Approaching coherent performance in differential detection via diversity

Adam Magen; Ofer Amrani

doi:10.1364/OE.23.004529

1. Introduction

In order to meet the increasing demand for higher data rates, fiber-optic communication systems have been evolving rapidly over the past decade. The development of these systems was achieved thanks to many advances, among them several detection schemes, such as differential direct detection (DDD) [1], digital coherent detection [2, 3] and digital self-coherent detection [4, 5]. Digital coherent detection (DCD) attracted considerable attention due to its ability to cope with transmission impairments such as chromatic dispersion (CD) and polarization -mode dispersion (PMD) with the aid of appropriate digital signal processing (DSP) techniques [7, 8]. However, In DCD, an additional local laser source is needed.

Targeting a receiver scheme that maintains most of the benefits of coherent detection, but does not necessitate the use of a local oscillator, resulted in a new detection scheme, Digital self-coherent detection (DSCD) [4, 5]. For employing DSCD, analog-to-digital converters (A/D), optical delay interferometers (ODI), and sometimes an additional intensity detector are used along with a DSP in order to reconstruct the received electrical fields almost as if they were detected by a coherent DCD Receiver. Notably, the transmitted signal is not required to be differentially modulated. Although DSCD shares some of the good qualities of DCD it still suffers from a major drawback being its inability to reconstruct the phase when the intensity of the received signal is below the A/D resolution, i.e. zero intensity samples [9, 10].

Following a brief description of the DSCD scheme, we introduce in Section 2 an approach for mitigating the influence of zero intensity samples. In Section 3 we derive analytical bounds on the probability of synchronization loss for both the standard DSCD scheme and the newly proposed X-DSCD scheme. Section 4 provides simulation results, while in Section 5 we examine the implications of X-DSCD for overall receiver performance.

2. The proposed scheme: X-DSCD

Figure 1 shows a schematic diagram of a standard DSCD receiver. A small fraction of the incident electric field enters a power monitoring port used for tracking the instantaneous field power, whereas the remaining field is injected into a structure identical to that of a standard optical differential quadrature-phase-shift-keying (DQPSK) receiver consisting of two ODIs, which are balanced delay-line interferometers, followed by a differential detection unit. The differential group delays of the two interferometers are identical and equal to the sampling duration that is to be applied at the analog to digital convertor following detection. The differential phases of the two interferometers are equal to ±45 degrees such that the signals generated by the two differential detectors are given by

u_{I} (t) = | E (t) | | E (t - τ) | \cos (Δ ϕ (t))

u_{Q} (t) = | E (t) | | E (t - τ) | \sin (Δ ϕ (t)),

where E(t) = |E(t)|exp[iϕ(t)] is the received electric field, τ is the differential group delay of the ODIs and Δϕ(t) = ϕ(t) − ϕ(t − τ). Provided that both |E(t)| and |E(t − τ)| are sufficiently different from 0 — a condition on which we elaborate in what follows — the phase increment Δϕ(t) can be extracted through the relation

Δ ϕ (t) = \tan^{- 1} (u_{Q} (t) / u_{I} (t)) .

Fig. 1 The standard DSCD receiver.

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Following sampling at a constant rate τ, which takes place in the A/D conversion units, and denoting by ϕ(0) the phase of the electric field at some arbitrary initial time t₀, the samples of the complex electric field E(n) = E(t₀ + nτ) are reconstructed using the relation

E(n) = |E(n) | exp(iϕ (n)) where the intensity |E(n)| = |E(t₀ + nτ) | is obtained from the power monitoring port, denoted P(t), and where $ϕ (n) = ϕ (0) + \sum_{j = 1}^{n} Δ ϕ (n τ)$ . In practice, the initial phase ϕ(0) is unknown to the receiver, which therefore sets its value arbitrarily. From a data transmission perspective, the overall phase uncertainty caused by the inability to reconstruct ϕ(0), is insignificant, provided that a sufficiently large number of samples (relative to the optical bandwidth of the received electric field) can be accumulated without synchronization loss (Typically, a so-called pilot signal known to the receiver is transmitted in predefined positions). As pointed out in a recent work [11], a major obstacle to DSCD receiver implementation is the synchronization loss, which occurs whenever the intensity of the field is too small to allow extraction of the phase increments via Eq. (3). Overcoming this obstacle is the aim of the scheme that we propose in what follows.

2.1. X-DSCD

According to the proposed scheme, the transmitter modulates two different symbol streams, one stream on each of the two cross polarizations. The symbol streams are, in general, statistically independent. A schematic diagram of the proposed polarization-diverse DSCD receiver, termed X-DSCD, is shown in Fig. 2. Its front-end consists of two standard DSCD receiver units fed with the two orthogonal polarization components of the electric field, and a third balanced detection unit where the polarization components are beat against each other. The two outputs of this unit are

u_{x y, I} (t) = | E_{x} (t) | | E y (t) | \cos (δ ϕ_{x y} (t))

u_{x y, Q} (t) = | E_{x} (t) | | E y (t) | \sin (δ ϕ_{x y} (t)),

where δϕ_xy(t) = ϕ_x(t) − ϕ_y(t) is the instantaneous phase difference between the two field components. Note that the two electric fields, referred to generally as E_x(t) and E_y(t), in Eq. (4) and (5), are assumed to incorporate any channel (or signal) impairments, e.g. PMD. Based on these equations, δϕ_xy(t) can be extracted through

δ ϕ_{x y} (t) = \tan^{- 1} (u_{x y, Q} (t) / u_{x y, I} (t)),

provided that both |E_x(t)| and |E_y(t)| are sufficiently different from 0. Equations (4)–(6) provide an alternative path for extracting the phases of the two fields and thus introduce diversity. For example (in discrete representation) the sample E_x(n) can be reconstructed even if |E_x(n)| is too small to be measured, provided that E_y(n), and the beating samples u_xy,Q(n) and u_xy,I(n) are known: first calculate δϕxy(n) using Eq. (6), then extract the phase ϕ_x(n) = δϕxy(n) + ϕy(n). The intensity |E_x(n)| is then easily calculated from either Eq. (4), or (5).

Fig. 2 The proposed, polarization-diversity, X-DSCD scheme.

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It follows from this example that the associated DSP complexity is negligible - Eq. (6) and extraction of a missing intensity from Eqs. (4) or (5), can be realized by 2 lookup table (LUT) operations (trigonometric functions) plus 3 multiplication/devision operations.

Clearly, in the scheme proposed in Fig. 2, synchronization loss occurs only in the event where both polarized signals can no longer be reconstructed.

In practice, a condition of the field amplitudes being greater than 0, needs to be replaced by the condition that the field amplitude is sufficiently large to be reliably detected. Assuming that the A/D conversion unit encodes signals ranging from -A and A using N_b bits, the requirement that the measured quantity is greater than 0 is translated into its being greater than $A / 2^{N_{b} - 1}$ , and hence the probability of synchronization loss is dependent upon the values A and N_b. Thermal noise in the A/D unit itself may also impact the results, but its contribution can be neglected in most cases of interest assuming that optical pre-amplification is employed.

While the presented X-DSCD scheme is roughly ”twice” more complex than the standard DSCD scheme (hardware-wise), recall that X-DSCD also doubles the information rate as it provides two independent data streams. Hence the added hardware complexity is only due to the ”beating” branch. The latency associated with both DSCD and XDSCD schemes is the same.

3. X-DSCD receiver: performance evaluation

Herein we evaluate the performance of the X-DSCD scheme and compare with a standard DSCD scheme.

For the purpose of the analysis we consider complex white Gaussian noise as the input signal. We further assume that the A/D input range is between −0.5 and 0.5, and hence the resolution of the A/D, denoted as Δ, is given by:

Δ = 2^{- N_{b}} .

A so-called, zero-intensity (ZI) sample is characterized by having an amplitude smaller than the resolution of the A/D, which in turn results in phase-synchronization-loss. The probability P_ZI of the delayed and multiplied signal to have zero-intensity is given by:

P_{Z I} = P r (| R e (z_{1} \cdot z_{2}^{*}) | \leq Δ, | I m (z_{1} \cdot z_{2}^{*}) | \leq Δ),

where z₁ and z₂ represent a sample of the incoming electric field and a delayed sample of the same incoming electric field, respectively. Since complex white Gaussian noise is the input signal in this example, the following model for z₁ and z₂ is chosen:

\begin{array}{l} z_{1} = r_{1} + i q_{1}, \\ z_{2} = r_{2} + i q_{2}, \end{array}

where r₁, r₂, q₁, and q₂ are samples of low-pass filtered white Gaussian noise. We define the Gaussian random vector

\underline{V} = {[r_{1} r_{2} q_{1} q_{2}]}^{T}

whose distribution is naturally given by its mean vector

E (\underline{V}) = 0

, and covariance matrix

E ({\underline{V} \underline{V}}^{T}) = (\begin{matrix} σ^{2} & ρ σ^{2} & 0 & 0 \\ ρ σ^{2} & σ^{2} & 0 & 0 \\ 0 & 0 & σ^{2} & ρ σ^{2} \\ 0 & 0 & ρ σ^{2} & σ^{2} \end{matrix}),

with ρ being the correlation between two samples at the output of the ODI.

According to Eq. (8) P_ZI is derived by integrating the joint probability density function (PDF) of $R e (z_{1} \cdot z_{2}^{*})$ and $I m (z_{1} \cdot z_{2}^{*})$ over a square in the $(R e (z_{1} \cdot z_{2}^{*}), I m (z_{1} \cdot z_{2}^{*}))$ plane. An upper bound, $P_{Z I}^{(U)}$ , on P_ZI can be derived by integrating the PDF over the smallest-radius circle containing this square. Similarly, a lower bound on P_ZI can be derived by integrating the PDF over the largest-radius circle contained inside the same square. The proposed bounds are given by the following equations:

\begin{array}{l} P_{Z I}^{(U)} = P r (| z_{1} \cdot z_{2}^{*} | \leq \sqrt{2} \cdot Δ), \\ P_{Z I}^{(L)} = P r (| z_{1} \cdot z_{2}^{*} | \leq Δ), \end{array}

The proposed bounds could be easily evaluated for the case of low-pass filtered complex white Gaussian noise. The following equations provide a general expression which can be used for their calculation:

\begin{array}{l} P r (| z_{1} \cdot z_{2}^{*} | \leq a) = \\ = \frac{1}{(2 π) σ^{2}} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} f (r_{1}, q_{1} | r_{2}, q_{2}) e^{- \frac{r_{2}^{2}}{2 σ^{2}} - \frac{q_{2}^{2}}{2 σ^{2}}} d r_{2} d q_{2}, \end{array}

\begin{array}{l} f (r_{1}, q_{1} | r_{2}, q_{2}) = \\ = \frac{1}{(2 π) (1 - ρ^{2}) σ^{2}} \iint_{D} e^{- \frac{{({\tilde{r}}_{1} - ρ r_{2})}^{2}}{2 (1 - ρ^{2}) σ^{2}} - \frac{{({\tilde{q}}_{1} - ρ q_{2})}^{2}}{2 (1 - ρ^{2}) σ^{2}}} d {\tilde{r}}_{1} d {\tilde{q}}_{1}, \end{array}

D = {(r_{1}, q_{1}) | \sqrt{r_{1}^{2} + q_{1}^{2}} \leq \frac{a}{\sqrt{r_{2}^{2} + q_{2}^{2}}}},

with a ≥ 0.

$P_{Z I}^{(U)}$ and $P_{Z I}^{(L)}$ can be utilized for formulating bounds on the receiver performance in terms of synchronization-loss probability.

3.1. Synchronization loss

We define synchronization loss as an event where phase reconstruction of both orthogonally polarized signals is no longer possible due to low intensity samples received in both polarizations. Notably, after such an event, field reconstruction of the two signals may resume only when a known reference phase is received.

By employing the bounds given in Eq. (11) we obtain for the standard receiver:

2 P_{Z I}^{(L)} - P_{Z I}^{{(L)}^{2}} \leq P_{S L} \leq 2 P_{Z I}^{(U)} - P_{Z I}^{{(U)}^{2}} .

For the proposed X-DSCD receiver, there are two case types that result with synchronization loss: Case 1) when both polarized signals are of zero intensity simultaneously; Case 2) when the synchronization of one of the polarizations has been lost, and the second polarization reaches a zero intensity sample. The second case type includes multiple different events out of which we consider the event of highest occurrence probability as shown in the approximation below:

\begin{array}{l} P_{S L} ≃ P (X_{n} = Z I, Y_{n} = Z I) + 2 P (X_{n} = Z I, Y_{n} = \bar{Z I}, Y_{n + 1} = Z I, X Y_{n} = Z I) \\ \leq P (X_{n} = Z I, Y_{n} = Z I) + 2 P (X_{n} = Z I, Y_{n} = \bar{Z I}, Y_{n + 1} = Z I, X Y_{n} = Z I) \\ + 2 P (X_{n} = Z I, Y_{n} = \bar{Z I}, Y_{n + 1} = Z I, X Y_{n} = Z I) \\ = P (X_{n} = Z I, Y_{n} = Z I) + 2 P (X_{n} = Z I, Y_{n} = \bar{Z I}, Y_{n + 1} = Z I) \\ = P_{Z I}^{2} + 2 P_{Z I} \cdot P_{Z I, Z I}, \end{array}

where X_n, Y_n, XY_n, represent the n’th sample of the corresponding polarization branch (or the beating branch XY),

P_{\bar{Z I}, Z I}

represents the probability of an event where a zero-intensity sample follows a non zero-intensity sample. Hence,

\begin{array}{l} P_{S L} ≃ P_{Z I}^{2} + 2 P_{Z I} \cdot P_{\bar{Z I}, Z I} \\ = P_{Z I}^{2} + 2 P_{Z I} \cdot (P_{Z I} \cdot P_{\bar{Z I} | Z I}) \leq 3 P_{Z I}^{2} \\ \leq 3 P_{Z I}^{{(U)}^{2}} . \end{array}

A single-letter lower bound is straight forward:

P_{Z I}^{{(L)}^{2}} \leq P_{Z I}^{2} \leq P_{S L} .

The bounds derived herein have been validated by means of simulations as shown in Section 4.4.2.

4. Simulation results

The performance of the dual-polarized X-DSCD scheme proposed in Section 2 was evaluated by means of simulations. In the simulations, two orthogonally polarized signals were generated, transmitted, received, and finally reconstructed as detailed in Section 2.2.1.

The probability of synchronization loss, P_SL, of the X-DSCD scheme was computed as a function of the number of bits, N_b, employed by the A/Ds. A comparison with the standard DSCD scheme is also provided.

We consider reconstruction of two fundamentally different types of signals: an unpolarized Gaussian noise signal, meant to represent the case of generic field reconstruction, as introduced earlier, and a standard QPSK signal, with randomly generated data in each of the two polarizations. In each case we took the parameter A, representing the maximum amplitude detected by the A/D unit, to be equal to 3 times the average transmitted power, thereby limiting the probability of A/D saturation.

4.1. Reconstruction of a digital communication signal

In this case, two orthogonally-polarized streams of signals, each consisting of 2²⁴ NRZ -QPSK symbols, are generated. Assuming baud rate of 28G[Symbols/Sec] yields total bit rate of 112G[bit/Sec].

The signals are fed into a 3^rd order Bessel-type filter of approximate constant group delay up to an angular frequency of 200G[Rad/Sec] followed by Mach-Zehnder modulators (MZM) employed for generating the QPSK signals.

PMD is then inserted. Two different values are considered for the PMD parameter τ_DGD: τ_DGD = T and τ_DGD = 2T, representing small and significant distortion conditions, respectively. This parameter represents the PMD-induced differential group-delay (DGD) between the two principle states of polarization axes of a fiber PMD [10]. A uniformly random axes rotation is also incorporated. Next, chromatic dispersion (CD) is added to the two signals via an FIR filter corresponding to the transfer function of a fiber CD, $e^{\frac{j D λ^{2} ω^{2}}{4 π c}}$ , with D being the dispersion tolerance, $D = 1400 [\frac{p s}{n m}]$ , and c denotes the speed of light. White gaussian noise is finally added so as to provide OSNR of 12[dB]. At the receiver side, the input signals are first filtered with an optical Gaussian-shaped filter whose bandwidth is 35G[Hz]. The sampling rate τ is set to T/2 where T = 35.7[psec] is the symbol time.

The curves plotted in Fig. 3 reveal the potential gain offered by the new scheme. The probability achieved by the standard DSCD receiver using a 9-bit A/D, P_SL ~ 10⁻³, is realized by the proposed dual-polarized X-DSCD scheme with as little as N_b = 5 bits. When both schemes employ N_b = 8 bits A/Ds, the proposed scheme provides some 3 orders of magnitude improvement in Loss-of-Sync probability.

Fig. 3 Probability of synchronization loss as a function of the number of bits N_b employed by the A/Ds: reconstruction of QPSK communication Signal.

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Figure 3 also suggests that the reconstruction process is insensitive to PMD (the curves for different PMD values practically coincide). Finally, note that the slope of the curve associated with X-DSCD is steeper than the one associated with the standard DSCD scheme, a behavior that can be attributed to the increased diversity provided by the proposed scheme.

The proposed scheme exhibits similar behavior also for a square 16QAM constellation as shown in Fig. 4. Two scenarios are presented in the Figure. The solid curves amount to a scenario where the average power of the 16QAM constellation, as well as the OSNR used in the simulation, are the same as in the QPSK case. Clearly, this OSNR (12[dB]) does not constitute a practical operation point for 16QAM; it was chosen merely to demonstrate the influence of increasing the modulation order on P_SL. While the gain of X-DSCD over DSCD is significant even for this scenario, we show that the gain further improves when the average received power is increased (to match the power of the Gaussian signal discussed in the next subsection). The latter scenario is given by the dashed curves (denoted in the figure by X-DSCD, Normalized).

Fig. 4 Probability of synchronization-loss as a function of the number of A/D bits: 16QAM constellation reconstruction.

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4.2. Reconstruction of a random Gaussian signal

Rather than demonstrating the performance of the proposed scheme for digital signal constellations of ever increasing modulation orders, we now consider reconstruction of a random Gaussian signal.

Reconstruction of two orthogonally polarized signals consisting of 2²⁴ samples of complex white Gaussian noise was examined via simulation. The real and imaginary parts of each signal were both characterized by a power spectral density N₀ of 2 and were used as inputs to a Gaussian shaped optical filter with a bandwidth of 35G[Hz] in the receiver. Again here, τ = T/2.

The advantage of the X-DSCD scheme is clearly demonstrated in Fig. 5 also for this type of signals. While the standard DSCD receiver achieves P_SL ~ 8.7 ⋅ 10⁻⁴ when using a 9-bit A/D, the proposed scheme achieves roughly the same performance using only 6 bits. When both receiver schemes employ 9-bit A/Ds, about 4 orders of magnitude improvement is achieved by the proposed scheme. The bounds derived in Eqs. (15) and (17) are also presented in the figure for comparison.

Fig. 5 Probability of synchronization-loss as a function of the number of A/D bits: complex white Gaussian noise reconstruction.

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The results obtained with Gaussian noise as input signal suggest that the proposed scheme is suitable for arbitrary digital constellation formats (of different point arrangements and large cardinality) as encountered for example in orthogonal frequency division multiplexing (OFDM).

5. Impact on receiver performance

While our analysis of the X-DSCD scheme reveal significant improvement in the synchronization-loss probability, one must also verify that the proposed scheme provides good error performance. Error correcting codes have been successfully implemented in digital (optical) communication systems to provide adequate error rate performance with high spectral efficiency. In particular, the ITU-T G.975 recommends using Reed-Solomon (RS) [255,239,17] code, which is commonly used in long-haul optical communication systems.

Consider the X-DSCD receiver detailed in Section 2 and assume that the transmitter employs two RS codes to independently encode the bit streams associated with each polarization. Using this code, up to t = 8 RS symbol errors in random positions can be corrected per stream (The code used is defined over GF(2⁸); thus each RS symbol corresponds to 8 bits). An output bit error rate (BER) of 10⁻¹⁵ can be achieved given input BER of P_b = 10⁻⁴ [12]. Let us now evaluate the impact of synchronization loss on the error rate in the X-DSCD scheme.

Synchronization loss will always result with a burst of errors until synchronization is regained. A re-synchronization symbol is hence transmitted in predefined intervals. Assume that a single synchronization symbol is transmitted per RS symbol, i.e. one QPSK symbol per 4 QPSK data-bearing symbols. Thus, synchronization loss will affect but a single RS symbol. Since we used 2 samples per QPSK symbol in the simulations, the probability of RS symbol error due to synchronization loss, P_RS,SL, is given by

P_{R S, S L} = 1 - {(1 - P_{S L})}^{8} ≃ 8 P_{S L} .

Now, let P_b denote the probability of bit error due to all impairments other than synchronization loss. One bit error per RS symbol suffices for this symbol to be in error. In this case, the probability of RS symbol error, denoted $P_{R S, \bar{S L}}$ ,

P_{R S, \bar{S L}} = 1 - {(1 - P_{b})}^{8} ≃ 8 P_{b} .

For the simulated parameters, < τ_DGD > = T and $D = 1400 [\frac{p s}{n m}]$ , it follows from Fig. 3 that the probability of synchronization loss when employing 8-bit A/Ds is P_SL ≃ 1 ⋅ 10⁻⁶.

By substituting P_SL = 1 ⋅ 10⁻⁶ and P_b = 10⁻⁴ into Eqs. (19) and (20), respectively, one obtains that the symbol error probabilities are P_RS,SL = 8 ⋅ 10⁻⁶, and $P_{R S, \bar{S L}} = 8 \cdot 10^{- 4}$ - some two-orders of magnitude apart. These values indicate that synchronization loss events have negligible effect on overall system performance, which in turn suggests that the X-DSCD scheme may approach the performance of a coherent receiver. For comparison, the standard DSCD scheme, even with as many as 9-bit A/Ds, can only provide P_RS,SL ≈ 1 ⋅ 10⁻³, which is unacceptable.

We note that the overall normalized information rate of the above X-DSCD scheme is $\frac{4}{5} \cdot \frac{239}{255} = 0.7498$ . System throughput is clearly affected by the insertion of synchronization symbols. This influence, however, can be easily alleviated by using inter-block interleaving, see e.g. [12].

6. Conclusions

A cross-polarization DSCD scheme, termed X-DSCD, was introduced. The X-DSCD receiver employs differential-type receivers for each of the polarizations, a third balanced detection unit where the polarization components are beat against each other, and some further digital signal processing. X-DSCD provides the following benefits: 1. no optical local oscilator is required at the receiver side (via DSCD); 2. it alleviates the problem of zero-intensity samples associated with DSCD (via diversity); 3. coherent performance is approached; 4. throughput is doubled compare to a standard DSCD scheme. Thus, X-DSCD may provide a viable and practical candidate for coherent optical receiver implementation.

Acknowledgments

The authors wish to thank Mark Shtaif for many helpful discussions. The authors also thank the anonymous referee whose comments greatly helped improving the clarity of the paper.

References and links

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Approaching coherent performance in differential detection via diversity

Abstract

1. Introduction

2. The proposed scheme: X-DSCD

2.1. X-DSCD

3. X-DSCD receiver: performance evaluation

3.1. Synchronization loss

4. Simulation results

4.1. Reconstruction of a digital communication signal

4.2. Reconstruction of a random Gaussian signal

5. Impact on receiver performance

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (20)

Optics Express