## 2 Signal and System Models
### Wireless Channel Model for LED-ID System  
It is assumed that an LED has a Lambertian radiant intensity:  
\[R_{\mathrm{0}}(\theta)=\left[\left(m+1\right)/2\pi\right]\mathrm{cos}^{\mathrm{ \infty}}(\theta) \tag{4}\]  
where \(m\) is the order of Lambertian emission, and is related to the transmitter semi-angle (at half power) \(\theta_{{}_{1/2}}\), as \(m=-\ln 2/\ln(\mathrm{\infty}\theta_{{}_{1/2}})\)[20]. \(P_{\mathrm{LED}}\) is the power emitted by an LED. \(\theta\) and \(\psi\), shown in Figure 1, are the irradiance (or zenith) and incidence angles, respectively. After modulation, the optical signal from an LED is given by:  
\[P_{\mathrm{LED}}[k]=P_{\mathrm{T}}(1+M_{\mathrm{1}}f[k]) \tag{5}\]  
Figure 4: Pulse Width Modulation.  
where \(P_{\mathrm{T}}\) is the LED output power without modulation, \(M_{1}\)is the modulation index, and \(f[k]\) is the normalized modulating signal. The modulation index is defined as the ratio of the LED's maximum current variation caused by the modulating signal to the LED bias current [21]. Because the walls are assumed to act as large reflecting surfaces, the light beams propagate from a LED transmitter to the LED reader via two main paths, _i.e._, line of sight (LoS) and diffuse channels. Since the rate of change of the channel impulse response is very slow compared to the frequency of the optical signal, it is usually sufficient to use a static parameter (channel DC gain) to represent the channel [14]. The DC gain can be estimated accurately by considering only the LoS path [21]. The channel transfer function of the DC gain [the Fourier transform of the LoS part of \(h[k]\) in (2)] at \(f=0\) is expressed as:  
\[H_{\mathrm{LoS}}(0)= \left\{\begin{aligned} &\frac{A}{d^{2}}\,R_{0}(\theta) \cos\psi,& 0\leq\psi\leq\psi_{c}\\ & 0,&\psi>\psi_{c}\end{aligned}\right. \tag{6}\]  
where \(A\) is the detector area, \(d\) is the distance between the LED transmitter and the LED reader, and \(\psi_{c}\) is the field-of-view (FOV) of the LED reader.  
For a diffuse channel, the impulse response of the diffuse signal \(h_{\mathrm{diff}}[k]\)shown in Figure 3 in [21] consists of some initial peaks due to the first reflections in the room and then a smooth exponential decay caused by the higher order reflections. This was modeled by an integrating sphere model [22, 23]. When both the LoS path and the diffuse channel are considered together, \(h[k]\) is modeled as a Rician _K_-factor distributed random variable, which is widely used in radio communications. The average received optical power at the reader is given by \(P_{\mathrm{R}}=P_{\mathrm{T}}H_{\mathrm{LoS}}(0)\). After the photodetection in the reader, the DC component of the detected signal is removed, and the output signal is given by:  
\[P_{\mathrm{F}}[k]=FP_{\mathrm{R}}M_{1}f[k]+W_{\mathrm{F}}[k] \tag{7}\]  
where \(W_{\mathrm{F}}[k]\)is the noise after the filter and is assumed to be AWGN with zero mean and variance \(\sigma_{W_{r}}^{2}\). Thus, the output SNR \(\gamma_{o}\) is given by [23]:  
\[\gamma_{o}= \frac{\left(FH_{\mathrm{LoS}}(0)P_{\mathrm{T}}M_{1}\right)^{2} \left\langle f[k]^{2}\right\rangle}{\sigma_{W_{r}}^{2}} \tag{8}\]  
where \(\left\langle\cdot\right\rangle\) is the time average operator.  
The BER performance of the signal is expressed as:  
\[\mathrm{BER} = Q\left(\sqrt{2\gamma_{o}}\right) \tag{9}\] \[= Q\left(\frac{\sqrt{2}FH_{\mathrm{LoS}}(0)P_{\mathrm{T}}M_{1}}{ \sigma_{W_{r}}}\right)\]