The schematic of an adaptive TDE with LMS-based method for coefficient updating is shown in Fig. 1. $rk$ is the received signal and $m\u02dck$ is the output after equalization. $\u03f5k=m\u02dck\u2212m^k$ is the error signal used as a feedback in the adaptation. In practical high-speed optical fiber communication systems, higher-order modulation schemes like QPSK together with polarization-division multiplexing are used to increase the spectrum efficiency. At the digital optical coherent receiver, a butterfly-structured TDE with four subequalizers is commonly used for simultaneous dispersion compensation and polarization-division demultiplexing. Assume that the four subequalizers are named Hxx, Hxy, Hyx, and Hyy, the outputs of the subequalizers can be written as a convolution between the received signals and the tap coefficients of the equalizer as Display Formula
$m\u02dcxx,k=\u2211l=\u2212LLCxx,lrx,k\u2212l,m\u02dcyx,k=\u2211l=\u2212LLCyx,lry,k\u2212lm\u02dcxy,k=\u2211l=\u2212LLCxy,lrx,k\u2212l,m\u02dcyy,k=\u2211l=\u2212LLCyy,lry,k\u2212l.$(1)
In Eq. (1), $rx,k=Ix,k+jQx,k$ is the received signal on the $X$-polarization and $ry,k=Iy,k+jQy,k$ is the received signal on the Y-polarization. $m\u02dcx,k=m\u02dcxx,k+m\u02dcyx,k$ and $m\u02dcy,k=m\u02dcxy,k+m\u02dcyy,k$ are the final outputs from the butterfly TDE on the two polarizations for the $k$’th time slot. Here, it is assumed that the four subequalizers all have the same number of the tap coefficients, $2L+1$.