# Probabilistically-Shaped DMT for IM-DD Systems with Low-Complexity Fast WHT-Based PDSP
###### Abstract  
Transmission capacity and receiver sensitivity of an intensity-modulation direct detection (IM-DD) optical discrete multi-tone (DMT) system can be improved by using the probabilistically shaping (PS) technique. However, different probabilistic distributions will be required owing to the unbalanced signal-to-noise ratio (SNR) among data-carrying subcarriers (SCs) induced by the imperfect frequency response of optical/electrical devices, which can increase the implementation complexity of the PS-DMT transceiver. In this work, different signal pre-processing schemes including pre-equalization, Walsh-Hadamard transform (WHT)-based full data-carrying SCs precoding (FDSP) and fast WHT-based partial data-carrying SCs precoding (PDSP) are investigated for SNR equalization in a short-reach PS-DMT transmission system. After transmission over 50 km single-mode fiber, the experimental results indicated that three pre-processed signals have almost the same generalized mutual information (GMI) performance and receiver sensitivity improvements. The proposed fast WHT-based PDSP scheme may be a good option for the implementation of the PS-DMT transmission systems with a large SC SNR fluctuation regarding computational complexity. intensity-modulation and direct-detection (IM-DD); discrete multi-tone (DMT); probabilistically shaping (PS); full data-carrying SC precoding (FDSP); partial data-carrying SC precoding (PDSP); Walsh-Hadamard transform (WHT) +
Footnote †: journal: _Photonics_  
scheme named the pre-emphasis (or pre-equalization) technique was also widely used to effectively compensate for power fading in transmission systems [10]. However, the traditional adaptive loaded-DMT or pre-emphasis technique is channel-dependent and requires the channel state information (CSI) with the reverse link, which is complex and time consuming. Nowadays, a channel-independent precoding scheme is seen as another effective way to compensate for unbalanced impairments. Thus, by employing this technique, only one symbol modulation scheme and one kind of CCMD are required in PS-enabled DMT transmission systems. Some precoding matrices such as the Walsh-Hadamard transform (WHT) matrix [11], constant amplitude zero autocorrelation sequence (CAZAC) matrix [12], and discrete Hartley transform (DHT) matrix [13] have been employed to realize precoding for DMT transmission systems. The precoding technique can also be applied to reduce the peak-to-average power ratio (PAPR) owing to the improvement of the autocorrelation performance of signal symbols [14], and mitigate nonlinear distortions induced by electrical/optical devices. In [15], [PERSON] et al. proposed and experimentally demonstrated an orthogonal circular matrix transform (OCT)-based precoding scheme in a short-reach IM-DD transmission system, which can effectively reduce the PS-enabled transceiver implementation complexity. Moreover, the precoding schemes based on seven commonly precoding matrices were comparatively investigated [16], and their computational complexities were theoretically analyzed and compared as well. It also indicated that, compared with other precoding matrices (e.g., OCT, DHT, CAZAC, etc. ), there is no need for multiplication operations in WHT precoding, which can be considered as a suitable scheme to compensate unbalanced impairments in an optical IM-DD system regarding the computational complexity. In the above-mentioned precoding works, all of the data-carrying SCs in DMT or OFDM symbols are used for precoding, and we call this method full data-carrying SC precoding (FDSP). Since the Hermitian symmetry (HS) constraint is required for the inverse fast Fourier transform (IFFT) to obtain the real-valued DMT signal, the number of data-carrying SCs is generally not an integer power of two. In this case, the precoding techniques cannot be directly implemented with the corresponding FFT-based fast algorithms. Moreover, when a large number of SCs are employed for data delivery, the FDSP exhibits high complexity from a hardware implementation point of view.