Statistical correlation and independence among parallel outputs from delay-interferometer based direct detection multilevel optical DPSK receivers

A delay-interferometer based direct detection multilevel optical differential phase-shift keying receiver is theoretically analyzed. Using the Karhunen–Loeve series expansion method, we employ the eigenfunction expansion of signal and spontaneous emission noise in the frequency domain and arrive at conditions to be satisfied if the parallel outputs from the delay-interferometers are to be statistically uncorrelated and independent. Noting that statistically uncorrelated random variables need not be statistically independent in general, our analysis leads to the fact that while statistical uncorrelatedness can be achieved among the parallel outputs by appropriately selecting the delay interferometer and optical filter bandwidth parameters, statistical independence is impossible to achieve. Through a qualitative argument based on the characteristic function of the parallel outputs from the delay interferometers, we show that as long as the parallel outputs are statistically uncorrelated, the statistical dependency among them is significantly reduced compared to the case where the parallel outputs are statistically correlated.