We review recent progresses on digital self-coherent detection of differential phase-shift keyed (DPSK) signal using orthogonal differential direct detection followed by high-speed analog-to-digital conversion and digital signal processing (DSP). Techniques such as data-aided multi-symbol phase estimation for receiver sensitivity enhancement, unified detection scheme for multi-level DPSK signals, and optical field reconstruction are described. The availability of signal field information brings the possibility to compensate for some linear and nonlinear transmission impairments through further DSP. An adaptive DSP algorithm for simultaneous electronic polarization de-multiplexing and polarization-mode dispersion compensation is also presented.
©2008 Optical Society of America
Optical transmission system based on self-homodyne differential phase-shift keying (DPSK) [1–4] has emerged as an attractive vehicle for supporting high-speed optical transport networks by offering lower requirements on optical signal-to-noise ratio (OSNR) and higher tolerance to system impairments such as certain fiber nonlinear effects as compared to traditional on-off-keying (OOK) based systems. Multilevel DPSK formats such as differential quadrature phase-shift keying (DQPSK) additionally offer high spectral efficiency and high tolerance to chromatic dispersion (CD), polarization-mode dispersion (PMD), and optical filtering, particularly when polarization-division multiplexing (PMUX) is also applied. Self-homodyne DPSK signals are received by differential direct detection that does not need an optical local oscillator (OLO) as required in coherent detection [5–8]. To generate coherent gain without the actual presence of a physical OLO, self-coherent detection was recently proposed, based either optical signal processing [9–11] or digital signal processing (DSP) [12,13]. With the help of high-speed analog-to-digital conversion (ADC) and DSP following orthogonal differential direct detection, the phase and even the field of a received optical signal can be digitally reconstructed [12,13]. Adaptive equalization of transmission impairments such as nonlinear phase noise, CD, and PMD could then be subsequently performed, in a similar way as digital coherent detection [14–16]. Such DSP-assisted self-homodyne detection is herein referred to as digital self-coherent detection (DSCD). These new capabilities bring opportunities to make transport systems more versatile, flexible, and ultimately cost-effective. With advances in high-speed electronic circuits, digital coherent and self-coherent detections are expected to find a wide range of applications to meet the ever-increasing demand of capacity upgrade and cost reduction in future optical networks.
This paper is organized as follows. In Section 2, we describe the architecture of DSCD. Section 3 briefly discusses a data-aided multi-symbol phase estimation (MSPE) scheme for receiver sensitivity enhancement [17–19]. Section 4 presents the detection of multi-level DPSK signals . The reconstruction of signal field and compensation of transmission impairments are discussed in Section 5. Section 6 presents a dual-polarization version of DSCD for electronic polarization de-multiplexing and PMD compensation (PMDC). A simple adaptive DSP algorithm for simultaneous electronic polarization de-multiplexing and PMDC is also presented. Section 7 concludes this paper.
2. Architecture of digital self-coherent detection
A schematic DSCD architecture is shown in Fig. 1. The optical complexity of the DSCD is similar to that of conventional direct-detection for differential quadrature phase-shift keying (DQPSK). The received signal, r(t)=|r(t)|exp[j·ϕ(t)], is first split into two branches that are connected to a pair of optical delay interferometers (ODIs) with orthogonal phase offsets θ and θ-π/2, where θ is an arbitrary phase value. Note that the phase orthogonality is assumed to be guaranteed, e.g., via the design reported in Ref. . This simplifies the control of the pair of ODIs to a single phase control. The delay in each of the ODI, τ, is set to be approximately T/sps, where T is the signal symbol period and sps is the number of samples per symbol of the analog-to-digital converters (ADCs) that convert the two detected analog signal waveforms, referred to as the I and Q components, to digitized waveforms uI(t) and uQ(t), which follow
In the special case with sps=1, the delay in the orthogonal ODI pair equals to the symbol period, and the I and Q decision variables for an m-ary DPSK signal can be directly obtained by setting θ=π/m, as to be discussed later. Any demodulator phase error ϕe=θ-π/m can be readily compensated by using the following simple electronic demodulator error compensation (EDEC) process 
3. Receiver sensitivity enhancement via data-aided MSPE
There is a well-known differential-detection penalty in receiver sensitivity for DPSK as compared to PSK. This penalty can be substantially reduced by using a data-aided MSPE that utilizes the previously recovered data symbols to recursively extract a new phase reference that is more accurate than that provided by the immediate past symbol alone, and its analog implementations have been proposed for optical DQPSK , DQPSK/ASK , and m-ary DPSK . The MSPE concept was recently extended to the digital domain in Refs.  and . An improved complex decision variable for m-ary DPSK can be written as 
where u(n) is the directly detected complex decision variable for the n-th symbol, m is the number of phase states of the m-ary DPSK signal, N is the number of past decisions used in the MSPE process, w is a forgetting factor, and Δϕ(n-q)=ϕ(n-q)-ϕ(n-q-1) is the optical phase difference between the (n-q)-th and the (n-q-1)-th symbols, which can be estimated based on the past decisions. For optical DQPSK, using recovered I and Q data tributaries, cI and cQ, we have 
where ⊕ denotes XOR logic operation.
One advantage of the digital implementation is that w can be conveniently set to 1, and N can be small (e.g., <5) to obtain most of the sensitivity enhancement . The computational complexity of the data-aided MSPE, in terms of the number of complex multiplications and additions, increases roughly linearly with N. Note that in digital coherent detection, carrier phase estimation, instead of the MSPE, is needed [8,22,23].
In a recent 40-Gb/s DQPSK experiment with offline DSP, the benefits of the MSPE and EDEC were confirmed. Figure 2 shows the BER performance with the data-aided MSPE activated in both back-to-back and nonlinear transmission configurations . At BER=10-3, the MSPE improves the back-to-back receiver sensitivity by 0.5 dB and 1 dB with N=1 and N=3, respectively. The forgetting factor w was set to 1. The performance difference between 215-1 and 27-1 patterns is negligible. The achieved back-to-back sensitivity is -41.5 dBm for BER=10-3, which is close to that obtained with coherent-detection QPSK (-42 dBm for BER=10-3) . After transmission over the 320-km fiber link, the signal performance was severely degraded by nonlinear phase noise and polarization-dependent frequency shift (PDFS) of the demodulator. Remarkably, with the combined use of EDEC and MSPE, the required OSNR for BER=10-3 is reduced by 3.2 dB, indicating the improved performance of DSCD over conventional differential direct detection that uses binary decision circuitries.
It is worth mentioning that the MSPE scheme can be applied to advanced modulation formats that involve simultaneous differential-phase and amplitude modulation, such as DQPSK/ASK . For quadrature amplitude modulation (QAM), such as 16-QAM, coherent detection, rather than differential detection, is usually used. In digital coherent detection, the decision-feedback aided carrier phase estimation  is effectively equivalent to the data-aided MSPE used here for DSCD.
4. Detection of m-ary DPSK
The DSCD can be used to receive high spectral-efficiency m-ary DPSK signals . An mary DPSK signal has log2(m) binary data tributaries that are usually obtained from m/2 decision variables associated with m/4 ODI pairs having the following orthogonal phase offsets, . With DSP, the last (m/2-2) decision variables can be derived by linear combinations of the first two decision variables, uI and uQ. This dramatically reduces the optical complexity associated with the detection of m-ary DPSK. The decision variables associated with phase offset πp/m (p=3,5,…,m/2-1), can be expressed as
Similarly, we can express their orthogonal counterparts as
The data tributaries of an m-ary DPSK signal can then be retrieved by .
When the data-aided MSPE is applied, uI and uQ need to be replaced with their corresponding improved decision variables. In effect, the complex decision variable u(n) or x(n) contains complete information on the differential phase between adjacent symbols, and is sufficient statistic, allowing to derive all the required decision variables. Note that a similar approach based on analog signal processing, rather than DSP, was reported in Ref. . The above formulas form the basis of a simple yet universal DSCD receiver platform for m-ary DPSK using only one pair of orthogonal demodulators as shown in Fig. 1.
5. Reconstruction of signal optical field and compensation of transmission impairments
5.1 Field reconstruction principle
The optical phase difference between adjacent sampling locations can be obtained from
With the differential phase information being available, a digital representation of the received signal field can be obtained by
where t0 is an arbitrary reference time, ϕ(t0) is a reference phase that can be set to 0, and the amplitude of the receiver signal can be obtained by an additional intensity detection branch  or approximated as below 
We note, however, that care needs to be taken at sampling locations where the signal amplitude is close to zero, particularly when the sampling resolution is limited [12,13]. In addition, the required digitization resolution is higher in DSCD than in digital coherent receiver.
When the inter-symbol interference caused by distortions such as dispersion and PMD is reasonably small, synchronous sampling with sps=1 may be used and the signal amplitude may be approximated as a constant. Note that DSCD can be designed to be polarization-independent to readily receive a single-polarization signal in an arbitrary polarization state, while digital coherent detection requires accurate polarization alignment between the signal and the OLO or polarization diversity. Once the received optical signal field is digitally available, advanced signal processing techniques, similar to those used in digital coherent receivers, may be applied to mitigate transmission impairments.
5.2 Reconstruction of signal constellation
Recently, in a proof-of-concept experiment, DSCD was shown to accurately reconstruct the constellation diagrams of a 40-Gb/s DQPSK signal and a 60-Gb/s 8ary-DPSK signal, and reveal quality degradations due to amplified spontaneous emission (ASE) noise and fiber nonlinearity . Figure 3 shows the experimental setup. A tunable laser locked at 1553 nm was used as the CW source. A 40-Gb/s return-to-zero (RZ) DQPSK signal was generated by modulating the CW source through a nested LiNbO3 Mach-Zehnder modulator (MZM) driven by the two 20-Gb/s data tributaries, followed by pulse carver, which is an x-cut LiNbO3 MZM driven sinusoidally at 20 GHz. The RZ pulses had a duty-cycle of 50%. The 20-Gb/s drive signals were generated by suitably multiplexing 10-Gb/s pseudo-random bit sequences (PRBSs) of length up to 215-1. To generate 8ary-DPSK, an additionally phase modulator, driven by another 20-Gb/s signal whose peak-to-peak amplitude is ~¼Vπ was used. The generated RZ-DQPSK signal or RZ-8ary-DPSK signal was then optionally transmitted over a transmission link consisting of a pre dispersion compensation module (DCM) providing -510 ps/nm dispersion, 4 80-km SSMF spans, each of which was followed by a 2-stage EDFA having a DCM inserted between its two stages, and post-DCM that compensated the overall dispersion to about zero. The span dispersion was under-compensated by 33, 37, 56, and 40 ps/nm (at 1550 nm), respectively, for the four spans. The span losses ranged from 18 to 21 dB. The power launched into each span can be varied for evaluating signal distortions under different conditions of ASE noise and fiber nonlinearity. After fiber transmission, the signal was sent into an optical pre-amplifier, followed by a bandpass filter with a 3-dB bandwidth of 0.3 nm.
The received signal was detected by a DSCD setup with offline DSP. The I/Q waveforms were then digitized at a sampling rate of 40 Gsamples/s using two 8-bit ADCs embedded within a Tektronix TDS6154C real-time oscilloscope. Care was taken to ensure that the two digitized waveforms were temporally aligned within 2 ps. The ADCs had low-pass filter characteristics with a cutoff frequency of about 15 GHz. The sampled I/Q waveforms were recorded and processed offline to reconstruct signal constellation diagrams under various link conditions.
From the digitized waveforms I(t) and Q(t), we can obtain the differential phase between adjacent symbols. The amplitude of the receiver signal can be obtained by an intensity detection branch  or approximated from I(t) and Q(t) . Figure 4 shows the reconstructed differential-phase constellation diagrams of the 40-Gb/s DQPSK signal at different optical signal-to-noise ratios (OSNRs). Each constellation contains about 5,000 symbols. As expected, the variance of the differential-phase distribution, σ 2 Δϕ, increased with the decrease of OSNR. Figure 4 also shows the measured differential-phase variance σ 2 Δϕ as a function of OSNR. The dashed line is calculated using the following relation 
where the noise effective bandwidth of the monitor NEB is determined by the optical bandpass filter to be 0.3 nm. The measured σ 2 Δϕ is in excellent agreement with the calculated one over an OSNR range from 10 dB to 25 dB. For OSNR>25 dB, the intrinsic differential-phase variance due to imperfect modulation starts to affect the measurement.
Figure 5 shows the reconstructed differential-phase constellations of the 40-Gb/s DQPSK signal after the 320-km SSMF transmission at different signal powers. The presence of the Gordon-Mollenauer nonlinear phase noise  becomes apparent at 6-dBm and 10-dBm signal powers, which corresponds to mean nonlinear phase shifts of 0.8 and 2 rad., respectively. Figure 5 also shows a reconstructed differential-phase constellation diagram of the 60-Gb/s RZ-8ary-DPSK signal. The modulator bandwidth limitation induced phase pattern dependence is more pronounced than in DQPSK, indicating that care needs to be taken in the modulation of multi-level DPSK formats.
5.3 Electronic dispersion compensation
With the availability of signal field, electronic dispersion compensation (EDC) may be performed to restore the original signal via, e.g., a multi-stage digital finite impulse response (FIR) filter that approximates the inverse function of the dispersion experienced by the signal during fiber transmission. EDC with DSCD was reported, both experimentally  and numerically , in typical optically repeated transmission links with the consideration of ASE noise and fiber nonlinearity. We note, however, that special care needs to be taken at sampling locations where the signal amplitude is close to zero [12,13]. For example, the effective sampling resolution needs to be sufficiently high.
5.4 Compensation of nonlinear phase noise
In optical fiber transmission, phase modulated signals may be degraded by the Gordon-Mollenauer nonlinear phase noise  resulting from the interaction between the self-phase modulation (SPM) and the ASE noise. It was found that the Gordon-Mollenauer nonlinear phase noise can be substantially compensated by a lumped post compensation process , which can be achieved by replacing the directly measured complex decision variable, u(n), with a compensated complex variable ν(n)
where cNL is the average nonlinear phase shift experienced by the signal over the fiber transmission, P(n) is the normalized power of the n-th symbol, and the factor of ½ is for the 50% under-compensation that was found to be optimum in the lumped post-compensation scheme . Post nonlinear phase noise compensation was recently demonstrated in digital coherent detection [27,28] and DSCD .
6. Dual-polarization digital self-coherent detection
Polarization multiplexing provides a straightforward way to double spectral efficiency, relax transmitter/receiver bandwidth requirement, and increase signal tolerance to CD and PMD. To receive a polarization-multiplexed DPSK signal, polarization diversity is needed for DSCD. Figure 6 shows the schematic of a dual-polarization DSCD. The received optical signal is first split by a PBS into two orthogonally polarized components, which are demodulated by two orthogonal ODI pairs, and detected and sampled according to the single-polarization DSCD architecture described previously. Figure 6 also shows an alternative implementation where four PBSs are placed after a single orthogonal ODI pair. The optical fields of these two polarization components are then reconstructed, and used to recover the fields of the original polarization components.
6.2 Simultaneous electronic polarization de-multiplexing and PMDC
Due to fiber birefringence, the two orthogonal polarization components of the optical signal as reconstructed at the receiver after fiber transmission are generally not the original polarization components of the polarization-multiplexed signal, so electronic polarization de-multiplexing (EPDMUX) is needed to recover the original polarization components. In addition, fiber birefringence induced signal polarization changes are usually time varying due to fluctuations in ambient temperature and mechanical stress, so adaptive EPDMUX is needed. In the presence of PMD, signal is further distorted. We discuss below how simultaneous EPDMUX and PMDC may be realized with DSCD.
Figure 7 illustrates the polarization evolution of a polarization-multiplexed optical signal in a typical fiber transmission link having PMD. Under the first-order PMD assumption, the two orthogonal polarization components reconstructed in the DSCD, Ex’ and Ey’, can be expressed as
where Ex and Ey are the original polarization components at the transmitter, matrix T represents the polarization transformation of the fiber link, R1 is the rotation matrix associated with the projection of the original signal on the principle state of polarization (PSP) axes of the fiber PMD, R2 is the rotation matrix associated with the projection of the signal components along the PMD PSP axes of the fiber PMD on the polarization axes of the PBS(s) used in the receiver, PMD(ω) is the PMD matrix, and P is a matrix representing the addition phase delay between the two reconstructed fields after the polarization beam splitting at the receiver.
Using the notations shown in Fig. 7, the above matrixes can be further expressed as
where θ 1 and θ 2 are rotation angles as illustrated in Fig. 7, τDGD is the PMD-induced differential group-delay (DGD) between the two PSP axes, δϕPMD is the phase delay caused by fiber PMD or birefringence, Δf is the frequency offset from the center frequency of the signal, and δϕPBS is the addition phase delay between the two reconstructed fields after the polarization beam splitting. When the polarization transformation matrix is known, the original signal polarization components can then be derived from the reconstructed polarization components through
In general, there are five parameters that specify the polarization transformation matrix, θ 1, τDGD, δϕPMD, θ 2, δϕPBS, and they need to be determined. When PMD is sufficiently small, e.g., the PMD-induced DGD is much smaller than the signal symbol period, τDGD may be set to zero, leaving four parameters to be determined. Since these parameters are generally time varying, it is needed to find the values of these parameters dynamically. For high-speed optical signal, parallelization in signal processing is needed to lower the speed requirement for EPDMUX and electronic PMDC. Figure 8 shows a parallel DSP arrangement with a multiplexer and demultiplexer pair and multiple (M) processing units (PUs) each operating on a block-by-block basis. Overlap of data processed by adjacent PUs is needed to address the PMD induced inter-symbol interference.
Figure 9 shows the block diagram of a PU. The PU inputs two blocks of samples that represent the two reconstructed optical fields, and outputs two blocks of samples that represent the original signal polarization components. The block size is N. Note that the field reconstruction process can share the same parallel structure used for EPDMUX and PMDC. Since the changes of parameters θ 1, τDGD, δϕPMD, and θ 2 are generally much slower than the signal symbol rate, a feed-forward path is used to estimate the best-guess values of these four parameters through a search process, and provide them to a real-time path. For example, the search process can be based on a global search that samples all possible sets of values of the matrix parameters, computes the corresponding Ex and Ey values for each set according to Eq. (16), and finds the set of best-guess values that give the minimized variances of |Ex(t)|2 and/or |Ey(t)|2 since each of the original signal polarization components is of DPSK format, which is has a constant amplitude characteristics. In effect, this is similar to the constant modulus algorithm (CMA) widely used in blind equalization of wireless DPSK signals , and recently used in digital self-coherent receivers . The feed-forward path constantly updates the best-guess values at a speed that is sufficient to track the physical changes of these parameters (e.g., 10 kHz). To save computational effort, a new set of best guess values can be obtained from those that are the nearest neighbors (in a multi-dimensional space constructed by these parameters) of the preceding set of best guess values.
The real time path uses the best guess values provided by the feed-forward path, computes Ex and Ey values for each of a set of possible δϕPBS values, and finds the Ex and Ey values that gives the minimized variance of |Ex(t)|2 and/or |Ey(t)|2. Note that in DSCD, δϕPBS needs to be found in real time (on a block by block basis) since there is an uncertainty in the relative phase difference between the reconstructed signal fields Ex’ and Ey’, as indicated in Eq. (9). For digital coherent detection, the OLO provides a common phase reference for both polarization components, so δϕPBS can be advantageously estimated in the feed-forward path to save computational effort. Simultaneous EPDMUX and PMDC have recently been demonstrated with digital coherent detection through offline DSP [14–16].
We have reviewed recent progresses on digital self-coherent detection. Techniques such as data-aided multi-symbol phase estimation (MSPE) for receiver sensitivity enhancement, unified detection of multi-level DPSK signals, and optical field reconstruction have been briefly discussed. Adaptive DSP methods for the compensation of linear and nonlinear transmission impairments, PMD in particular, are also described. With real-time implementations on the horizon [32–34], digital self-coherent detection and digital coherent detection in general are expected to find interesting applications in future high-speed optical transport systems.
The authors wish to thank Y. K. Chen, A. R. Chraplyvy, C. R. Giles, and R. W. Tkach for their support.
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