## Abstract

Mode-division multiplexing over 33-km few-mode fiber is investigated. It is shown that 6×6 MIMO processing can be used to almost completely compensate for crosstalk and intersymbol interference due to mode coupling in a system that transmits uncorrelated 28-GBaud QPSK signals on the six spatial and polarization modes supported by a novel few-mode fiber.

©2011 Optical Society of America

## 1. Introduction

As optical transmission systems operating over single strands of legacy single-mode fiber (SMF) are reaching their fundamental capacity limits [1], space-division multiplexing (SDM) is likely to become a critical technology to meet future traffic demands. The multiple parallel transmission paths required for SDM can initially be based on already installed fiber bundles, but will eventually ask for the deployment of additional fiber infrastructure, possibly using new, SDM-optimized fiber types. In order to make SDM systems feasible with respect to cost and energy consumption, photonic and electronic integration technologies need to be explored that make SDM scale better than through simple hardware duplication [2]. In this respect, mode-division multiplexing (MDM) over few-mode fiber (FMF) can be a promising approach if reliable long-haul MDM transmission can be realized and modal crosstalk can be dealt with using multiple-input-multiple-output (MIMO) signal processing. In [3] it has been pointed out that in a SDM MIMO system with multiple coupled modes, the full capacity potential of the SDM waveguide can be realized at low outage probabilities if the transmitter is able to individually address all propagation modes supported by the SDM waveguide and if the receiver is able to mode-selectively and coherently detect all propagation modes.

Previous work on MIMO transmission over multimode fiber (MMF) has mainly focused on mode-group-division multiplexing (MGDM) in fibers that allow the propagation of hundreds of modes. Using various kinds of coupling techniques, such as offset launch, groups of several tens of modes could be independently excited and received [4–6]. However, the inability to individually excite and detect each propagation mode lets modal dispersion within each mode group still limit the bandwidth-distance product of such systems, which makes them unattractive for long-haul transmission. In [7–9], MDM transmission over fibers that support three spatial propagation modes has been demonstrated, but not all propagation modes were simultaneously excited and received. In [10], the first experimental demonstration of full coherent 6×6 MIMO transmission over a FMF with six propagation modes has been presented.

In this paper the results presented in [10] are extended to 28 GBaud and 33 km of fiber. Furthermore, due to refinements in digital-signal processing algorithms, considerably reduced penalties are obtained with an improved MIMO equalization scheme. In Sec. 2 the experimental setup used to investigate MDM transmission performance is introduced, and signal generation, fiber design, mode coupling, as well as receiver architecture are discussed. In Sec. 3 an algorithm for system identification is introduced, allowing us to extract the 36 impulse responses of the 6×6 MIMO channel. The improved MIMO equalization scheme is elucidated in Sec. 4 and the MDM transmission performance is assessed in Sec. 5. Finally, conclusions are drawn in Sec. 6.

## 2. Experimental setup

In order to investigate the performance of an MDM transmission system, an experiment was set up consisting of four main building blocks:

- Generation of transmit signals and noise loading
- FMF designed to support three spatial modes
- Mode multiplexer and demultiplexer
- Three polarization diversity coherent receivers that jointly detect the fully spatially resolved optical field in the fiber

Figure 1 gives a detailed overview of the setup; the four building blocks are described in detail in the following subsections.

#### 2.1. Signal generation

The three decorrelated and polarization multiplexed transmit signals are generated as follows: First, a non-return-to-zero signal carrying a De Bruijn binary pattern with a period of 2^{11} = 2048 symbols is generated at 28 Gb/s using a programmable pattern generator (PPG). This signal is split into two copies that drive a double-nested Mach-Zehnder modulator (DMZM). A delay of 16 symbols each of duration *T* = 1/28 ns is inserted in order to decorrelate Inphase (I) and Quadrature (Q) components of the resulting quadrature phase-shift keyed (QPSK) signal. An external cavity laser at a wavelength of 1555 nm with a nominal linewidth of 100 kHz is used as optical source. Polarization multiplexing is emulated using a polarization rotated copy of the optical signal with a delay of ∼327 symbols and a polarization beam combiner (PBC). Noise loading is performed at the transmitter using a variable optical attenuator (VOA), an Erbium doped fiber amplifier (EDFA), and an optical-bandpass (OBP) filter with a 3-dB bandwidth of 1.3 nm. The resulting signal is then split into three copies, two of which are decorrelated by another ∼740 symbols and ∼1442 symbols, respectively, to generate a total of 3 × 2 = 6 delay decorrelated data streams.

#### 2.2. Few-mode fiber

The 33-km long FMF was designed to allow propagation of the LP_{01} mode as well as the two-times degenerate LP_{11} mode, subsequently referred to as the LP_{11a} and the LP_{11b} modes. The FMF has a depressed cladding index profile with a normalized frequency of *V* ≈ 5. The index profile was optimized in order to minimize and equalize the differential group delay (DGD) to less than 60 ps/km across the C-band and to effectively cut off the LP_{21} and LP_{02} modes. At 1550 nm, a loss of 0.21 dB/km was measured for both LP_{01} and LP_{11} modes, and the effective areas of the LP_{01} and LP_{11} modes were calculated to be approximately 155 *μ*m^{2} and 320 *μ*m^{2}, respectively.

The intensity profiles of the spatial modes after 33-km fiber propagation are shown in Figs. 2(a)–2(c). Either the LP_{01} mode or the LP_{11}
* _{a}* mode was launched, and the intensity profile at the fiber end facet was captured with an InGaAs infrared camera (Note that some pixels of the camera were malfunctioning). For the LP

_{01}mode the image profile was found to be independent of polarization or fiber arrangement. When one of the LP

_{11}modes was launched, a linear combination of LP

_{11a}and LP

_{11b}was observed after propagation, and by moving the fiber, e.g., by using an arrangement of loops similar to a manual polarization controller, it was possible to generate intensity profiles similar to the expected theoretical LP

_{11}mode profiles [see Figs. 2(b) and 2(c)]. During these measurements, strong polarization dependence of the coupling between the two degenerate LP

_{11}modes was evident. Note that the arrangement of loops is only used while recording the intensity profiles with the infrared camera and not in the further measurements reported. Without the loops random linear mode combinations are captured. The crosstalk between the LP

_{01}and the LP

_{11}modes after 33-km fiber was fluctuating over time in a range of −15 to −20 dB, indicating a clear mode separation even over longer distances. In contrast, the two LP

_{11}modes are continuously mixing inside the fiber as a result of being linear combinations of the “true” fiber modes. In particular, the LP

_{01}mode corresponds to the HE

_{11}mode that can be linearly polarized either horizontally or vertically. The LP

_{11}modes are linear combinations of the TE

_{01}mode, the TM

_{01}mode, and the two times degenerate HE

_{21}mode. For a more detailed discussion of the fiber modes and their mixing properties, we refer to [11, 12].

#### 2.3. Mode multiplexer

In order to selectively excite all spatial modes propagating in the FMF, a mode multiplexer (MMUX) is required. The MMUX used in this work is based on free space optics, and its principle is shown in Fig. 2(d). Three polarization multiplexed signals are fed into the MMUX using three SMFs that are terminated by collimators resulting in a nominal beam diameter of 500 *μ*m. These collimated beams are then combined using beam splitters. Mode selectivity is achieved using phase plates. The phase plates are made of 0.7-mm thick Borosilicate glass, and the phase structure (a *π* phase jump between two half planes) was etched using a photolithographic process. A similar principle has been previously described in [13, 14]. However, in our arrangement the phase plates are located in the image plane with respect to the end-faced of the FMF, whereas in [13] and [14] they are located in the Fourier plane. Finally, the combined beam is imaged on the FMF facet using lenses *f*
_{1} and *f*
_{2} with focal lengths of 75 mm and 3.9 mm, respectively. As the MMUX design is reciprocal, the same device can also be used as mode demultiplexer. In this case the SMFs act as mode filters [15].

Coupling losses of 8.3 dB, 10.6 dB, and 9.0 dB were measured using a configuration of two MMUXs connected by a 2-m long FMF for the LP_{01}, the LP_{11a}, and LP_{11b} modes, respectively. The theoretically minimum coupling loss of this configuration is 5.5 dB for all three spatial modes. Crosstalk induced by the two MMUXs was measured by launching the LP_{01} mode and measuring the power at both LP_{11} outputs. The crosstalk, defined as the power measured at one of the two LP_{11} ports divided by the power measured at the LP_{01} port of the receiving MMUX was below −28 dB for both LP_{11} modes. Thus a strong mode selectivity of the MMUX is confirmed. Further details on the MMUX design and performance are given in [16].

#### 2.4. Receivers

Each of the three demultiplexed signals is preamplified using an EDFA and received by a polarization diversity coherent optical receiver. Each receiver is composed of a polarization-diversity optical 90-degree hybrid followed by four pairs of balanced photodetectors. In this work, the transmit laser is also used as local oscillator in a self-homodyne arrangement. The waveforms received from all supported propagation modes are simultaneously sampled and stored using three different four-channel real-time oscilloscopes (Scopes 1–3) that are co-triggered using a trigger setup similar to [17]. The oscilloscopes have effective resolutions of 4.5 bit, 4–5 bit (frequency dependent), and 5.5 bit; bandwidths of 30 GHz, 20 GHz, and 32 GHz; and sampling rates of 80 GS/s, 50 GS/s and 80 GS/s (locked with a 10 MHz reference). The waveforms were resampled to two samples per symbol and bulk chromatic dispersion compensation of 665 ps/nm was performed on a personal computer prior to off-line MIMO equalization.

## 3. System identification

During propagation in the FMF, signals couple between modes in a random manner. As the LP_{01} and the LP_{11} modes propagate with different group velocities, crosstalk and intersymbol-interference (ISI) spreading over multiple symbols can occur. Crosstalk builds up if light is coupled from one mode to another and remains there upon detection. ISI is induced if this crosstalk is coupled back to the original mode after some amount of propagation in a mode with different group velocity. This effect is similar to polarization mode dispersion (PMD) and can be fully compensated using linear MIMO equalization as long as mode dependent loss (MDL), the analogon to polarization dependent loss (PDL) in SMF, is negligible. The effects of PMD in SMFs are well understood today [18], but the nature of mode coupling in various kinds of FMF still needs to be investigated. At low enough optical power levels, the mode coupling processes are linear; nonlinear coupling [19] is expected to occur at power levels beyond the ones used in this work.

In this section we analyze mode coupling in a 33-km long FMF by identifying the 36 impulse responses describing linear coupling between all six transmit signals. Using the *a priori* knowledge of the complex transmit pattern, the channel impulse response **h** = [*h*
_{0}, *h*
_{1},..., *h*
_{2N−1}]* ^{T}* can be estimated using the least-mean square (LMS) algorithm by minimizing the error
${|{x}_{k}\hspace{0.17em}-\hspace{0.17em}{\widehat{\mathbf{\text{h}}}}_{k}^{T}{\mathbf{\text{s}}}_{k}|}^{2}$ where for each sampling instant

*k*the estimate

**ĥ**

*of*

_{k}**h**is updated according to [20]

*x*is the

_{k}*k*-th received sample and

**s**

*= [*

_{k}*a*, 0,

_{k}*a*

_{k}_{+1}, 0, ... ,

*a*

_{k+N−1}, 0]

*is a vector of*

^{T}*N*twofold oversampled complex transmit symbols

*a*that contribute to

_{k}*x*;

_{k}*μ*is the LMS adaptation gain.

Several data sets were captured, from which two are evaluated in this work. The first one uses a slightly suboptimum setting of the MMUX to highlight the effects of modal crosstalk at the transponders. The second data set uses optimized MMUX settings to minimize this type of crosstalk. Figure 3 shows the magnitude of the complex impulse responses for all six received waveforms at an optical signal-to-noise ratio per polarization (OSNR_{Pol}) of 23 dB over the full pattern period of 2048 symbols for the first data set with 0 dBm optical power launched into the FMF per spatial mode (OSNR_{Pol} is defined as the optical signal power of one polarization and spatial mode divided by the noise power within 0.1 nm per spatial and polarization mode). Six groups of peaks can be identified for each waveform. By correlating the pattern delays described in Sec. 2.1, individual transmit modes and thus all 36 impulse responses can be identified. Strong coupling between all modes is evident for this data set, as expected from the suboptimal settings of the MMUXs. For the received LP_{01} modes (RX LP_{01x} and RX LP_{01y}) one strong peak and several lower peaks with a spacing of 16 symbols is observed. This spacing corresponds to the delay inserted at the transmitter to decorrelate the two quadratures. We hence identify these side peaks as artefacts from using a delayed copy for modulating I and Q components. The origin and impact of this artefact will be discussed in more detail in a separate paper [21]. Furthermore, two strong peaks that are spaced about 40 symbols apart are observed for the received LP_{11b} modes (RX LP_{11bx} and RX LP_{11by}). This delay does not correspond to a multiple of the quadrature delay and is therefore attributed to a suboptimal adjustment of the two MMUXs separated by the 33 km of fiber. Thus, signal parts from the LP_{01} as well as the LP_{11} modes are detected. From the 40-symbol separation of the peaks, a DGD of 1.43 ns is concluded while 2.0 ns were expected from the fiber’s nominal modal dispersion of 60 ps/km. Before recording the second data set the adjustment of the mode multiplexer and demultiplexer was carefully optimized and the FMF input power per spatial mode was increased to 4 dBm. The resulting impulse responses are shown in Fig. 4. It can clearly be seen that the cross-coupling between the LP_{01} mode and the LP_{11} modes is considerably reduced, while the LP_{11a} and LP_{11b} modes are still strongly coupled. The artificial side peaks due to the quadrature delay are still evident.

## 4. MIMO equalization

In coherent optical transmission systems, MIMO signal processing is already widely used in its 2×2 realization in order to enable polarization-division multiplexing (PDM) and to compensate for PMD. In the case of MDM over six modes, the respective algorithms need to be scaled to 6×6 MIMO, requiring 36 adaptive finite-impulse-response (FIR) filters. In comparison, a SDM system using PDM over three uncoupled waveguides requires three 2×2 adaptive FIR filters. Hence, the equalizer complexity increases by a factor of three, when allowing for coupled SDM waveguides, assuming an equal number of taps per adaptive FIR filter and equal complexity of the adaptation algorithm. In general, going from *M* uncoupled SDM waveguides using PDM to a full 2*M* × 2*M* MIMO system on *M* coupled waveguides results in a complexity increase of 4*M*
^{2}/(4*M*) = *M*. Figure 5 shows a block-diagram of the network of 6×6 adaptive linear equalizers used to approach the minimum-mean-square error independently for the received soft symbols *y _{m}*(

*k*) according to

**w**

*(*

_{m,n}*k*) is the

*L*-element column vector of equalizer coefficients that undo coupling between the

*m*-th transmitted mode and the

*n*-th received mode at the

*k*-th symbol time-step and

**x**

*(*

_{n}*k*) is the vector of

*T*/2 spaced received samples contributing to

*y*(

_{m}*k*). For each of the six channels, carrier phase estimation (CPE) of

*ϕ*(

_{m}*k*) is independently performed using the fourth power algorithm [22]. Finally, hard decision of the Gray-mapped QPSK symbols is performed, and the bit-error ratio (BER) is estimated over 10

^{6}bits. For each of the 36 equalizers the estimation of the optimum coefficients is continuously updated using the LMS algorithm [20] where

*d*(

_{m}*k*) =

*â*(

_{m}*k*) is the

*k*-th received symbol of the

*m*-th MDM channel after hard decision, and Δ is a delay for estimating the carrier phase

*ϕ*(

_{m}*k*) over causal and anti-causal components. Figure 6 shows a detailed block diagram of the MIMO equalization algorithm described above.

In order to achieve convergence of the above decision-directed LMS algorithm, proper initialization of the equalizer coefficients is required. In conventional PDM-QPSK systems this so-called “pre-convergence” can typically be obtained blindly, e.g., by using the constant-modulus algorithm (CMA). However, as up to 120 taps per equalizer are used in this work, the data-aided LMS algorithm is used for the first 5 × 10^{5} symbols in order to initialize the coefficients [23]. This algorithm is identical to the decision-directed LMS, except that *a priori* knowledge of the transmitted symbols *d _{m}*(

*k*) =

*a*(

_{m}*k*) is used and the carrier phase

*ϕ*(

_{m}*k*) is estimated from the averaged deviation with respect to the absolute phase. The algorithm’s capability to switch between data-aided and decision-directed modes of operation is indicated by the switch in Fig. 6. In order to realize this algorithm in a practical transmission system, a training pattern is required during start-up.

## 5. Transmission performance

Using the MIMO equalization scheme described above, the transmission performance over 33-km FMF is studied in this section. The equalizer parameters, such as the LMS adaptation gain *μ* and the averaging length and delay Δ for CPE are optimized for each point. Figure 7(a) shows the minimum achieved BERs vs. OSNR_{Pol} using the first data set (Fig. 3) with suboptimal adjustment of the mode couplers and *L* = 120 taps per equalizer (filled symbols). As a reference, the BERs for the back-to-back measurement of each of the six receiver channels without PDM and SDM are plotted as open symbols. Variations of these curves characterize variations of the receiver hardware (optical 90-degree hybrid, balanced photodetectors, and oscilloscope front-end). Furthermore, the theoretical limits for 28-GBaud QPSK with (dashed) and without (solid) correcting for I/Q correlation are shown. For low signal-to-noise ratios and if the equalizer length exceeds the quadrature delay some correlation is present in the signal, and the LMS algorithm can exploit this additional I/Q correlation information, leading to a reduction of the theoretical BER limit. For further discussion we refer to [21]. The results of Fig. 7(a) show that even with a suboptimal alignment of the mode couplers, BERs below 10^{−3} can be achieved with penalties of about 2 dB for the LP_{11} modes and about 4 dB for the two polarizations of the LP_{01} mode, respectively.

Figure 7(b) shows the same results for the second data set (Fig. 4) with optimized coupler adjustment, again with *L* = 120 taps per equalizer. In this case the transmission penalties are considerably reduced to about 1 dB for all modes. This result demonstrates the effectiveness of MIMO equalization and the importance of accurate excitation and reception of all individual modes. Furthermore, Fig. 7(c) shows the results for the second data set with *L* = 80 taps per equalizer. Figure 7(d) reveals that large penalties with an error floor for the LP_{01} mode occur for *L* = 20. We contribute this to the fact that the LP_{01} mode (group delay *τ*
_{01}) suffers from residual uncompensated crosstalk caused by the two LP_{11} modes (group delay *τ*
_{11}) if *L* ≲ |*τ*
_{01} − *τ*
_{11}|/(2*T*). In contrast, each of the LP_{11} modes receives crosstalk from one mode experiencing *τ*
_{11} and one mode experiencing *τ*
_{01}. Hence, one interfering mode is compensated even for small values of *L*, resulting in a reduced crosstalk. Finally, the required OSNR_{Pol} for a BER of 10^{−3} is plotted vs. the number of equalizer taps in Fig. 7(e), showing that almost optimum performance is achieved for 100 taps corresponding to an equalizer length that exceeds the maximum DGD of 1.43 ns found in Sec. 3.

## 6. Conclusions

In this work, we assessed MDM over a FMF customized to allow propagation of six spatial and polarization modes. Our analysis of the 36 channel impulse responses and the transmission performance shows that even with optimized coupler adjustment, crosstalk occurs not only between the LP_{11} modes but also to some extent between the LP_{11} modes and the LP_{01} mode. A maximum DGD of 1.43 ns for the 33-km long FMF is found, corresponding to a required length of *L* > 80 *T*/2-spaced taps, for each of the 36 equalizers at a symbol rate of 28 GBaud. Our experiments prove the principle of scaling capacity using MDM in FMF in combination with MIMO signal processing. However, a host of open questions, such as, efficient amplification schemes, optimal fiber design, hardware efficient MIMO equalization schemes, and the impact of fiber nonlinearity remain to be answered.

## References and links

**1. **R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. **28**, 662–701 (2010). [CrossRef]

**2. **P. J. Winzer, “Energy-efficient optical transport capacity scaling through spatial multiplexing,” IEEE Photon. Technol. Lett. **23**, 851–853 (2011). [CrossRef]

**3. **P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express (to be published). [PubMed]

**4. **A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent optical MIMO (COMIMO),” J. Light-wave Technol. **23**, 2410–2419 (2005). [CrossRef]

**5. **C. P. Tsekrekos and A. M. J. Koonen, “Mitigation of impairments in MGDM transmission with mode-selective spatial filtering,” IEEE Photon. Technol. Lett. **20**, 1112–1114 (2008). [CrossRef]

**6. **S. Schöllmann, N. Schrammar, and W. Rosenkranz, “Experimental realisation of 3 x 3 MIMO system with mode group diversity multiplexing limited by modal noise,” in *National Fiber Optic Engineers Conference*, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWA68.

**7. **N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10 km two-mode fiber with mode coupler,” in *Optical Fiber Communication Conference*, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWA4.

**8. **A. Al Amin, A. Li, X. Chen, and W. Shieh, “LP_{01}/LP_{11} dual-mode and dual-polarisation CO-OFDM transmission on two-mode fibre,” Electron. Lett. **47**, 606–608 (2011). [CrossRef]

**9. **C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2x100Gb/s, over 40km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer and demultiplexer,” Opt. Express (to be published). [PubMed] [PubMed]

**10. **R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, R. Essiambre, P. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6 × 6 MIMO processing,” in *Optical Fiber Communication Conference*, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB10.

**11. **D. Marcuse, *Theory of Dielectric Optical Waveguides* (Academic, 1974).

**12. **H. Kogelnik and P. J. Winzer are preparing a manuscript to be called “Modal birefringence in weakly guiding fibers.”

**13. **W. Q. Thornburg, B. J. Corrado, and X. D. Zhu, “Selective launching of higher-order modes into an optical fiber with an optical phase shifter,” Opt. Lett. **19**, 454–456 (1994). [CrossRef] [PubMed]

**14. **W. Mohammed, M. Pitchumani, A. Mehta, and E. G. Johnson, “Selective excitation of the LP_{11} mode in step index fiber using a phase mask,” SPIE Opt. Eng. **45**, 074602 (2006).

**15. **O. Wallner, W. R. Leeb, and P. J. Winzer, “Minimum length of a single-mode fiber spatial filter,” J. Opt. Soc. Am. A **19**, 2445–2448 (2002). [CrossRef]

**16. **R. Ryf, C. Bolle, and J. von Hoyningen-Huene, “Optical coupling components for spatial multiplexing in multimode fibers,” in Proceedings of European Conf. Opt. Commun . (2011), paper Th.12.B.1 (to be published).

**17. **P. J. Winzer, A. H. Gnauck, G. Raybon, M. Schnecker, and P. J. Pupalaikis, “56-Gbaud PDM-QPSK: coherent detection and 2,500-km transmission,” in *Proceedings of European Conf. Opt. Commun.*, (VDE-Verlag, 2009), paper PD2.7.

**18. **J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA **97**, 4541–4550 (2000). [CrossRef] [PubMed]

**19. **C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in mode division multiplexed transmission over few-mode optical fiber,” IEEE Photon. Technol. Lett. (to be published).

**20. **N. Benvenuto and G. Cherubini, *Algorithms for Communications Systems and their Applications* (Wiley, 2002). [CrossRef]

**21. **A. Sierra, S. Randel, P. J. Winzer, R. Ryf, and R.-J. Essiambre are preparing a manuscript to be called “Analysis of test sequences for systems with MMSE-based equalization.”

**22. **A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory **29**, 543–551 (1983). [CrossRef]

**23. **M. Kuschnerov, M. Chouayakh, K. Piyawanno, B. Spinnler, E. de Man, P. Kainzmaier, M. S. Alfiad, A. Napoli, and B. Lankl, “Data-aided versus blind single-carrier coherent receivers,” IEEE Photon. J. **2**, 387–403 (2010). [CrossRef]