(\sigma _n^2\textbf{I} + \sigma _s^2{\gamma ^2}{\hat{\textbf{H}}^T}\hat{\textbf{H}})^{ - 1}} \approx \sigma _n^{ - 2}\textbf{I}. $$

Based on Eqs. (15) and (16), the detection metric of the LSD algorithm can be given by:

(17)$${M_{LSD}} = {\textbf{x}^T}\hat{\textbf{H}}{\hat{\textbf{H}}^T}\textbf{x}. $$

Accordingly, the detection probability and the false alarm probability of the LSD algorithm with threshold K_LSD are, respectively, given by:

(18)$$\begin{aligned} {P_{D,\textrm{ }LSD}} &= 1 - {F_{{\chi ^2}}}\left( {\frac{{{K_{LSD}} \cdot \textrm{tr}(\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}\boldsymbol{\Sigma})}}{{\textrm{tr}({{(\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}\boldsymbol{\Sigma})}^2})}};\frac{{{{\textrm{[tr}(\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}\boldsymbol{\Sigma})]}^2}}}{{\textrm{tr(}{{(\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}\boldsymbol{\Sigma})}^2})}}} \right)\\ {P_{F,\textrm{ }LSD}} &= 1 - {F_{{\chi ^2}}}\left( {\frac{{{K_{LSD}}N||\hat{\textbf{h}}||_{}^2}}{{\sigma_n^2||\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2}};\frac{{{N^2}||\hat{\textbf{h}}||_{}^4}}{{||\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2}}} \right) \end{aligned}. $$

Here, || · ||_Fro denotes the Frobenius norm of a matrix. Next, based on the NP criterion and Eq. (18), the decision threshold of the LSD algorithm can be derived as:

(19)$${K_{LSD}} = {N^{ - 1}}\sigma _n^2{{||\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2} / {||\hat{\textbf{h}}||_{}^2}} \cdot F_{{\chi ^2}}^{ - 1}(1 - {P_{F,\textrm{ }LSD}};{{{N^2}||\hat{\textbf{h}}||_{}^4} / {||\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2}}). $$

Based on Eqs. (17) and (19), it is evident that LSD can not only avoid matrix inverse operations, but also does not require a priori information of signal variance σ_s². The computational complexities of M_LSD and K_LSD are O(mN) and O(m²), respectively, significantly reduced from O(N³) of LLD. Although LSD is significantly simplified compared with LLD, we will show later that LSD can perform very close to LLD.

2.4 Location-aware near-blind detection (LND)

The LSD algorithm still requires a priori information of noise variance σ_n², which cannot be accurately estimated by an SU in practice, especially in a multipath channel. Thus, we further simplify the LSD algorithm and propose a location-aware near-blind detection (LND) algorithm, which does not require signal variance σ_s² or noise variance σ_n² as a priori knowledge. The detection metric of the LND algorithm is given by:

(20)$${M_{LND}} = {{({\textbf{x}^T}\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}\textbf{x})} / {({\textbf{x}^T}\textbf{x})}}. $$

By using this detection metric, the false alarm probability will not be affected by the uncertainty of noise variance σ_n² as long as the detection threshold is fixed. In other words, σ_n² is not required for an SU to determine its detection threshold. Based on Eq. (20), the detection probability and the false alarm probability with threshold K_LND are, respectively, given by:

(21)$$\begin{aligned} {P_{D,\textrm{ }LND}} &= 1 - {F_F}\left( {{{\sum\nolimits_{i = 1}^N {\lambda_i^ + } } / {\sum\nolimits_{i = 1}^N {\lambda_i^ - } }};{{{{\left( {\sum\nolimits_{i = 1}^N {\lambda_i^ + } } \right)}^2}} / {\sum\nolimits_{i = 1}^N {{{(\lambda_i^ + )}^2}} }},{{{{\left( {\sum\nolimits_{i = 1}^N {\lambda_i^ - } } \right)}^2}} / {\sum\nolimits_{i = 1}^N {{{(\lambda_i^ - )}^2}} }}} \right)\\ {P_{F,\textrm{ }LND}} &= 1 - {F_F}\left( {{{\sum\nolimits_{i = 1}^N {\omega_i^ + } } / {\sum\nolimits_{i = 1}^N {\omega_i^ - } }};{{{{\left( {\sum\nolimits_{i = 1}^N {\omega_i^ + } } \right)}^2}} / {\sum\nolimits_{i = 1}^N {{{(\omega_i^ + )}^2}} }},{{{{\left( {\sum\nolimits_{i = 1}^N {\omega_i^ - } } \right)}^2}} / {\sum\nolimits_{i = 1}^N {{{(\omega_i^ - )}^2}} }}} \right) \end{aligned}.$$

Here, F_F (x; d₁, d₂) denotes the CDF of an F-distribution with parameters d₁ and d₂, and λ_i and ω_i (i = 1, …, N) are the eigenvalues of D and P, respectively, where D = (ĤĤ^T – K_LNDI)Σ and P = ĤĤ^T – K_LNDI. The operators ‘⁺’ and ‘⁻’ are defined by a⁺ = max{a, 0} and a⁻ = max{–a, 0}.

Unlike LSD, we cannot explicitly derive K_LND from P_{F, LND} in Eq. (21). However, because P_{F, LND} is an increasing function of K_LND, we can adopt an iterative approach (e.g., bisection method [36]) to find K_LND given a target P_{F, LND}. The initial iteration range can be set as [µ_min, µ_max], where µ_min and µ_max are the smallest and the largest eigenvalues of ĤĤ^T, respectively. Generally, it is time-consuming to directly calculate the eigenvalues of P in each iteration. Thanks to the relationship Eig(P) = Eig(ĤĤ^T) – K_LND, where Eig(·) denotes all the eigenvalues of a matrix, we only need to calculate the eigenvalues of ĤĤ^T to find the eigenvalues of P. We further observe that ĤĤ^T is a banded symmetric Toeplitz matrix. In [37], an efficient numerical method has been provided to calculate the eigenvalues for such a type of matrix, whose computational complexity is O(mN²). Thus, in the LND algorithm, the total computational complexities of M_LND and K_LND are O(mN) and O(mkN²), respectively, where k is the number of iterations. By experimentally testing the LND algorithm, we note that in our study k = 10 is sufficient to achieve a target P_{F, LND}. Also, we will show later that the performance of the LND algorithm is almost not affected by the noise uncertainty when ignoring a priori information of noise variance.

Note that, in this work, we assume that PUs have high SNRs to guarantee their transmission quality. When an SU is close to PUs with a sufficiently high SNR, all the LLD, LSD, and LND algorithms can achieve good detection performance. When an SU is far from PUs with a low SNR, LLD, LSD, and LND are still useful for practical applications. For this, let us consider the following two practical scenarios: first, for a mobile SU with a low SNR, when it detects the absence of PUs, it can move closer to the LEDs for better SNR and access a high-speed transmission link; second, for a fixed SU with a low SNR, the transmitters at the LEDs can adjust their configurations to a lower level, e.g., reduce the transmission data rate or change the modulation format, to at least satisfy the SU’s transmission demand (even with a low-speed link).

2.5 Incorporation of cyclic-prefix-based detection (CD)

In an OFDM-based CVLC system, cyclic prefix (CP) can not only alleviate ISI, but also offer a cyclostationary property to facilitate spectrum sensing based on cyclic-prefix-based detection (CD) [13]. By incorporating CD in the proposed algorithms, we can further enhance the spectrum sensing accuracy. We assume that each sample vector x contains D OFDM symbols, and the number of samples in CP and data in each OFDM symbol are L_C and L_D, respectively. First, we represent x as:

(22)$$\textbf{x} = \textbf{HQ}\mathrm{\mathbf{\tilde{s}}}, $$

where

(23)$$\textbf{Q} = \left[ {\begin{array}{cccc} {{\textbf{Q}_0}}&{}&{}&{}\\ {}&{{\textbf{Q}_0}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\textbf{Q}_0}} \end{array}} \right],\quad {\textbf{Q}_0} = \left[ {\begin{array}{ll} {{\textbf{O}_{{L_C} \times {L_D}}}}&{{\textbf{I}_{{L_C} \times {L_C}}}}\\ {{\textbf{I}_{{L_D} \times {L_D}}}}&{{\textbf{O}_{{L_D} \times {L_C}}}} \end{array}} \right], $$

and $\mathrm{\mathbf{\tilde{s}}}$ is an L_d × D dimensional vector comprising data in D symbols, which approximately follows a normal distribution. According to the derivations in Sections 2.2 to 2.4, to incorporate CD, we only need to substitute H by HQ (or Ĥ by ĤQ) in all the algorithms. Taking the LSD algorithm as an example, the incorporation of CD in LSD will result in the following metric:

(24)$${M_{LSD + CD}} = {\textbf{x}^T}\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{\hat{\textbf{H}}^T}\textbf{x}. $$

Accordingly, the detection probability and the false alarm probability of the LSD algorithm incorporated with CD are, respectively, given by:

(25)$$\begin{aligned} {P_{D,\textrm{ }LSD + CD}} &= 1 - {F_{{\chi ^2}}}\left( {\frac{{{K_{LSD + CD}} \cdot \textrm{tr(}\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}\boldsymbol{\Omega})}}{{\textrm{tr}({{(\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}\boldsymbol{\Omega})}^2})}};\frac{{{{\textrm{[tr}(\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}\boldsymbol{\Omega})]}^2}}}{{\textrm{tr}({{(\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}\boldsymbol{\Omega})}^2})}}} \right)\\ {P_{F,\textrm{ }LSD + CD}} &= 1 - {F_{{\chi ^2}}}\left( {\frac{{{K_{LSD + CD}}||\hat{\textbf{H}}\textbf{Q}||_{\textrm{Fro}}^2}}{{\sigma_n^2||\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2}};\frac{{||\hat{\textbf{H}}\textbf{Q}||_{\textrm{Fro}}^4}}{{||\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2}}} \right) \end{aligned}, $$

where K_LSD_+CD is the decision threshold and Ω = σ_n²I + σ_s²γ ²HQQ^TH^T. Furthermore, the decision threshold of the LSD algorithm incorporated with CD can be derived as:

(26)$${K_{LSD + CD}} = \sigma _n^2\frac{{||\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2}}{{||\hat{\textbf{H}}\textbf{Q}||_{\textrm{Fro}}^2}} \cdot F_{{\chi ^2}}^{ - 1}(1 - {P_{F,\textrm{ }LSD + CD}};\frac{{||\hat{\textbf{H}}\textbf{Q}||_{\textrm{Fro}}^4}}{{||\hat{\textbf{H}}\textbf{Q}{\textbf{Q}^T}{{\hat{\textbf{H}}}^T}||_{\textrm{Fro}}^2}}). $$

Since the algorithm incorporated with CD can take advantage of CP, it is expected to outperform its original version. Of course, this is at the cost of more a priori knowledge and a higher computational complexity. To highlight the main differences between the proposed algorithms, Table 1 compares the algorithms in the aspects of a priori knowledge required, complexity, and detection performance.

Table 1. Main differences between the proposed algorithms.

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2.6 Influence of location error and receiver tilt

In the previous derivations, we assumed that an SU has the perfect information of its receiver location and orientation angle, which can ensure the estimated channel matrix Ĥ to be very close to a real one H. However, in practice, there can be a location error because the estimated coordinate is not always the same as the real coordinate. Also, when an SU’s receiver tilts randomly, there exists the error of its polar angle. These can cause the error of channel estimation. To express the influence of location error and receiver tilt, we construct an error matrix E and then add it to H:

(27)$$\hat{\textbf{H}} = \textbf{H} + \textbf{E}. $$

In E, there are two types of estimation errors. The first is the error of the amplitude of nonzero elements h_j (j = 1, …, m) in H, which is caused by both location error and receiver tilt according to Eqs. (6) and (9). The second is the error of the position of nonzero elements h_j (j = 1, …, m) in H, which is caused by location error according to Eqs. (6) and (9). In general, the second type of error is larger in the channel estimation, i.e., a larger ||E||, because it impacts more elements in H.

Based on the error matrix E, we can derive how the values of detection metrics M_LLD, M_LSD, and M_LND in Eqs. (12), (17), and (20) are, respectively, affected by the estimation error E (see Appendix B):

(28)$$|\Delta {M_{LLD}}|\le \frac{{\sigma _s^2}}{{\sigma _n^2}}||\textbf{x}|{|^2}||\textbf{E}||(2||\textbf{E}||+ ||\textbf{H}||){(1 - \frac{{\sigma _s^2}}{{\sigma _n^2}}||\textbf{H}{\textbf{H}^T}||)^{ - 1}}{(1 - \frac{{\sigma _s^2}}{{\sigma _n^2}}||\hat{\textbf{H}}{\hat{\textbf{H}}^T}||)^{ - 1}}, $$

(29)$$|\Delta {M_{LSD}}|\le ||\textbf{x}|{|^2}||\textbf{E}||(2||\textbf{E}||+ ||\textbf{H}||), $$

(30)$$|\Delta {M_{LND}}|\le ||\textbf{E}||(2||\textbf{E}||+ ||\textbf{H}||). $$

Here, ΔM denotes the change in M due to the estimation error E. According to Eqs. (28), (29), and (30), we note that a subtle error in H will lead to a small change in M (i.e., ||E||→0 ⇒ |ΔM|→0) for all the proposed LLD, LSD, and LND algorithms. Thus, a small receiver tilt will not cause severe performance degradation for all the three algorithms. Meanwhile, because LND only uses the multipath feature to decide PUs’ status with its energy level of received data samples normalized (according to Eq. (20)), while LLD and LSD both use the energy level of received data samples and the multipath feature to decide PUs’ status (according to Eqs. (12) and (17)), LND tends to be less robust against location error than LLD and LSD when location error induces a large ||E||.

3. Simulations and discussions

To evaluate the performance of the proposed spectrum sensing algorithms, we consider various spectrum sensing schemes in Table 1, in which the three proposed algorithms and two conventional schemes ED and CD are included.

For these schemes, we conducted simulation studies based on the following assumptions. In a considered CVLC cell, there are multiple LEDs, one or multiple PUs, and one SU. The SU senses whether the spectrum resource is in use by the PUs. All the LEDs are assumed to have a Lambertian radiation pattern [16]. The channel between LEDs and the SU follows the multipath characteristic, wherein the first indoor reflection is assumed. Here, although the multipath delay can often be equalized at the receiver with the aid of pilots or cyclic prefixes, as discussed in Section 1, when an SU is performing spectrum sensing, it cannot employ the conventional approaches to equalize the multipath signal. In addition, the multipath delay (around 0∼8 ns in our considered model) is not negligible compared with SU’s sampling interval t₀, which is 2 ns in our simulations. In this case, the delay and the sampling interval are of the same magnitude. Therefore, the multipath effect cannot be neglected in our spectrum sensing system.

When a PU is present, all the LEDs are assumed to simultaneously transmit the same DCO-OFDM signals with quadrature phase shift keying (QPSK) symbols over 64 subcarrier channels. For the scenarios with different numbers of LEDs, Figs. 2(a), 2(b), and 2(c) show the layouts of the LEDs on the ceiling based on the indoor coordinate. Figure 2(d) shows the side and corner locations of the SU on the receiving plane based on the indoor coordinate. The SU’s receiver is assumed to face upward in Figs. 3–9, and can be tilted in Fig. 10. The key parameters of the considered CVLC system are listed in Table 2. Other parameters for the SU’s receiver are the same as those in [15] and [17]. In our simulations, for each test point, we conducted Monte-Carlo simulations 5000 times to calculate the detection and false alarm probabilities.

Fig. 2. Schematic diagrams of: (a) one-LED layout; (b) two-LED layout; (c) four-LED layout; and (d) SU’s locations at the side and corner of the receiving plane.

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Fig. 3. ROC curves of the different schemes under the one-LED layout when SU is at (a) side (N = 240); (b) corner (N = 4000).

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Fig. 4. ROC curves of the different schemes under the two-LED layout when SU is at (a) side (N = 160); (b) corner (N = 3200).

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Fig. 5. ROC curves of the different schemes under the four-LED layout when SU is at (a) side (N = 80); (b) corner (N = 1600).

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Fig. 6. Comparison of detection probabilities under different SNRs when SU is at (a) side; (b) corner. (P_F = 0.1)

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Fig. 7. Influence of noise uncertainty on different spectrum sensing schemes when SU is at corner. (N = 2000, P_F = 0.1)

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Fig. 8. Distribution of detection probability vs. location error when SU is at side: (a) LLD (N = 80, P_F = 0.1); (b) LSD (N = 80, P_F = 0.1); and (c) LND (N = 1600, P_F = 0.1).

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Fig. 9. Distribution of detection probability vs. location error when SU is at corner: (a) LLD (N = 2000, P_F = 0.1); (b) LSD (N = 2000, P_F = 0.1); and (c) LND (N = 6000, P_F = 0.1).

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Fig. 10. Influence of receiver tilt on detection probabilities of LSD and LND when SU is at side: (a) α = 90^°, β = 30^°; and (b): α = 90^°, β = 60^°. (P_F= 0.1)

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Table 2. Key parameters of the CVLC system.

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3.1 Spectrum sensing with accurate location information

We first evaluate the spectrum sensing performance based on the assumption of accurate location information. Considering the different LED layouts, Figs. 3, 4, and 5 show both analytical and simulated receiver operating characteristic (ROC) curves [39] for the different schemes when the SU is at side and corner, respectively. An ROC curve demonstrates the relationship between the detection probability P_D and the false alarm probability P_F. A larger area below the curve represents a better detection accuracy. In each figure, we assume different sampling sizes N which are related to received SNR. In general, a lower SNR requires a larger N to achieve a desired detection probability. We note that all the simulated ROC curves can well match their corresponding analytical results, thereby verifying the effectiveness and accuracy of the analytical models of Eqs. (13), (18), (21), and (25).

Figure 3 shows the ROC curves of the one-LED layout. We note that LLD, LSD, and ED achieve the same sensing performance when the SU is at either corner or side. This is because, with a single LED, there is no multipath effect in the LOS channel. Thus, according to Eqs. (6) and (9), the estimated channel matrix Ĥ in LLD and LSD changes to a scalar matrix. Then, based on Eqs. (12) and (17), the detection metrics M_LLD and M_LSD change to the form of ax^Tx, where a is a constant. Here, these detection metrics are just equal to the detection metric of ED [15]; therefore, they achieve the same performance. In addition, both CD and LSD + CD algorithms show the highest detection probabilities because they exploit the information of CP. In contrast, LND shows the lowest detection probability. This is because, according to Eq. (20), the metric M_LND changes to a constant, which fails to provide any useful information for spectrum sensing. In summary, LLD and LSD are more suitable to a CVLC system with a single LED.

Figure 4 depicts the ROC curves of the two-LED layout. In Fig. 4(a), when the SU is at side, LLD and LSD achieve the same detection probability as that of ED, while LND shows the lowest detection probability, which is similar to the result of the one-LED layout. This is because here the two LEDs have the same distance from the SU, and thus there is no multipath effect for the LOS channel. Figure 4(b) shows the results when the SU is at corner, in which optical path difference between the two LEDs leads to multipath effect. When the false alarm probability P_F is 0.1, the detection probabilities P_{D, ED}, P_{D, LLD}, P_{D, LSD}, P_{D, LND}, and P_{D, CD} are 0.56, 0.61, 0.61, 0.26, and 0.68, respectively. Moreover, if LSD is incorporated with CD, its detection probability P_{D, LSD+CD} increases to 0.73. Thus, it is evident that the proposed location-aware spectrum sensing scheme can effectively improve sensing accuracy under a two-LED multipath channel.

Figure 5 shows the ROC curves of the four-LED layout. In Fig. 5(a), the multipath effect is relatively weak when the SU is at side. The corresponding detection probabilities P_{D, ED}, P_{D, LLD}, P_{D, LSD}, P_{D, LND}, P_{D, CD}, and P_{D, LSD+CD} are 0.76, 0.78, 0.78, 0.21, 0.84, and 0.86, respectively, when P_F = 0.1. When the SU is at corner, where multipath effect is strong, the detection probabilities P_{D, ED}, P_{D, LLD}, P_{D, LSD}, P_{D, LND}, and P_{D, CD} are 0.61, 0.71, 0.71, 0.40, and 0.72, respectively (see Fig. 5(b)), when P_F = 0.1. With the incorporation of CD in LSD, P_{D, LSD+CD} increases to 0.80. Compared with ED, the proposed LLD, LSD, and LSD + CD algorithms can improve the detection probabilities by 10%, 10%, and ∼20%, respectively, when multipath effect is strong. Meanwhile, compared with CD, LSD + CD can improve the detection probabilities by 8%. In addition, although LSD does not require a priori knowledge of signal noise and is simpler to implement, it achieves almost the same performance as that of LLD (see Figs. 3, 4, and 5). Thus, due to their simplicity, we will mainly focus on LSD and LND for performance evaluation in the following. Moreover, although CD outperforms LSD in a multipath channel, LSD is still competitive for CVLC since it is computationally simpler than CD and not limited to the modulation formats that require CP.

Figure 6 compares the detection probabilities of the different schemes under different SNRs when the SU is located at side and corner. For ED, LSD, and CD, the sampling size N and the LED layout are the same as those in Fig. 5. For LND, we specifically consider two different values of N. One is the same as the other schemes, and the other is multiplied by 20 and 3, respectively, at side and corner. In Fig. 6(a), when the SU is at side, LND with N = 1600 can achieve almost the same performance as that of CD with N = 80. Specifically, the detection probability of LND P_{D, LND} exceeds 0.8 when SNR is larger than −4 dB. Figure 6(b) shows the results when the SU is at corner. We see that here, LSD can improve the detection probability by 10% compared with ED when SNR is −13 dB. This means that a significant improvement can be achieved by the proposed spectrum sensing schemes. Moreover, LND with N = 6000 outperforms ED with N = 2000. Specifically, P_{D, LND} exceeds 0.8 when SNR is larger than −11 dB. Because LND is a near-blind detection algorithm, it requires a larger sampling size N and therefore takes a longer sensing time. With a sufficiently long sensing time, LND can achieve a sensing accuracy close to those of the other schemes with the least a priori knowledge.

Next, we compare the performance of ED, LLD, LSD, LND, and CD under different noise uncertainties. We consider the scenario when the SU is at corner and there are four LEDs. Here for case studies, we assume that the noise uncertainties are 0.5 dB and 1 dB. Other parameters are the same as those in Fig. 6(b). Figure 7 shows the detection probabilities under different SNRs. We note that, due to the noise uncertainty, the detection probabilities of all the schemes excluding LND degrade significantly when SNR is low. This is because here, noise accounts for a dominant part in the energy level of received data samples. All the schemes excluding LND use the energy level to detect the presence of PUs. Thus, the imprecise knowledge of noise variance can severely influence the energy level, consequently degrading the sensing accuracy. Specifically, for the conventional ED, CD, and the proposed LLD and LSD, to achieve a detection probability of 0.8, their SNR penalties in comparison with LND are around 4∼5 dB and 6∼7 dB respectively for 0.5-dB and 1-dB noise uncertainties. In contrast, because LND does not require any a priori knowledge of noise variance, it is immune to the noise uncertainty and its performance does not degrade. Specifically, under 0.5-dB and 1-dB noise uncertainties, LND can outperform all the other schemes at SNR ≤ −7 dB and SNR ≤ −5 dB, respectively. Thus, LND is promising for spectrum sensing when an SU cannot obtain accurate noise information.

From Figs. 3–7, we see that the channel matrix Ĥ can help characterize the multipath feature from the received signal, thereby improving the sensing accuracy effectively. With the help of Ĥ, LLD and LSD can significantly improve the detection probability for a strong multipath channel, and LND can effectively detect a PU’s signal without any a priori knowledge of the signal. Therefore, the location information is important for our proposed algorithms.

3.2 Spectrum sensing with inaccurate location information

For location-aware spectrum sensing, location error and receiver tilt can affect the accuracy of the estimated CSI, leading to degraded sensing accuracy. In this section, based on the four-LED layout, we simulate to evaluate how the location error and the receiver tilt can impact the sensing performance of the proposed spectrum sensing schemes.

Figure 8 shows the distribution of detection probabilities of LLD, LSD, and LND when the SU is at side, in which the horizontal location error is assumed to be distributed in a 1 m × 1 m square centered at the receiver location. That is to say, the location error occurs in both X- and Y- directions, which ranges from −0.5 m to 0.5 m. In Figs. 8(a) and 8(b), we note that both LLD and LSD can achieve a detection probability of 0.78 when the SU’s location information is precise. However, the detection accuracy is subtly degraded with the increase of location error. For example, when the location error deviates from (0, 0) to (0, 0.5), the detection probabilities of LLD and LSD gradually decrease from 0.78 to 0.76 and 0.75, respectively. Within the 1 m × 1 m square, for LLD, about 64% and 100% areas can reach a detection probability of 0.77 and 0.70, respectively; for LSD, about 50% and 100% areas can reach a detection probability of 0.75 and 0.70, respectively. Thus, both LLD and LSD are robust to different degrees of location error, which confirms our prediction in Section 2.6. In Fig. 8(c), we note that LND can achieve a detection probability of 0.83 when its location information is precise. When the location error is small and deviates from (0, 0) to (0, 0.1), the detection probability slightly decreases from 0.83 to 0.82. However, when the location error becomes larger and deviates from (0, 0.1) to (0, 0.5), the detection probability decreases significantly from 0.82 to 0.11. We also note that only about 35% and 44% areas of the square can reach a detection probability of 0.80 and 0.70, respectively. Thus, LND is only robust to a small location error, but sensitive to a large location error, which also confirms our prediction in Section 2.6. Similar observations can be made for the results in Fig. 9, where the SU is at corner. Therefore, it is confirmed that LND is only robust to a small location error, but sensitive to a large location error, while LLD and LSD are robust to different degrees of location error.

Based on results in Figs. 8 and 9, we further compare their average detection probabilities in Table 3, in which the horizontal location error is distributed within a 0.2 m × 0.2 m, 0.4 m × 0.4 m, and 1 m × 1 m square centered at the receiver location. We also include ED for comparison. We find that, even when the location error is within a 1 m × 1 m square, LLD and LSD can still outperform ED. Moreover, when an SU is at side, the average detection probabilities of LLD, LSD, and LND are 0.77, 0.77 and 0.70, respectively, when the location error is within a 0.4 m × 0.4 m square. Since indoor positioning techniques have been mature to reach a positioning accuracy of 0.2 m [3,40], all LLD, LSD, and LND are effective at side when the location error in X- or Y- coordinate is less than 0.2 m. However, when an SU is at corner, the average detection probabilities of LLD, LSD, and LND are 0.66, 0.66 and 0.48, respectively, when the location error is within the 0.2 m × 0.2 m square. Therefore, at the corner, LLD and LSD are more effective, and LND requires a longer sensing time to ensure a high detection accuracy.

Table 3. Average detection probabilities of LSD and LND when location error is within a certain range.

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We also evaluate how an SU’s receiver tilt impacts the spectrum sensing performance of the different schemes. Here, an SU is assumed to be at side and its receiver’s polar angle β changes from 0^° to 30^° and 0^° to 60^°. In addition, for performance comparison, we consider two scenarios: the SU does not know the value of β and the SU knows the value of β. Figure 10 shows the detection probabilities of LSD and LND under different SNRs. We note that, when the SU’s receiver orientates towards the direction of β = 30^° or 60^°, the spectrum sensing performance of both LSD and LND are almost the same for the above two scenarios, where the difference of the detection probabilities is smaller than 0.02. This means that to achieve a high detection probability, it is not necessary for the SU to know its receiver’s polar angle. This is because the receiver tilt only causes a small estimation error in H, which will not severely reduce the effectiveness of the selected detection metrics and decision thresholds in both algorithms. Thus, even if an SU has no a priori information of its receiver’s polar angle, LSD and LND are still robust to the receiver tilt with almost no performance degradation.

4. Conclusion

We proposed a location-aware spectrum sensing scheme for a CVLC system, where the multipath characteristic of the CVLC channel can be effectively estimated based on the location information of an SU. Based on different a priori information, we proposed three algorithms to detect PUs’ signals and also developed their corresponding analytical models to optimize their spectrum sensing performance. Simulation studies showed that, compared with the conventional ED scheme, the LLD and LSD algorithms can significantly improve the detection probabilities by up to 10% when an SU is at corner. Moreover, the sensing accuracy can be further enhanced when LSD is incorporated with CD, which can improve the detection probabilities by up to 20% and 8% compared with ED and CD, respectively. Also, when an SU cannot obtain accurate noise information, the LND algorithm is efficient to outperform the other algorithms when the noise uncertainty is over 0.5 dB. Finally, the evaluation on the influence of location error and receiver tilt indicated that both LSD and LND are robust to the receiver tilt, and LSD is robust to the location error.

Appendix A. Derivations of Eqs. (13), (18), (21), and (25)

To derive Eqs. (13), (18), (21), and (25), we need the following lemma.

Lemma 1 [41]: If y ∼ $\mathcal{N}$ (0, Σ) and A is a positive definite matrix, then the random variable y^TAy approximately follows a chi-square distribution of gχ² (h), where g = tr((AΣ)²) / tr(AΣ) and h = [tr(AΣ)]² / tr((AΣ)²).

Since the derivations of Eqs. (13), (18), and (25) are similar, here we just representatively derive Eq. (18). According to Eq. (9), Ĥ has a full-row rank, so ĤĤ^T is positive definite. Thus, Lemma 1 is applicable to the random variable x^TĤĤ^Tx. The expression of P_D in Eq. (18) can be derived by applying Lemma 1 directly. Note that, since tr(ĤĤ^T) = ||Ĥ||² Fro and ||Ĥ||²Fro = N||ĥ||², the expression of P_F is also obtainable.

We proceed next to derive the expression of P_D in Eq. (21). Note that the derivation of P_F is similar. Recall that P_D = Pr(M > K | $\mathcal{H}_1$) and x | $\mathcal{H}_1$ ∼ $\mathcal{N}$ (0, Σ), where Σ = σ_n²I + σ_s²γ ²HH^T. It is obvious that Σ is positive definite and can be decomposed as Σ = Σ^1/2Σ^1/2, where Σ^1/2 is also positive definite. Thus, Σ^−1/2x follows a normal distribution (i.e., Σ^−1/2x ∼ $\mathcal{N}$ (0, I)). Let the random vector y = Σ^−1/2x, so we have:

(31)$${{({\textbf{x}^T}\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}\textbf{x})} / {({\textbf{x}^T}\textbf{x}}}) > K \Leftrightarrow {\textbf{x}^T}(\hat{\textbf{H}}{\hat{\textbf{H}}^T} - K\textbf{I})\textbf{x} > 0 \Leftrightarrow {\textbf{y}^T}{\boldsymbol{\Sigma}^{\frac{1}{2}}}(\hat{\textbf{H}}{\hat{\textbf{H}}^T} - K\textbf{I}){\boldsymbol{\Sigma}^{\frac{1}{2}}}\textbf{y} > 0. $$

Since Σ^1/2(ĤĤ^T – KI)Σ^1/2 is symmetric, after spectrum decomposition, we have Σ^1/2(ĤĤ^T – KI)Σ^1/2 = C^TΛC, where C is orthogonal and Λ = Diag{λ₁, …, λ_s, λ_s₊₁, …, λ_N}. Here, λ₁, …, λ_s are all positive, and λ_s₊₁, …, λ_N are all negative. Let the random vector z = Cy, then it follows that:

(32)$$\begin{aligned} {\textbf{y}^T}{\boldsymbol{\Sigma}^{\frac{1}{2}}}(\hat{\textbf{H}}{{\hat{\textbf{H}}}^T} - K\textbf{I}){\boldsymbol{\Sigma}^{\frac{1}{2}}}\textbf{y} > 0 \Leftrightarrow {\textbf{z}^T}\Lambda \textbf{z} > 0 &\Leftrightarrow {\textbf{z}_1}^T{\Lambda ^ + }{\textbf{z}_1} > {\textbf{z}_2}^T{\Lambda ^ - }{\textbf{z}_2}\\ &\Leftrightarrow {{({\textbf{z}_1}^T{\Lambda ^ + }{\textbf{z}_1})} / {({\textbf{z}_2}^T{\Lambda ^ - }{\textbf{z}_2})}} > 1 \end{aligned}. $$

Here, Λ⁺ = Diag{λ₁, …, λ_s}, Λ⁻ = Diag{−λ_s₊₁, …, −λ_N}, z₁ and z₂ are sub-vectors of z, which are comprised of the first s elements and the last N – s elements of z.

After applying Lemma 1, z₁^TΛ⁺ z₁ and z₂^TΛ⁻ z₂ follow chi-square distributions, given by:

(33)$$\begin{aligned} &{\textbf{z}_1}^T{\Lambda ^ + }{\textbf{z}_1}\sim \frac{{\sum\nolimits_{i = 1}^s {\lambda _i^2} }}{{\sum\nolimits_{i = 1}^s {\lambda _i^{}} }}{\chi ^2}\left( {\frac{{{{\left( {\sum\nolimits_{i = 1}^s {\lambda_i^{}} } \right)}^2}}}{{\sum\nolimits_{i = 1}^s {\lambda_i^2} }}} \right)\\ &{\textbf{z}_2}^T{\Lambda ^ - }{\textbf{z}_2}\sim \frac{{\sum\nolimits_{i = s + 1}^N {\lambda _i^2} }}{{ - \sum\nolimits_{i = s + 1}^N {\lambda _i^{}} }}{\chi ^2}\left( {\frac{{{{\left( {\sum\nolimits_{i = s + 1}^N {\lambda_i^{}} } \right)}^2}}}{{\sum\nolimits_{i = s + 1}^N {\lambda_i^2} }}} \right) \end{aligned}. $$

According to the definitions of y and z, z₁ and z₂ are independent. Thus, it follows that:

(34)$$\frac{{{\textbf{z}_1}^T{\Lambda ^ + }{\textbf{z}_1}}}{{{\textbf{z}_2}^T{\Lambda ^ - }{\textbf{z}_2}}}\sim \frac{{ - \sum\nolimits_{i = s + 1}^N {\lambda _i^{}} }}{{\sum\nolimits_{i = 1}^s {\lambda _i^{}} }}F\left( {\frac{{{{\left( {\sum\nolimits_{i = 1}^s {\lambda_i^{}} } \right)}^2}}}{{\sum\nolimits_{i = 1}^s {\lambda_i^2} }},\frac{{{{\left( {\sum\nolimits_{i = s + 1}^N {\lambda_i^{}} } \right)}^2}}}{{\sum\nolimits_{i = s + 1}^N {\lambda_i^2} }}} \right). $$

Finally, note that Σ^1/2(ĤĤ^T – KI)Σ^1/2 and (ĤĤ^T – KI)Σ own the same eigenvalues. Based on Eqs. (32) and (34), the expression of P_D in Eq. (21) is obtained.

Appendix B. Derivations of Eqs. (28), (29), and (30)

We first show the derivation process of Eq. (28):

(35)$$\begin{aligned} |\Delta {M_{LLD}}| &=|{\textbf{x}^T}[\sigma _n^{ - 2}\textbf{I} - {(\sigma _n^2\textbf{I} + \sigma _s^2\hat{\textbf{H}}{{\hat{\textbf{H}}}^T})^{ - 1}}]\textbf{x} - {\textbf{x}^T}[\sigma _n^{ - 2}\textbf{I} - {(\sigma _n^2\textbf{I} + \sigma _s^2\textbf{H}{\textbf{H}^T})^{ - 1}}]\textbf{x}|\\ & =|{\textbf{x}^T}[{(\sigma _n^2\textbf{I} + \sigma _s^2\textbf{H}{\textbf{H}^T})^{ - 1}} - {(\sigma _n^2\textbf{I} + \sigma _s^2\hat{\textbf{H}}{{\hat{\textbf{H}}}^T})^{ - 1}}]\textbf{x}|\\ & =|{\textbf{x}^T}{(\sigma _n^2\textbf{I} + \sigma _s^2\hat{\textbf{H}}{{\hat{\textbf{H}}}^T})^{ - 1}}[(\sigma _n^2\textbf{I} + \sigma _s^2\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}){(\sigma _n^2\textbf{I} + \sigma _s^2\textbf{H}{\textbf{H}^T})^{ - 1}} - \textbf{I}]\textbf{x}|\\ & =|{\textbf{x}^T}{(\sigma _n^2\textbf{I} + \sigma _s^2\hat{\textbf{H}}{{\hat{\textbf{H}}}^T})^{ - 1}}[\sigma _s^2(\hat{\textbf{H}}{{\hat{\textbf{H}}}^T} - \textbf{H}{\textbf{H}^T}){(\sigma _n^2\textbf{I} + \sigma _s^2\textbf{H}{\textbf{H}^T})^{ - 1}}]\textbf{x}|\\ & =\sigma _s^2 |{\textbf{x}^T}{(\sigma _n^2\textbf{I} + \sigma _s^2\hat{\textbf{H}}{{\hat{\textbf{H}}}^T})^{ - 1}}[(\textbf{E}{\textbf{H}^T} + \textbf{H}{\textbf{E}^T} + \textbf{E}{\textbf{E}^T}){(\sigma _n^2\textbf{I} + \sigma _s^2\textbf{H}{\textbf{H}^T})^{ - 1}}]\textbf{x}|\\ &\le \frac{{\sigma _s^2}}{{\sigma _n^2}} ||\textbf{x}|{|^2}||\textbf{E}||(2||\textbf{E}||+ ||\textbf{H}||)||{(\textbf{I} + \frac{{\sigma _s^2}}{{\sigma _n^2}} \hat{\textbf{H}}{{\hat{\textbf{H}}}^T})^{ - 1}}||||{(\textbf{I} + \frac{{\sigma _s^2}}{{\sigma _n^2}} \textbf{H}{\textbf{H}^T})^{ - 1}}||\\ &\le \frac{{\sigma _s^2}}{{\sigma _n^2}}||\textbf{E}||||\textbf{x}|{|^2}(2||\textbf{E}||+ ||\textbf{H}||){(1 - \frac{{\sigma _s^2}}{{\sigma _n^2}}||\textbf{H}{\textbf{H}^T}||)^{ - 1}}{(1 - \frac{{\sigma _s^2}}{{\sigma _n^2}}||\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}||)^{ - 1}} \end{aligned}$$

where the last step is based on Banach’s Lemma [36].

The derivations of Eqs. (29) and (30) are straightforward. We only show the former:

(36)$$\begin{aligned} |\Delta {M_{LSD}}| = |{\textbf{x}^T}\hat{\textbf{H}}{{\hat{\textbf{H}}}^T}\textbf{x} - {\textbf{x}^T}\textbf{H}{\textbf{H}^T}\textbf{x}| &= |{\textbf{x}^T}(\textbf{H}{\textbf{E}^T} + \textbf{E}{\textbf{H}^T} + \textbf{E}{\textbf{E}^T})\textbf{x}| \\ & \le ||\textbf{x}|{|^2}||\textbf{E} ||(2||\textbf{E}||+ ||\textbf{H}||) \end{aligned}. $$

Funding

National Natural Science Foundation of China (62001319); Open Fund of IPOC (BUPT) (IPOC2020A009); Priority Academic Program Development of Jiangsu Higher Education Institutions; Support from Jiangsu Engineering Research Center of Novel Optical Fiber Technology and Communication Network.

Acknowledgments

Most of this work was conducted at Soochow University when Zile Jiang was visiting there.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Schemes	References	σ_n²	σ_s²	Location Information	L_C and L_D	Computational Complexity		Detection Performance
Schemes	References	σ_n²	σ_s²	Location Information	L_C and L_D	M	K	Detection Performance
Proposed LLD	Section 2.2	√	√	√	×	O(N³)	O(N³)	Optimal, with σ_n², σ_s², and location information
Proposed LSD	Section 2.3	√	×	√	×	O(mN)	O(m²)	Sub-optimal, with reduced a priori knowledge
Proposed LND	Section 2.4	×	×	√	×	O(mkN²)	O(mN²)	Moderate, but immune to noise uncertainties
Proposed LSD + CD	Section 2.5	√	×	√	√	O(mN)	O(m²)	Outperform LSD and CD with extra a priori knowledge
ED	Our work [15]	√	×	×	×	O(N)	O(1)	Moderate, with the lowest complexity
CD	RF-based work [13]	√	√	×	√	O(N³)	O(N³)	Optimal, with σ_n², σ_s², L_C, and L_D

Room size (length × width × height)	7 m × 7 m × 3 m
Height of receiving plane	0.85 m
Launch power of each LED	16 Watt (layout 1)
	8 Watt (layout 2)
	4 Watt (layout 3)
Modulation index	0.1 [38]
LED Lambertian order	1
LED semi-angle at half power	60°
*FOV at the receiver (ψ_c)*	150°
*Effective area of photo-detector (A_r)*	10⁻⁴ m² [16,18]
Receiver responsivity (γ)	0.35 A/W
Reflection coefficient of wall	0.83
*Gain of the optical filter (T_s)*	1
Refractive index of the optical concentrator (g)	1.5
OFDM bit rate	500 Mb/s
Sampling interval (t₀)	2 ns

Schemes	References	σ_n²	σ_s²	Location Information	L_C and L_D	Computational Complexity		Detection Performance
Schemes	References	σ_n²	σ_s²	Location Information	L_C and L_D	M	K	Detection Performance
Proposed LLD	Section 2.2	√	√	√	×	O(N³)	O(N³)	Optimal, with σ_n², σ_s², and location information
Proposed LSD	Section 2.3	√	×	√	×	O(mN)	O(m²)	Sub-optimal, with reduced a priori knowledge
Proposed LND	Section 2.4	×	×	√	×	O(mkN²)	O(mN²)	Moderate, but immune to noise uncertainties
Proposed LSD + CD	Section 2.5	√	×	√	√	O(mN)	O(m²)	Outperform LSD and CD with extra a priori knowledge
ED	Our work [15]	√	×	×	×	O(N)	O(1)	Moderate, with the lowest complexity
CD	RF-based work [13]	√	√	×	√	O(N³)	O(N³)	Optimal, with σ_n², σ_s², L_C, and L_D

Room size (length × width × height)	7 m × 7 m × 3 m
Height of receiving plane	0.85 m
Launch power of each LED	16 Watt (layout 1)
	8 Watt (layout 2)
	4 Watt (layout 3)
Modulation index	0.1 [38]
LED Lambertian order	1
LED semi-angle at half power	60°
*FOV at the receiver (ψ_c)*	150°
*Effective area of photo-detector (A_r)*	10⁻⁴ m² [16,18]
Receiver responsivity (γ)	0.35 A/W
Reflection coefficient of wall	0.83
*Gain of the optical filter (T_s)*	1
Refractive index of the optical concentrator (g)	1.5
OFDM bit rate	500 Mb/s
Sampling interval (t₀)	2 ns

Location-aware spectrum sensing for cognitive visible light communications over multipath channels

Abstract

1. Introduction

2. Location-aware spectrum sensing

2.1 Working principle

2.2 Location-aware likelihood-ratio-test detection (LLD)

2.3 Location-aware semi-blind detection (LSD)

2.4 Location-aware near-blind detection (LND)

2.5 Incorporation of cyclic-prefix-based detection (CD)

2.6 Influence of location error and receiver tilt

3. Simulations and discussions

3.1 Spectrum sensing with accurate location information

3.2 Spectrum sensing with inaccurate location information

4. Conclusion

Appendix A. Derivations of Eqs. (13), (18), (21), and (25)

Appendix B. Derivations of Eqs. (28), (29), and (30)

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (3)

Equations (36)

Optics Express

Range of Location Error	ED		LLD		LSD		LND
Range of Location Error	Side	Corner	Side	Corner	Side	Corner	Side	Corner
No location error	0.76	0.61	0.78	0.72	0.78	0.72	0.83	0.65
0.2 m × 0.2 m square			0.78	0.66	0.77	0.66	0.82	0.48
0.4 m × 0.4 m square			0.77	0.64	0.77	0.64	0.70	0.43
1 m × 1 m square			0.77	0.62	0.76	0.62	0.50	0.35