ML in Visible light communication

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aDepartment of Electrical Engineering, University of Guilan, Rasht, Iran

bOptical Communications Research Group, Faculty of Engineering and Environment, Northumbria University, Newcastle, UK

A R T I C L E I N F O

Keywords:

Visible light communications (VLC) Carrier-less amplitude and phase (CAP) modulation

Deep learning (DL)

A B S T R A C T

We propose in this paper a carrier-less amplitude and phase visible light communications (VLC) system with deep learning (DL)-based post-equalizer (EQ) to significantly increase the transmission data rate. The proposed system is analyzed for various conditions including modulation order, transmitted signal bandwidth, and non- line of sight VLC channel. Results show that the highest data rate and spectral efficiency of 100 Mb/s and 4.67 b/s/Hz are achieved for the modulation order and signal bandwidth of 64 and 25 MHz, respectively.

In addition, we compare the performance and complexity of the proposed system with different types of EQs including least mean square and Volterra series. The study shows the DL-based EQ is qualified to mitigate mixed linear and nonlinear impairments by providing improved bit error rate performance compared to the other EQs for all modulation orders and the transmitted signal bandwidth.

1. Introduction

Optical wireless communications (OWC) use visible light (VL), in- frared (IR), and ultraviolet (UV) spectral bands to meet some of the demands for wireless connectivity in fifth-generation (5G) and 6G networks. OWC, as a complementary technology to the radio frequency (RF) wireless systems, has unique features such as almost unlimited bandwidth (BW), no spectrum authorization and regulations, much safer to the environment, higher security in the physical layer (PHY), higher energy efficiency and improved sustainability [1]. In the VL band OWC, known as VL communications (VLC), uses white light emitting diodes (LEDs) and photodetectors (PDs) as the transmitter (Tx) and the receiver (Rx); this offers simultaneous illumination and data communications [2]. However, white LEDs have some limitations that may cause VLC systems to underperform in practical applications.

Blue LEDs with phosphor and RGB (red, green, and blue) LEDs are the two most widely used methods of producing white light. Even though, RGB LEDs offer higher data rates,𝑅_𝑑, blue LEDs with phosphor coat- ing have simpler implementation and lower cost. However, the long responsivity of phosphor limits the LED modulation bandwidth,𝐵_LED, to only a few MHz, which in turn greatly constrains the achievable transmission capacity of the system. Moreover, white LEDs are another source of nonlinearity in VLC systems, leading to signal distortion and intersymbol interference (ISI) [3,4].

A common solution to mitigate these limitations is to employ linear and nonlinear equalizers (EQs) including adaptive least mean square

∗ Corresponding author.

E-mail address: [email protected](G. Baghersalimi).

(LMS) – the most widely used, Volterra series, and deep learning (DL)- based filters. As (i) in [5] it was shown how an adaptive EQ with the LMS algorithm could suppress the ISI in indoor VLC systems; (ii) in [6], it was shown that a Rx with a decision feedback EQ (DFE) with nonlinear Volterra feed-forward section could effectively mitigate the effects of LED’s nonlinearity with improved performance by up to 5 dB in terms of optical power compared with a simple DFE; and (iii) in [7]

it was demonstrated that higher𝑅_𝑑 of 170 Mb/s could be achieved by employing artificial neural network (ANN)-based equalization using white LEDs.

DL is a powerful subfield of machine learning (ML), which can learn, recognize, and predict complex patterns among high-dimensional input information data [8]. The DL methods have been used in many application domains including speech recognition, computer vision, self-driving cars, natural language processing, predictive forecasting, and communication systems [9–11]. For example, authors in [12,13]

introduced several DL applications and improvements in the PHY of communication links. In [14], a deep neural network (DNN) was employed as an accurate tool for channel estimation to reduce both distortion and interference. Authors in [15] reviewed the challenges and applications of using DL in VLC, whereas in [16] DL was utilized to reduce flickering and increase the dimming level in a VLC system. In [17], the authors determined an autoencoder to mitigate the nonlinearity of the LED and enhance the system bit error rate (BER) performance.

In [18], a DNN-based signal detector namely bi-directional long-short term memory for VLC links with 25 Mbps non-line of sight (NLOS)

https://doi.org/10.1016/j.optcom.2022.128741

Received 7 April 2022; Received in revised form 18 June 2022; Accepted 5 July 2022 Available online 8 July 2022

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Fig. 1. Schematic block diagram of the proposed CAP-VLC system with DL-based post-EQ.

and 50 Mbps NLOS using the second-order and first-order reflection, respectively, were reported with the BER of < 1 × 10⁻⁴. In [19], an orthogonal frequency division multiplexing (OFDM)-based optical quadrature spatial modulation for multiple-input multiple-output OWC with DNN-aided detection was proposed with a successful outcome.

There are different widely used spectrally efficient modulation schemes in intensity modulation/direct detection (IM/DD) optical systems. In this work, we have adopted the carrier-less amplitude and phase (CAP) modulation scheme for VLC. To the best of the authors’

knowledge, the DL-based post-EQ has not been used in CAP-VLC systems. Therefore, inspired by many of the aforementioned research works, the effectiveness of applying DL for post-EQ in an indoor CAP- VLC system is investigated in this work to increase𝑅_𝑑 and improve the BER performance. In our study, we successfully demonstrate 𝑅_𝑑 of 100 Mb/s, which to our knowledge is the highest 𝑅_𝑑 for a single band CAP-VLC system using a single white LED so far. In addition, we compare the BER performance of the proposed system with two common equalization techniques of Volterra series and adaptive LMS algorithm.

The rest of the paper is organized as follows. In Section 2, an overview of the DL-based indoor VLC-CAP system is explained. In Section 3, the most important results are presented and discussed.

Finally, Section4is allocated to conclude the paper.

2. System overview

Fig. 1illustrates the schematic block diagram of the proposed CAP- VLC system with DL-based post-EQ which is explained comprehensively as follows. Note, we assume that each LED follows the Lambertian radiation pattern.

2.1. The Tx

At the Tx, a random data bit stream {𝐼_𝑛} is mapped into the quadrature amplitude modulation (QAM) format prior to being applied to the CAP modulator. CAP is a spectrally efficient and low-cost technique, which can be implemented with ease, due to its simplicity and the lack of need for fast Fourier transform (FFT) and inverse FFT (IFFT) as required in OFDM. Like the QAM format, except that instead of using a local oscillator, CAP generates carrier frequencies using two orthogonal digital filters whose impulse responses are Hilbert transform pairs [20–

23]. The mapped complex data are then up-sampled by a specific factor 𝑛_𝑠 =⌈2 (1 +𝛽)⌉, where⌈.⌉is the ceiling function and𝛽is the roll-of factor of the square root raised cosine (SRRC) filter. Next, the real and imaginary elements of the up-sampled signal are separated and sent to

the digital in-phase (I) and quadrature (Q) SRRC filters, respectively, which are given as [24]:

𝑓_𝐼(𝑡) =

⎡⎢

⎢⎢

⎢⎣

sin [𝛾(1 −𝛽)] + 4𝛽(_𝑡

𝑇𝑠

) cos [𝛾𝛿]

𝛾 [

1 −(_4𝛽𝑡

𝑇𝑠

)2]

⎤⎥

⎥⎥

⎥⎦

.cos [𝛾𝛿] (1)

𝑓_𝑄(𝑡) =

⎡⎢

⎢⎢

⎢⎣

sin [𝛾(1 −𝛽)] + 4𝛽(

𝑡 𝑇𝑠

) cos [𝛾𝛿]

𝛾 [

1 −(_4𝛽𝑡

𝑇_𝑠

)2]

⎤⎥

⎥⎥

⎥⎦

.sin[𝛾𝛿] (2)

where𝑇_𝑠is the symbol duration,𝛾=𝜋𝑡∕𝑇_𝑠, and𝛿= 1 +𝛽.

The outputs of the digital finite impulse response (FIR) filters are summed together to produce the desired CAP signal, which is given by [24]:

𝑠(𝑡) =𝑠_𝐼(𝑡)⊗ 𝑓_𝐼(𝑡) −𝑠_𝑄(𝑡)⊗ 𝑓_𝑄(𝑡) (3) where𝑠_𝐼(𝑡)and𝑠_𝑄(𝑡)are up-sampledIandQcomponents, respectively and⊗indicates the time domain convolution. Finally,s (t)is scaled and a direct current (DC)-bias is added to it which makes it nonnegative within the dynamic range of the LED prior to IM of the light source as follow [25]:

𝐼_𝑖𝑛(𝑡) =𝐼_𝐷𝐶+𝛼𝑠(𝑡), (4)

where𝛼is the scaling factor and defined as: 𝛼= 𝑀 𝐼×𝐼_max

(𝑀 𝐼+ 1) × max{𝑠(𝑡)} (5)

where𝐼_max=𝐼_DC+ 0.5𝐼_PPis the maximum value of𝐼_in,𝐼_DC is the DC- bias current,𝐼_PPis the peak-to-peak current, and𝑀 𝐼is the modulation index, which is defined as:

𝑀 𝐼=𝐼_max−𝐼_DC

𝐼_DC (6)

The Hammerstein method is adopted to model LED characteristics in this study. It includes a nonlinear function and a linear component to represent the LED with the limited bandwidth, which are characterized as a third-order polynomial and a first-order low pass Butterworth filter, respectively, as given by [25]:

𝑓_NL(𝑥) =

⎧⎪

⎨⎪

⎩

0.1947, 𝑥 <0.1

0.2855𝑥³− 1.0886𝑥²+ 2.0565𝑥− 0.0003, 0.1≤𝑥 <1 1.2531, 𝑥≥1

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ℎ_LED(𝑡) =𝑒^−2𝜋𝑓^LED^𝑡 (8)

where𝑓_𝐿𝐸𝐷is the 3-dB cut-off frequency of the LED. The LED transmit output power is defined as:

𝑃_out(𝑡) =𝑓_NL( 𝐼_in(𝑡))

⊗ ℎ_LED(𝑡) (9)

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the Tx and the Rx, respectively.𝑇_𝑠(𝜓)and𝑔(𝜓)are the optical filter and non-imaging concentrator gain, respectively. For the NLOS paths, every wall is divided into𝐿_𝑅small squares, which can be assumed as both the Tx and Rx.ℎ(

𝑡, 𝑠, 𝜖_𝑛^𝑟)

represents the NLOS impulse response between the main Tx and the reflecting element𝜖_𝑛^𝑟, which acts as a Rx.

ℎ( 𝑡, 𝜖_𝑛^𝑆, 𝐷)

is the impulse response between𝜖_𝑛^𝑆 (i.e., acting as a Tx) and the main Rx,𝐷. It is worth to mention that, only a single reflection per wall is considered in this study and other higher order reflections are ignored since they contribute very little to the total received optical power [24]

2.3. The Rx

At the Rx side, the received signal is detected by an optical Rx composed of a single PD and a trans-impedance amplifier (TIA). The regenerated electrical received signal is given by:

𝑦(𝑡) =𝑠(𝑡) ∗ℎ_𝑐(𝑡) +𝑛(𝑡) (13)

where𝑛(𝑡)is the additive white Gaussian noise with the power𝑃_𝑛 = 𝑁₀𝐵_Rx, 𝑁₀ is the noise power spectral density, and𝐵_Rx is the bandwidth of the Rx. Note,𝑛(𝑡)is mostly dominated by the ambient light induced shot noise. Following DC removal, a DL-based EQ is employed, which is explained in Subsection D in details. The equalized signal is applied to a set of inverse matched filters 𝑔_𝐼(𝑡) = 𝑓_𝐼(−𝑡) and 𝑔_𝑄(𝑡) = 𝑓_𝑄(−𝑡) to split theI andQ components of the CAP signal, respectively. Next, the detached signals are down-sampled prior to QAM demodulation.

2.4. DL-based EQ

As mentioned before, DL is a powerful subfield of ML, which em- ploys DNN to solve complex problems. DNN is a class of ANNs with two or more hidden layers of nodes [27], see Fig. 2. As shown, it consists of four layers including the input layer, two hidden layers, and output layer. The number of nodes, shown in circles, 120, 300, and 400 for both the input and output layer, and the 1st and 2nd hidden layers, respectively. All layers are fully connected, which means that every node in each layer is connected to all nodes in the next layer. The DL method splits the modeling dataset into training and test sets. The process for the training set is offline and includes two steps: (i) forward; and (ii) backward propagation [28]. First, in forward propagation, a subset of training dataset named minibatch crosses from the input layer to the hidden layers and then the output layer. The mathematical computations of forward propagation in the 1st hidden layer is given by [29]:

𝑍^[1]=𝑊^[1]𝑋+𝑏^[1] (14)

𝐴^[1]=𝑔^[1](𝑍^[1]) (15)

where[.]is the number of layer,𝑋and𝐴^[1]are the input and output of the 1st layer, respectively,𝑊^[1],𝑏^[1]and𝑔^[1](.)are the weights, biases,

𝑋=

⎢⎢

⎣

1 … ₁

𝑥^(1)[1]

2 …𝑥^(128)[1]

2

. . . 𝑥^(1)[1]

120 …𝑥^(128)[1]

120

⎥⎥

⎦(𝑛^[0]×𝑁)=(120×128)

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𝑏^[1]=

⎡⎢

⎢⎢

⎢⎣ 𝑏^(1)[1]

1 …𝑏^(128)[1]

1

𝑏^(1)[1]

2 …𝑏^(128)[1]

2

𝑏^(1)[1]

3 …𝑏^(128)[1]

3

. . . 𝑏^(1)[1]

300 …𝑏^(128)[1]

300

⎤⎥

⎥⎥

⎥⎦(𝑛^[1]×𝑁)=(300×128)

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𝑍^[1]=

⎡⎢

⎢⎢

⎢⎣ 𝑧^(1)[1]

1 …𝑧^(128)[1]

1

𝑧^(1)[1]

2 …𝑧^(128)[1]

2

𝑧^(1)[1]

3 …𝑧^(128)[1]

3

. . . 𝑧^(1)[1]

300 …𝑧^(128)[1]

300

⎤⎥

⎥⎥

⎥⎦(𝑛^[1]×𝑁)=(300×128)

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𝐴^[1]=

⎡⎢

⎢⎢

⎢⎣ 𝑎^(1)[1]

1 …𝑎^(128)[1]

1

𝑎^(1)[1]₂ …𝑎^(128)[1]₂ 𝑎^(1)[1]₃ …𝑎^(128)[1]₃

. . . 𝑎^(1)[1]

300 …𝑎^(128)[1]

300

⎤⎥

⎥⎥

⎥⎦(𝑛^[1]×𝑁)=(300×128)

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where𝑛^[𝓁] is the number of nodes in𝓁^thlayer. The matrix dimension for𝑊^[𝓁],𝑏^[𝓁],𝑍^[𝓁], and𝐴^[𝓁]are(

𝑛^[𝓁]×𝑛^[𝓁−1])

,(𝑛^[𝓁]×𝑁),(𝑛^[𝓁]×𝑁), and(𝑛^[𝓁]×𝑁), respectively, where𝑁 is the size of minibatch, which is assumed to be 128 in this work. Generally, the forward propagation process for all layers is given as:

𝑍^[𝓁]=𝑊^[𝓁]𝐴^[𝓁−1]+𝑏^[𝓁] (21)

𝐴^[𝓁]=𝑔^[𝓁](𝑍^[𝓁]) (22)

where𝐴^[𝓁−1]is the input of the𝓁th layer.

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Fig. 2. The structure of the proposed DL-based post-EQ.

Fig. 3. The BER performance against (𝐸_b∕𝑁₀)_testwith the DL-based EQ for a range of (𝐸b∕𝑁0)trainfor 128 QAM and signal BW of 5 MHz.

The rectified linear unit (ReLU) and Sigmoid values are used for the intermediate and output layers, respectively, which are given as [28]:

ReLU (𝑥) =

{𝑥, 𝑥 >0

0, 𝑥≤0, (23)

Sigmoid= 1

1 + exp(−𝑥) (24)

There are two types of learning process for ML algorithms: su- pervised and unsupervised. In the former, which is used here, the network is trained based on the input/output pairs known as the labeled data [27]. The labeled data (𝑥, 𝑦) is shown as portsAandB inFig. 1. PortA, i.e., the input to DNN, is the received signal with the LED and channel induced impairment, and portBis the desired CAP signal, which is used as the prediction target.

Accordingly, a loss functionL is defined to determine the error, which indicates the difference between the outputs of DNN and the correct answers.Lis expressed by various metrics [30], and here we have used the mean square error (MSE), which is defined as:

MSE(𝑎, 𝑦) = 1 𝑁

∑𝑁

𝑖=1

(𝑎^𝑖−𝑦^𝑖)², (25)

where𝑎is the estimated real output of DNN, and𝑦is the desired labeled value.

In the backward propagation phase, the main aim of DL is to find the correct weights and biases parameters of (𝑤,𝑏), respectively, for minimizing L using different optimization algorithms. Stochastic gradient descent (SGD) [31] is the most popular optimization algorithm in DL, which is also adopted in this study. First,𝑤and𝑏are initialized to a random values and the gradients of differentiable𝐿are calculated based on𝑤 and 𝑏using the chain rule. Here the gradients of cross entropy function are calculated as a useful example. Note, we have the followings:

𝑑𝐿

𝑑𝑥=𝑑𝑥, (26)

𝑑𝑎^[2]= 𝑑𝐿 𝑑𝑎^[2] = −𝑦

𝑎^[2]+ 1 −𝑦

1 −𝑎^[2], (27)

𝑑𝑧^[2]= 𝑑𝐿 𝑑𝑧^[2] = 𝑑𝐿

𝑑𝑎^[2]×𝑑𝑎^[2]

𝑑𝑧^[2] =𝑑𝑎^[2]×𝑔^′( 𝑧^[2])

= (−𝑦

𝑎^[2]+ 1 −𝑦 1 −𝑎^[2]

)

×( 𝑎^[2](

1 −𝑎^[2]))

=𝑎^[2]−𝑦, (28)

𝑑𝑤^[2]= 𝑑𝐿 𝑑𝑤^[2] = 𝑑𝐿

𝑑𝑎^[2] × 𝑑𝑎^[2]

𝑑𝑤^[2]

= (−𝑦

𝑎^[2] + 1 −𝑦 1 −𝑎^[2]

)

×( 𝑎^[2](

1 −𝑎^[2])

×𝑎^[1]𝑇)

=( 𝑎^[2]−𝑦)

×𝑎^[1]𝑇=𝑑𝑧^[2]×𝑎^[1]𝑇, (29)

𝑑𝑏^[2]= 𝑑𝐿 𝑑𝑏^[2] = 𝑑𝐿

𝑑𝑎^[2]×𝑑𝑎^[2]

𝑑𝑏^[2] =𝑑𝑧^[2]=𝑎^[2]−𝑦, (30)

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Fig. 4. Flowchart of training and test process.

Fig. 5. BER performance of the proposed system for 64- and 128-QAM for a range of transmit signal BW.

Generally, the equation for the backward propagation for𝑁training example is given as:

𝑑𝑍^[𝓁]=𝑑𝐴^[𝓁]×𝑔^′[𝓁](𝑍^[𝓁]), (31) 𝑑𝑊^[𝓁]= 1

𝑁 ×𝑑𝑍^[𝓁]×𝐴^{[𝓁−1]𝑇}, (32)

𝑑𝑏^[𝓁]= 1

𝑁 ×𝑑𝑍^[𝓁], (33)

𝑑𝐴^[𝓁−1]=𝑊^[𝓁]𝑇×𝑑𝑍^[𝓁], (34)

where the dimension of matrices𝑑𝑊^[𝓁],𝑑𝑍^[𝓁],𝑑𝑏^[𝓁], and𝑑𝐴^[𝓁−1]equal to(𝑛^[𝓁]×𝑛^[𝓁−1]),(𝑛^[𝓁]×𝑁),(𝑛^[𝓁]×𝑁), and(𝑛^[𝓁]×𝑁), respectively. The updated and optimized weights and biases in each layer are given as:

𝑤∶ =𝑤−𝛼𝑑𝑤, (35)

𝑏∶ =𝑏−𝛼𝑑𝑏, (36)

where,𝛼 is the learning rate. The backward propagation algorithm is repeated for𝑚∕𝑁times to complete one epoch, where𝑚is the number of complete training features for each node in the input layer. An epoch is completed when all dataset is applied to the DL algorithm. Although determining the number of epochs is different due to the convergence of algorithm, but it is considered to 100 in this study.

As shown inFig. 2, prior to the use of activation function, the batch normalization method is applied, which speeds up the optimization process to provide higher𝛼. Note, (i) according to [32], the demand for the regularization technique is reduced noticeably; and (ii) the adaptive moment estimation (Adam) optimizer is employed due to faster convergence. More detailed information can be found in [33]. The pseudocode for the utilized learning algorithm is explained in Algorithm 1. To avoid overfitting and make the model more generalized, the dataset is divided into two sets of training and validation [34]. In the validation set, the hyperparameters can be changed and tuned to minimize the error in the produced model, which is fitted on the training dataset. In the test

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Table 1 DNN parameters.

Symbol Parameter Value

𝑛^[0] Number of nodes for input layer 120

– Number of hidden layers 2

𝑛^[1] Number of nodes for 1st hidden layer 300 𝑛^[2] Number of nodes for 2nd hidden layer 400

𝑛^[3] Number of nodes for output layer 120

𝑔(.) Activation function for hidden layers ReLU 𝑔(.) Activation function for output layer Sigmoid

– Number of Batch normalization layers 2

– Optimizer SGD with Adam

– (𝐸_b∕𝑁₀)_train 5, 10, 15, 20 dB

– Default (optimum) (𝐸b∕𝑁0)train 10 dB

L Loss function MSE

𝛼 Learning rate 0.01

𝑁 Size of minibatch 128

𝑚 Number of training samples for each node 10⁶

– Size of training dataset 120×10⁶

– Size of test dataset 30000

– Number of epoch 100

Table 2 System parameters.

Symbol Parameter Value

– Room size 5×5×3 m³

𝜙_1∕2 Semi-power half angle 70^◦

𝜓_𝑐 PD field of view 60^◦

𝐴 PD detector area 1 cm²

𝑅_PD Responsivity of PD 0.54 A/W

𝑇_𝑠(𝜓) Optical filter gain 1 𝐵_𝑚 LED modulation bandwidth 4.5 MHz 𝛽 Roll of factor ofI/Qfilters 0.5

𝑛_𝑠 Up-sampling factor 6

𝐿_𝑓 I/Qfilter length 10 sym.

𝐵_𝑡 Total bandwidth 5–25 MHz

– Modulation type QAM

𝑀 Modulation order 64 & 128

𝐺_TIA TIA gain 50–60 dB

– Tx position Center of ceiling

– Rx position LOS scenario: Center of floor

NLOS scenario: (1.15, 1.15, 0.85) m

phase, which is done online, the new and never-before-seen received CAP signals are used as the input dataset in the DNN EQs. The trained network attempts to recover them and minimize the error based on the gained knowledge. All the parameters for DNN are summarized in Table 1.

3. Result and discussion

In this section, we investigate the performance of the proposed CAP-VLC system with a DL-based post-EQ by means of computer simu- lations. MATLAB and Python are used to simulate VLC subsystem and the DL-based EQ, respectively. All key system parameters adopted are summarized inTable 2. To train the network, we have used a range of energy per bit to noise ratios (𝐸_b∕𝑁₀)_train. Due to the computation time, we considered only (𝐸_b∕𝑁₀)_train values of 5, 10, 15, and 20 dB.

Fig. 3depicts the BER performance against the (𝐸_b∕𝑁₀)_testfor a range of (𝐸_b∕𝑁₀)_trainvalues, 128-QAM and a signal bandwidth of 5 MHz. It is shown that, the best (optimum) BER performance is achieved for (𝐸_b∕𝑁₀)_trainof 10 dB, with the (𝐸_b∕𝑁₀)_testgains of 0.2, 0.8, and 5.7 dB compared with the (𝐸_b∕𝑁₀)_trainvalues of 15, 20 and 5 dB, respectively, at the 7% forward error correlation (FEC) BER limit of 3.8 ×10⁻³. Considering the trade-off between the simulation time and estimation of the BER, the coefficients of the DNN network corresponding to the optimum (𝐸_b∕𝑁₀)_train of 10 dB is adopted for the rest of evaluation during the test phase.Fig. 4 illustrates the flowchart of training and test processes.

Fig. 5 compares the BER performance of the proposed system for 64- and 128-QAM for a range of transmit signal BW. Referring to the

7% FEC BER limit, 128-QAM with a signal BW of 5 MHz offers the best performance with the (𝐸_b∕𝑁₀)_testgains of 1, 1.8, 6, and 6 dB compared with 10, 15, and 20 MHz 128-QAM, and 25 MHz 64-QAM, respectively.

𝑅_𝑑of 93.4 Mb/s and the spectral efficiency of 4.67 b/s/Hz are attained for 20 MHz 128-QAM with respect to the 7% FEC BER limit. To achieve higher𝑅_𝑑, the proposed system is also examined for 64-QAM and a signal BW of 25 MHz. Results demonstrate that𝑅_𝑑is improved by up to∼6 Mb/s reaching 100 Mb/s with respect to the 7% FEC BER limit.

At the Rx side, the small value of peak-to-peak injection current𝐼_pp leads to lower𝐸_b∕𝑁₀ deterioration, which degrades the BER performance. Furthermore, large values of 𝐼_pp may exceed the LED linear range, thus resulting in nonlinear distortions.Fig. 6depicts the BER performance against𝐼_pp for (𝐸_b∕𝑁₀)_train = 10dB,𝑀 = 64and BW of 25 MHz, respectively. As can be seen, the lowest BER of∼2.26×10⁻³ is achieved at 0.7≤𝐼_ppof≤0.8 A. Therefore,𝐼_ppof 0.8 A is considered as the default value in this study.

Fig. 7(a) depicts the BER performance of the proposed system with 64-QAM and a BW of 25 MHz for three different post-EQ schemes. It is clearly seen that, at the 7% FEC BER limit DL offers (𝐸_b∕𝑁₀)_testgains 2 and 8 dB compared with Volterra and LMS EQs, respectively, which can be used to trade-off transmission range against𝑅_𝑑. Note, reducing the bandwidth results in improved BER performance as shown inFig. 7(b) with the DL-based EQ outperforming LMS and Volterra. For instance, at the BER of 3.8×10⁻³, the (𝐸_b∕𝑁₀)_testpenalties are∼6 and 10 dB for Volterra and LMS, respectively compared with the DL method.

Next, we evaluated the BER performance of LOS and LOS + NLOS 64-QAM VLC links with a signal BW of 25 MHz. As can be seen in Fig. 8, the overall BER performance of pure LOS and LOS + NLOS links are almost the same.

Finally, the complexity of the DL-based EQ is compared with LMS and the 2nd order Volterra series EQ as summarized inTable 3[35].

As it is expected, the computational complexity of the DL-based EQ is approximately 10⁴ times larger than other two EQs. However, in this work the proposed system with only two hidden layers is considered, which is a simple DNN using DL technique, while more complex networks can be implemented using field programmable gate array (FPGA) boards. As an example, in [36] a DNN with two hidden layers was implemented using FPGA (Xilinx Zynq UltraScale) with the complexity of about 182007, thus demonstrating the potential use of FPGA in DL- based EQs in wireless communications. Note, 𝑀_𝐿 and𝑀_NL are the linear and nonlinear memory depths, respectively.𝑀₁, 𝐻₁,𝐻₂, and 𝐻₃are the number of nodes in input, first hidden layer, second hidden layer, and output layer, respectively.

4. Conclusion

In this paper, we introduced a deep learning-based post-EQ for

‘‘single’’ CAP-VLC system considering both the LED’s bandwidth lim- itation and its nonlinearity effects. The results showed that, the highest data rate of about 100 Mb/s is achieved for 64-QAM, the signal bandwidth of 25 MHz and the LED modulation bandwidth of 4.5 MHz (i.e., standard white LED). Finally, we compared the performance of the proposed scheme with two common EQs of LMS and Volterra series, demonstrating improved BER results.

Algorithm 1. Pseudo-code of DNN training algorithm Input:Number of layers𝐿, number of neurons for all layers, activation functions, loss function, number of epochs𝐸, number of transmitted batches in one epoch𝑁, number of training example in each minibatch𝑚, learning rate𝛼= 0.01,𝛽₁= 0.9,𝛽₂= 0.999, 𝜂= 10⁻⁸

Output:A trained DNN

1:Initializeweights and biases for all layers randomly 2: forepoch = 1 to𝐸do

3: for𝑡= 1to𝑁do

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Fig. 6. The BER against𝐼_ppfor the 64-QAM CAP-VLC system with the DL-based EQ.

Fig. 7.The BER performance against (𝐸b∕𝑁0)testfor three different EQs schemes for the signal bandwidth of: (a) 25 MHz, and (b) 20 MH.

Table 3

Comparison of complexity for three different EQs.

Type of EQs 𝑀_𝐿 𝑀_{𝑁 𝐿} 𝑀₁ 𝐻₁ 𝐻₂ 𝐻₃ Complexity Value of complexity

DL-based – – 120 300 400 120 𝑀₁×𝐻₁+𝐻₁×𝐻₂+𝐻₃ 156400

LMS 6 – – – – – 𝑀_𝐿 6

Volterra series 6 6 – – – – 𝑀_𝐿+𝑀_{𝑁 𝐿}(𝑀𝑁 𝐿+ 1)∕2 27

Algorithm 1. Pseudo-code of DNN training algorithm 4: for𝓁= 2 to𝐿do

5: doBatch normalization as follow:

6:𝜇𝛽← ¹

𝑚

∑𝑚 𝑖=1𝑥_𝑖 7:𝜎_𝛽²← ¹

𝑚

∑𝑚

𝑖=1(𝑥_𝑖−𝜇𝛽)² 8:𝑥^∧_𝑖 ←𝑥_𝑖-√^𝜇𝛽

𝜎²_𝛽+𝜖

9:𝑠_𝑖←𝛾 𝑥^∧_𝑖 +𝛽

10: doforward propagation 11:end for

12: calculateloss function 13: for𝓁= 2to L

14: dobackward propagation

15: initialize𝑉_𝑑𝑤,𝑉_𝑑𝑏,𝑆_𝑑𝑤,𝑆_𝑑𝑏to zero 16:𝑉_𝑑𝑤=𝛽₁𝑉_𝑑𝑤+(

1 −𝛽₁) 𝑑𝑤 17:𝑉_𝑑𝑏=𝛽₁𝑉_𝑑𝑏+(

1 −𝛽₁) 𝑑𝑏

Algorithm 1. Pseudo-code of DNN training algorithm 18:𝑆_𝑑𝑤=𝛽₂𝑆_𝑑𝑤+(

1 −𝛽₂) 𝑑𝑤² 19:𝑆_𝑑𝑏=𝛽₂𝑆_𝑑𝑏+(

1 −𝛽₂) 𝑑𝑏² 20:𝑉^{𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑}

𝑑𝑤 = ^V^dw

1−𝛽₁^𝑡

21:𝑉^{𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑}

𝑑𝑏 =_1−𝛽^V^db𝑡 1

22:𝑆^{𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑}

𝑑𝑤 = ^S^dw

1−𝛽^𝑡 2

23:𝑆_𝑑𝑏^{𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑}= ^S^db

1−𝛽^𝑡 2

24: updateall weights and biases as follow:

25:𝑤=𝑤−𝛼 ^𝑉

𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑

√𝑑𝑤 𝑆^{𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑}_𝑑𝑤 +𝜀

26:𝑏=𝑏−𝛼 ^𝑉

𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑

√𝑑𝑏 𝑆^{𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑}_𝑑𝑏 +𝜀

27:end for 28:end for 29:end for

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Fig. 8. The BER performance against (𝐸b∕𝑁0)testfor LOS and LOS + NLOS 64-QAM VLC systems.

Declaration of competing interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.

Acknowledgments

The authors would like to thank anonymous reviewers for their helpful comments.

Funding

We have not received any funding support.

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