Self-coherent detection for optical OFDM via polarization diversity

Omer Onn; Ofer Amrani

doi:10.1364/OSAC.2.001600

1. Introduction and background

Self-Coherent receiver schemes allow reconstruction of the transmitted field, without the need for a local-oscillator on the receiver side. Previous work has limited Digital Self-Coherent Detection (DSCD) usage to treating differential Phase-Shift Keying (PSK) signals [1,2], or discrete Quadrature-Amplitude Modulation (QAM) signals [3], while in the current work we shall focus on coherent modulation such as 16QAM-based optical Orthogonal Frequency Division Multiplexing (OFDM). Optical OFDM has been chosen due to its efficient spectral utilization, the narrowband sub-channels that make frequency dependent impairments (such as Chromatic Dispersion) easier to mitigate, the simple introduction of time-domain channel-estimation pilots, and temporal guard intervals that reduce inter-symbol interference caused by dispersion. The OFDM’s susceptibility to transmitter-receiver frequency mismatch becomes irrelevant in self-coherent receivers, which makes it an ideal candidate for our analysis. A different approach to self-coherence detection, as detailed in [4], is not addressed in the current work as it is based on the transmission of the optical carrier signal accompanied by optical filtering for extracting the carrier at the receiver side.

The DSCD receiver [1] detects phase differences between adjacent symbols by interfering the received signal with a delayed version of itself. The coherent signal is reconstructed in a Digital Signal Processing (DSP) unit by detecting the signal power in a dedicated branch, and by summing differential-phase samples. This approach requires dealing with two issues: 1. all the differential-phases must be successfully detected in order to maintain synchronization of the phase reconstruction process; 2. summing the differential-phases also means that the noise is summed, resulting with phase-noise accumulation and potential drift of the reconstructed phase. Herein we discuss methods to mitigate these sources of performance degradation. While the emphasis in this work is on optical OFDM, when addressing the phase-noise accumulation problem in Section 3, we begin by modeling the problem and its solution for the single-carrier case and then switching to OFDM to illustrate the differences between the two.

1.1. Receiver architecture: cross-DSCD (X-DSCD)

In order to reliably detect optical signals, the so-called X-DSCD architecture assumes the transmission of two independently modulated optical signals on two orthogonal polarizations, X and Y. Detailed architecture of the X-DSCD receiver can be found in [5] (See Fig. 6).

The instantaneous power of each polarization, $P(t )$, is detected by a dedicated photo detector, also used to facilitate Clock-Data Recovery (CDR) [6], and automatic gain-control.

Phase detection is realized by sampling the outputs of two Optical Delay-Interferometers (ODI) that produce an optical signal proportional to the phase difference between 2 adjacent samples, henceforth – Differential-Phase (DP):

(1)$$\begin{array}{l} u_x^I(t )\equiv {\mathop{\rm Re}\nolimits} \{{{{\bar{E}}_x}(t ){{\bar{E}}^\ast }_x({t - \tau } )} \}= {E_x}(t ){E_x}({t - \tau } )\cos ({\Delta {\phi_{\tau ,x}}(t )} )\\ u_x^Q(t )\equiv {\mathop{\rm Im}\nolimits} \{{{{\bar{E}}_x}(t ){{\bar{E}}^\ast }_x({t - \tau } )} \}= {E_x}(t ){E_x}({t - \tau } )\sin ({\Delta {\phi_{\tau ,x}}(t )} )\end{array},$$

where τ is the differential delay of the ODI, set to the duration of a transmitted symbol, and

(2)$${\phi _x}(t )- {\phi _x}({t - \tau } )\equiv \Delta {\phi _{\tau ,x}}(t )= {\tan ^{ - 1}}({u_x^Q(t )/u_x^I(t )} ), $$

is the DP in τ-intervals. Phase and power are then used to provide Field Reconstruction (FR) of the received field, as follows (description given only for the X-polarization branch)

(3)$${\bar{E}_x}({t = {t_0} + k\tau } )= {\bar{E}_{k,x}} = \sqrt {{P_{k,x}}} \cdot {e^{\,\,\,j{\phi _{k,x}}}} = \sqrt {{P_{k,x}}} \cdot {e^{\,\,\,j\left( {{\phi_{0,x}} + \sum\limits_{i = 1}^k {\Delta {\phi_{i,x}}} } \right)}}.$$

The receiver’s Analog-to-Digital Converter (ADC) quantizes the outputs of the ODIs given by Eq. (1). When the ODI output value is less than the sampling threshold (the smallest quantization level) in the in-phase or quadrature branches, the corresponding ADC yields a Zero-Intensity Sample (ZIS) [5]. In this case, a DP sample is lost and as a result, one may experience Loss-of-Synchronization (LOSy) in the FR process. Magen and Amrani [5] introduced the, so-called, cross-polarization branch (XY-branch) of the X-DSCD, in order to exploit the diversity in the power of the two polarizations composing the received signal. By interfering the two polarizations, the polarization branch with the low-power is amplified by the power of the other branch, thus providing the samples:

(4)$$\begin{array}{l} u_{k,xy}^I \equiv {E_{k,x}}{E_{k,y}}\cos ({\Delta {\phi_{k,xy}}} )\\ u_{k,xy}^Q \equiv {E_{k,x}}{E_{k,y}}\sin ({\Delta {\phi_{k,xy}}} )\end{array},$$

with

(5)$${\phi _{k,x}} - {\phi _{k,y}} \equiv \Delta {\phi _{k,xy}} = {\tan ^{ - 1}}({u_{k,xy}^Q/u_{k,xy}^I} ),$$

being the instantaneous phase-difference between the two polarizations. Note that this value is only dependent on time k, and hence it can be obtained even when the low-power polarization has lost synchronization due to ZIS. Provided that only a single polarization branch (in this example, X) loses a sample at time k, it follows from (5) that it is still possible to recover its phase via

(6)$${\phi _{k,x}} = \Delta {\phi _{k,xy}} - {\phi _{k,y}}, $$

and similarly, for ZIS in the Y-polarization. While it follows from the above description that the proposed architecture cannot recover from an event where both polarized signals are of zero intensity, simultaneously, the probability of such event is very small [5]. While the cross-polarization branch is constantly monitored, its output need only be utilized by the DSP when a ZIS is detected in one of the single polarization branches. Throughout this work, we assume no polarization-mode dispersion as a polarization maintaining channel is used, or otherwise a method of correcting the rotation of the state of polarization is applied.

2. Influence of quantization and noise on phase reconstruction

The sources for degradation in DP detection can be categorized into channel-related and receiver-specific distortions, which are unique to the differential FR of DSCD. Herein we consider one polarization but the results are easily reproducible for the orthogonal one.

The optical channel can also introduce Chromatic Dispersion (CD) which is more pronounced as the signal bandwidth increases (also more problematic at 1550 nm compared to lower wavelength such as 1310 nm). In this work, it is assumed that CD is either negligible, or otherwise mitigated by optical means at the receiver or digitally at the transmitter side. The latter may be realized, for OFDM transmission, by means of adequate phase compensation per subcarrier, provided that the channel is known at the transmitter side The transmitter electronics and amplifiers located along the channel also contribute Additive White-Gaussian Noise (AWGN) [7] to the received signal.

Receiver-specific distortion arises from the filter at the receiver frontend, sampling timing (which can be mitigated with the above mentioned CDR on the power branch) and quantization and the reconstruction process itself. Combined, these receiver-specific impairments introduce noise to the DP detection process, to be denoted as $n_k^{\Delta \phi }$. It is defined as the difference (in radians) between the transmitted DP and the detected DP at discrete-time k,

(7)$$n_k^{\Delta \phi } = \Delta {\hat{\phi }_k} - \Delta {\phi _k}, $$

with the caret hat ($^{\wedge}$) denoting a value calculated by the DSP from the sampled and quantized received analog signal. The DP-samples are summed so as to obtain the Accumulated-Phase (AP) for the FR. The AP-noise is the difference between the resultant FR phase and the transmitted phase. It is denoted $n_k^\phi $ and defined as the accumulated DP-noise until time k. The final expression for the FR phase is given by:

(8)$$\angle {\hat{E}_k} = {\hat{\phi }_k} = {\hat{\phi }_0} + \sum\limits_{i = 1}^k {\Delta {{\hat{\phi }}_i}} = {\phi _0} + {n^{{\phi _0}}} + \sum\limits_{i = 1}^k {({\Delta {\phi_i} + n_i^{\Delta \phi }} )} = {\phi _k} + n_k^\phi .$$

2.1. Additive noise

The first contribution to DP is the AWGN originating from the channel and receiver hardware, and is given by:

(9)$$n_k^{\Delta \phi } = {\tan ^{ - 1}}\left( {\frac{{\tilde{u}_k^Q}}{{\tilde{u}_k^I}}} \right) - \Delta {\phi _k}, $$

with the tilde hat (∼) denoting the analog outputs of the ODI with AWGN. It manifests itself as zero-mean random AP-noise whose variance is directly-proportional to the AWGN variance, and where high correlation exists between adjacent samples in time.

2.2. Quantization noise

By design, the ADC has a limited number of quantization levels, determined by the number of bits employed for coding its output (a.k.a. quantization depth). The quantization operation (denoted by $\lfloor.\rfloor$) distorts the detected DP as follows:

(10)$$n_k^{\Delta \phi } = {\tan ^{ - 1}}\left( {\frac{{\lfloor{\tilde{u}_k^Q} \rfloor }}{{\lfloor{\tilde{u}_k^I} \rfloor }}} \right) - \Delta {\phi _k}.$$

Through simulation, we verified that adjacent samples of this noise are uncorrelated and identically distributed, hence the AP-noise in Eq. (8) behaves as a random-walk-process whose variance is obtained via accumulation of the DP noise samples:

(11)$${\mathop{\rm var}} ({n_k^\phi } )= k \cdot {\mathop{\rm var}} ({n_1^{\Delta \phi }} ). $$

The variance of the DP noise can be minimized by optimizing the ADC to reduce the quantization noise, thus minimizing the contribution of the quantization-related DP-noise to the AP. Phase random-walk is known in coherent optical receivers from the laser-linewidth phase-drift [8–10]; herein, DP noise is dependent on the quantization of the received signal, and not on a local light source (as the latter is avoided in the proposed receiver structure).

Consequently, the variance (11) at the end of a short block of, say, L = 100 samples is limited to 100 times the variance of a single DP-noise sample (10), and $n_k^\phi $ is centered around a local mean value, namely, the phase-drift. This drift translates into a phase rotation of the reconstructed symbols, which introduces errors into the detection of the coherent symbols.

2.3. Estimation of the accumulated-phase noise

We can conclude that the behavior of the AP noise in the FR process is: 1. causal random walk with increasing variance, continuously rotating the phase of the reconstructed signal, as a result of quantization noise. 2. random variations around the long term phase drift, due to AWGN with zero-mean.

To estimate the phase-drift, a linear minimum Mean-Square-Error (MSE) estimator of the AP-noise, ${\hat{\phi }^{drift}}$, is realized by averaging over a short block-of-samples (window). In the DSP, the AP can be de-rotated based on this estimation to counter the phase drift and obtain a zero-mean error between the corrected AP and the transmitted signal phase:

(12)$$E[{{\phi_k} - \hat{\phi }_k^c} ]= 0, $$

where superscript “c” represents the post-correction AP. With the phase drift mitigated, the DSCD provides performance close to that of a coherent receiver as both receivers operate on a zero-mean phase-noise signal with variance proportional to the received SNR.

In the design of the system, the window length L is chosen so as to optimize a trade-off: while averaging over many samples may typically improve the accuracy of estimation, considering fewer samples means, in our case, smaller variance, and a smaller MSE per block, smaller MSE for the entire FR and eventually lower overall Bit-Error-Rate (BER).

To estimate the drift as described above, knowledge of the transmitted signal is required. Nazarathy [3] suggests using agreed-upon training-symbols (pilots) equi-spaced along the data stream. In this case, the phase-drift can be estimated as the average phase-difference between the known phase of the pilots and the AP. Using pilot symbols reduces the throughput of the system, as they occupy 5-20% of the transmission.

As a possible alternative, herein, we shall also consider a data-aided approach [1,8,9,11–13], whereby the detected symbols, ${\hat{s}_k}$, act as reference for estimating the AP noise. This is realized by taking the average difference between the detected-symbol phase, ∡${\hat{s}_k}$, and the AP, ∡${\hat{E}_k}$. This approach entails no throughput loss. A single estimator is calculated to represent the drift of a complete block of L samples:

(13)$${\hat{\phi }^{drift}} = \frac{1}{L}\sum\limits_{i = k - L + 1}^k {({\angle {{\hat{s}}_i} - \angle {{\hat{E}}_i}} )}. $$

Clearly, this approach can only be applied when the processed symbols belong to a discrete modulation scheme, for which hard decision can be made and the phase is well-defined. Yet another downside is that detection errors degrade the accuracy of the estimator.

3. Phase error correction

3.1. Single-carrier signal

Correction is carried out in consecutive disjoint blocks of M samples (termed correction block). Each block is corrected with an estimator based on a fraction of the samples corrected in the preceding block. Thus, for an arbitrary block spanning over discrete time indices {m + 1:m + M}, the new estimator is based on the estimation window comprising of the last L samples of the preceding block, or indices {m-L + 1:m}:

(14)$$\left\{{\hat{\phi }_k^{drift}} \right\}_{k = m + 1}^{m + M} = \frac{1}{L}\sum\limits_{i = m - L + 1}^m {({\angle {{\hat{s}}_i} - \angle {{\hat{E}}_i}} )}. $$

This single estimator (14) is then used to counter the AP-drift of the block of samples {m + 1:m + M}:

(15)$$\hat{E}_k^c \equiv {\hat{E}_k} \cdot {e^{ - j\hat{\phi }_k^{drift}}}. $$

As each block is corrected upon reception, the estimator is calculated using corrected symbols, ${\hat{s}_i}$, which can therefore be considered as reliable reference for estimating the drift. This method, therefore, relies on the successful estimation and correction of the drift in previous blocks, which will be thoroughly discussed in the next section. Incorrect detection will degrade the estimator and the correction process, thus propagating errors through the detection process. To improve the accuracy of the estimation, we define a binary indication variable Di, used for partitioning the estimation window into, so called, usable and unusable symbols as defined below:

(16)$$\left\{{\hat{\phi }_k^{drift}} \right\}_{k = m + 1}^{m + M} = \frac{1}{{\sum\limits_{i = m - L + 1}^m {{D_i}} }}\sum\limits_{i = m - L + 1}^m {({\angle \hat{s}_i^c - {{\hat{\phi }}_i}} ){D_i}},\qquad {D_i} = \left\{ \begin{array}{ll} 1 & \hat{s}_i^c \in h\\ 0 & \hat{s}_i^c \notin h \end{array} \right.,$$

where h denotes the set of usable symbols among the L-symbol estimation window; these may very well be the symbols having relatively-high instantaneous power (e.g. all QPSK symbols, or the outer-ring symbols in 16QAM).

For a given value of M, increasing the ratio M/L reduces the computational complexity required for drift-estimation for the price of degrading its accuracy. When the constellation is dense, the system is more sensitive to phase noise. Namely, one shall need to tune the phase more often before the noise variation accumulates to the point of error (this calls for small M). Through simulation, we identified that M/L = 80/40 provides a good performance-complexity tradeoff.

A detailed block diagram depicting the complete drift estimation and correction process is given in Fig. 1.

Fig. 1. Detailed block diagram of the drift estimation and correction process.

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Notably, the proposed phase correction method may allow us to deal with the (transmitter) laser line-width as it, too, manifest itself as AP noise whose statistical behavior is that of a random walk process. Figure 2 depicts AP-noise and the corresponding data-aided estimation and correction. In both figures, the left-most region - bounded between the y-axis and the dashed line running parallel to it - represent early samples that hardly suffer any phase drift. The right region - bounded between the dashed lines for samples 4800-4900 - suffers significant phase drift of -0.8[rad] in the left-hand-side (LHS) figure. Also shown in the LHS figure, the estimation (red curve), which is seen to successfully track the phase drift. Consequently, the phase drift is eliminated in the right figure and the AP is shown to correctly center at the desired zero-mean.

Fig. 2. AP noise of a QPSK signal with 9 bit quantizer and 12 dB SNR, noise and drift estimation (left), and noise after correction of drift (right).

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3.2. Optical OFDM

Inverse Fast Fourier Transform (FFT), employed by the OFDM transmitter, transforms a discrete set of inputs taken from a high-order signal constellation, such as 16QAM, into a discrete-time signal whose samples obey a Gaussian distribution. These samples are variables of practically continuous distribution for typical inverse-FFT sizes. FR of an OFDM signal by the X-DSCD receiver is subject to the same AP drift-behavior as discussed in Section 2.2. Yet, unfortunately, due to its continuous nature, the time-domain version of the OFDM signal cannot be employed as reference for extracting the phase drift as suggested by Eq. (14).

On the other hand, the frequency-domain OFDM signal consists of discrete QAM symbols, which may potentially serve as a reference signal for the estimation. This is possible due to a duality in the phase behavior of the time-domain and the frequency-domain samples. As detailed in Section 2, the time domain AP error is composed of drift n^drift, (causing average AP-error ${\hat{\phi }^{drift}}$ (14)), and instantaneous variations (residual zero-mean AP-noise n^random).

To demonstrate this duality, consider a time-domain sample q of an arbitrary OFDM symbol of size Q (i.e. DFT length) spanning over indices q={m + 1,m + Q}. A time-domain sample can therefore be expressed as

(17)$${\hat{E}_q} = \sqrt {{{\hat{P}}_q}} {e^{j({{\phi_q} + n_q^\phi } )}} = \sqrt {{{\hat{P}}_q}} {e^{j{\phi _q} + j({{n^{drift}} + n_q^{random}} )}}\,\, = {e^{j{n^{drift}}}}\sqrt {{{\hat{P}}_q}} {e^{j({{\phi_q} + n_q^{random}} )}}, $$

consequently, the frequency-domain samples (indexed k) are given by

(18)$$\begin{array}{c} \left\{{DFT\left[{\{{{{\hat{E}}_q}} \}_{m + 1}^{m + Q}} \right]} \right\}_{k = m + 1}^{m + Q} = \sum\limits_{q = m + 1}^{m + Q} {{{\hat{E}}_q}{e^{jqk\frac{{2\pi }}{Q}}}} = {e^{j{n^{drift}}}}\sum\limits_{q = m}^{m + Q} {\sqrt {{{\hat{P}}_q}} {e^{j({{\phi_q} + n_q^{random}} )}}{e^{jqk\frac{{2\pi }}{Q}}}} \\ = {e^{j{n^{drift}}}}\{{{{\hat{s}}_k}{e^{jn_m^\ast }}} \}_{k = m + 1}^{m + Q}, \end{array}$$

with n*_m denoting the zero-mean random-phase component at the output of the DFT.

The right most term in Eq. (18) shows that the time-domain phase drift translates to the same drift in the frequency-domain (provided that the drift is almost-constant for a period of Q samples; i.e. within a single OFDM symbol). Thus, utilizing the recovered QAM symbols, ${\hat{s}_k}$, the drift can be estimated in the frequency-domain and mitigated in the time-domain - where the actual FR process is carried out.

Finally, the Q samples of a single OFDM symbol can be used for the previously discussed correction blocks of length M = Q. Here, the trade-off concerning the estimation window length coincides with the trade-off over the number of sub carriers of the transmitted signal – more sub-carriers means narrower sub-carriers which allow to better cope with frequency dependent phenomena such as CD, while the detection becomes more susceptible to the AP-drift for longer correction blocks.

4. Attending to zero intensity samples

The other impairment of the X-DSCD scheme that we address in this work is LOSy due to a ZIS, as discussed in Subsection 1.1. A ZIS in a given sample k occurs when the output of both ODI's, Eq. (1), of a detection branch instantaneously sample a signal smaller than the ADC detection threshold (resolution):

(19)$$\delta_{LSB} \equiv \left\{{{{\max }_{x \in \Re }}x|\lfloor x \rfloor = 0} \right\}, $$

with $\delta_{LSB}$ denoting the maximum sample voltage for which the binary representation (quantization level) is zero; (“LSB” represents that this voltage is associated with the Least-Significant Bit of the quantizer).

In the case of ZIS, the XY-branch can be utilized for providing the missing phase, as suggested by Eq. (6), so as to assist the FR process recover from LOSy, provided that the other polarization is still synchronized. Alas, this recovery process now introduces phase-noise into the polarization suffering a ZIS (e.g. X), originating in the synchronized polarization (e.g. Y) and the XY-branch, denoted $n_{k,y}^\phi $ and $n_{k,xy}^{\Delta \phi }$ respectively, hence:

(20)$${\hat{\phi }_{k,x}} = \Delta {\phi _{k,xy}} + {\phi _{k,y}}\; + n_{k,xy}^{\Delta \phi } + n_{k,y}^\phi. $$

This total phase-noise, $n_{k,y}^\phi + n_{k,xy}^{\Delta \phi }$, is centered around a phase-drift that is uncorrelated with the AP-noise in polarization X, accumulated until the ZIS in sample k, namely $n_{k - 1,x\; }^\phi $. As such, the phase correction term evaluated by the preceding estimation window is no longer relevant for the symbols of the current block starting from sample k. Moreover, for the following block, phase drift estimation shall begin at sample k so as to feed the estimation process in Subsection 3.1 with correct symbols.

Anyway, it is clear that one can benefit from reducing the probability of a ZIS event. This, in turn, may be achieved by customizing a quantizer such that its quantization-level-distribution takes into account the characteristics of the signal to be sampled.

As mentioned earlier, the time-domain signal associated with QAM-based OFDM is typically of a Gaussian distribution with zero-mean. The output of an ODI fed with an OFDM signal follows a narrow distribution around zero mean - as it is a product of two OFDM (time-domain) samples - with distribution obeying the modified Bessel function of the second kind [14]. To this end, it is suggested to employ a non-uniform quantizer that assigns more levels in the ranges of values that occur more often and fewer quantization levels to ranges that are infrequent. This will 1. reduce the ZIS threshold (19) (and probability), and 2. reduce the average quantization noise and the phase-drift associated with it (Subsection 2.2). This approach is equivalent to companding used to treat high peak-to-average power-ratio in OFDM transmitters [15–17].

The following quantization function is applied at the receiver side:

(21)$${\left\lfloor{u_k^{\{{I,Q} \}}} \right\rfloor _{\log }} = \frac{{{M_{\max }}}}{F} \cdot \left\{ {\exp \left[ {n \cdot \frac{{\ln({F + 1} )}}{{({{2^N} - 1} )/2}}} \right] - 1} \right\}$$

(22)$$n = floor\left( {\frac{{\ln ({|{u_k^{\{{I,Q} \}} \cdot F/{M_{\max }}} |+ 1} )}}{{{\raise0.7ex\hbox{${\ln({F + 1} )}$} \!\mathord{\left/ {\vphantom {{\ln({F + 1} )} {[{({{2^N} - 1} )/2} ]}}} \right.}\!\lower0.7ex\hbox{${[{({{2^N} - 1} )/2} ]}$}}}}} \right),$$

where N denotes the number of ADC bits, n the index of the ADC’s quantization level to which a specific sample corresponds (spanning from (1-N)/2 to (N-1)/2), M_max the maximum quantization level, corresponding to n=½(N-1), F denotes an arbitrary density-factor that determines the density of the quantization levels at low power. F was optimized through simulation to be 1000, to minimize BER.

The suggested quantizer reduces the threshold of the uniform quantizer, given by:

(23)$${\delta _{uniform}} = 2 \cdot M \cdot {2^{ - ({N - 1} )}},$$

to the logarithmic one:

(24)$${\delta _{\log }} = {{\mathop{\rm e}\nolimits} ^{\log (F )/{2^{ - ({N - 1} )}}}} \cdot {\raise0.7ex\hbox{$M$} \!\mathord{\left/ {\vphantom {M F}} \right.}\!\lower0.7ex\hbox{$F$}},$$

with F = 1000 and N = 9 bit, the threshold of the logarithmic quantizer is more than 2 orders of magnitude smaller than a 9-bit uniform one. This translates to a significantly reduced probability of a ZIS as well as reduced average quantization noise. Provided in the next section are simulation results of an OFDM receiver with a uniform quantizer (note: BER = 1e-3 @ 14 bits, in Fig. 3) and the logarithmic quantizer (BER = 1e-3 @ 9 bits, in Fig. 4).

Fig. 3. BER Performance vs. quantization depth; variety of receivers with a uniform-scale quantizer. 16QAM (left) and 16QAM-OFDM (right) signals at SNR = 18 dB.

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Fig. 4. BER Performance vs. quantization depth; variety of receivers with a logarithmic scale quantizer.

16QAM-OFDM signal transmitted over a channel with SNR of 12 dB (left) and 18 dB (right).

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5. Simulation results

Herein, we compare the performance of a coherent optical receiver for 16QAM-based optical-OFDM, with 5 self-coherent receiver architectures: 1. DSCD, 2. DSCD with data-aided phase correction, 3. and 4. X-DSCD with and without phase correction, 5. X-DSCD with pilot-based phase correction (with 20 pilots per 100 symbols). The simulation considers a 128 sub-carrier OFDM signal, using the 16 outer (8 leftmost and 8 rightmost) sub-carriers as guard-bands. For all receivers, we assume a perfect transmitter, with a zero-linewidth laser on the transmitter side, to isolate the phase-drift caused by the receiver. Likewise, for the coherent receiver provided for comparison, we assumed a zero-linewidth Local oscillator. Data is divided into blocks of 10⁵ samples and the initial phase is assumed to be perfectly estimated from a synchronization symbol at the beginning of each block.

A pseudo-random sequence of symbols (in the frequency domain) is generated and converted to a time-domain OFDM signal using IFFT. White Gaussian noise is added to simulate the channel. For restoring the received signal strength, the receiver samples the absolute value of the (noisy) time-domain samples using the ADC; the ADC converter provides the digital representation of the nearest available analog quantization level (rounded-down). Similarly, for the phase reconstruction, the simulation assumes an ADC sampling of the signals at the interferometer output as given by Eq. (1). Equations (2) and (3); these are then employed to complete the reconstruction of the instantaneous phase of the received signal. In the event that the differential phase is not available due to the very low power of the input signal (i.e. ZIS), the receiver takes advantage of diversity to restore the lost phase using another set of interferometers described in Eq. (4), and finally the phase is reconstructed using Eq. (5). (Note: when the XY channel also encounters a ZIS, the algorithm arbitrarily assumes 45-degree phase for continuing the operation.)

The reconstructed temporal signal is converted back to the frequency domain via digital FFT processing. The impact of phase noise on the frequency-domain samples is mitigated using the algorithm described in Section 3. The received bit sequence is then extracted and compared to the one transmitted for evaluating the Bit error rate. 10⁵ runs, of 10⁵-bit long blocks were executed for a set of input parameters (noise-level/ADC resolution/quantization).

Figure 3 (left) shows that for a 16QAM signal, at SNR = 18 dB, a uniform-scale ADC approaches coherent performance with 11 ADC bits, with X-DSCD performing worse than the DSCD. Note that the DSCD receiver assumes arbitrary phase in case of LOSy.

For a 16-QAM-based OFDM signal with a uniform quantizer (Fig. 3 - right), even at the simulated maximum of 18 bits, none of the compared self-coherent receivers approach the performance of the coherent receiver. On the other hand, the customized logarithmic-scale quantizer (Fig. 4 - right), approaches coherent performance with a 14-bit ADC.

Figure 4 also shows that for both 12 dB and 18 dB SNR, the receiver with phase-correction achieves coherent performance with 12 bits, while 14 bits are required to achieve this without phase correction. These results reiterate the conclusion from [3] stating that a large enough quantizer (14 bits, in our case) is sufficient to achieve an error-rate similar to that of a coherent receiver, for an 18 dB SNR channel, with or without phase-correction.

Finally, Fig. 5 provides a different look at the results obtained with a logarithmic quantizer and promotes the following conclusions: down to BER=∼5*10⁻³ (SNR = 16 dB), where the dominant source of errors is the AWGN, a 12bit logarithmic quantizer suffices for approaching coherent performance, while phase-correction can further extend the capabilities of the X-DSCD scheme down to BER=∼3*10⁻⁵ (SNR = 18 dB). Pilot-aided X-DSCD provides BER below 10⁻⁶ (SNR = 18 dB) yet with a minor gap (of 0.4 dB) from coherent performance (note that it also entails some rate loss, yet pilots are typically also used in coherent systems). As the SNR improves, the quantization noise becomes the dominant source of errors and prevents the X-DSCD from achieving the performance of the coherent receiver. Increasing the number of ADC bits will push the error floor seen in Fig. 5 down, and the performance gap between the different approaches is reduces. At 18 bits, the performance of all receivers practically coincide with those of the coherent receiver down to BER=∼10⁻⁷ (SNR = 20 dB).

Fig. 5. Performance vs. SNR. Different receiver schemes, for 12 quantization bits, compared in terms of BER for a 16QAM-OFDM signal.

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Fig. 6. The analyzed, polarization-diversity, X-DSCD receiver scheme.

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6. Conclusions

In self-coherent receiver implementations, the quantizer type and number of bits affect the ZIS probability and the phase error accumulated in the process of reconstructing the received signal. Increasing the number of bits, reduces the accumulated error at the cost of ADC complexity. As a viable tradeoff, we offer a mechanism for tracking and correcting the phase error dynamically.

Herein, for an optical QAM-based OFDM modulated signal, tracking of the phase error is carried out in the frequency domain, while the actual phase correction is performed in the time domain. Additionally, we recommend using a non-uniform quantizer for allowing a self-coherent receiver to practically approach the performance of a coherent receiver. Minimizing the long streak of errors that may follow LOSy events promotes effective use of error correction codes in self-coherent receivers, thus contributing to closing the performance gap between coherent and self-coherent receivers.

A designer of a practical system may wish to further improve system performance by either tailoring the M/L ratio to the given channel SNR, design a quantizer to accurately match the 2^nd order Bessel distribution of the ODI outputs, or increase the number of ADC bits, to match a BER goal.

We have shown that the self-coherent architecture presented in this work offers a feasible alternative to coherent receivers for reliable detection of OFDM modulated signals.

Acknowledgements

The authors wish to thank the editor and anonymous reviewer for helpful comments.

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Self-coherent detection for optical OFDM via polarization diversity

Abstract

1. Introduction and background

1.1. Receiver architecture: cross-DSCD (X-DSCD)

2. Influence of quantization and noise on phase reconstruction

2.1. Additive noise

2.2. Quantization noise

2.3. Estimation of the accumulated-phase noise

3. Phase error correction

3.1. Single-carrier signal

3.2. Optical OFDM

4. Attending to zero intensity samples

5. Simulation results

6. Conclusions

Acknowledgements

References

Cited By

Figures (6)

Equations (24)

OSA Continuum