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We present a new class of optical modulation formats based on the combination of m-ary pulse-position modulation (m-PPM) or m-ary frequency-shift keying (FSK) with additional polarization and/or phase modulation, which is applied on the information carrying pulses in the case of m-PPM or on the information carrying frequency carriers in the case of m-FSK. We describe the principle and implementation of this class of optical modulation formats, and formulate their theoretical receiver sensitivities in optically pre-amplified receivers. Pilot-assisted frequency-domain equalization, similar to that used in coherent optical orthogonal frequency-division multiplexing (CO-OFDM), is used for reliable channel estimation and compensation. CO-OFDM also allows m-FSK to be implemented with high spectral efficiency. As a particular format in this class, m-PPM in combination with polarization-division-multiplexed quadrature phase-shift keying (PDM-QPSK), termed as PQ-mPPM, offers superior receiver sensitivity in optically pre-amplified receivers at bit error ratios (BERs) around the thresholds of common forward-error correction codes. Record receiver sensitivities of 3.5 photons per bit (ppb) at BER = 10−3 and 2.7 ppb at BER = 1.5 × 10−2 are experimentally demonstrated at 2.5 Gb/s and 6.23 Gb/s using PQ-16PPM and PQ-4PPM, respectively. We further demonstrate the transmission of a 6.23-Gb/s PQ-4PPM signal over a 370-km unrepeatered ultra-large-area-fiber span with 71.7-dB total loss budget.
©2011 Optical Society of America
1. Introduction
High-sensitivity optical transmission is of importance to many optical communication applications, particularly to deep-space optical communications [1,2] and unrepeatered fiber transmission [3]. High receiver sensitivity reduces the required signal photons per bit (ppb) for achieving a given bit error ratio (BER), and usually leads to improved transmission link performance. Photon-counting receivers, shot-noise limited coherent receivers, and optically pre-amplified receivers are commonly used optical receivers in these scenarios. Since highly sensitive photon-counting receivers rely on complex hardware such as superconducting nanowires, and currently have limited bandwidths [4], shot-noise limited coherent receivers and optically pre-amplified receivers are commonly used for high-speed (>1 Gb/s) optical transmission [1,2,5]. Recently, binary differential phase-shift keying (DPSK) has been used to achieve high-sensitivity and high-data-rate optical communication with optically pre-amplified receivers [6,7]. More recently, digital coherent detection has been introduced to fiber optical communication [8–10]. The most widely studied modulation format with digital coherent detection is polarization-division-multiplexed quadrature phase-shift keying (PDM-QPSK) [9,10], which offers higher sensitivity and spectral efficiency (SE) than DPSK.
In an attempt to find the most sensitive (or most power-efficient) modulation format in optical links, polarization-switched QPSK (PS-QPSK) was recently proposed [11], which provides ~1 dB higher sensitivity than binary phase-shift keying (BPSK) and PDM-QPSK at BER = 10−3, a typical threshold of low-overhead forward-error correction (FEC) codes [12]. This improvement was experimentally confirmed in a coherent optical orthogonal frequency-division multiplexing (CO-OFDM) experiment [13]. More recently, a new power-efficient format based on the combination of PDM-QPSK and m-PPM, termed as PQ-mPPM, was proposed [14,15], and shown to offer sensitivity advantages over BPSK, PDM-QPSK, and m-PPM. A 2.5-Gb/s PQ-16PPM signal was experimentally generated and received, using a novel low-overhead pilot-assisted single-carrier frequency-domain-equalization (PA-SC-FDE) scheme, achieving a record receiver sensitivity of 3.5 ppb at BER = 10−3, more than 3 dB better than previous Gigabit/sec-class records [14,15]. More recently, a 6.23-Gb/s PQ-4PPM signal was generated and received with a further improved receiver sensitivity of 2.7 ppb [16] assuming the use of a 19.25%-overhead FEC code having a BER threshold of 1.5 × 10−2 [17]. Furthermore, this 6.23-Gb/s PQ-4PPM signal was transmitted over an unrepeatered 370-km ultra-large-area fiber (ULAF) link with a total link loss budget of 71.7 dB using only Erbium-doped fiber amplifiers (EDFAs) at the transmitter and receiver sites [14–16], showing its promise in optical transmission applications that require high receiver sensitivity.
In this paper, we systematically present the principle, implementation, and performance of this new class of optical modulation formats that uses a combination of m-PPM or m-FSK with additional polarization and/or phase modulation (A) applied on the information carrying pulses in the case of m-PPM or on the information carrying frequency carriers in the case of m-FSK. We refer this class of modulation as A-mPPM/FSK, standing for additionally modulated m-PPM or m-FSK. The paper is organized as follows. In Section 2, we describe the principle and implementation of A-mPPM/FSK. In Section 3, we analyze the theoretical receiver sensitivity and bandwidth expansion factor of A-mPPM/FSK. The bandwidth expansion factor is a measure for the potential spectral efficiency of a format as well as for the required bandwidth of transponder components. Section 4 presents the experimental results obtained with PQ-4PPM [16]. Concluding remarks are given in Section 5.
2. Principle of A-mPPM/FSK
The principle of the generation and detection of an A-mPPM/FSK signal is as follows. At the transmitter, each A-mPPM/FSK symbol carries log2(m) + p bits, in which the log2(m) bits are encoded through m-PPM or m-FSK, and the remaining p bits are encoded through additional polarization and/or phase modulation. When the additional modulation involves two orthogonal polarizations, e.g., in the case of PDM or polarization-shift keying (PolSK), a PDM I/Q modulator is used. The two polarization components of this A-mPPM/FSK signal are modulated on an optical carrier through the use of four digital-to-analog convertors (DACs) and two I/Q modulators followed by a polarization-beam combiner (PBC). Figure 1 illustrates the encoding concept in the context of PQ-4PPM/FSK. In each PQ-4PPM/FSK symbol, there are 6 bits, the first two of which are encoded through 4-PPM or 4-FSK and the remaining four are encoded through PDM-QPSK; 4-PPM (4-FSK) encodes 00, 01, 10, 11 at time (frequency) slot position 1, 2, 3, and 4, respectively. For example, the first symbol contains six bits, “010111”. The first two bits “01” are encoded through 4-PPM (4-FSK) so the pulse (frequency) in the 4-PPM (4-FSK) symbol is located in the second time (frequency) slot. The remaining 4 bits “0111” are encoded via PDM-QPSK, i.e., “01” and “11” are encoded on the x- and y-polarization components, respectively. The above process repeats for the remaining PQ-4PPM/FSK symbols.
At the receiver, the PQ-mPPM/FSK signal is detected by a digital coherent receiver [14–16], including processing of the recovered signal fields by a digital signal processor (DSP). The first DSP step is frame synchronization. Then the time slot or frequency slot that has the highest energy out of the m slots of each PQ-mPPM/FSK symbol is found. The location of the highest-energy slot is used to recover the first log2(m) bits associated with m-PPM or m-FSK, and the recovered optical field in this slot is used to recover the remaining 4 bits associated with the PDM-QPSK modulation. More details on the receiver signal processing can be found in Refs [14–16].
It should be noted that multi-pulse PPM [18], where multiple optical pulses are transmitted in different times slots in each PPM symbol, can be applied to improve the throughput (or data rate) of the transmitter. For 1-pulse m-PPM, the number of pulse patterns in each symbol is m, and log2(m) bits can be transmitted per symbol. For k-pulse m-PPM, the number of pulse patterns in each symbol is and the number of bits per symbol is . For m = 16, 2-pulse 16-PPM carries log2 (120) = 6.9 bits per symbol, which is ~73% higher than that carried by 16-PPM. Note that the increase data rate is at the expense of reduced receiver sensitivity or reduced immunity to noise. Using more than 2 pulses per PPM symbol further increases the data rate carried by the PPM, but at further reduced receiver sensitivity. The A-mPPM/FSK concept can be extended by using multiple pulses (carriers) in each symbol and additionally modulate these pulses (carriers) in their polarizations/phases.
Note also that the idea of having a hybrid modulation format starting from an orthogonal modulation format (such as m-PPM or m-FSK) and enriching it with an m-PSK format dates back to 1966 in the context of digital communications [19], and the performance of the hybrid modulation format was discussed by Lindsey and Simon in 1972 [20]. The contribution of our work is the special/unique implementations of hybrid modulation by using optical m-PPM or m-FSK in combination with optical polarization modulation or switching, and sometimes additionally optical quadrature amplitude modulation (QAM). Finally, the class of orthogonal modulation formats also includes mode orthogonal modulation (MOM) where the pulse location is represented in the space domain through different optical modes or different spatial locations.
3. Theoretical performance
3.1 General formularization
If all time or frequency slots of m-PPM or m-FSK are equally likely to be errored, the relation between BER and symbol error ratio (SER) of an m-PPM or m-FSK signal is [1]
The BER of an A-mPPM/FSK signal can then be expressed as
where p is the number of bits carried by the two polarizations and quadratures and SERm-PPM and BERAddM are respectively the SER of m-PPM and BER of the additional modulation format at a given signal-to-noise ratio per symbol (SNRsym). The first term on the R.H.S. of Eq. (2) accounts for the bit errors caused by incorrectly identifying the m-PPM pulse or m-FSK frequency, which on average leads to p/2 bit errors in the decoding of the additional modulation, and m/[2(m-1)]log2(m) errors in m-PPM or m-FSK decoding. The second term on the R.H.S. of Eq. (2) accounts for the bit errors caused by incorrectly decoding the PDM-QAM, even when the m-PPM time slot or m-FSK frequency slot is correctly identified.For large m, Eq. (2) can be approximated as
Regarding SERm-PPM, we have
where f1(SNRsym,v) is the probability density function of a filled (“1”) slot having an energy of v, and P(v0>v) is the probability that there is at least one empty (“0”) slot having an energy higher than v, which can be further expressed aswhere p0(SNRsym,v) is the probability that an empty (“0”) slot has an energy higher than v. Both p0 and f1 can be analytically obtained for both single-polarization noise and dual-polarization noise cases [7]. Note that the above formulas are applicable to the case with m-FSK. For A-mPPM/FSK, the SNR per bit, SNRb is related to SNRsym byFor an ideal optically pre-amplified receiver, the amplifier noise figure (NF) is 3 dB, and ppb equals SNRb [7]. The above formulas provide a basis to analytically calculate the BER performance of an A-mPPM/FSK signal as a function of SNRb. Note that the above BER performances presented in this paper are practically realizable through essentially a 2-stage detector, which may be suboptimal [20].
In addition to receiver sensitivity, another key performance indicator is the so-called bandwidth expansion factor (BWEF), which is defined as the ratio between the needed modulation speed and the signal bit rate. The smaller the BWEF of a format, the higher the spectral efficiency (SE) the format may support and the lower the speed requirements on the underlying transponder hardware components at a given bit rate. For m-PPM or m-FSK, we have [1]
and for Add-mPPM/FSK, we haveObviously, Add-mPPM/FSK has a smaller BWEF (i.e., supports higher SE) than m-PPM or m-FSK for the same slot number m.
3.2 A-mPPM/FSK with the additional modulation being PDM-QPSK
An important subset of A-mPPM/FSK formats is formed with the additional modulation being PDM-QPSK (p = 4), referred to as PQ-mPPM/FSK. Substituting p with 2 in Eq. (2), we obtain the BER of a PQ-mPPM/FSK as [14–16]
where the BER performance of PDM-QPSK is related to SNRsym as [21]Figure 2 shows the theoretical BER performance of m-PPM as a function of SNRb assuming no polarization filtering (PF) of the amplified spontaneous emission (ASE) noise at the receiver. At BER = 10−3, the required ppb for 16-PPM is 6.7 dB. Figure 3 shows the theoretical BER performance of PQ-mPPM as a function of SNRb. The required ppb at BER = 10−3 for PQ-16PPM is 3.9 dB, which is 2.8 dB better than 16-PPM.
Fig. 2 Theoretical receiver sensitivity performance of m-PPM assuming that the ASE noise is not polarization filtered before detection.
To verify the above theoretical results, we conducted numerical simulation for the case of PQ-16PPM. The signal recovery was based on PA-SC-FDE [14,15]. Figure 4 shows the BER performance of PQ-16PPM as a function of SNRb. The analytical results obtained using the approximation of Eq. (2) match well with those obtained using Eq. (1). The simulated results also match well with the analytical results with a small constant additional SNR penalty of ~0.2 dB, probably due to imperfect channel compensation. The theoretical results were further verified through experiments reported in Refs [14] and [15]. Figure 5 shows the receiver sensitivity improvement of PQ-mPPM over BPSK (or PDM-QPSK) at BER = 10−3 as a function of m, where m = 2i (i = 1,2,3,…10). Evidently, PQ-mPPM outperforms PDM-QPSK for all the values of m, and the improvement increases with the increases of m, and approaches ~4 dB at m = 1024.
Fig. 4 Comparison between the BER performances of PQ-16PPM obtained by the analytical study and those by numerical simulations.
Fig. 5 Receiver sensitivity improvement of PQ-mPPM over BPSK and PDM-QPSK (at BER = 10−3) versus log2(m).
Figure 6 shows the BWEF of m-PPM and PQ-mPPM as a function of m. In addition to sensitivity improvement, PQ-mPPM also improves bandwidth efficiency (or reduces the BWEF) by carrying more bits for the same modulation symbol rate.
3.3 A-mPPM/FSK with the additional modulation being PS-QPSK
It is of interest to consider the subset of A-mPPM/FSK where the additional modulation is the highly-sensitive PS-QPSK (p = 3). Using the same methodology as above, we can express the BER of a PS-QPSK-mPPM signal as
where BERPS-PQSK is the BER of PS-QPSK, which itself can be expressed as [15]where BERQPSK is the BER of QPSK, and SERPS is the SER of polarization switching (equivalent to 2-PPM with PF to obtain single-polarization ASE noise), both of which can be analytically calculated [7,21]. The BER performances of both PS-QPSK and PS-QPSK-4PPM as a function of SNRb are shown in Fig. 3. PS-QPSK and PS-QPSK-4PPM require ~6 dB and ~4.9 dB ppb at BER = 10−3, respectively, which are ~2 dB and ~1 dB worse than PQ-16PPM. We note that the asymptotic BER performance of PS-QPSK-4PPM (e.g., at BER<10−9) is better than PQ-16PPM, which is in agreement with a recent study conducted on PS-QPSK-PPM by Karlsson and Agrell [22]. In the BER range between 10−2 and 10−3, around the BER thresholds of commonly used FEC codes for optical communications, the sensitivity of PS-QPSK-4PPM is similar to that of PQ-4PPM, but is much less bandwidth efficient.3.4 A-mFSK with CO-OFDM
Being the frequency-domain equivalent of m-PPM, m-FSK has the same theoretical performance as m-PPM [1]. The identification of the modulated frequency carriers in an m-FSK signal has been realized through either optical filters before detection [23] or digital filters after digital coherent detection [24]. With the use of CO-OFDM [25], m-FSK can be implemented with high spectral efficiency [23]. In principle, we can combine m-FSK with the same additional modulation formats (A-mFSK) as in the case of A-mPPM to achieve better sensitivity and bandwidth efficiency than with m-FSK alone.
When the additional modulation is a constant-power format, A-mFSK has the advantage (over A-mPPM) of having a constant power profile. It is known that m-PPM has high peak-to-average-power ratio (PAPR) for large m and may encounter fiber nonlinearity issues, e.g., in optical booster amplifiers [1]. So, A-mFSK may be more attractive than A-mPPM for large values of m when the PAPR issue becomes a concern.
With the increase of m, the symbol duration for A-mFSK increases, making it difficult to track the phase offset between the transmit laser and the receive optical local oscillator (OLO). One particular additional modulation of interest that does not require phase tracking is PolSK. Figure 7 illustrates the encoding of a PolSK-4FSK signal. PolSK encodes bit 0 (1) by aligning the output along x (y) polarization. In effect, PolSK-mFSK can be treated as (2m)-PPM with single-polarization noise. PolSK-mFSK offers higher receiver sensitivity than m-PPM due to doubled effective slot number and the inherent PF. Note that PF can also be implemented with m-PPM, but this requires optical polarization tracking (of the signal) at the receiver, which adds receiver complexity. With the use of pilot-assisted CO-OFDM channel estimation and compensation [26], the decoding of a PolSK-mFSK signal can be readily realized in the digital domain by finding the frequency slot and polarization state (i.e., x or y) that has the highest power among the 2m possible choices (m frequency slots, each with 2 polarization states) for each PolSK-mFSK symbol.
Figure 8 shows the theoretical receiver sensitivity performance of PolSK-mFSK. Compared to m-PPM without PF (shown in Fig. 2), PolSK-mFSK provides better receiver sensitivity at any given m value. Figure 9 shows the BWEF of PolSK-mFSK as compared to m-PPM. PolSK-mFSK outperforms m-PPM in terms of bandwidth efficiency, especially at small m values.
4. Experimental performance of PQ-mPPM
Figure 10 shows the schematic of the experimental setup used to evaluate the receiver sensitivity of PQ-mPPM. At the transmitter, an external cavity laser (ECL) at 1550 nm with a linewidth of ~100 kHz was used as the laser source, followed by a PDM-I/Q modulator. Each PQ-4PPM symbol contained 6 bits, of which the first 2 bits were encoded through 4-PPM and the remaining 4 bits were encoded through PDM-QPSK. The data bit sequence was a PRBS of length 215-1. The four field components of the encoded PQ-4PPM signal, corresponding to the I and Q components of both x- and y-polarizations, were stored in two synchronized arbitrary waveform generators (AWGs), each having two 10-GS/s digital-to-analog converters (DACs). Twofold oversampling was used, leading to a 4-PPM slot rate of 5 GHz (a symbol rate of 1.25 GHz), which resulted in a raw data rate of 7.5 Gb/s for PQ-4PPM. The DAC outputs were amplified to a peak-to-peak voltage swing of 3.5 V before driving the modulator. The signal has a 3-dB spectral bandwidth of ~6 GHz.
Fig. 10 Schematic of the experimental setup. Insets: (a) power waveform of the 6.23-Gb/s PQ-4PPM signal; (b) signal constellation before PPM demodulation; (c) signal constellation after PPM demodulation and phase compensation.
To measure the receiver sensitivity, the generated 7.5-Gb/s PQ-4PPM signal was attenuated by a variable optical attenuator (VOA) before being split by a 50:50 optical coupler (OC) into two parts, one entering an EDFA with a noise figure (NF) of 3.4 dB, and the other entering a power meter. An optical spectrum analyzer (OSA) was used to measure the optical signal-to-noise ratio (OSNR) of the optically pre-amplified signal. The signal was then filtered by a 0.5-nm optical filter before being received by a digital coherent receiver.
To study the transmission performance of the PQ-4PPM signal in unrepeatered transmission, another EDFA was used to boost the signal power right after the transmitter and the first OC was removed. A VOA was used to adjust the signal power launched into a 370-km ULAF, which had a loss of 69 dB (corresponding to a loss coefficient of 0.187 dB/km) and an effective area of 120 μm2.
The digital coherent receiver frontend consisted of a 100-kHz-linewidth ECL serving as the OLO, a polarization-diversity optical hybrid, four balanced detectors, and four 50-GS/s analog-to-digital converters (ADCs) in a real-time sampling scope. The four sampled waveforms were stored and down-sampled to 10 GS/s before being processed offline. The offline digital signal processing (DSP) was similar to that described in Refs [14,15]. To ensure reliable channel estimation (CE) even in the presence of PPM errors, PA-SC-FDE was used, where the CE is based on known pilot symbols [14,15]. Inset (a) in Fig. 10 shows a received signal power waveform, indicating that the random locations of the input 4-PPM pulses were correctly identified at the receiver. Inset (b) shows signal constellations in the two original polarization states after channel compensation, but before PPM demodulation, where the points near the origin represent slots without PPM pulses. Inset (c) shows signal constellations after PPM demodulation and phase compensation. Clear QPSK constellations are recovered. Figure 11 shows the frame structure of the PA-SC-FDE scheme used for PQ-4PPM. We used similar pilot sequences (T1, T2, and T3) as those used in OFDM [26] for frame synchronization and CE. To minimize overhead, no guard interval (GI) was used for payload symbols although a GI is used in each pilot sequence for accurate CE. The overlap-and-add technique was used during the channel compensation process. For unrepeatered transmission, electronic dispersion compensation [26] was first applied prior to channel synchronization. Additional pilot symbols, each occupying one time slot after every 160 slots, were inserted to assist phase estimation (PE). Similar to Refs [14,15], the pilot symbols caused a negligible rate overhead of 0.925%. Compared to Refs [14,15], the power penalty due to the pilots used for synchronization and CE remained at 0.2 dB, but the power penalty due to the pilots used for PE was reduced from 0.4 dB to 0.1 dB owing to the four-fold reduction in PPM symbol size. Assuming the use of a 19.25% overhead for hard-decision forward error correcting codes (FEC) such as those in [17], which correct BER from 1.5 × 10−2 to below 10−15, the net data rate of the signal is 6.23 Gb/s.
Figure 12 shows the experimentally measured BER performance of the PQ-4PPM signal as a function of ppb (referred to the photons per net information bit). The theoretical performance is also shown for comparison. At BER = 1.5 × 10−2, the system requires 2.7 ppb, which is 1.8 dB away from theory. In the 1.8-dB overall penalty, ~0.8 dB is due to the 19.25% FEC overhead, 0.4 dB is due to the excess EDFA noise, 0.2 dB is due to the pilot sequences used for synchronization and CE, and 0.1 dB is due to the pilot symbols used for PE, leaving only ~0.3 dB to account for the hardware implementation penalty. As a reference, PDM-QPSK performance was also measured by turning off the PPM modulation. The net data rate of the PDM-QPSK signal was 4.15 Gb/s. At BER = 2 × 10−2, the system requires ~4 ppb for PDM-QPSK, which is essentially the same as that recently reported in Ref [27]. The 2.7-ppb sensitivity obtained for the PQ-4PPM signal is better than that obtained for the PDM-QPSK signal by 1.9 dB. The theoretical performance of PQ-4PPM and PDM-QPSK are also shown in Fig. 12 for comparison. It can be seen that PQ-4PPM suffers ~0.6 dB less implementation penalty than PDM-QPSK, probably because the implementation penalty associated with demodulating PPM is lower than with demodulating PDM-QPSK. Compared to the PQ-16PPM demonstration at 2.5 Gb/s [14], this PQ-4PPM demonstration offers a 2.5-fold increase in net data rate and a 1.1-dB reduction in ppb.
Figure 13 shows the BER performance as a function of signal launch power for the unrepeatered 370-km transmission system. Digital self-phase modulation with 10 back-propagation steps compensation was applied for high input powers. The measured BER is below 1.5 × 10−2 when the signal launch power is between 12.5 dBm and 17.5 dBm, indicating a substantial power margin of 5 dB. The optimum signal launch power was 15.5 dBm, corresponding to a mean nonlinear phase shift of 2.7 radians for each PPM pulse. At the optimum power, the signal Q factor (derived from the BER) is 8.5 dB, which is 1.7 dB higher than the FEC threshold (1.5 × 10−2). To assess the maximum allowable link loss, we added 2.7 dB of additional loss after the 370-km ULAF, increasing the total link loss to 71.7 dB. The optimum signal launch power was found to be 16.5 dBm, at which the measured BER is just below the BER threshold. At the optimum power, the transmission penalty is estimated to be ~1.5 dB. This allowable link loss budget (71.7 dB) compares reasonably well with the 71.5 dB recently obtained with return-to-zero PDM-BPSK and third-order Raman pumping [3]. This experiment has shown good nonlinear tolerance of the PQ-4PPM signal in single-channel transmission, and it would be of interest to study its nonlinear tolerance in wavelength-division multiplexed (WDM) transmission, which is beyond the scope of this paper. We expect that with the return-to-zero (RZ) like pulse format for PQ-mPPM, it could have high nonlinear tolerance to WDM nonlinear effects, e.g., inter-channel cross-phase modulation, when the dispersion-induced pulse broadening is sufficiently high.
Fig. 13 Measured BER performance as a function of signal launch power after transmission over a 370-km ULAF link.
5. Summary
We have systematically presented the principle, implementation, and performance of a new class of power-efficient optical modulation formats called A-mPPM/FSK that combine m-PPM or m-FSK with additional polarization and/or phase modulation. As a particular format in this class, m-PPM in combination with PDM-QPSK, or PQ-mPPM, offers superior receiver sensitivity in optically pre-amplified receivers at a BER around the thresholds of common FEC codes. A record receiver sensitivity of 2.7 ppb at BER = 1.5 × 10−2 has been experimentally demonstrated at 6.23 Gb/s using PQ-4PPM. We have also demonstrated the transmission of a 6.23-Gb/s PQ-4PPM signal over a 370-km unrepeatered ULAF span with a total loss budget of 71.7 dB. Another particular format, m-FSK in combination with PolSK, is also of interest due to its constant-power profile and its ability to operate without the need for phase tracking. This class of power-efficient modulation formats is expected to be attractive in applications where photon efficiency is of critical importance, such as in deep-space communications and unrepeatered fiber transmission.
Acknowledgments
The authors wish to thank OFS Labs for providing the ULAF used in the experiment.
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