Simple optoelectronic frequency-offset estimator for coherent optical OFDM

Jokhakar Jignesh; Bill Corcoran; Chen Zhu; Arthur Lowery

doi:10.1364/OE.25.032161

1. Introduction

Coherent optical OFDM (CO-OFDM) allows for high spectral efficiency by supporting high-order complex modulation schemes and by providing a compact signal spectral [1,2]. This allows for ultra-high speed, spectrally efficient signaling in fiber communication links. However, CO-OFDM is highly sensitive to the carrier frequency offsets (CFO) that causes a loss of orthogonality of the subcarriers, leading to errors.

Digital signal processing (DSP) at the receiver side of a fiber optic link is often used to estimate and then compensate for the CFO. There are several different approaches taken to achieve this. A spectral peak search method can be used [3], which finds the peak in the absolute Fast-Fourier transform (FFT) spectral of a large number of samples from the coherently received signal. Xinwei et al. proposed a technique that estimates the CFO based on the power variations in the null subcarriers [4] and Ming et al. proposed a likelihood function [5] to be maximized. An Iterative frequency offset estimation method has been proposed in [6] for OFDM signals. Digital autocorrelation based technique that uses the cyclic prefix of OFDM to estimate the CFO can also be used [7]. Training-based techniques have been proposed by Schmidl and Cox in [8] and by Minn et al. in [9]. Such training-symbols and pilot-subcarrier based CFO estimation techniques have also been proposed by Fan et al., which achieve dynamic tracking of the CFO [10]. Digital phase-locked loops (PLLs) have also been explored for CFO estimation and correction by using a carrier phase estimator [11]. Of all of these techniques, the most widely used techniques are the spectral peak search method [3] and the digital autocorrelation method [7]. Both techniques require a large number of samples to be processed which drastically adds to the computational latency in the DSP. Any attempts to reduce the computational cost/ latency by processing less number of samples will result in loss of estimate accuracy or resolution that has deleterious effect on the OFDM performance. The techniques proposed by Schmidl and Cox in [8] and by Minn et al. in [9] are also widely used but, as the proposed CFO estimation method in this paper is also blind, we choose the blind spectral-peak search method for fair comparison.

The latency, as a figure of merit for comparison, is a decisive parameter for a processing system. A small improvement in latency is important owing to its impact on certain business models. For example, the latency over the internet was widely exploited by few Wall Street traders who managed to get faster access to the stock prices than the rest by using dedicated optical fiber internet connections. A latency of a few milliseconds was translated to multi-million-dollar stock deals in favour of these few traders with high speed access. This, consequently, has fueled parallel and real-time processing research over decades. This paper intends to address this latency issue in data-recovery by reducing the load on DSP.

Instead of replacing these algorithms with their parallel-processing counterparts, we propose to completely remove the estimation algorithm. While parallelization can reduce the processing time by a given factor related to the number of parallel processes, this will increase the required high-speed (GHz-scale) digital hardware resources. To break the hardware/latency trade off from parallelization, we instead propose to minimize latency by performing CFO estimation with an analog optoelectronic system and low-speed (MHz-scale) digital processing. This aim provides another step toward all-analogue signal processing for low latency coherent communications [12].

To this end, we propose an optoelectronic frequency offset estimator (OEFOE) that operates in parallel to the coherent receiver, as shown in Fig. 1, and allows for removal of the computationally expensive estimation algorithms from the DSP.

Fig. 1 Receiver implementing the proposed estimator. ADC: Analog-to-Digital converter; LO: Local oscillator.

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All-optical or electro-optical techniques can also be employed, to avoid placing a processing burden on the DSP. These include optical injection locking (OIL) [13], optical phase-locked loops (OPLL) [14] and electro-optic phase-locked loops (EOPLL) [15]. Although these methods have their own advantages, such as simultaneous phase noise mitigation and sequential architectures, but they are restricted by the complexities in OPLL, bandwidths of the components in EOPLL or requirement of a guard band in OIL case. The proposed OEFOE gives accurate estimates for the CFO of < 4% error without requiring high speed, expensive components, while keeping the design simpler than OIL, EOPLL or an OPLL. While we have shown that it is possible to get accurate frequency offset estimates from an OEFOE using off-the-shelf components (as presented at ECOC’16 [16]), in this paper we use simulations to perform an in-depth analysis of the effects of various system parameters on estimate accuracy. The system takes the received signal and the local oscillator (LO) signal as inputs and gives an electrical output corresponding to the frequency offset as shown in Fig. 1. This output is provided as a feedback to the local oscillator laser to perform the offset correction.

Section 2 outlines the mathematical underpinning for the operation of the proposed OEFOE. Section 3 presents two versions of the OEFOE depending on the implementation scenario and gives simulation results considering parameters specific to each version design. Section 4 shows the effects of system parameters common to both designs, and investigates the Q performance after transmission over various distances with QPSK or 16-QAM modulated signals. Finally, the conclusions are made in Section 5.

2. Concept of optoelectronic frequency offset estimation for OFDM

The underlying concept for the working of the OEFOE lies in the cyclostationarity property of OFDM signals with cyclic prefixes [7]. We show mathematically how this leads to frequency offset estimation. Consider an OFDM signal E_s(t) with a cyclic prefix of period T_CP. Let $τ_{1}$ be the OFDM symbol period without the cyclic prefix as shown in the inset of Fig. 2(a).

Fig. 2 a) Proposed optoelectronic CFO estimator (OEFOE) design. Inset: OFDM symbol frame structure. b) System output with photodiode integration time for CFO = 410 MHz.

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Thus,

E_{s} (t) = E_{s} (t + τ_{1}) for t = 0 to T_{C P} .

Now consider the autocorrelation R(

τ

) of the signal E_s(t) with delay

τ

.

∴ R (τ) = \int_{0}^{\infty} E_{s} (t) E_{s} {(t + τ)}^{*} d t .

Let the carrier frequency of the transmitted signal be

ω_{0}

and carrier frequency offset

Δ f

. Thus, the received signal Y_s and the local oscillator signal Y_L are

Y_{s} (t) = E_{s} (t) e^{j ω_{0} t + ϕ_{1}} .

Y_{L} (t) = E_{L} (t) e^{j (ω_{0} + 2 π Δ f) t + ϕ_{2}} .

where E_L(t) is the amplitude of the local oscillator signal and

ϕ_{1}

,

ϕ_{2}

are the phase offsets. The autocorrelations of Y_s(t) and Y_L(t) are

\begin{matrix} R_{s} (τ) = \int_{0}^{\infty} Y_{s} (t) Y_{s} {(t + τ)}^{*} d t \\ = e^{- j ω_{0} τ} \int_{0}^{\infty} E_{s} (t) E_{s} {(t + τ)}^{*} d t \\ = e^{- j ω_{0} τ} R (τ) . \end{matrix}

\begin{matrix} R_{L} (τ) = \int_{0}^{\infty} Y_{L} (t) Y_{L} {(t + τ)}^{*} d t \\ = e^{- j ω_{0} τ} e^{- j 2 π Δ f τ} \int_{0}^{\infty} E_{L} (t) E_{L} {(t + τ)}^{*} d t \\ = e^{- j ω_{0} τ} e^{- j 2 π Δ f τ} L (τ) . \end{matrix}

where. The phase offsets

ϕ_{1}

and

ϕ_{2}

are affected by the laser phase noise but the integration pro

L (τ) = \int E_{L} (t) E_{L} {(t + τ)}^{*} d t

cess and the Wiener process nature of the phase noise negates their effect. This will be explained further in section 5.1. Note that Rs(

τ

) and R_L(

τ

) are not dependent on the time parameter, t, but remain constant for a fixed value,

τ

, for a fixed CFO,

Δ f

.

The ratio of R_s( $τ$ ) and R_L( $τ$ ) is:

D (τ) = \frac{R_{s} (τ)}{R_{L} (τ)} = \frac{e^{j 2 π Δ f τ} R (τ)}{L (τ)} .

Now, if the delay

τ = τ_{1}

from Eqs. (1) and (2),

R (τ_{1}) = \int_{0}^{\infty} E_{s} (t) E_{s} {(t + τ_{1})}^{*} d t

.

The $R (τ_{1})$ can be written as summation of multiple OFDM symbols as follows

R (τ_{1}) = \sum_{n = 0}^{\infty} \int_{t = 0}^{t = T_{O F D M}} E_{s} (n T_{O F D M} + t) E_{s} {(n T_{O F D M} + t + τ_{1})}^{*} d t .

where, n determines the OFDM symbol index and T_OFDM is the OFDM symbol period given as T_OFDM = T_CP +

τ_{1}

. For realistic system, n ranges up to a finite value N_int that reflects the number of OFDM symbols that need to be integrated for the proposed system to converge to the required estimate.

R (τ_{1})

can then be extended as

R (τ_{1}) = \sum_{n = 0}^{N_{int}} (\int_{0}^{T_{C P}} E_{s} (n T_{O F D M} + t) E_{s} {(n T_{O F D M} + t)}^{*} d t + \int_{t > T_{C P}}^{t = T_{O F D M}} E_{s} (n T_{O F D M} + t) E_{s} {(n T_{O F D M} + t + τ_{1})}^{*} d t)

Since the OFDM symbols are uncorrelated for t > T_CP, the second term will reduce to zero. As a result, for

τ = τ_{1}

,

R (τ) = \sum_{n = 0}^{N_{int}} \int_{0}^{T_{C P}} {| E_{s} (n T_{O F D M} + t) |}^{2} d t

and is real. It is important to note here that the strength of the autocorrelation is proportional to the length of cyclic prefix. Thus, longer cyclic prefixes will improve the system accuracy, as will be shown in next section. Similarly, since the amplitude of the local oscillator signal E_L(t) can be taken as constant,

L (τ) = \int_{0}^{T_{C P}} {| E_{L} (t) |}^{2} d t

and is also real. Thus, the only complex term in Eq. (7) is

e^{j 2 π Δ f τ}

and the CFO can be calculated as

Δ f = \frac{∠ D (τ_{1})}{2 π τ_{1}} .

So, by mixing both the signal & local oscillator with delayed copies of themselves and then integrating the resulting waveforms, we are able to gain an estimate of the frequency. This mixing and integration can be done using 90° hybrids and slow photodiodes respectively and whole estimation can thus be performed in analog domain.

3. System design and simulations

In this paper, we propose two sub-system designs to provide the required mixing for the OEFOE, where each design has its own benefits in different implementation scenarios.

3.1 OEFOE version 1

3.1.1 System design

The first design of the proposed optoelectronic CFO estimator (OEFOE 1) is shown in Fig. 2(a). This design is conceptually closest to the method described in the previous section, with the delayed signal and delayed LO mixing occurring physically separately.

OEFOE 1 takes the received OFDM signal as input and splits it using a 3-dB coupler. The signal is delayed in one of the arms by a specific delay $τ_{1}$ , equal to the OFDM symbol period without the cyclic prefix (as shown in the inset of Fig. 2(a)). We assume that the structure of the cyclic prefix of the signal in the optical network or a link is standardized and known to the receiver. Thus, the delay line’s length ( $τ_{1}$ ) can be spliced accordingly. Just to make the proposed OEFOE flexible, the delay line fiber can be made detachable such that it can be replaced appropriately whenever the cyclic prefix length is changed. Problems may arise only when the system dynamically changes the cyclic prefix length.

The signal and the delayed version are then fed to a 90°-hybrid and the outputs of the hybrid are detected by slow photodiodes in balanced configuration. The output of the configuration will converge to R_s( $τ_{1}$ ) [17] as shown in Fig. 2(a). Simultaneously a similar setup generates a signal that converges to R_L( $τ_{1}$ ) as shown in Fig. 2(a). The integration operation required for R_s( $τ$ ₁) and R_L( $τ$ ₁) is performed by the slow photodiodes [18], with a response time governed by the parameter N_int. The integration time of the photodiodes has to be more than (N_{int $\times$}T_OFDM). Figure 2(b) shows the OEFOE’s output against time from which the convergence can be extrapolated to find the number of OFDM symbols to be integrated (N_int).

Simulations were performed in VPItransmissionMaker for a 28-Gbaud QPSK modulated OFDM signal and 15% cyclic prefix. The experimental verification was shown previously in [16]. In this paper we perform simulations to understand the effects of various parameters involved in the sytem. It was observed that the output converges to the actual CFO of 410 MHz after 1.5 $μ$ s. Hence, the number of OFDM symbols that need to be integrated to converge to the required estimate can be given as N_int = 1.5e-6/T_OFDM. This dependence of convergence on the cyclic prefix is discussed in Section 4. In the case of fast photodiodes, the integration can be provided by bandwidth-limiting filters. However, this is not advised as it increases the cost. The remaining operations to calculate the CFO (Eq. (9)) can be performed using a simple microprocessor. To do so, the electrical signals at the output of the balanced configuration need to be sampled after convergence. A single sample taken after 1.5 $μ$ s is sufficient to calculate the CFO using Eq. (9). Note that the actual sampling takes place after the balanced photodiodes in Fig. 2(a); the trace in Fig. 2(b) just helps to visualize the convergence time.

As depicted in Fig. 2(a), OEFOE 1 could be implemented with passive off-the-shelf components such as 3-dB couplers, 90°-hybrids, and optical delay lines. Additionally, the electronic components can have low bandwidth, as only slow photodiodes and ADCs with MHz sampling rate are required. Moreover, only a small number of float-point operations are required on the samples to give the desired frequency offset estimates, which can readily be performed by a simple microcontroller. Given these desirable qualities, we next simulate OEFOE 1.

3.1.2 Simulations and results for the OEFOE 1

The OEFOE 1 design in Fig. 2(a) was simulated in VPItransmissionMaker software (Version 9.7). A 28-Gbaud OFDM signal of FFT length 156, 100 subcarriers and 15% cyclic prefix was generated and oversampled at 40-GSa/s sampling rate. With this configuration, the delay $τ_{1}$ can be calculated to be 0.8 ns. Substituting $τ_{1}$ = 0.8 ns in Eq. (9) with $∠ D (τ_{1}) \in [\begin{matrix} - π & π \end{matrix}]$ , we find the estimation range of the OEFOE to be [-625 MHz + 625 MHz].

Figure 3(a) shows the scatterplot of the CFO estimated by the OEFOE in back-to-back setup compared with the spectral-peak search method. The CFO was randomly chosen within the range [-1250 MHz + 1250 MHz] for 1000 runs and estimates were calculated by both methods. As expected, the OEFOE gives accurate results with estimate errors < 1% up to $\pm$ 625 MHz. The slope of the scatter plot reverses beyond 625 MHz because of phase reversal of $∠ D (τ_{1})$ beyond $π$ radians for positive CFO and beyond $- π$ radians for negative CFO, creating an ambiguity in estimating CFO beyond $\pm$ 625 MHz (Fig. 3(a)). To resolve this ambiguity we add a known phase shift $Δ φ = 2 π f_{k} τ$ digitally to R_s( $τ$ ) where f_k is known ( + 2 MHz in our case) giving $R_{s}^{k} (τ) = R_{s} (τ) e^{j 2 π f_{k} τ}$ and leading to $Δ f^{k}$ . If, $Δ f$ < 625 MHz, $Δ f^{k} - Δ f$ will be positive owing to positive frequency shift and if $Δ f$ > 625 MHz, $Δ f^{k} - Δ f$ will be negative due to the negative slope in this region. Hence, with one additional multiplicative and subtracting operation in the microcontroller, the ambiguity can be resolved to give full estimation range of [-1250 + 1250] MHz as shown in Fig. 3(b). As these computations are to be performed on a low-clock rate microcontroller, the cost of providing this computations is orders of magnitude lower than for GHz-clock DSP ASICs used in conventional coherent communication systems. This method to resolve ambiguity will fail within the range [623 625] MHz of the absolute actual CFO, excluding the limiting values. However, despite of this failure in ambiguity resolution, the error in CFO calculation will be a maximum of 4 MHz which is still very low (< 6.4%). Apart from that, the chance of the actual CFO falling within this range is low. So, in practical systems that have the CFO drifting frequently, the proposed OEFOE will have a negligible outage probability.

Fig. 3 Scatterplots of Estimated CFO vs. Actual CFO (MHz) for a) with ambiguity beyond 625 MHz CFO and b) ambiguity resolved.

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3.1.3 Implementation issues for the OEFOE 1

In practical implementation of the OEFOE with discrete components as suggested above, the device’s performance is affected by various parameters such as:

1) different delays in the optical delay lines before the hybrids in Fig. 2(a) instead of a single delay, $τ_{1}$ .
2) different phase differences between signals at the outputs of the 3-dB couplers in Fig. 2(a).

These parameters could be precisely controlled if OEFOE 1 was to be implemented on an integrated 'photonic chip', by providing a phase-stable platform. In the case when the OEFOE needs to be prototyped with discrete components, the OEFOE 1 fails to give accurate result owing to the parameters mentioned above that cannot be controlled or calibrated any more. For such scenario of implementation with discrete components, we propose a different design, Version 2, as discussed in the next section.

3.2 OEFOE version 2

3.2.1 System design

We propose a modified version of the OEFOE setup that can be prototyped with discrete components. This second version (OEFOE 2) uses only one optical delay line and one 3-dB coupler for both received and LO signals and exploits a dual-polarization coherent receiver to estimate the CFO. The received signal Y_s(t) and the local oscillator signal Y_L(t) are orthogonally polarized, placed on x and y polarizations respectively using polarization controllers (PC) and combined using a 3-dB coupler. Thus, the 3-dB coupler will have signal $Y_{s} (t) \hat{x} + Y_{L} (t) \hat{y}$ on one output arm and a delayed version $Y_{s} (t + τ_{1}) \hat{x} + Y_{L} (t + τ_{1}) \hat{y}$ on the other arm after passing through an optical delay line, $τ_{1}$ . These signals, after splitting by polarization beam splitters (PBS) are received as $Y_{s} (t) \hat{x}$ and $Y_{s} (t + τ_{1}) \hat{x}$ at the inputs of one 90°-hybrid and $Y_{L} (t) \hat{y}$ and $Y_{L} (t + τ_{1}) \hat{y}$ at the inputs of other hybrid, similar to the OEFOE 1 setup in Fig. 2(a). Thus, the rest of the system is the same as in Fig. 2(a), and we can acquire the CFO estimates at the output. We can then show that this system is robust to phase and delay perturbations that effected the OEFOE 1 design. The proposed systems in Figs. 3 and 4 resemble a coherent receiver, which is usually costly; however, we do not use the high-speed photodiodes or high-speed ADCs, which are the major contributors to the high cost of the coherent receivers.

Fig. 4 Modified setup. PC: polarization controller; PBS: polarization beam splitter.

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3.2.2 Effects of the phase difference between signals at outputs of the 3-dB coupler

Let the phase mismatch between the two output arms of the 3-dB coupler be $Δ ϕ$ . Thus, the inputs of the PBSs in Fig. 4 are

P_{1} (t) = e^{j Δ ϕ} (Y_{s} (t) \hat{x} + Y_{L} (t) \hat{y}) .

P_{2} (t + τ_{1}) = Y_{s} (t + τ_{1}) \hat{x} + Y_{L} (t + τ_{1}) \hat{y} .

After being split by the PBSs, from Eqs. (5) and (6), the autocorrelation values will be

R_{s} (τ) = e^{- j ω_{0} τ} e^{j Δ ϕ} R (τ) and R_{L} (τ) = e^{- j ω_{0} τ} e^{- j 2 π Δ f τ} e^{j Δ ϕ} L (τ) .

As a result, after taking a ratio of R_s(

τ

) and R_L(

τ

) in the microprocessor, the

e^{j Δ ϕ}

term is cancelled and the value of D(

τ

) remains the same as in Eq. (7). So, any phase mismatch between the signals in the output arms of the 3-dB coupler should have no effect on OEFOE 2.

3.2.3 Effects of the delay variations in the optical delay line ( $τ_{1}$ )

For commercial SMF-28e fibers, the temperature variations in the fiber lengths we use to implement our system can cause a delay variation up to $\pm$ 30 ps [19], corresponding to a ~4% variation away from the required delay value. Again, we use a 28-Gbaud, QPSK encoded OFDM signal of FFT length 156, 100 subcarriers and 15% cyclic prefix as a test signal. As observed in Fig. 5, the delay variations cause a linear increase in the error and degradation of the Q performance of the system. This is observed despite of assuming perfect extinction ratio PBS whose effect will be added in the next section. With the delay variations increasing from 0 ps to 36 ps (4.5% or τ₁), the % error increases from 0.16% to 3.6% causing the Q of the recovered signal to drop from 20.6 dB to 19.35 dB i.e. by 1.25 dB. This degradation, however, can be nulled by keeping the system in a temperature-controlled environment or by integration. The delay variations are more problematic in the OEFOE 1 as it uses two optical delay lines and the paths are independent. On the other hand, OEFOE 2 uses a single optical delay line and thus the shifts are common for both the OFDM and the LO signals. Thus, OEFOE 2 is less affected by the delay variations than OEFOE 1, if both versions are implemented in lab using discrete components. Still, shifts in delay away from $τ_{1}$ cause inaccuracies.

Fig. 5 Q (dB) and estimate % error vs. delay variations (ps) in optical delay line.

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3.2.4 Effect of the polarization misalignment in the polarization controllers

As the OEFOE 2 uses polarization diversity, the polarization alignment of the signal becomes an important factor in the performance. For accurate CFO estimations, the polarization controllers in Fig. 4 need to be aligned properly such that the Ys(t) and the local oscillator signal Y_L(t) are orthogonally polarized on x and y polarizations respectively. In case of polarization misalignment, the system gives errors. As shown in the Fig. 6(b) the estimate errors increase nonlinearly with a decrease in the polarization extinction ratio (ER). In simulations, the polarization beam splitter is modelled to have a perfect ER, and so the change in the ER determined by the polarization misalignment θ, where ER = −10log₁₀(sin[θ]).

Fig. 6 a) Scatterplots of the CFO estimates for different polarization extinction ratio (ER) and b) Estimate error (%) vs. ER (dB).

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In reality, the limited ER of the polarization beam splitters would need to be taken into account. It can be observed that the errors increase rapidly from almost 0% to 50% when the polarization misalignment θ is increased from 0° (ER = −20.5 dB) to 10° (ER = −8 dB). This is reflected in Fig. 6(a) wherein the scatterplot shows perfect estimation for ER = - $\infty$ i.e. when θ = 0 but the errors increase rapidly when the ER is slightly increased from −16 dB to −12 dB. This shows the proposed OEFOE 2 is also sensitive to the polarization alignment of the polarization controllers and should be properly taken care of while performing the experiments.

A specific OEFOE design can be chosen according to the application scenario. The OEFOE 1 design is accurate with < 0.2% errors when fabricated on an integrated photonic chip with controllable design parameters. On the other hand, when implemented with discrete components, the OEFOE 1 performance is majorly affected by the delay variations and phase difference between outputs of 3-dB couplers. Consequently, the OEFOE 2 design is proposed that is robust to the 3-dB coupler phase differences and less sensitive to the delay variations compared with the OEFOE 1 design. However, this design is sensitive to the polarization extinction ratio of the PBSs in the design and needs to be carefully taken care of for accurate results. The next sections discuss the parameters in the system that are common to both designs and their effects on the estimation accuracy.

4. Variation of performance against cyclic prefix length

As seen in Eq. (8), the strength of the autocorrelation depends on the limits of the integral; i.e. on the duration of the cyclic prefix (T_CP) that in turn determines the accuracy of the system. In other words, from Eq. (8), R( $τ$ ) will be more accurate when the cyclic prefix is longer. Intuitively, longer cyclic prefix, T_CP means that the autocorrelation is calculated over more symbols that increases its strength and thus the accuracy of the system. However, a longer cyclic prefix reduces spectral efficiency, showing a trade-off between Q performance and spectral efficiency.

The effect of changing cyclic prefix length can be seen in Fig. 7(a)-7(c) where the OEFOE converges to the actual CFO faster for the signals with longer CP. From Fig. 7(a)-7(d) it can be observed that the convergence is affected by the CP proportion (a-c) and not by the FFT size (Fig. 7(d)), giving us freedom when choosing the FFT size according to the channel frequency response. Moreover, the magnitude of the frequency offsets also affects the convergence as can again be observed from Fig. 7(a)-7(c); with greater frequency offsets causing slower convergence.

Fig. 7 CFO estimate (MHz) vs. time (μs) for: a) actual CFO = 200 MHz and different cyclic prefix (CP); b) actual CFO = 500 MHz and different cyclic prefix; c) actual CFO = 1250 MHz and different cyclic prefix and d) 15% CP with actual CFO = 200 MHz and different FFT size (NFFT).

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The convergence helps us to determine the sampling rate of the ADCs in the proposed OEFOE. We observe that for maximum trackable CFO (1250 MHz) and 15% CP the estimation signal converges at 1.35 μs. Any CP lower than 15% causes the convergence to occur after 2.4 μs. To ensure the convergence, we sample the electrical signals after 1.5 μs for CP set at 15%. Conversely, this means that for a sampling period of 1.5 μs, the CP must be at least 15%.

Figure 8 shows the effect of varying the CP length while retaining a sampling period of 1.5 μs. The estimate error of the OEFOE remains less than 0.2% for cyclic prefix of 15% or more, while it increases rapidly if the CP is reduced below 15%. Thus, for the rest of our investigations, we will work with a CP of 15% and a sampling period of 1.5 μs.

Fig. 8 Estimate error (%) vs. Cyclic prefix (%).

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5. Simulated system performance

We next investigate OEFOE performance against system parameters, such as laser linewidth and OSNR, before simulating performance after transmission through 200 to 700 km fiber links.

5.1 Laser linewidth

The linewidth of the lasers used in a system determine the variance of the phase noise for that system. Figure 9 shows the OEFOE estimates for a CFO of 300 MHz at OSNR of 20 dB, irrespective of the linewidths investigated (from 100 kHz to 60 MHz), accurate to < 1 MHz. Thus, the proposed OEFOE system is robust to phase noise. This can be explained by the laser phase noise following a Wiener model [20]. Let $φ (t)$ be the phase noise added to the signal at time t. Following the Wiener model [20],

φ (t) = φ (t - 1) + Δ φ .

where,

φ (t) - φ (t - 1) = Δ φ_{t}

is Gaussian distributed with zero mean and finite variance

σ^{2}_{t}

. Continuing further,

φ (t - 1) - φ (t - 2) = Δ φ_{t - 1} \sim N (0, σ_{t - 1}^{2}) .

∴ φ (t) - φ (t - 2) = Δ φ_{t - 2} \sim N (0, (σ_{t - 1}^{2} + σ_{t}^{2})) .

Eventually,

Fig. 9 CFO estimate vs. laser linewidth (MHz) for CFO = 300 MHz.

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∴ φ (t) - φ (t - T_{C P}) = Δ φ_{t - τ_{1}} \sim N (0, (σ_{t}^{2} + σ_{t - 1}^{2} + ...... + σ_{t - τ_{1} - 1}^{2})) .

Hence, while calculating R( $τ$ ), the phases of E_s(t) and E_s(t + $τ_{1}$ ) subtract and the integration process converges to the mean of $Δ φ_{t - τ_{1}}$ that is zero, nulling the effect of phase noise on the CFO estimation.

5.2 AWGN loading

To test the robustness of the system to the channel noise, the OFDM signal was QPSK modulated and noise loaded with varying optical signal to noise ratio (OSNR). Figure 10 compares the Q performance of the signal recovered using OEFOE with that of the signal recovered using the conventional spectral peak search method with OSNR sweep. We again use a test signal of QPSK encoded OFDM at a rate of 28-Gbaud, with FFT length 156, 100 occupied subcarriers and a 15% cyclic prefix. The CFO is set to 300 MHz in the simulations. The phase mismatch between the output arms of the 3-dB coupler and the delay variations in the optical delay line are varied randomly within the ranges [- $π$ , + $π$ ] and [0, 30 ps] respectively.

Fig. 10 a) CFO estimates vs. OSNR (dB) and b) Q (dB) vs. OSNR (dB) for 28-Gbaud QPSK, FFT length = 156, number of subcarriers = 100, 15% cyclic prefix and carrier frequency offset (CFO) = 300 MHz.

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The mean CFO estimate plot from the OEFOE shows random variations independent of the OSNR. These variations are caused by the delay fluctuations in the optical delay line. Regardless, the mean estimates of the CFO from the OEFOE shows errors less than 1 MHz. The Q performance varies by maximum 0.8 dB around the mean Q at different OSNR. The plot of mean Q of signal recovered by OEFOE closely follows that of signal recovered by the spectral peak search method. Thus, it can be inferred that the system is robust to the noise added to the signal. This can be explained by the integration operation performed by the slow photodiodes that average out the additive white Gaussian noise with zero mean.

5.3 Chromatic dispersion and nonlinear effects of the fiber

The OEFOE system is added in parallel to the coherent receiver setup. As a result, the input signal is perturbed with chromatic dispersion and nonlinearities when transmitted over a length of optical fiber. As such it becomes important to test the OEFOE over a transmission link. Additionally, the modulation format flexibility is also an important characteristic for the modern 400G optical networks handling signals with different modulation formats and needs to be examined.

5.3.1 Simulated transmission system: QPSK

To understand the effects of chromatic dispersion and nonlinearities of the optical fiber, the signal was simulated over transmission links of lengths from 200 km to 700 km. A recirculating loop was simulated to achieve these link lengths by varying the number of recirculations. Each loop consisted of one span of 100 km. Figure 11 shows the mean OEFOE and spectral peak search estimates with varying launch powers for various transmission lengths with actual CFO fixed at 300 MHz. The plots show no particular pattern with increasing transmission lengths or launch powers. The mean estimates show < 1 MHz errors compared with the spectral peak search.

Fig. 11 Simulated CFO estimates vs. Launch power (dBm) for QPSK with launch power = −4 dBm and carrier frequency offset (CFO) = 300 MHz.

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Figure 12 shows the plots of the Q of the signal recovered by the OEFOE compared with that of the signal recovered by the spectral-peak search method with launch power sweeps and different transmission distances. The system using the proposed OEFOE gives a negligible penalty of mean 0.3-dB with 0.7-dB fluctuations due to delay variations in optical delay line and maximum penalty of 0.6-dB. These low penalties are independent of chromatic dispersion and the self-phase modulation (SPM) effect evident in the high launch power regime. Thus, the OEFOE maintains the performance despite of these impairments.

Fig. 12 Simulated Q (dB) vs. Launch power (dBm) for 28-Gbaud QPSK signal with FFT length = 156, number of subcarriers = 100 and 15% cyclic prefix for link distances 200 km, 300 km, 500 km and 700 km.

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Since the CFO estimation by OEFOE is performed in parallel to the coherent reception, the system receives a signal that is not dispersion compensated. The fiber dispersion causes pulse spreading that leads to the inter-symbol interference (ISI). Since the interfering parts are uncorrelated, the autocorrelation removes their effects. Thus, the OEFOE is robust to the chromatic dispersion as well as the nonlinearities in the optical fiber. The parameters of the optical fiber chosen for simulations are given in Table 1.

Table 1. Simulation Parameters

View Table

5.3.2 Simulated transmission system: 16-QAM

Higher-order QAM modulation formats have been tested for long-haul optical communication links to increase the throughput and meet the rapidly increasing needs for data rates. From Eq. (8), the autocorrelation R( $τ$ ) calculates $\int {| E_{s} (t) |}^{2}$ i.e. the power of the modulating signal for the period t = 0 to T_CP. For higher-order QAMs, this power can be the same or higher than the QPSK modulating signal, depending on the data in the period. Thus, the autocorrelation R( $τ$ ) will not be weaker than that for QPSK, and may often be stronger. Hence, moving to a higher QAM order format should not cause any degradation in the performance. This is verified in Fig. 13 where a 28-Gbaud 16-QAM modulated OFDM signal is noise loaded and recovered using the OEFOE estimates. The system performance is compared with that of a system using the spectral peak search method. As expected, both the systems again perform similarly with marginal mean-Q degradation of 0.3 dB for the OEFOE-recovered signal. Like QPSK case, the Q value varied by maximum 0.7-dB due to the fluctuations in optical delay line with maximum penalty of 0.8-dB.

Fig. 13 Simulated a) CFO estimates vs. OSNR (dB) and b) Q (dB) vs. OSNR (dB) for 28-Gbaud 16-QAM with FFT length = 156, number of subcarriers = 100, 15% Cyclic prefix and carrier frequency offset (CFO) = 300 MHz.

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The 16-QAM modulated signal transmission over fiber links was simulated for different lengths from 200 km to 700 km and again the Q performance of both the systems were found to be similar to each other as can be observed in Fig. 14. The robustness of the system can be intuitively extended to the cross-phase modulation (XPM) for a multichannel WDM system; however, this remains to be investigated. Thus, the proposed designs are robust to the modulation format, laser phase noise, the additive noise, chromatic dispersion and the fiber nonlinearities on transmission, with the system implementing OEFOE giving Q performance similar to a system using spectral-peak search method in DSP. However, as Ref [7]. showed that the required cyclic prefix (CP) length for accurate CFO estimation using the digital autocorrelation increases with the order of the M-QAM format. We can, thus, conclude that despite of similar performance, the OEFOE will require longer CP for higher-order M-QAM when M>64.

Fig. 14 Simulated Q (dB) vs. Launch power (dBm) for 28-Gbaud 16-QAM with FFT length = 156, number of subcarriers = 100, 15% Cyclic prefix for link distances 200 km, 300 km, 500 km and 700 km.

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Finally, we discuss the equalization-enhanced phase noise (EEPN). The EEPN is caused by the electronic dispersion-compensation and adds to the impairments from the laser phase noise [21]. The digital CFO estimators have been noted to suffer in performance due to this EEPN. On the other hand, OEFOE’s performance remains unaffected by EEPN as it operates in the analogue domain before electronic dispersion compensation, the EEPN will not be present at the input of the OEFOE. Thus, the OEFOE is robust to EEPN too, a quality that is not present in DSP-based CFO compensation methods.

For practical implementation in deployed optical networks, the OEFOE system requires fabrication as an integrated package along with the coherent receivers. Considering this, the OEFOE 1 proves to be a better option in practical scenario as it provides a simpler design and does not rely on the polarization diversity. On the other hand, OEFOE 2 can be used for proof of concept demonstrations and to test any modifications or extension work. Additionally, both the systems rely on the time domain characteristic of the OFDM signals i.e. the cyclic prefix addition that can be emulated in a single carrier system as well by adding known training symbols on regular interval ( $τ_{1}$ ). As a result, with modification in the transmitted signal, the system can be extended to single-carrier and Nyquist-WDM systems. This can be taken as a future work. The OEFOE designs have the same topology as a dual-polarization (DP) coherent receivers which are expensive that may seem demotivating. However, we use slow photodiodes and low-speed ADCs with bandwidths in range of MHz (instead of GHz bandwidth components in DP coherent receiver system). Thus, the manufacturing cost can be expected to be much lower than that of the DP coherent receivers and could be further reduced if integration is used. The proposed system can be considered as a step towards achieving a DSP-free, analog-processing system that is a matter of interest for many researchers recently to reduce the power consumption by the high-speed processors. For example, an all-analog chip performing a constant modulus based adaptive equalization was proposed recently [12]. Combined with the OEFOE carrier recovery, the analog chip can achieve a power-efficient all-analog signal-processing solution that can target DSP-free coherent optical data-center interconnects (DCI) market.

6. Conclusions

An optoelectronic carrier-frequency offset estimator that replaces the computationally expensive estimation DSP algorithms is designed and tested using simulations. The system was designed in two versions, one suitable to integration on a photonic chip while the other suitable to be implemented in a lab with discrete equipments. Both the versions use passive optical components and avoid the use of high-speed photodiodes or high-speed analog-to-digital converters. For a 28-Gbaud OFDM signal (N = 156, 100 subcarriers), the system can estimate CFO up to $\pm$ 1250 MHz with estimate errors < 4%. The simulations with varying cyclic prefix lengths suggested the CP length to be fixed $\geq$ 15% to maintain the errors $<$ 1%. The system was found to be robust to the chromatic dispersion, laser linewidth, self-phase modulation, AWGN noise in the channel and the launch powers in a transmission link. Hence, the system proves to be an alternative to computationally expensive digital CFO estimation algorithms, providing a way towards reducing the DSP latency.

Funding

Australian Research Council’s Laureate Fellowship scheme (FL130100041).

Acknowledgment

We thank VPIphotonics (www.vpiphotonics.com) for their support under the university program.

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Parameter	Value
Attenuation	0.2 dB/km
Dispersion	16 $\times$ 10⁻⁶ s/m²
Dispersion slope	0.08 $\times$ 10³ s/ m³
Group refractive index	1.47
Nonlinear index	2.6 $\times$ 10⁻²⁰ m²/W
Core area	80 $\times$ 10⁻¹² m²
EDFA noise figure	4-dB
Span length	100 km

Simple optoelectronic frequency-offset estimator for coherent optical OFDM

Abstract

1. Introduction

2. Concept of optoelectronic frequency offset estimation for OFDM