# Low-Complexity Chromatic Dispersion Equalization FIR Digital Filter for Coherent Receiver
## 1 Introduction  
In wired communication systems, the traditional coaxial cable communication is limited by the transmission distance and gradually replaced by optical fiber communication with strong anti-electromagnetic interference and long transmission distance. For optical signals with different frequency components or different model components, the signal pulses are broadened due to different group velocities, which causes signal distortion after being transmitted through the optical fiber. This physical phenomenon is called chromatic dispersion (CD) [1]. As the CD accumulates, the front and rear optical pulses overlap, causing inter-symbol interference (ISI) and increasing the bit error rate (BER). Therefore, the CD is one factor that restricts the development of optical fiber transmission systems to ultra-high-speed, ultra-large capacity, and ultra-long distance. Although dispersion compensation fibers (DCF) or dispersion-compensating modules (DCM) allow CDE in the optical domain, this increases the complexity and cost of technical implementation. The digital coherent receivers perform CDE in the electrical domain, restore the broadening and distortion of the pulse signal, achieve dynamic compensation, and have been widely used in high-speed coherent optical communication systems [2; 3; 4; 5] The digital filters used for CDE involving the time-domain chromatic dispersion equalizer (TD-CDE) filter and frequency-domain chromatic dispersion equalizer (FD-CDE) filter [6; 7; 8]. The CDE module occupies a large power consumption in the DSP, and a large numberof calculation units also limit the development of the coherent receiver to a compact size [9] Therefore, the CDE module should reduce the algorithm complexity and hardware implementation conditions to improve the efficiency of coherent receivers [10; 11]. Although the FD-CDE method is considered the best choice for commercial coherent receivers, due to the need for time-frequency conversion of the input sequence, this may cause time aliasing and weak interaction with other TD modules limiting the reduction of energy consumption [12]. The whole process of TD-CDE is carried out in TD, which avoids the problems of FFT block size and time aliasing. So in this paper, we mainly discuss TD-CDE and its optimization methods. The traditional truncation process can be regarded as a rectangular window function to truncate the infinite and non-causal TD impulse response function. The ratio of the main lobe energy of the window function to the total energy determines the filtering performance after truncation to a certain extent. The spectrum side lobe of the rectangular window is high, and high-frequency interference is easily brought into the window transformation [13]. To further improve the performance of the window function, a series of classic window functions such as the Hanning window, Blackman-Harris window, and Rife-Vincent window is proposed [13]. However, these window functions still cannot meet the requirements of the narrow main lobe, small side lobe, and fast attenuation simultaneously. [PERSON] proposed the use of rectangular shape pulse self-convolution to design a triangular-shaped pulse [14]. On this basis, Hanning convolution windows, triangular self-convolution windows, and hybrid convolution windows composed of rectangular windows and cosine windows have been successively proposed [15; 16]. Although the method mentioned above has a certain improvement effect, application of the TD-CDE still needs to be verified. The number of taps in the TD-CDE filter is positively correlated with the accumulative CD in the channel. For ultra-high-speed long-distance coherent optical communication systems, achieving CDE requires lots of TD-CDE filter taps, and the amount of calculation is very large. In this paper, we propose a lower-complexity TD-CDE filter structure, which is shown in the next Section. The main contributions of this paper are in two aspects.