x channel y channel are combined and form channels of complex signals e g i jq and the equalization is performed mathematically based on the complex numbers as will be appreciated the recovered output signals 412 a thru 412 d from the coherent receiver 410 can be written as v x y t d x y t h link u00b7 e j u00b7 2 u00b7 u03c0 u00b7 u03b4f u00b7 t u03c6 t equation 1 d x y t is the differential coded information bits on the x and y polarization signals h link is the combined transfer function of the link and hardware u03b4f is the lofo as compared to the transmitter carrier frequency and u03c6 t is the random phase noise introduced by laser and link propagation the distortions from the communication channel line and hardware impairments are compensated for in the adaptive equalizer 430 after the equalizer 430 the output signals s x and s y pass through the digital delay interferometer 440 which generates output signals 442 a and 442 b u x u y the output of the digital interferometers 440 is u x y nt s nt u00b7 s nt t u00b7 e j 2 u00b7 u03c0 u00b7 u03b4f u00b7 t u03c6 nt u2212 u03c6 nt t equation 2 as mentioned above with the dqpsk format information bits are coded as transient between adjacent symbols s nt.