1 Introduction

The utilization of commercially available light-emitting diodes (LEDs) by visible light communication (VLC) to facilitate data transmission over short distances, all while adhering to lighting standards, positions VLC as a complementary solution for indoor wireless access (Karunatilaka et al. 2015). The data is conveyed through visible light by modulating the intensity of it and is subsequently captured using a photodiode(PD) via direct detection (i.e. intensity modulation/direct detection (IM/DD)). Achieving high data rate communication via VLC presents a considerable challenge due to the need to efficiently use the constrained spectral bandwidth of white LEDs. The challenge of achieving high data rates with VLC, due to the limited spectral bandwidth of white LEDs, has led to the common utilization of orthogonal frequency division multiplexing (OFDM)-distinguished by its high spectral efficiency-in the literature (Armstrong 2009; Zhang et al. 2021; Shieh 2011; Jiang et al. 2021). In addition to its primary benefits, the use of OFDM in VLC offers other advantages, such as its resistance to inter-symbol interference (ISI), adaptability in diverse channel conditions, and requiring just a simple one-tap equalizer at the receiving end. However, conventional OFDM inherently utilizes the fast Fourier transform (FFT) and produces bipolar and complex symbols, which are incompatible with the requirement of VLC, requiring real and unipolar symbols. Hermitian symmetry is employed in conjunction with two predominant techniques to ensure unipolar and positive symbols: DC-biased optical OFDM (DCO-OFDM) (Armstrong and Schmidt 2008), which utilizes a positive DC bias, and asymmetrically clipped optical OFDM (ACO-OFDM) that clips negative symbols to zero (Armstrong and Lowery 2006).

The high peak-to-average power ratio (PAPR), a predominant issue with employing OFDM in wireless communication, can push the LED beyond its designated power range, operating it in a nonlinear zone. This not only results in energy inefficiencies and a reduced LED lifespan but it also induces signal distortion that deteriorates the bit error rate (BER) in VLC(Inan et al. 2009). PAPR mitigation techniques for optical OFDM have been extensively studied previously and the suggested methods can be categorized into three: signal distortion techniques, multiple signalling approaches, and pre-coding schemes.

Signal distortion-based methods, including clipping high signal peaks (Xu et al. 2014), using non-linear companding (i.e. \(\mu\)-law, A-law) to compress substantial peaks (Zhang et al. 2017), and constellation stretching (Doblado et al. 2015), are relatively easy to implement. However, they can introduce distortive noise, potentially compromising the performance of the system. In multiple signalling approaches (i.e. selective mapping (SLM) (Xiao et al. 2015), partial transmit sequence (PTS) (Tan et al. 2015), pilot-assisted (PA)(Popoola et al. 2014), and tone reservation (TR) (Bai et al. 2017)) generate multiple potential signals that all encapsulate the identical data and subsequently select the one with the lowest PAPR for transmission. However, these methods often necessitate additional data, compromising bandwidth efficiency and elevating computational complexity.

In PAPR reduction strategies that utilize pre-coding (i.e. discrete cosine transform (DCT), zadoff-chu transform (ZCT), discrete Fourier transform (DFT) (Ranjha and Kavehrad 2013), walsh-hadamard transform (WHT) (Salama et al. 2022), discrete hartley transform (DHT) (Zhou and Qiao 2015), and vandermonde-like matrix (VLM) transform (Sharifi 2019)), the transmitter multiplies the modulated data with a pre-coding matrix prior to the IFFT block within the OFDM framework. Following this, the receiver employs the inverse pre-coding matrix post the FFT block to retrieve the original data. One of the standout merits of pre-coding is its multiple benefits, which encompass signal independence, the elimination of signaling overhead necessities, and minimal impact on BER performance(Slimane 2007).

In this study, the pre-coding process of conventional DCO-OFDM within a VLC system is facilitated through the utilization of a Gaussian matrix (GM). In literature, GM is employed as a post-coding PAPR reduction scheme in DCO-OFDM, achieving approximately 3 dB improvement in PAPR performance (Sharifi and Azarnia 2020). Furthermore, the post-coding GM method is integrated with DCT pre-coding in Taha et al. (2022), resulting in a performance improvement of approximately 1 dB compared to DCT pre-coding. These studies operate under the assumption that the channel is an additive white Gaussian noise (AWGN). However, the utilization of GMs as a post-coding technique after the IFFT undermines one of the primary advantages of OFDM, which is to transform the frequency-selective channel into a single-tap flat fading channel. In this work, a designated set of GMs, M, is generated to function as candidate pre-coding matrices, aiming to retain all the benefits of OFDM in the presence of a frequency-selective channel while simultaneously reducing PAPR. These matrices are created once and remain constant throughout the communication process, with both the transmitter and receiver possessing awareness of their existence. On the transmitter side, the PAPR performance of each candidate Gaussian pre-coding matrix is assessed independently. Subsequently, the pre-coding matrix exhibiting the lowest PAPR is selected, and its corresponding index is transmitted to the receiver as side information. It is revealed that the PAPR performance achieved through the proposed Gaussian pre-coding method outperforms that of DCT pre-coding. Moreover, the integration of the Gaussian pre-coding method with \(\mu\)-Law companding yields further enhancements in PAPR performance.

The study is structured as follows: Section 2 outlines the system model and the application of DCT pre-coding, \(\mu\)-Law companding, and the proposed Gaussian pre-coding to the system. Section 3 presents PAPR and BER performance obtained through simulations. Section 4 concludes the study.

Notations: The bold small letters are denoted as vectors and bold uppercase letters to refer to matrices. \(\left[ \cdot \right] ^{T}\) represents the transpose of a vector or a matrix, \(z^{*}\) indicates the complex conjugate of z, and \(E\left[ \cdot \right]\) refers to expectation value.

2 System model

2.1 DCO-OFDM through DCT pre-coding

The conventional DCT pre-coded DCO-OFDM system (Dissanayake and Armstrong 2013), as illustrated in Fig. 1a, conducts an initial mapping of data to complex-valued symbols (\(s_n; n \in [0,1,\dots ,\left( N/2-2\right) ]\)) on the transmitter side, utilizing complex-valued modulation schemes such as QAM. The pre-coding is employed prior to storing mapped symbols, \(\textbf{s}=\left[ s_0, s_1, \ldots , s_{\left( N/2-2\right) }\right] ^{T}\), into \(\textbf{x}\) in order to achieve a Hermitian symmetry property. The pre-coded symbols, denoted as \(\textbf{s}_p\) and given in (1), are obtained as

$$\begin{aligned} \textbf{s}_p = \textbf{C} \times \textbf{s}= \left[ s_{p_0}, s_{p_1}, \ldots , s_{p_{\left( N/2-2\right) }}\right] ^{T}, \end{aligned}$$
(1)

where the DCT pre-coding matrix \(\textbf{C}\), having dimensions \((N/2-1) \times (N/2-1)\), is defined as \(\textbf{C} = \left( c_{(k,l)} \right) _{0 \le k,l \le (N/2-2)}\). \(c_{\left( k,l\right) }\) is the \(\left( k,l\right) ^{th}\) element of \(\textbf{C}\) and expressed as

$$\begin{aligned} c_{\left( k,l\right) }={\left\{ \begin{array}{ll} \sqrt{\frac{1}{\left( N/2-1\right) }}, &{} k=0 \\ \sqrt{\frac{1}{\left( N/2-1\right) }} \cos \left[ \frac{\pi \left( 2l+1\right) k}{N-2}\right] , &{} k=1, \dots , \left( N/2-2\right) . \end{array}\right. } \end{aligned}$$
(2)
Fig. 1
figure 1

Block diagram of a DCT pre-coded DCO-OFDM (without pink boxes) and b DCT pre-coded and \(\mu\)-law companded DCO-OFDM (with pink boxes)

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Then the pre-coded symbols, \(\textbf{s}_p\), are organized into the N-length vector \(\textbf{x}\) as in (3)

$$\begin{aligned}&\textbf{x}=\left[ 0, \textbf{s}_p^T, 0, \left( \textbf{K} \times \textbf{s}_p^{*} \right) ^T \right] ^{T} \end{aligned}$$
(3)

where \(\textbf{K}\) is an \(N \times N\) exchange matrix such that the elements on the antidiagonal are 1 and the others are 0. The input array for the IFFT is explicitly given as

$$\begin{aligned} \textbf{x} = \left[ 0, {s_p}_0, {s_p}_1, \ldots , {s_p}_{\left( N/2-2\right) }, 0, {s_p}_{\left( N/2-2\right) }^{*}, \ldots , {s_p}_0^{*}, {s_p}_1^{*} \right] ^{T}, \end{aligned}$$
(4)

which shows the Hermitian symmetry property to ensure a real-valued output (Jiang et al. 2020), \(\textbf{x}_f\), as given in (5), after the application of IFFT.

$$\begin{aligned} \textbf{x}_f = \left( \textbf{F}^{T}\right) ^{*} \times \textbf{x} \end{aligned}$$
(5)

Here, the \(N \times N\) FFT transformation matrix is represented as \(\textbf{F} = \left( f_{\left( k,l\right) }\right) _{0 \le k,l \le N-1 }\), with \(f_{\left( k,l\right) }\) denoting the \((k,l)^{th}\) element and is defined as:

$$\begin{aligned} f_{\left( k,l\right) } = \frac{1}{\sqrt{N}} e^{\left( - j2\pi kl /{N} \right) }. \end{aligned}$$
(6)

Before transmission, a cyclic prefix (CP) of length not shorter than the channel impulse response (CIR) tap number is appended. This \(\vartheta\)-tap CIR is expressed as

$$\begin{aligned} \textbf{h} = \left[ h_0, h_1, h_2, \ldots , h_{\vartheta -1}\right] . \end{aligned}$$
(7)

The samples are then serialized and directed to a digital-to-analogue converter (D/A). In DCO-OFDM, a DC bias is integrated into the resultant electrical signal to negate its bipolar nature and enable IM/DD. It is imperative to clip any residual negative peaks before employing the LED to transition this electrical signal to an optical one.

At the receiving terminal, subsequent to the traversal of the optical signal, a PD intercepts the signal. Following this, the signal is then converted back to its analogue electrical form. Upon traversing the optical channel, the emitted visible light signal is intercepted by a PD at the receiver and subsequently transformed into an analogue electrical signal. This signal is then digitized using an analogue-to-digital converter (A/D). Following this, the CP is excised from the samples, and the data undergoes a serial to parallel (S/P) transformation. The functioning of the system is predicated on the assumption that the response of the channel is accurately delineated with the utilization of the \(\vartheta\)-taps. The received N samples are represented as \(\textbf{y}\), defined in (8),

$$\begin{aligned} \textbf{y} = \textbf{H} \times \textbf{x}_f + \textbf{g}, \end{aligned}$$
(8)

where \(\textbf{g}\) signifies additive white Gaussian noise (AWGN) comprising N independent and identically distributed (i.i.d) noise components and \(\textbf{H}\) is the circulant channel matrix of dimensions \(N \times N\) as specified in Cinemre and Hacioglu (2021). The elements of \(\tilde{\textbf{x}}\) can be approximated by:

$$\begin{aligned} \tilde{\textbf{x}} = \textbf{F} \times \textbf{y} = \textbf{F} \times \textbf{H} \times \underbrace{\left( \left( \textbf{F}^{T}\right) ^{*} \times \textbf{x} \right) }_{\textbf{x}_f} + \textbf{F} \times \textbf{g}= \mathbf {\Lambda } \times \textbf{x} + \textbf{F} \times \textbf{g}, \end{aligned}$$
(9)

where \(\mathbf {\Lambda }=\textbf{F} \times \textbf{H} \times \left( \textbf{F}^{T}\right) ^{*}\) is an \(N \times N\) diagonal matrix. Subsequently, unbiased estimates of \(\textbf{s}_p\), situated in \(\textbf{x}\) as per (4), are represented as \(\tilde{\textbf{s}}_p\). Because of zero mean AWGN, the expected value of \(E\left[ \tilde{\textbf{s}}_p\right]\) is equal to \(\textbf{s}_p\). The \(n^{th}\) constituents of \(\tilde{\textbf{s}}_p\) are derived from:

$$\begin{aligned} \tilde{s}_{p_n} = \frac{\tilde{x}_n}{\Lambda _{(n, n)}}, \end{aligned}$$
(10)

with n ranging from 0 to \(N-1\). The inverse pre-coding takes place after equalization

$$\begin{aligned} \hat{\textbf{s}}=\textbf{C}^T \times \tilde{\textbf{s}}_p \end{aligned}$$
(11)

by utilizing IDCT matrix, \(\textbf{C}^T\), on the receiver side. The estimation of the transmitted N symbols is retained in the \(\hat{\textbf{s}}\) vector. Decisions regarding the \(n^{th}\) transmitted symbol, \(s_n\), are made using the maximum likelihood decision (MLD) criterion:

$$\begin{aligned} \tilde{s}_{n} = \arg \min _{p_{i}} \left( |\hat{s}_{n} - p_{i}|^{2} \right) , \end{aligned}$$
(12)

where \(p_{i}\) is one among the \(P\) constellation points and \(\tilde{s}_{n}\) signifies the \(n^{th}\) element of the detected symbols.

2.2 DCO-OFDM through proposed method using GM pre-coding

The proposed DCO-OFDM with the GM pre-coding scheme is given in Fig. 2.

Fig. 2
figure 2

Block diagram of a the proposed GM pre-coded DCO-OFDM (without pink boxes) and b GM pre-coded DCO-OFDM and \(\mu\)-law companded DCO-OFDM (with pink boxes)

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In this methodology, the orthogonalized-M square matrices are employed, represented as \(\textbf{G}^{\dagger }_m\), where \(m\in \{1, \dots , M\}\). The constituents of GM, symbolized by \(g_{\left( k,l\right) }\), are determined based on pseudorandom values that adhere to the standard normal distribution characterized by a mean of zero and a variance of one. For symbols \(\textbf{s}\) designated for transmission with a length of \(\left( N/2-1\right)\), the associated \(\left( N/2-1\right) \times \left( N/2-1\right)\) GMs are denoted as \(\textbf{G}_m = \left( g_{\left( k,l\right) } \right) _{0 \le k,l \le \left( N/2-1\right) }\), where the \((k,l)^{th}\) element, \(g_{\left( k,l\right) }\), adheres to the distribution \(g_{\left( k,l\right) } \sim \mathcal {N}(0,1)\). The robustness of the subsequent multiplicative operations at the receiver end is enhanced by ensuring that the pre-coding matrices maintain orthogonality. This objective is achieved using the Gram-Schmidt method (Raj et al. 2021), which effectively ensures the orthogonality of the matrices by replacing the original columns with orthogonalized counterparts. The GMs, \(\textbf{G}_m\), are orthogonalized to define pre-coding matrices, \(\textbf{G}^{\dagger }_m\), and then, this enables to define inverse pre-coding GMs as \((\textbf{G}^{\dagger }_m)^T\).

Algorithm 1
figure a

Selection of Optimum GM

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The selection of the optimum GM is outlined in Algorithm 1. Candidate pre-coding GMs, \(\textbf{G}^{\dagger }_m\), are independently generated and then multiplied with the time-domain modulated symbols, \(\textbf{s}\). The pre-coded symbols, \(\textbf{s}_p\) given in (13),

$$\begin{aligned} \textbf{s}_p=\textbf{G}^{\dagger }_m \times \textbf{s} \end{aligned}$$
(13)

where \(\textbf{s}=\left[ s_0, s_1, \ldots , s_{\left( N/2-2\right) }\right] ^{T}\) is the array of mapped symbols. The PAPR values of the resulting sequences are calculated, and the GM that produces the minimal PAPR is selected as the optimum matrix. Initially, the PAPR of DCO-OFDM without pre-coding, \(PAPR_o\), is utilized in the first step. To assess the PAPR performance of the GMs, the complementary cumulative distribution function (CCDF) is employed, as outlined in Cinemre and Hacioglu (2021). The PAPR of \(\textbf{x}_{f}\) can be calculated as

$$\begin{aligned} {\text {PAPR}}(\text{dB})=10 \log _{10} \frac{\max _{0 \le n \le N-1}\left\{ \left| x_{f_n}\right| ^2\right\} }{E\left\{ \left| x_{f_n}\right| ^2\right\} } \end{aligned}$$
(14)

Then, the CCDF provides insights into the probability of the PAPR surpassing a given threshold \(PAPR_0\) (Jiang et al. 2008);

$$\begin{aligned} CCDF(PAPR_{0})=P_{r}\left( {\text {PAPR}}>PAPR_{0}\right) . \end{aligned}$$
(15)

Notably, the PAPR performance of the proposed method is influenced by the number of candidate GM pre-coding matrices. The index of the selected matrix, represented as \(m^*\), is transmitted as side information to aid in signal reconstruction at the receiving end. Through this method, a potential reduction in PAPR is achieved, which enhances the effective transmission of the signal. However, it comes with the trade-off of incurring a transmission overhead of \(\log _2 M\) bits, which is necessary for the unique representation of all matrices.

2.3 DCO-OFDM through \(\mu\)-law companding

The \(\mu\)-law companding scheme, a logarithmic-based nonlinear method, is employed to reduce PAPR by enlarging small signal amplitudes, thereby minimizing the disparity between peaks and lower values (Wang et al. 1999). The symbols subjected to \(\mu\)-law companding adhere to the identical procedure illustrated in Figs. 1b–2b and are denoted as \(\textbf{x}_{\mu _n}\) in (16):

$$\begin{aligned} x_{\mu _n} = \frac{\max (\textbf{x}_{f})}{\log (1+\mu )} \log \left( 1+\frac{\mu }{\max (\textbf{x}_{f})}|x_{f_n}|\right) {\text {sgn}}(x_{f_n}); \quad \forall n \in [0,1,\dots ,\left( N-1\right) ] \end{aligned}$$
(16)

where \(\mu\) is the level of companding, and \({\text {sgn}}\) is the sign function (Vallavaraj et al. 2010). In the receiver, symbol decompanding is carried out through the utilization of (17) prior to the detection process.

$$y_{n} = \frac{{{\text{max}}\left( {{\mathbf{y}}_{\mu } } \right)}}{\mu }\left( {e^{{\frac{{\left| {y_{{\mu _{n} }} } \right|\log \left( {1 + \mu } \right)}}{{max\left( {{\mathbf{y}}_{\mu } } \right)}} - 1}} } \right){\text{sgn}}(y_{{\mu _{n} }} );\quad \forall n \in \left[ {0,1, \ldots ,\left( {N - 1} \right)} \right]$$
(17)

3 Numerical results

The model room with the size of \(5 \times 5 \times 3 \text {m}\) is considered for simulation, with its top-down perspective illustrated in Fig. 3. It is equipped with four transmitter LEDs with semi-angle at half power of \(55^{\circ }\), positioned at the (x, y, z) coordinates: \((1.25, 3.75, 3 \text {m})\), \((3.75, 3.75, 3 \text {m})\), \((1.25, 1.25, 3 \text {m})\), and \((3.75, 1.25, 3 \text {m})\). For the purpose of this simulation, it is assumed that light reflections occur solely from the side walls, ceiling, and floor, with respective reflection coefficients of 0.8, 0.7, and 0.2. The receiver PDs located in different positions (\(PD_1, PD_2, PD_3, PD_4\)) in the model room have a field of view (FOV) of \(85^{\circ }\) and encompass a physical area of \(10^{-4} \, \text {m}^{2}\).

Fig. 3
figure 3

Top view of model room

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Fig. 4
figure 4

Channel impulse response at the receiver

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The CIRs at specific PD locations are derived using the ray tracing method outlined in Lee et al. (2011), accounting for both line of sight (LOS) and two-reflection NLOS scenarios, as depicted in Fig. 4. Within this figure, the precise coordinates of the PDs are also detailed. For simplification purposes, LEDs are presumed to follow Lambertian radiation patterns, with reflecting surfaces treated as Lambertian reflectors. In this study, a comparative analysis is conducted to evaluate the performance of BER and PAPR of three distinct DCO-OFDM methods defined in Sect. 2. This evaluation examines the PD placement at location \(PD_4\) which exhibits a frequency-selective channel response that is used for defining the matrix \(\textbf{H}\).

Fig. 5
figure 5

Comparative BER performance of the different methods for \(N=256\) and 16-QAM

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Fig. 6
figure 6

Comparative PAPR performance of the different methods for \(N=256\) and 16-QAM

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This study compares the Gauss pre-coding approach, employing varying candidate numbers of matrices, against DCT pre-coding and the \(\mu\)-law companding method. Furthermore, an analysis of the combined application of pre-coding and companding methodologies is conducted. As depicted in Fig. 5, the utilization of \(\mu\)-law companding methods has a noticeable impact on the BER performance of the DCO-OFDM system, whereas the DCT and Gauss pre-coding methods demonstrate slight improvements in BER. The proposed Gauss pre-coding method exhibits superior PAPR performance compared to both DCT pre-coding and \(\mu\)-law companding, as shown in Fig. 6. Moreover, it reveals that augmenting the number of candidate GMs from 10 to 40 results in an approximate 1 dB enhancement in PAPR. However, it is important to note that this improvement comes at the expense of an increased complexity in the system. The combination of DCT and Gauss pre-coding methods with \(\mu\)-law companding results in a significant improvement of approximately 2 dB in PAPR compared to using solely DCT and Gauss pre-coding methods. However, it is noteworthy that this improvement in PAPR comes at the expense of a notable degradation in BER performance as in Fig. 5.

The efficiency of the method in terms of signal power at the transmitter has been analyzed by considering the parameters of an LED in relation to the signal output power and the PAPR. The threshold (\(V_{\text{ tov } }\)) and saturation (\(V_{\text{ sat } }\)) voltage levels can be formally delineated as the dynamic range boundaries of the LEDs. In order to guarantee that the values acquired at the output of the IFFT remain within the dynamic range suitable for LED illumination, it is imperative to apply a scaling factor, \(\alpha\), to these values (Yu et al. 2013). Additionally, a DC bias value (\(V_{\text{ dc }}\)) should be added to the IFFT output to be transmitted. The scaled and DC-added version of the IFFT output to be transmitted can be defined as

$$\begin{aligned} \textbf{x}_{f_{scaled}} = \alpha \textbf{x}_{f} \times + V_{\text{ dc }}. \end{aligned}$$
(18)

Note that the determination of this scaling factor is performed individually for each OFDM block, with the aim of preserving the operational integrity of the LED within its dynamic range and proactively averting any risk of signal clipping. The signal power is computed as \(E[\alpha ^2\textbf{x}_{f}^2]\), yielding the results shown in Table 1. The calculations were performed consistently across all methods within each OFDM block, utilizing identical transmission symbols. A total of 20,000 OFDM blocks were randomly generated, each containing random symbols, and the average signal power obtained from these 20,000 blocks is presented in Table 1.

Table 1 Output signal powers after scaling (\(V_{\text {sat}}=3.15 \mathrm {~V}\), \(V_{\text {tov}}=2.625 \mathrm {~V}\), \(V_{\text {avg}}=2.88 \mathrm {~V}\))
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Table 1 presents the adjusted output signal powers for \(N=256\), ensuring they remain within the LED dynamic range. Application of companding leads to increased scaled signal powers across all methods. When combining the proposed Gauss precoding method with companding, the resulting output signal power exceeds that of DCT precoding with companding by \(0.1381 \mathrm {~dB}\) and \(0.3962 \mathrm {~dB}\) for \(M=10\) and \(M=40\), respectively. Moreover, the proposed method demonstrates a signal power at least \(0.7692 \mathrm {~dB}\) higher than companding alone. Additionally, it can achieve signal power levels up to \(0.6814 \mathrm {~dB}\) higher than DCT precoding and \(1.5751 \mathrm {~dB}\) higher than DCO-OFDM without companding. These results suggest that the proposed method should attain a superior signal-to-noise ratio (SNR) level under equivalent noise conditions. Companding enhances signal power across all methods, resulting in SNR levels at least \(1.3647 \mathrm {~dB}\) higher for the same receivers. Increased signal power generally correlates with higher SNR, thereby reducing BER and enhancing throughput. Furthermore, it should be noted that the PAPR can exert a considerable influence on the BER performance within practical systems (Lu et al. 2018). Therefore, it is anticipated that the proposed method will exhibit improved BER performance while maintaining higher throughput in real-world scenarios. Even when employing companding and the proposed method simultaneously, the forward error correction (FEC) limit of \(3.8 \times 10^{-3}\) is expected to be reached with a reasonable SNR level below \(20 \mathrm {~dB}\).

4 Conclusion

This article introduces a novel pre-coding method utilizing Gaussian matrices as a strategic approach to mitigate the challenges associated with PAPR in VLC systems operating under frequency-selective channels. Within the DCO-OFDM framework, the proposed Gaussian pre-coding method demonstrates notable enhancements in PAPR performance, outperforming the traditional DCT pre-coding method and the \(\mu\)-law companding method in PAPR reduction. Simulation results underscore the effectiveness of increasing the number of candidate GMs, although this enhancement escalates system complexity. The study highlights a trade-off between PAPR and BER, revealing that the collaboration of DCT and Gauss pre-coding methods with \(\mu\)-Law companding achieves a 2 dB improvement in PAPR. As part of future investigations, an exploration into determining the optimal number of candidate GMs can be undertaken and a systematic approach, involving the utilization of previously tested GMs from the simulations, can be explored for the creation of new GMs instead of employing random GMs. Furthermore, the methodology can be tested within practical indoor lighting conditions, such as a typical working office space, to evaluate its power efficiency accurately. This real-world testing may allow for insights into energy optimization and identify any challenges for implementation.