Pulse amplitude modulation (PAM) is considered an excellent low-cost solution to implement multi-level
modulation in short-reach systems. For high-baud rates, these systems suffer from severe effects of chromatic dispersion (CD).
On the other hand, the Kramers-Kronig (KK) direct-detection receiver enables the reconstruction of the electrical field of the
received optical signal under a strict condition on the signal to be minimum-phase . Using KK with PAM enables the
compensation of CD with reduced cost and complexity. To meet the minimum-phase condition, the signal needs to be single
sideband (SSB) with a high enough carrier-to-signal power ratio (CSPR). One way to produce the SSB signal is to use optical
filtering of the modulated optical PAM signal. Due to the limited sharpness of optical filters, the signal is vestigial
sideband (VSB) rather than SSB, which limits the performance of the KK receiver.
In this example, a 53 Gbaud PAM4 signal is generated electrically, and raised-cosine pulse shaping is used with a roll-off
factor of 0.01 to limit the electrical spectrum to about 26.8 GHz. The optical signal is generated using a Mach-Zehnder
modulator and transmitted over 80 km single-mode fiber. At the receiver side, the signal is amplified and passed through
an optical filter. The center frequency and 3-dB bandwidth of the filter are optimized to remove most of the lower sideband
(LSB) and suppressing the amplification noise, simultaneously. An optical attenuator is used to model the insertion loss of
the filter, and the signal is then directly detected. The detected signal is digitized, and VPI's DSP library is used to
apply the KK algorithm followed by CD compensation. Finally, symbol error ratio (SER) is estimated.
To examine the tolerance of the system towards the position of the optical filter relative to the signal spectrum, the center
frequency of the filter is swept, and the system performance is assessed.
shows the spectra of the optical signal before and after filtering.
shows the SER for different values of shifts in the center frequency of the filter. The optimum performance is obtained with a shift of 18 GHz.
shows the CSPR for different values of the shift. The optimum CSPR for this system is found to be about 12 dB.
Higher CSPR are achievable when shifting the filter to the left on the frequency axis which also results in
increasing the residual LSB. This affects the field reconstruction process adversely. On the other hand, lower
values of CSPR also worsen the performance of the KK receiver . The eye diagram for filter shift = 18 GHz is shown in
and the simulation schematic is shown in
Keywords: PAM4, Kramers-Kronig, direct detection, vestigial sideband
Similar demonstrations are available in VPItransmissionMaker Optical Systems and on the VPIphotonics Forum.
 X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer. "Kramers-Kronig receivers for 100-km datacenter interconnects." Journal of Lightwave Technology 36, no. 1 (2018): 79-89.