## 1 Introduction  
Recently, emerging services such as 4K/8K, 5G and virtual reality (VR)/ augmented reality (AR) have imposed challenging demands on passive optical networks (PONs) [1; 2; 3; 4; 5; 6]. The future PONs should be able to provide on-demand modifications of traffic transmission policies through network re-configurability, flexibility and elasticity. For this purpose, software-defined networking (SDN) emerged as an effective technique to meet the demands [7]. When extending the SDN control to the physics layer of the network, it is able to offer highly desirable network functionalities. The effective use of SDN in PON systems showed its great potential in re-configurability, backward compatibility and elasticity [2]. In order to obtain higher throughput and larger capacity, researchers have tried different techniques for advanced modulation in PON transmissions, such as the quadrature amplitude modulation (QAM), pulse amplitude modulation (PAM) or the phase shift keying (PSK) based sub-carrier modulation (SCM) [8], the orthogonal frequency division multiplexing (OFDM) [5; 9],and the carrier-less amplitude/phase modulation (CAP) [2; 3; 4; 10; 11; 12]. Among these different methods, multi-band CAP (multi-CAP) using intensity modulation and direct detection was proposed as a feasible solution for next generation PON due to its low cost and simple structure. By combining the dynamic software-reconfigurable embedded digital orthogonal filters in the digital signal processing (DSP) unit of optical transceivers, the functionality, upgradability and flexibility of the PON transmission link can be considerably enhanced through a centralized SDN controller. However, conventional CAP preferred double side band (DSB) modulation and the transmission distance were limited due to frequency selected power fading introduced by chromatic dispersion (CD) [3]. The transmission of single sideband (SSB) optical signal was considered to be more robust against fiber dispersion and power fading, but the signal-signal beat interference (SSBI) generated by square-law detection and aggravated by accumulated dispersion [2; 13; 14; 15; 16] degrades the performance of the SSB system. Recently, different methods have been proposed to mitigate the SSBI effects. Refs. [2; 13] introduced a large guard band between the optical carrier and the modulated signal, but this method will limit the achievable spectral efficiency (SE) significantly. For higher SE, DSP methods were applied to cancel the SSBI without guard bands. Ref. [14] used two-stage Hilbert superposition cancellation construction to cancel the 1 st order and 2 nd order interference. The transmission performance could be improved by this construction, but the computation complexity was intolerable. Ref. [15] proposed a novel digital receiver-based two-stage linearization filter for SSBI cancelling. The DSP structure of this filter allowed low computation complexity, but the method itself would introduce extra unwanted SSBI terms that limit the compensation gain. In 2016, a direct detection coherent receiver based on the Kramers-Kronig (KK) relations was proposed which showed predominant SSBI mitigation capability [16]. Under the minimum phase requirement, the KK-receiver could reconstruct an optical field from the detected amplitude waveform, which made it very similar to a coherent receiver. Compared with the SSBI mitigation schemes mentioned above, the KK-receiver can provide more effective SSBI mitigation. The reduction in computation complexity also indicates the scheme's great potential in low cost applications. Another shortcoming of the SSB system is that a portion of the bandwidth of digital-to-analog converters (DACs) is wasted since it only carries one piece of sideband information. In order to make full utilization of DAC, a twin-SSB system was proposed [13; 17; 18], in which two groups of independent signals were modulated onto the right-sideband (RSB) and the left-sideband (LSB), respectively.