## 1 Introduction  
Driven by the rapid evolution of digital technologies and the emergence of innovative applications, including cloud computing, smart home systems, and autonomous driving, there has been a substantial increase in data traffic within short-reach optical networks [1, 2, 3]. Despite the benefits of coherent detection (CD), such as high spectral efficiency and enhanced sensitivity over long-haul transmissions, its adoption in short-reach optical networks has been constrained by the necessity for costly narrow-linewidth local oscillator (LO) lasers. In comparison, conventional DD eliminates the need for an LO, offering a simpler architecture and cost-effective implementation. However, traditional DD technology primarily relies on intensity modulation and direct detection (IM-DD), which is limited to amplitude modulation and detection, thereby resulting in low spectral efficiency (SE). Additionally, IM-DD systems are incapable of optical field recovery, preventing the digital compensation of linear signal impairments induced by fiber links, which further restricts the achievable transmission distance [4, 5, 6]. In recent years, to leverage the benefits of both DD and CD, a range of self-coherent detection (SCD) schemes have been introduced. These schemes are designed to enhance SE and mitigate the frequency-selective fading induced by dispersion [7]. At the transmitter, the carrier and signals are fed into the optical fiber. At the receiver, the desired signal can be retrieved from the carrier-signal beat term after photodiode (PD) detection. However, some schemes are limited to SSB modulation and thus cannot make full use of receiver bandwidth compared with CD, such as the Kramers-Kronig (KK) receiver [8] and iterative cancellation (IC) receiver [9]. To improve spectral efficiency, numerous DSB-SCD schemes have been proposed, such as carrier-assisted differential detection (CADD) [10, 11, 12, 13], carrierless phase retrieval (CLPR) [14], asymmetric self-coherent detection (ASCD) [15], and carrier-assisted phase retrieval (CAPR) [16], as well as a filter-assisted direct detection (FADD) architecture that uses an OBPF or optical band-stop filter (OBSF) to obtain complex-valued double-sideband (CV-DSB) signals, as presented in our previous research [17]. PDM should be the next target to be achieved since it doubles the system capacity and spectral efficiency. The polarization fading phenomenon induced by random polarization rotation is a fundamental obstacle for PDM-SCD compared with PDM coherent detection, which seriously reduces system performance. Polarization fading occurs because the optical carrier cannot be split equally into X- or Y-polarization using a simple polarization beam splitter (PBS) without active polarization control [18, 19], since the carrier undergoes random polarization state rotation in the fiber channel. Additionally, the SCD scheme relies on the reference optical carrier to reconstruct the optical field digitally [20]. As a result, the single-polarization optical field recovery is no longer fulfilled without a sufficiently strong carrier. To solve this problem, [PERSON] et al. proposed a filter-based PDM-SSB scheme [21]. At the transmitter, the generated PDM-SSB signals with two orthogonal carriers are located at opposite sides. At the receiver side, dual optical filters implementing polarization diversity are employed to suppress the undesired carrier on the opposite side. The scheme has been fully validated in other SSB systems [22, 23]. In addition, the real-value DSB modulation format is also verified, but a multi-input-multi-output (MIMO) equalization algorithm with high complexity is required to eliminate the sideband crosstalk caused by OBPF [24]. In [25], a minimal coherent receiver architecture with Alamouti encoding and single-ended PD was proposed to reduce receiver complexity and cost, but it sacrificed half of the SE. To further increase the SE, PDM CV-DSB modulation has attracted extensive attention.