## 3 The Effective SNR  
Due to both the high PAPRs of DMT signals and the finite dynamic ranges of digital-to-analog conversion (DAC) devices, digital clipping for the given upper and lower thresholds is usually performed before DAC. The clipped signal is given as [5]  
\[\overline{x}(n)=\left\{ \begin{array}{ll}x(n),&A_{l}\leq x(n)\leq A_{u}\\ A_{u},&x(n)>A_{u}\\ A_{l},&x(n)<A_{l}\end{array} \right.. \tag{8}\]  
For simplicity, we consider symmetric clipping (\(A_{u}=-A_{l}=A_{th}\)). The clipping ratio (CR) is defined as  
\[\gamma\ =\frac{A_{th}}{\sqrt{\mathbb{E}\Bigl{\{}\left|x(n)\right|^{2} \Bigr{\}}}}=\frac{A_{th}}{\sigma_{x}}. \tag{9}\]  
The output of clipping process can be considered as the sum of \(x(n)\) and the clipping noise \(c_{clip}(n)\):  
\[\overline{x}(n)=x(n)+c_{clip}(n). \tag{10}\]  
Nevertheless, based on the [PERSON] theorem [20] for Gaussian inputs \(\mathbf{x}\sim\mathcal{N}\bigl{(}0,\sigma_{x}^{2}\bigr{)}\), the clipped signal can also be decomposed statistically into two uncorrelated parts, given by  
\[\overline{x}(n)=\alpha\ x(n)+d(n), \tag{11}\]  
where \(d(n)\) is the component of clipping distortion that is statistically uncorrelated to \(x(n)\) (e.g., \(\mathbb{E}[x^{*}(n)d(n)]=0\)), and \(\alpha\) is a linear attenuation factor that can be calculated by  
\[\alpha\ =\ \frac{\mathbb{E}[x^{*}(n)\overline{x}(n)]}{\mathbb{E}\Bigl{[} \left|x(n)\right|^{2}\Bigr{]}}=1-erfc\biggl{(}\frac{\gamma}{\sqrt{2}}\biggr{)}, \tag{12}\]  
where \(erfc(\cdot)=1-erf(\cdot)\). The power of \(c_{clip}(n)\) can be obtained by  
\[\sigma_{clip}^{2}=\Biggl{\{}\Bigl{(}1+\gamma^{2}\Bigr{)}erfc\biggl{(}\frac{ \gamma}{\sqrt{2}}\biggr{)}-\sqrt{\frac{2}{\pi}}\gamma\ e^{-\frac{\gamma^{2}}{ 2}}\Biggr{\}}\sigma_{x}^{2}. \tag{13}\]  
Therefore, \(\sigma_{clip}^{2}\) is a function of \(\gamma\) and is independent of both the QAM modulation order and the size of IFFT/FFT. \(\overline{x}(n)\) is converted into the analog signal \(\overline{x}(t)\) by the DAC component. Due to the finite bit resolution \(\mathcal{Q}_{1}\) of the DAC, the quantization noise is then generated. The corresponding power can be calculated as  
\[\sigma_{DAC}^{2}=\frac{\max\lvert\overline{x}(n)\rvert^{2}}{3\cdot 4\mathcal{Q}_{1}} =\frac{2\cdot\gamma^{2}\sigma_{x}^{2}}{3\cdot 4\mathcal{Q}_{1}}erf\left(\frac{ \gamma}{\sqrt{2}}\right). \tag{14}\]  
To modulate the LED, \(\overline{x}(t)\) is added by DC current \(I_{DC}\) and obtain the final driving signals. We assume that the DMT signal is properly clipped, avoiding the LED nonlinearity. After optical channel transmission, the optical signal is detected by the optical detector, a combination of a photo diode (PD) and a trans-impedance amplifier (TIA). In most of the cases, the main sources of noise in the optical detector can be modeled as an additive white Gaussian noise (AWGN) following the distribution as \(\mathcal{N}\big{(}0,\sigma_{w}^{2}\big{)}\). Then, the received electrical signal is captured by analog-to-digital conversion (ADC) components with \(\mathcal{Q}_{2}\) bits of resolution. The power of the quantization noise in the ADC is given as  
\[\sigma_{ADC}^{2}=\frac{\gamma^{2}\cdot\rho_{h}^{2}\cdot\sigma_{x}^{2}}{3\cdot 4 \mathcal{Q}_{2}}, \tag{15}\]  
where \(\rho_{h}^{2}\) is the optical path gain coefficient. For simplicity, \(\mathcal{Q}_{1}=\mathcal{Q}_{2}=\mathcal{Q},\rho_{h}^{2}=1\) and \(I_{DC}=1\) are adopted in analysis. Finally, the total noise power in IM/DD channel link can be approximately measured with \(\sigma_{total}^{2}=\sigma_{ADC}^{2}+\sigma_{DAC}^{2}+\sigma_{clip}^{2}+\sigma_ {w}^{2}\).