## 5 Experimental Results
The radio-frequency (RF) amplifier with a gain of \(20\,\mathrm{dBm}\) is included for better extinction ratios between the multi-level signals. The directly modulated laser (DML) source, with \(P_{\text{s}}^{\text{(max)}}=1.1\,\mathrm{dBm}\) and operating wavelength of \(\lambda=1550\,\mathrm{nm}\), modulates the electrical data signal. With the help of a single-mode coupler and a delay line of \(167.27\,\mathrm{m}\), the data frames are segregated and launched into the MMF with the light  
Figure 7: Experimental setup for the joint-PPE scheme by using an optical (\(2\times 2\)) MIMO system with an MMF channel with a PFS configuration. launching conditions of 0 \(\upmu\)m and 15 \(\upmu\)m. In contrast to the setup with two different DMLs for two layers, the advantage of using the PFS setup is that both MIMO layers operate on an identical wavelength and the overall complexity is minimized. A variable optical attenuator (VOA) is incorporated in the setup for the purpose of BER performance evaluation over a distinct range of the average optical received powers. Please note that the power at both layers is constrained before transmission. The lower-order mode (LOM) groups are excited using the centric launching condition into the MMF and high-order mode (HOM) groups are generated with an eccentric launching condition of 15 \(\upmu\)m. These transmitted signals are multiplexed together using a custom-made fusion coupler (labeled as MUX) and launched into a 100 m long MMF. After propagating through the MMF channel, the transmitted signal is severely degraded with ISI and ICI as the modal dispersion is considerably higher compared to the chromatic dispersion in the SMF-based optical system. At the receiver's side, a de-multiplexer (DMUX) segregates the LOMs and HOMs. Eventually, the MMF photo-detector, labeled as a PIN diode, converts the optical signals into the electrical domain. These signals are further amplified using a trans-impedance amplifier. The large sensing areas of the PIN diodes lead to an inherent increase in capacitance. Finally, the electrical signals are captured by the digital storage oscilloscope (DSO), which are stored with 40 GSa/s and a bandwidth of 13.6 GHz. In the end, the stored signals are forwarded to the offline signal processing chain using one sample per symbol. The receive filtering is the first step, which is adapted according to the rectangular transmit filter with a similar impulse response. This step is followed by the symbol clock recovery, frame synchronization and channel estimation. Initially, an MMSE post-equalizer is employed. Afterwards, the post-equalizer is replaced by the joint-PPE filter. In the end, the BER performance is measured with respect to the average optical received power \(P_{\mathrm{r}}\). For maintaining the reliability of the results, 52 frames consisting of over a million bits on each layer are evaluated. Moreover, the results do not consider the forward error correction decoding. The BER in dependence on the average optical received power \(P_{\mathrm{r}}\) is demonstrated in Figure 8, where the two scenarios with different modulation formats are shown. The corresponding results are presented in a tabular form, where the optical received power gains are demonstrated at a BER of \(10^{-4}\) in Table 3. Evidently, the required optical received power is reduced when the joint-PPE scheme is employed as compared to PE-only. While using PAM-2 with a jointly computed 4-taps pre-equalizer and 21-taps MMSE post-equalizer, a \(P_{\mathrm{r}}\) gain of \(\approx\)1.93 dB is achieved in comparison to a PE-only. Correspondingly, when the PAM-4 format is used on both the MIMO inputs with the proposed joint-PPE filter, the \(P_{\mathrm{r}}\) gains of \(\approx\)0.96 dB and \(\approx\)1.84 dB are noted incorporating the two and four taps pre-equalizer, respectively. When the joint-PPE scheme with the PAM-4 MIMO inputs is analyzed, the \(\Delta P_{\mathrm{r}}\) gain of \(\approx\)0.88 dB with different pre-equalizer lengths is observed.