# Mixed-Carrier Communication for Technology Division Multiplexing
## 2 Transmitter and Receiver Design
A conventional PPM envelope follows Equation (6), where \(p(t)\) is a unity pulse of a duration of \(T_{b}\) and \(c_{k}\in\{c_{0},c_{1},...,c_{L_{1}-1}\}\) is the PPM symbol sequence, i.e., codeword:  
\[x_{\text{PPM}}(t)=\sum_{k=0}^{L_{1}-1}c_{k}p(t-\frac{kT_{r}}{L_{1}}) \tag{6}\]  
where \(T_{r}\) represents the envelope duration for \(L_{1}\)-PPM offering a bit rate of \(R_{b-\text{PPM}}\) and follows:  
\[T_{r}=\frac{log_{2}L_{1}}{R_{b-\text{PPM}}}=L_{1}T_{b}=L_{1}L_{2}T_{\text{PWM}} \tag{7}\]  
Based on Equation (6), the \(L_{1}\)-BPM waveform is explained by replacing \(p(t)\) in Equation (6) by the PWM modulated beacon, \(b_{\text{PWM}}\), in Equation (3), as shown in Equation (8):  
\[x_{\text{BPM}}(t)=2L_{1}P\sum_{k=0}^{L_{1}-1}c_{k}b_{\text{PWM}}(t-\frac{kT_{r}}{L_{1}}) \tag{8}\]  
Similar to reshaping the OFDM samples to formulate a PWM-sampled beacon in Equation (4), the OFDM samples are reshaped to compose the dimming control PWM cycles as follows:  
\[x_{\text{PWM}}(t)=\sum_{n=0}^{M}x_{n-\text{PWM}}(t-nT_{\text{PWM}}) \tag{9}\]  
\[x_{n-\text{PWM}}= \begin{cases}2L_{1}P+\sum_{k=1}^{m}x_{k-\text{NACO-OFDM}}(t ),&0\leq t\leq D\ T_{\text{PWM}}\\ \sum_{k=m+1}^{\gamma_{f}-m}x_{k-\text{ACO-OFDM}}(t),&D\ T_{\text{PWM}}<t\leq T _{\text{PWM}}\end{cases}  \tag{10}\]  
where \(M=T_{r}/T_{\text{PWM}}\) is the number of PWM cycles per an MCC envelope, \(D\) is MCC duty cycle responsible for dimming control, and \(m=D\ T_{\text{PWM}}/T_{\text{OFDM}}\) is the number of OFDM symbols per the on duration of a PWM cycle. Finally, an MCC waveform, \(x_{\text{MCC}}(t)\) that is generated by the digital-to-analog converter (DAC); to modulate a light-emitting diode (LED) after using a bias-tee for providing enough DC-bias for the LED operation, is shown in Equation (11):  
\[x_{\text{MCC}}(t)=x_{\text{PWM}}(t)(1-x_{\text{PPM}}(t))+x_{\text{BPM}}(t) \tag{11}\]  
One of the main novelties about MCC is that it can support both low- and high-end receivers, simultaneously. The high-end receivers are based on conventional OFDM reception and processing. A high-speed photodetector (PD) captures the optical MCC waveform and converts the signal to the electrical domain followed by a trans-impedance amplifier (TIA). After the signal is digitized using an ADC and the cyclic prefix is removed, FFT is used to inverse the IFFT process on the transmitter side. A QAM demodulator captures the high-speed bits from the FFT complex output, taking into consideration that only odd sub-carriers are active for ACO-OFDM, as shown on the upper side of Figure 2. A simpler low-speed and low-complexity receiver is designed to capture low-speed bits on the lower side of Figure 2. After the signal is captured by a PD and amplified using a TIA, the high-frequency harmonics other than the beacon frequency are filtered using a low pass filter. A low-speed PD can filter out the high frequencies, as well. The beacon phase and position within the MCC envelope are then demodulated using both BnPSK and BPM demodulators as shown in Figures 3 and 4, respectively. A BnPSK demodulator is built upon the conventional PSK demodulator shown in [23]. The sinusoidal waveform is multiplied \(L_{2}\) times with a sinusoidal waveform that has the same frequency but with a phase shift of \(2\pi/L_{2}\) at every multiplier stage, followed by \(L_{2}\) integrator and dump blocks. The maximum output is chosen to indicate the correct phase of the input beacon, and thus represents the BnPSK bit stream. According to [24], a PPM demodulator can be implemented using a matched filter that is matched to the beacon shape that samples every \(T_{b}\) sec and compares the output to a threshold to decide the pulse location usinghard decision decoding or using a soft decision decoder and maximum likelihood (ML) detection to decide the received bits. In the proposed approach, a simpler energy detector is implemented to measure the energy of the input waveform every \(T_{b}\) sec. The maximum energy indicates the beacon location within the MCC envelope, so the BPM bits are detected: