1994 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 13, JULY 1, 2010 Estimating the Performance of Direct-Detection DPSK in Optical Networking Environments Using Eigenfunction Expansion Techniques João J. O. Pires, Member, IEEE, and Luís G. C. Cancela, Student Member, OSA Abstract—In-band crosstalk, due to multiple interferers, has been identified as one of the most severe impairments in optical transparent networks, especially in the ones with a large number of nodes and a high wavelength density. Due to its robustness to in-band crosstalk differential phase-shift keying (DPSK) emerges as an attractive modulation scheme to be used in such environ- ments. This paper proposes a rigorous formulation to estimate the performance of direct-detection DPSK receivers using an eigen- function expansion technique in the time domain. The method takes into account both the in-band crosstalk, due to an arbitrary number of interfering terms, and the amplified spontaneous emission noise and is able to deal with any combination of optical and electrical filter shapes. Using this method the accuracy of an approximation based on the wideband optical filtering assump- tion was evaluated and shown that the approximation, although not providing reliable results for the error probabilities, can be used with confidence to compute power penalties due to in-band crosstalk. Furthermore, the crosstalk tolerance of DPSK over on-off keying was quantified and shown that this tolerance is reduced when the number of interferers increases. Index Terms—Differential phase-shift keying, eigenvalues and eigenfunctions, in-band crosstalk, optical networking. I. INTRODUCTION D IFFERENTIAL PHASE-SHIFT KEYING (DPSK) is employed as a modulation scheme in a wide variety of digital communication systems. This scheme entered in the optical arena in the late 1980s, in the context of a research effort that took place at that time in order to use coherent detection techniques as a way to improve receiver sensitivity and hence transmission distance [1], [2]. However, the ad- vent of erbium-doped fiber amplifiers refocused the interest on direct-detection techniques and, as a consequence, the possibility of using a Mach–Zehnder delay interferometer in conjunction with a dual-photodiode balanced receiver, to demodulate optical DPSK signals, started to attract atten- tion [3]–[5]. In the last years, direct-detection DPSK has regained momentum aiming mainly dense wavelength division multiplexing (DWDM) long-haul transmission applications Manuscript received March 12, 2009; revised December 04, 2009, January 29, 2010; accepted March 03, 2010. Date of publication May 27, 2010; date of current version July 05, 2010. J. J. O. Pires is with Instituto de Telecomunicações and Department of Elec- trical and Computer Engineering, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (e-mail: jpires@lx.it.pt). L. G. C. Cancela is with Instituto de Telecomunicações and Department of Science and Information Technology, Instituto Superior de Ciências do Trabalho e da Empresa, 1649-026 Lisboa, Portugal (e-mail: luis.cancela@iscte.pt). Digital Object Identifier 10.1109/JLT.2010.2051319 [6]–[8]. This renewed interest comes from the fact that DPSK outperforms the traditional binary on-off keying (OOK) in such aspects as receiver sensitivity, robustness to transmission impairments, and tolerance to signal power fluctuations [9], [10]. In addition, there is also a significant interest about the promises of this scheme, having in view optical networking applications. Besides the reasons invocated for long-haul transmission, this modulation format presents some properties particularly advantageous in the context of optical networks, including the following: 1) the constant-intensity nature of DPSK favors the use of semiconductor optical amplifier gates in optical switching applications [11], because the patterning effects and the dynamic interactions between DWDM channels due to carrier lifetimes can be greatly reduced [12], [13]; 2) the fact that the decision threshold is zero facilitates the design of optical receivers for burst applications in diverse areas, such as optical packet switching, optical burst switching, and passive optical networking, because it avoids the use of complex deci- sion threshold tracking circuits employed in traditional OOK receivers [14], [15]; 3) the robustness to narrow band optical filtering [16] facilitates the design of transparent optical net- works in the measure that contributes to mitigate the cascading filtering effects of optical add-drop multiplexers (OADMs); 4) a higher tolerance to in-band crosstalk than OOK [17] permits to relax the isolation requirements of optical devices, such as multiplexers, and optical switches. The improved tolerance of DPSK towards crosstalk is par- ticularly advantageous in optical networking environments, be- cause due to imperfections of optical devices used to build net- work elements, such as OADMs, optical cross-connects, optical burst switches, as well as from spurious reflections inside the network, there are many leakage signals that interfere with the desired signal originating crosstalk [18]–[20]. It is well known that this phenomenon can be particularly damaging when the interference and the signal have the same nominal wavelength, leading to the so-called in-band or homodyne crosstalk, since in this case the signal-crosstalk beatings originated at the receiver can not be filtered out, becoming, as a consequence, an impor- tant source of signal quality degradation. In the context of optical transparent networks its damaging effect is further enhanced due to the fact that the crosstalk ac- cumulates as the signal, corresponding to a given lightpath, tra- verses multiple network nodes [21]. In networks with a large number of nodes and a high wavelength density this impair- ment can become one of the most severe sources of performance degradation. Hence, the development of techniques capable of providing accurate performance estimations in the presence of 0733-8724/$26.00 © 2010 IEEE