# Probabilistically-Shaped DMT for IM-DD Systems with Low-Complexity Fast WHT-Based PDSP
## 3 Experimental Setup  
The experimental setup of the PS-DMT systems enabled by a fast WHT-based PDSP scheme is illustrated in Figure 3. At the transmitter (Tx), the digital PS-DMT signal is generated offline with digital signal processing (DSP) approaches in Matlab. Firstly, the pseudo-random binary sequence (PRBS) is generated and then sent to the PS module, which is clearly mentioned in Section 2.1. After that, the shaped 64-QAM symbols are precoded with the proposed fast WHT scheme. Note that only 96 and 64 positive-frequency SCs are, respectively, used for FDSP and PDSP schemes, and other SCs are filled with zeros. After the HS operation, the cyclic prefix (CP) with a length of eight points is added for each 256-point inverse fast Fourier transform (IFFT) output to resist ISI. Additionally, one training symbol (TS) is inserted in the front of each PS-DMT frame to realize both timing synchronization and channel estimation [22]. Finally, the precoded PS-DMT signal is digitally clipped to combat the digital-to-analog converter (DAC)-induced quantization noise [23] and reduce PAPR. The analog electrical DMT signals are converted by using a Tektronix arbitrary waveform generation (AWG, AWG7122C). The corresponding sampling rate and the resolution for D/A conversion are 12-GSa/s and 10-bits, respectively. Note that the average signal power for traditional/FDSP/PDSP PS-enabled DMT signals are set to equal in this experiment. The converted electrical DMT signals are suppressed by a low-pass filter (LPF) and then amplified by a Mini-Circuits electrical amplifier (EA, ZX60-14012L-S+) with a bandwidth of \(\sim\)6 GHz. The Mach-Zehnder modulator (MZM) works at the quadrature point to minimize the non-linear distortion. The wavelength and output power of the laser diode (LD) are 1550 nm and 11 dBm, respectively. Then, the 2.5 dBm optical PS-DMT signals are coupled into a 50 km single-mode fiber (SMF), and its dispersion and loss are \(\sim\)17 ps/nm/km and \(\sim\)0.2 dB/km, respectively. At the receiver (Rx), the optical PS-DMT signal is attenuated by a variable optical attenuator (VOA). An optical coupler (OC) is used to change the received optical power (ROP). Then, the output PS-DMT signal with 90% power is detected by a photodiode (PD). The recovered \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Precoding** & \multicolumn{2}{c}{**Refer to [16]**} & \multicolumn{2}{c}{**Fast Algorithm [18]**} \\ \cline{2-5}
**Scheme** & **Real Mult.** & **Real Add.** & **Real Mult.** & **Real Add.** \\ \hline FDSP & 0 & \(2F^{2}-2F\) & - & - \\ PDSP & 0 & \(2P^{2}-2P\) & 0 & \(2P\text{log}_{2}\) (\(P\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: WHT-based FDSP/PDSP computational complexity comparison. signal is amplified by an EA and captured by a digital storage oscilloscope (DSO, Lecroy Wavenaster 820-Zi-A, Teledyne Lecroy, Chestnut Ridge, New York, USA). The sampling rate of DSO and resolution for A/D conversion are 40 GSa/s and 8 bits, respectively. The sampled signals are post-processed with the Rx DSP flow, which consists of symbol timing synchronization, remove CP, FFT, ISFA-enhanced channel estimation [23], frequency domain equalization, fast WHT decoding, and PS demodulation. Finally, the different BER or GMI performances are calculated. When the rate of LDPC is set to 9/10, the different rates of CCDM, i.e., 1.75, 1.55 and 1.25 bits per one-dimensional symbol are discussed in this work. Therefore, according to Equation (2), the corresponding information rates (IRs) are 4.9, 4.5, and 3.9 per two-dimensional symbol (bits/QAM symbol). Meanwhile, the constellations of the three probability distributions for probabilistically shaped 64-QAM are also given in Figure 4. It shows clearly that, as the \(R_{DM}\) decreases, the probabilistically shaping for the DMT symbol can become more obvious. The bandwidth of the 64-QAM PS-DMT signal is constantly equal to 4.5 ((96 \(\times\) 12)/256) GHz.