Coherent optical communications have gained renewed interest due to the availability of high speed digital signal processing, low priced components as well as partly relaxed receiver requirements at high data rates. When the outputs of a coherent homodyne receiver are sampled at the Nyquist rate, the digitized waveform contains full information of the electric field, preserving the amplitude, phase and polarization from the optical domain to the electrical domain, enabling new potential of multi-level signaling (M-ary PSK and M-ary QAM modulation), as well as the possibility of exploring polarization multiplexing. Additionally, it enables quasi-exact compensation of linear transmission impairments such as Chromatic Dispersion (CD) and Polarization Mode Dispersion (PMD) by a linear filter. Digital Signal Processors (DSP) has been evolving as a practical solution for robust optical long-haul transmission, and it is expected that data converters will be able to satisfy the required high sampling rates in the near future. We will present the computer simulations results of algorithms for compensation of CD and PMD in a coherent optical system exploiting polarization multiplexing. The benefits of combining a phase estimation algorithm with a decision directed least-mean-square equalizer in a feedback configuration will be shown.