Analysis and iterative equalization of transient and adiabatic chirp effects in DML-based OFDM transmission systems

Chia-Chien Wei

doi:10.1364/OE.20.025774

1. Introduction

With exponentially increasing needs for Internet traffic, the development of short-range (<~100 km) optical communication systems, including metro and access networks, have been considered to be the key to the future broadband services. Unlike long-haul communication systems, however, short-range systems require low hardware cost and operation expenses. To meet the requirement in such cost-sensitive systems, intensity modulation and direct-detection (IMDD) without chromatic dispersion compensation has attracted a lot of research interest and development efforts. As a consequence, it is desirable to generate high-capacity signals by cost-effective transmitters, such as a directly modulated DFB laser (DML), of which the superiorities are not only potentially low cost, but also compact size, low power consumption, and high optical power. Alternatively, an electro-absorption modulated laser (EML), the monolithic integration of a DFB laser and an electro-absorption modulator (EAM), has similar superiorities, but low output power.

Moreover, thanks to the advance in digital-signal-processing (DSP) technology, orthogonal frequency-division multiplexing (OFDM) has been envisioned as a prominent modulation and multiplexing technique [1]. On the basis of using higher-order quadrature amplitude modulation (QAM) format, OFDM signals can effectively achieve high spectral efficiency and ultimately lower the bandwidth requirement of components. Utilizing commercially matured 10-GHz DMLs and EMLs, 40-Gbps OFDM signals have been demonstrated and transmitted over 5-km and 20-km standard single-mode fiber (SSMF), respectively [2]. Unlike a relative expensive Mach-Zehnder modulator (MZM), which can control the phase and amplitude of an optical field independently [3, 4], the optical OFDM signals generated by DMLs or EMLs are inherently double sideband (DSB) and frequency-chirped. Consequently, the transmission distance without dispersion compensation is severely limited by detrimental dispersion- and chirp-related power fading [2, 5]. Fortunately, power fading is periodic and predictable in fiber channel, and secondary low-fading bands have been proposed to carry supplementary OFDM subcarriers to achieve full utilization of transceiver bandwidth [6, 7]. Even though multi-band OFDM signalling could utilize bandwidth effectively under the influence of power fading, dispersion-induced distortion would deteriorate the transmission performance. This distortion has been theoretically revealed as subcarrier-to-subcarrier intermixing interference (SSII) [8], and been experimentally observed in an EML-based OFDM system [7]. Nevertheless, since a chirped intensity modulator is simplified by considering transient chirp but omitting adiabatic chirp in [8, 9], the model, unfortunately, would not agree with a DML-based transmission system well. This statement could be supported by many literatures that have thoroughly investigated the influence of the adiabatic chirp on a DML-based monochromatic wave or on-off keying (OOK) signal [10–12]. That is, the omission of the adiabatic chirp will make the theoretical transmission performance in [8] no longer conform the practical DML-based cases. As a consequence, theoretical analysis will be necessary to understand the effect of the adiabatic chirp on a DML-based OFDM system, and furthermore, the understanding can be helpful in dealing with related transmission impairment.

Applying small-signal approximation, this work analyses the effects of both transient and the adiabatic chirp on a DML-based IMDD OFDM transmission system. In addition to deriving power fading of each subcarrier as a simple formula, the small-signal model (SSM) can calculate the dispersion-induced distortion statistically and deterministically by approximating the distortion as the 2nd-order SSII. According to the SSM, the presence of the adiabatic chirp will ease power fading or even provide power gain, but will increase the SSII after dispersive transmission. To examine the feasibility of the SSM, large-signal simulation (LSS) based on the laser rate equations is carried out for comparison. Since the adiabatic chirp depends on the bias of drive current, the consistency between the SSM and the LSS is investigated under different bias currents. When the DML is operated properly such that the OFDM signals are not affected much by the laser resonance relaxation, the transmission distortion is mainly caused by the interaction between dispersion and chirp, i.e. the SSII discussed in this work. In such situations, the SSM can estimate the transmission performance of the LSS well. Furthermore, since the SSM can deterministically estimate the dispersion-induced distortion by calculating SSII, it is possible to eliminate the distortion with the knowledge of the transmitted data. As a result, this work also proposes an iterative equalization to mitigate the distortion without the knowledge of the transmitted data. Because the distortion is mostly eliminated by the proposed equalization, the maximum transmission distance of DML-based OFDM signals can be significantly improved. For instance, if the bias current is set as 30 mA in the LSS, the maximum SSMF transmission distance of a 30-Gbps DML-based OFDM signal can be extended from 10 km to more than 100 km by the proposed equalization. Moreover, because of power gain caused by the chirps at some distances, the iteratively equalized OFDM signals would exhibit negative power penalties.

2. Analysis of the chirp effects

In order to analyze the effects of laser chirp, the DML dynamics can be modeled by the extensively used rate equations of the carrier density of the gain medium, N(t), the photon density in the laser cavity, S(t), and the phase of the optical field, φ(t) [13–15],

\frac{d N}{d t} = \frac{I}{q V} - v_{g} g S - (A N + B N^{2} + C N^{3})

\frac{d S}{d t} = Γ v_{g} g S + Γ β_{s p} B N^{2} - \frac{S}{τ_{p}}

\frac{d φ}{d t} = \frac{α Γ v_{g} σ}{2} (N - N_{t h})

where I is the injection current, V is the active volume, q is the electronic charge, g is the optical gain, v_g is the group velocity, AN is the nonradiative recombination rate, BN ² is the radiative recombination rate, CN ³ is the Auger recombination rate, Γ is the optical confinement factor, β_sp is the spontaneous emission factor, τ_p is the photon lifetime, α is the linewidth enhancement factor, σ is the differential gain, and N_th is the carrier density at threshold. The output laser power is proportional to the photon density,

P = S \times V η h ν

∕

(2 Γ τ_{p})

, where η is the total quantum efficiency including the coupling efficiency from laser to fiber, and hν is the photon energy. In addition, the optical gain can be simplified as [13–15],

g = \frac{σ (N - N_{t r})}{1 + ε S}

where N_tr is the transparent carrier density, and ε is the gain suppression coefficient, which characterizes the spectral hole burning (SHB). Because the SHB generates a small roughly symmetrical dip of the gain spectrum around the laser frequency, the Kramers-Kronig relation ensures that the symmetrical feature of the imaginary index alone will not produce any change of the real index at the laser frequency [16]. Accordingly, the phase variation of Eq. (1c) is not affected by ε, and the frequency chirp above the laser threshold can be approximated as [16],

Δ ν = \frac{1}{2 π} \cdot \frac{d φ}{d t} ≅ \frac{α}{4 π} (\frac{1}{S} \frac{d S}{d t} + \frac{ε}{τ_{p}} S) = \frac{α}{4 π} (\frac{1}{P} \frac{d P}{d t} + κ P)

where the first term indicates the transient chirp, and the second term is the adiabatic chirp with a structure-dependent constant,

κ = 2 Γ ε / (V η h ν)

. In [17], an additional term is added to obtain better approximation of the frequency chirp. In general operating conditions with

τ_{p} κ P ≪ 1

, Eq. (3), however, is sufficient to accurately model the chirp.

To simplify the following analysis, each subcarrier of an OFDM signal is approximated as a monochromatic wave, since the subcarrier number is generally large and the bandwidth of subcarriers is relatively small. Consequently, the modulation current of a DML is simplified as $I = I_{B} + \sum_{m = 1}^{M} ℜ {i_{m} e^{j m ω t}}$ , where I_B is the bias current, M is the subcarrier number, $ω / (2 π)$ is the subcarrier spacing, $i_{m}$ is the complex current envelop of the m^th subcarrier, and $ℜ {\cdot}$ represents the real part. According to Eqs. (1) and (2), an SSM-based frequency- dependent transfer function, H(m), can be analytically derived such that the complex envelop of the output power, $p_{m}$ , equals to $i_{m} \times H (m)$ [13]. Then, the output power of a DML- based OFDM signal could be written as,

P = P_{B} + \sum_{m = 1}^{M} ℜ {p_{m} e^{j m ω t}} = P_{B} \cdot [1 + \sum_{m = - M}^{M} x_{m} e^{j m ω t}]

where P_B is the bias power,

x_{m} = p_{m} / (2 P_{B})

for positive integer m is the normalized power envelop,

x_{m} = x_{- m}^{*}

for negative integer m implies power is never complex, and

x_{m} = 0

for non-integer and zero m indicates the signal is discrete-tone and DC-free. Besides, the optical phase is needed to determine the envelop of the optical field, i.e.

E = \sqrt{P} e^{- j φ}

. Defining

X = \sum_{m = - M}^{M} x_{m} e^{j m ω t}

and integrating Eq. (3) inserted by Eq. (4), the optical phase becomes,

φ ≅ \frac{α}{2} [\ln (P_{B} (1 + X)) + κ P_{B} \int (1 + X) d t] = \frac{α}{2} \ln (1 + X) + \frac{Ω}{2 ω} \tilde{X} + \frac{α}{2} \ln P_{B} + \frac{Ω}{2} t

where

Ω = α κ P_{B}

and

\tilde{X} / ω = \int X d t

.

Ω / (4 π)

indicates the bias-dependent frequency shift and

\tilde{X} = \sum_{m = - M}^{M} {\tilde{x}}_{m} e^{j m ω t}

has the frequency components of

{\tilde{x}}_{m \neq 0} =

- j x_{m} / m

and

{\tilde{x}}_{0} = 0

. Notably, α and κ of commercial DMLs are about 2-5 and 10-15 GHz/mW, respectively, and therefore, Ω is about 20-75 GHz assuming P_B is 0 dBm [14]. Using Eqs. (4) and (5) and the 2nd-order Taylor expansion based on the small-signal assumption of

x_{m} < < 1

, the normalized field,

E^{'} = E / \sqrt{P_{B}}

, can be approximated as,

E^{'} ≅ 1 + \underset{S_{1}}{\underset{︸}{(\frac{1 - j α}{2} X - j \frac{δ}{2} \tilde{X})}} + \underset{S_{2}}{\underset{︸}{(- \frac{1 + α^{2}}{8} X^{2} - \frac{δ^{2}}{8} {\tilde{X}}^{2} - j \frac{δ (1 - j α)}{4} X \tilde{X})}}

where

δ = Ω / ω

, and the last two terms of Eq. (5) is neglected. The first bracket of Eq. (6), S₁, indicates the desired OFDM signal, but the second bracket, S₂, is the 2nd-order distortion in the optical field domain composed of the intermixing terms among subcarriers. After fiber transmission with the distance of L, the group-velocity dispersion parameter of β₂ and no fiber loss, the m^th frequency component of

S_{1} = \sum_{m = - M}^{M} s_{m} e^{j m ω t}

becomes,

s_{m} = \frac{1}{2} (\sqrt{1 + α^{2}} e^{- j θ_{α}} - \frac{δ}{m}) x_{m} e^{j m^{2} θ_{D}}

where

θ_{D} = β_{2} L ω^{2} / 2

and

θ_{α} = \tan^{- 1} α

. After square-law photo-detection and neglecting 3rd- and 4th-order terms, the detected signal can be written as,

{| E^{'} |}^{2} ≅ 1 + (S_{1} + S_{1}^{*}) + \underset{D_{R}}{\underset{︸}{S_{1} \times S_{1_{_{}}}^{*}}} + \underset{D_{T}}{\underset{︸}{(S_{2} + S_{2}^{*})}}

Because of

S_{1}^{*} = \sum_{m = - M}^{M} s_{m}^{*} e^{- j m ω t} = \sum_{m = - M}^{M} s_{- m}^{*} e^{j m ω t}

, the m^th subcarrier of the detected signal,

S_{1} + S_{1}^{*}

, is

s_{m} + s_{- m}^{*}

, and its power is

w_{m} = 〈 | s_{m} + s_{- m}^{*} |^{2} 〉

with the power fading of [11],

\frac{w_{m}}{〈 | x_{m} |^{2} 〉} = (1 + α^{2}) \cos^{2} (m^{2} θ_{D} - θ_{α}) + \frac{δ^{2}}{m^{2}} \sin^{2} (m^{2} θ_{D})

where

〈 | x_{m} |^{2} 〉

is the normalized received power without dispersion, and

〈 \cdot 〉

indicates the expectation value. If the adiabatic chirp is zero, the dispersion-induced power fading of Eq. (9) becomes

(1 + α^{2}) \cos^{2} (m^{2} θ_{D} - θ_{α})

, as the case of an EML-based system [7]. For a chirp-free signal, the power fading becomes

\cos^{2} (m^{2} θ_{D})

[18].

The theoretical dispersion-induced power fading described by Eq. (9) is plotted in Fig. 1 with different Ω to examine the influence of the adiabatic chirp. Notably, α is fixed to 3 in the following content, since the effect of the transient chirp has been studied in [8]. From Figs. 1(a) and 1(b), the second term of Eq. (9) induced by the adiabatic chirp will ease the first dip of power fading. The other dips at higher frequencies, however, are less affected by the adiabatic chirp, because the adiabatic term of Eq. (9) is inverse proportional to the square of frequency. As evidence of the theoretical results, the measured results, such as Fig. 6(b) in [6], show the much weaker 1st fading dip compared with the others at higher frequencies. In some cases, actually, the adiabatic chirp could not only result in less fading, but also provide power gain, such as the relative power at 3 GHz with Ω of 100 GHz in Fig. 1(b). Since dispersion could transfer phase modulation into intensity modulation to yield power gain, higher adiabatic chirp could provide more gain. However, this does not necessarily imply better transmission performance, since dispersion-induced distortion also depends on chirp [8].

Fig. 1 Relative power of subcarriers with the dispersion of (a) 320 ps/nm and (b) 1600 ps/nm.

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To study the effect of dispersion-induced distortion, the 2nd-order distortion of the detected OFDM signal is considered as SSII, since they come from the products among subcarriers. According to Eq. (8), the 2nd-order SSII consists of $D_{R} = S_{1} \cdot S_{1}^{*}$ and $D_{T} = S_{2} + S_{2}^{*}$ . The first one is intermixed at the receiver after transmission owing to square-law photo-detection, and the other is intermixed at the transmitter before transmission owing to square-root-law amplitude modulation (i.e. linear intensity modulation) and chirp-induced phase modulation. Since the bandwidths of the 2nd-order SSII are twice, the SSII can be represented as $D_{z} = \sum_{m = - 2 M}^{2 M} d_{z, m} e^{j m ω t}$ , where the subscript $z \in {R, T}$ and $d_{z, m} = d_{z, - m}^{*}$ . Besides, each term of the SSII comes from the convolution of subcarriers, and it follows that [Appendix],

d_{z, m > 0} = \frac{1}{2} ξ_{z} (\frac{m}{2}, m) x_{m / 2}^{2} + \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} ξ_{z} (k, m) x_{k} x_{k - m}^{*}

where

⌊ \cdot ⌋

denotes the integer part. The intermixing coefficients in Eq. (10) are,

ξ_{R} (k, m) = [\frac{1 + α^{2}}{2} + \frac{δ^{2} + j m α δ}{2 k (k - m)}] \cos ((2 k - m) m θ_{D}) - j \frac{(2 k - m) δ}{2 k (k - m)} \sin ((2 k - m) m θ_{D})

ξ_{T} (k, m) = - [\frac{1 + α^{2}}{2} + \frac{δ^{2} + j m α δ}{2 k (k - m)}] \cos (m^{2} θ_{D}) + j \frac{m δ}{2 k (k - m)} \sin (m^{2} θ_{D})

for

k \neq m

, and

ξ_{l} (k, m) = 0

for

k = m

. Then, the m^th frequency component of the total SSII,

D = D_{R} + D_{T} = \sum_{m = - 2 M}^{2 M} d_{m} e^{j m ω t}

, is,

d_{m > 0} = \frac{1}{2} [ξ_{R} (\frac{m}{2}, m) + ξ_{T} (\frac{m}{2}, m)] x_{m / 2}^{2} + \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} [ξ_{R} (k, m) + ξ_{T} (k, m)] x_{k} x_{m - k}

From Eqs. (11) and (12), without residual dispersion, the sum of ξ_R and ξ_T is zero, and the detected OFDM signal is SSII-free. Moreover, the subcarriers are assumed to have equal power of

w = 〈 | x_{m} |^{2} 〉

in the following discussion for simplicity, and therefore, the power of the total SSII,

μ_{m} = 〈 | d_{m} |^{2} 〉

, could be simplified as,

μ_{m > 0} ≅ \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} {[ξ_{R} (k, m) + ξ_{T} (k, m)]}^{2} w^{2}

where the first term of Eq. (12) is neglected due to large M, and the expectation values of the cross terms are zero. If the adiabatic chirp is zero, Eq. (13) could be further simplified as an analytical form [8]. Even if the subcarriers do not have equal powers, w ² in Eq. (13) can be replaced by

〈 | x_{k} |^{2} 〉 〈 | x_{m - k} |^{2} 〉

to calculate the power of the SSII. Moreover, the normalized power of the SSII,

μ_{m} / w

, is plotted in Fig. 2 , where the OFDM signal occupies 8-GHz bandwidth, and M and w are set as 400 and 2.5 × 10⁻⁵, respectively. Comparing Figs. 2(a) and 2(b), more dispersion roughly induces higher SSII power. Besides, although increasing the adiabatic chirp may not enlarge the SSII power at all the frequencies as shown in Fig. 2(a), the in-band (≤ 8 GHz) SSII power roughly increases with larger adiabatic chirp. To further investigate the effects of power fading and the SSII on the transmission performance, the signal-to-interference ratios (SIR) with different dispersion and adiabatic chirp are plotted in Fig. 3 , where the SIR is calculated by

w_{m} / μ_{m}

according to Eqs. (9) and (13). The parameters of the signals adopted in Figs. 3(a) and 3(b) are identical to those in Figs. 2(a) and 2(b), respectively. If the adiabatic chirp is zero, most subcarriers show better performance, but those at the frequencies around the first fading dip will be completely unusable. In short, from Figs. 1-3, the adiabatic chirp could eliminate the fading dip or even provide gain, but it would also generate much higher SSII to result in lower SIR, except for the subcarriers which suffer from strong fading with little adiabatic chirp.

Fig. 2 Relative power of the SSII with the dispersion of (a) 320 ps/nm and (b) 1600 ps/nm.

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Fig. 3 Theoretical SIR with the dispersion of (a) 320 ps/nm and (b) 1600 ps/nm.

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3. Numerical simulation

One of the main benefits of the SSM presented in Section 2 is to characterize the DML-based OFDM signals via the simple equations, such as the dispersion- and chirp-related power fading of Eq. (9) and the distortion represented as SSII of Eq. (13). Nonetheless, the SSM is derived through several assumptions, such as monochromatic subcarriers with relatively small power and the omission of higher-order terms, and therefore, it might not exactly meet practical cases. As a result, numerical LSS is carried out in this section to investigate the feasibility of the SSM, according to Eqs. (1) and (2). Table 1 summarizes the intrinsic parameters of the DML used in the LSS [12–15, 17], and the corresponding κ is 11.97 GHz/mW. Notably, even though the DML dynamics described by Eqs. (1) and (2) are the simplified mathematical forms, they have been broadly used to study various DML-based transmission systems and shown good agreement with experimental measurements [14, 15]. Moreover, instead of simulating the laser parasitic in detail, a low-pass filter with 7.5-GHz bandwidth is inserted in front of the laser [19]. As to the drive signal of the DML, it is composed of 400 subcarriers and 500 data blocks, and all the subcarriers are encoded by random binary data as 16-QAM format. With the subcarrier spacing of 19.53 MHz and the CP of 1/32, the assemble data rate is 30.3 Gbps. Because the purpose of the numerical LSS is to examine the SSM, instead of optimizing the system based on those specific parameters in Table 1, the drive current is intentionally adjusted to make all the subcarriers have the same optical power. Besides, to fix the peak-to-average power ratio (PAPR) of randomly generated drive currents, hard clipping is applied to enforce the maximum PAPR to 13 dB in all the following cases [19]. Furthermore, because the adiabatic chirp depends on the bias current, the effect of adiabatic chirp is investigated by adjusting the bias currents. The peak-to-peak drive currents, I_pp, are also adjusted to control the normalized received power, w, and the carrier-to-signal power ratio (CSPR).

Table 1. Laser parameters

View Table

Firstly, to investigate power fading, the SSM-based theoretical fading of Eq. (9) and the simulation results are plotted in Fig. 4(a) for comparison. The bias currents of 40, 30, 20 and 15 mA are adopted to result in different Ω of 129.4, 94.8, 60.3 and 43 GHz, respectively, and the peak-to-peak drive currents are set as 42, 30, 18.1 and 12.3 mA to generate identical w of 2.1 × 10⁻⁵. With the different residual dispersion of 320 and 1600 ps/nm, the SSM shows < 1-dB overestimation of the relative power implying that the SSM could predict power fading well, except the highly chirped case biased at 40 mA with 1600-ps/nm dispersion. The overestimation comes inherently from the Taylor expansion of Eq. (6). Secondly, without considering any noise, Figs. 4(b) and 4(c) exhibit the SIRs of the theoretical and simulation results based on the operation conditions discussed in Fig. 4(a). In these figures, the SIRs of the LSS at back-to-back (BtB) are also exhibited as a reference. The main reason of the finite SIRs at BtB is that the nonlinear DML dynamics will generate intrinsic interference around the laser resonance frequency to result in overshoots in the optical waveform. Consequently, the intrinsic interference at BtB will get worse, when the bias current is decreased to make the relaxation resonance frequency lower [13]. Thus, the theoretical SIR will underestimate the interference, especially when the interference is dominated by that caused by the nonlinear dynamics, i.e. the cases with lower bias current and/or lower residual dispersion, such as the cases with the bias current of 20 and 15 mA in Fig. 4(b). If the dispersion- and chirp-related SSII dominates the transmission performance, such as the cases with the bias current of 40 and 30 mA in Fig. 4(c), the SSM could predict the SIRs well. Moreover, Figs. 4(b) and 4(c) only account for the statistical consistency between the SSII powers of the SSM and the LSS. To further demonstrate the accurateness of the SSM, Fig. 4(d) plots the correlation coefficients between the simulation distortion and the theoretical SSII. As expected, the higher correlation is observed with larger adiabatic chirp and more dispersion. This fact implies not only the dispersion-induced distortion dominates in such conditions, but also the SSM can deterministically calculate the main distortion.

Fig. 4 (a) The relative subcarrier powers with 320- and 1600-ps/nm dispersion, (b) the SIRs with 320-ps/nm dispersion, and (c) the SIRs with 1600-ps/nm dispersion of the SSM and the LSS; (d) the correlation coefficients between the SSII of the SSM and the distortion of the LSS.

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4. Iterative equalization

Since the dispersion-induced distortion could be well estimated via the 2nd-order SSII of the SSM, it is possible to mitigate the distortion in a DML-based IMDD system, which, unlike a coherent system, cannot provide the full knowledge of optical field. The knowledge of the transmitted data, however, is required to calculate the SSII. To rebuild the SSII without the knowledge of the transmitted data, this work proposes an iterative equalization technique at the receiver to deterministically estimate and mitigate the SSII in a practical system. The concept of the iterative equalization at the receiver is shown in Fig. 5 , of which the similar concept has been used in an MZM-based amplitude-modulated OFDM system [4], and the required knowledge of the transmitted data is replaced by the demodulated data after linear frequency-domain data equalization and nonlinear hard decision. Although the decision errors will disturb the accuracy of the SSII estimation, the preliminary SSII estimation is still expected to lower the distortion and the decision errors. Then, the iterative process is adopted to further reduce the decision errors and to improve the estimation accuracy of the SSII. Moreover, because the SSII calculation of Eq. (12) is based on the power envelop at BtB, i.e. x_m in Eq. (4), the most challenge part of the proposed iterative equalization is the necessary knowledge of the transfer function from the drive current to the power envelop, i.e. H(m). This transfer function must be measured at BtB and sent to the receiver. Once the transfer function is known, the other important parameters including α and Ω can be estimated by fitting the relative power of each subcarrier according to Eq. (9). As to L, it can be measured in advance, for instance, by the optical time-domain reflectometer (OTDR) [20]. As a consequence, the demodulated data could be obtained as $r_{m}^{(l)}$ after hard decision, where the superscript l indicates the result of the l^th iterative process. Combined with the known transfer function and the estimated parameters, the SSII can be calculated by Eqs. (6)–(8), in which x_m is replaced by $r_{m}^{(l)} \times H (m)$ , i.e. the estimated power envelop without normalization. Notably, since $r_{m}^{(l)} \times H (m)$ is estimated in the frequency domain, the calculation of $S_{1} \times S_{1}^{*}$ , $X^{2}$ , ${\tilde{X}}^{2}$ and $X \tilde{X}$ , in Eqs. (6) and (8) involves the discrete convolution of the frequency components of the corresponding 1st-order terms. Thus, the discrete convolution will dominate the computational complexity of the SSII calculation, and it is computed through multiplying their inverse Fourier transform in the time domain. Employing fast-Fourier transform (FFT), each transform and inverse transform requires $O (\tilde{M} \log_{2} \tilde{M})$ operations, and the multiplication requires $O (\tilde{M})$ , where $\log_{2} \tilde{M}$ is the minimum integer such that $\tilde{M} \geq 2 M$ [21]. Thus, the overall extra computational complexity of $O (\tilde{M} \log_{2} \tilde{M})$ is required for each iteration. Furthermore, even without decision error, $r_{m}^{(l)} \times H (m)$ and x_m are different in their magnitude. Accordingly, the SSII weighting shown in Fig. 5 is required to provide proper weighting factors to the calculated SSII to reach the best SSII mitigation. Similar to the data equalization, the SSII weighting factors are found by training symbols at the receiver, and in this work, these factors equal to the regression coefficients of the linear least squares fitting between the calculated SSII and the received distortion of training symbols. Notably, although the weighting factors are irrelevant to frequency in principle, the factor for each frequency component is actually found individually to make the SSII mitigation more tolerant to the possible errors of the simplified model and the estimated parameters.

Fig. 5 The concept of the proposed iterative equalization

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To evaluate the transmission performance of the DML-based OFDM signal over SSMF with and without the iterative equalization, the LSS involving additional noises is carried out with the parameters used in Section 3. In short-distance DML-based transmission systems (< ~100 km), because only two or less in-line optical amplifiers and no pre-amplifier are required, optical signal-to-noise ratio (OSNR) is generally much higher than 40 dB [22]. The main noise source in such a system is electrical noises from a photo-detector, including thermal and shot noises. As a consequence, to focus on the dispersion- and chirp-related distortion and the electrical noises, the SSMF is modeled as a linear dispersive channel with β₂ of –21.66 ps²/km and no loss; the OSNR is fixed at 40 dB without overestimation, and a PIN photo-detector is used with shot noise, thermal noise of 13.6×10^–12_{$A/ \sqrt{Hz}$}, and the responsivity of 0.7 A/W, taking into consideration. These parameters are adopted such that the received power requirement of a 10-Gbps NRZ-OOK signal at BtB is about –19-dBm to reach the bit-error rate (BER) of 10⁻⁹. Furthermore, the additional phase noise caused by 1-MHz laser linewidth is included in the LSS, and it is simply modeled by the Wiener process [23].

Figure 6 shows the demodulated signal-to-interference-plus-noise ratio (SINR) after transmission with and without the proposed iterative equalization. The bias current and peak-to-peak drive current are both set to 30 mA, and the received power is 0 dBm. To investigate the effect of decision errors on the iterative equalization, the ideal equalization with the full knowledge of the transmitted data is also applied for comparison. If the interference plus noise is Gaussian distributed, the required SINR to reach the FEC limit (BER of 10⁻³) is about 16.5 dB [24], which is plotted by the black dashed lines in Fig. 6 as a reference. After 20-km SSMF transmission, more than one third subcarriers cannot reach the FEC limit without the iterative equalization, as shown in Fig. 6(a). In this case, the proposed equalization can apparently improve the SINR, and the 1st equalization can perform as well as the ideal equalization. Since decision errors indeed happen in the simulation, the overlap of two SINR curves indicates the decision errors affect the SSII calculation little. A few subcarriers around 5.5 GHz, however, still cannot reach the FEC limit after the SSII mitigation, because they suffer from fading to become less tolerant to the noises, as shown in Fig. 4(a). When the transmission distance is increased to 100 km, the received subcarrier power is increased due to the chirp-induced gain shown in Fig. 4(a), but the even larger increment of the SSII power leads to the much worse demodulated SINR, as shown in Fig. 6(b). Nonetheless, compared with Fig. 6(a), the benefit of the chirp-induced gain would be noted after the iterative equalization, and all the subcarriers can achieve the FEC limit with the ideal equalization. In addition, owing to the increment of decision errors, the practical equalization cannot perform as well as the ideal one until the 2nd iteration.

Fig. 6 The SINRs with and without the iterative equalization after (a) 20-km SSMF and (b) 100-km SSMF. (0-dBm received power, 30-mA I_B and 30-mA I_pp).

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Moreover, based on the operation condition of Fig. 6, Fig. 7 exhibits the BER performance as a function of received power, and the BERs are not only calculated by error counting but also estimated by Gaussian-approximated SINR [24]. That is, if the SINR of the m^th subcarrier is given as ρ_m, its BER is approximated as $(3 / 8) \times erfc (\sqrt{ρ_{m} / 10})$ , where $erfc (\cdot)$ denotes the complementary error function. The separate markers in Fig. 7 denote the BERs calculated by error counting, and the solid curves represent the Gaussian-approximated BERs. The good agreement between the BERs obtained from two methods implies that the Gaussian approximation works well, at least around the BERs of interest. Hence, it can be claimed that the FEC limit shown in Fig. 6 is a good reference. Corresponding to Fig. 6(a), the SINRs of more than one third subcarriers are below the FEC limit to result in that the average BER cannot achieve the BER of 10⁻³ in Fig. 7(a), but it can be achieved after applying the proposed equalization. Similarly, the BER curves of the signals with the 1st and ideal equalization are almost the same. Compared with the case at BtB, the power penalty at the FEC limit after equalization is about 3 dB mainly contributed by power fading. After 100-km transmission, the demodulated signal suffers from much more serious SSII to show the BER floor of ~0.03 in Fig. 7(b). Furthermore, as the received power is 0 dBm, applying the iterative equalization cannot make the SINR and the BER better than those at BtB owing to the presence of residual distortion, as shown in Figs. 6(b) and 7(b), respectively. Nevertheless, if electrical noises dominate the BER performance at lower received power, the signal with the iterative equalization could outperform the signal at BtB due to the chirp-induced gain. Hence, compared with the case at BtB, the signal after 100-km transmission can behave about 0.9-dB sensitivity improvement, after applying the iterative equalization.

Fig. 7 The BERs with and without the iterative equalization after (a) 20-km SSMF and (b) 100-km SSMF. (30-mA I_B and 30-mA I_pp)

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Figure 8 exhibits the sensitivities to reach the FEC limit as functions of transmission distance. The sensitivities based on the bias currents of 15, 20, 30 and 40 mA are plotted in Figs. 8(a), 8(b), 8(c) and 8(d), respectively, and their peak-to-peak drive currents are set identical to those adopted in Fig. 4. Since the SIRs of these cases at BtB shown in Fig. 4(b) or 4(c) are mostly much higher than the FEC limit of 16.5 dB, the sensitivities are mainly determined by electrical noises. Accordingly, their identical CSPRs make the sensitivities at BtB about the same. Furthermore, after short transmission, the SIRs are still higher than the FEC limit for most subcarriers, such as the 20-km cases shown in Fig. 4(b), but the signals biased on 15, 20 and 30 mA cannot achieve the BER of 10⁻³, as shown in Figs. 8(a)-8(c). The reason is that power fading shown in Fig. 4(a) will decrease the received signal-to-noise ratios (SNRs). As to the case with 40-mA bias current, not only the subcarriers suffer from the power fading of less than 1 dB, but also about one fourth subcarriers are provided the chirp-induced gain, so that the FEC limit can be achieved after 20-km transmission. Similarly, the lower adiabatic chirp caused by the lower bias current would generate worse fading, and therefore, decrease the maximum transmission distances. The maximum transmission distances without the proposed equalization in Figs. 8(a)-8(d) are about 7, 8, 10 and 23 km, respectively. On the other hand, applying the proposed iterative equalization to mitigate the SSII could improve the transmission performance, but the improvement depends on the bias current and the distance. Within a short distance of ~10 km, the main issue is bad SNRs caused by power fading, and the SSII mitigation does not help much. However, larger adiabatic chirp and even more dispersion might ease power fading, as shown in Fig. 4(a). Consequently, it is possible to improve the performance significantly after longer transmission and SSII mitigation. Unfortunately, SSII mitigation only slightly improves the sensitivity and transmission distance in Fig. 8(a), because not only power fading but also the nonlinear DML response dominates. In Fig. 8(b), similar situation can be observed within 10 km, but it can achieve the FEC limit after the transmission distance between 80 and 115 km owing to more chirp-induced gain and less nonlinear DML response. In Figs. 8(c) and 8(d), because the nonlinear DML response affects the signal little, the proposed iterative equalization could extend the transmission distance beyond 90 km. Meanwhile, since a large portion of distortion can be modeled as SSII and be eliminated by the iterative equalization well, the performance is mainly determined by the SNRs, or equivalently, by power fading. Hence, negative sensitivity penalties are observed after transmission, and the case with the 40-mA bias current can outperform the case with the 30-mA bias current after the transmission of less than 80 km. Nonetheless, because the distortion cannot be completely removed by the 2nd-order SSII mitigation, residual distortion would eventually limit the maximum transmission distance. Hence, the maximum transmission distances in Figs. 8(b)-8(d) decrease with the larger adiabatic chirp, or equivalently, with more deviation from the SSM. On the other hand, corresponding to Fig. 8(c), Figs. 8(e) and 8(f) plot the sensitivities of the signals with the same bias current of 30 mA, but the different peak-to-peak drive currents of 20 and 45 mA, respectively. This indicates the OFDM signals of Figs. 8(e), 8(c) and 8(f) have the same Ω of 94.8 GHz, but they have different w of 9.5 × 10⁻⁶, 2.1 × 10⁻⁵ and 4.8 × 10⁻⁵, respectively. The same Ω represents the OFDM signals would suffer from similar power fading, and the larger w implies larger CSPR and better SNR at the same received power. Accordingly, when the electrical noises dominate the SINR, whether the distortion is inherently small or mitigated mostly by the proposed equalization, the signal with larger w could perform better. For instance, applying the proposed equalization, the OFDM signal of Fig. 8(f) can outperform those of Figs. 8(e) and 8(c) after the transmission of < 50 km. However, the larger w also implies larger distortion, including 2nd-order and higher-order SSIIs, such that there would be more residual distortion after the iterative equalization. As a result, the maximum transmission distance decreases with larger w, as shown in Figs. 8(e), 8(c) and 8(f). In addition, for the many cases of Fig. 8, the 1st equalization can generate apparent improvement, even compared with the ideal equalization, and therefore, the iterative process might be neglected to lower computational complexity. Such good performance of the 1st equalization is because the SSII is contributed by a lot of subcarriers, and a few decision errors may not affect the calculated SSII much around the BER of interest. Even if the iterative process is applied, the 2nd equalization could perform as well as the ideal one, and triple or more iteration might not provide further improvement for the most cases.

Fig. 8 The sensitivities as functions of the transmission distance with the drive currents of (a) I_B = 15 mA, I_pp = 12.3 mA, (b) I_B = 20 mA, I_pp = 18.1 mA, (c) I_B = 30 mA, I_pp = 30 mA, (d) I_B = 40 mA, I_pp = 42 mA, (e) I_B = 30 mA, I_pp = 20 mA, and (f) I_B = 30 mA, I_pp = 45 mA.

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Because the chirp parameters of α and Ω are required to calculate the SSII, they need to be estimated at the receiver. Although the estimation errors of these parameters in Figs. 6-8 are less than ± 5%, it seems necessary to study how the estimation errors affect the equalization performance. Actually, the proposed equalization is quite insensitive to the estimation errors. To show this, the sensitivity penalties caused by the estimation errors of α and Ω are exhibited in Fig. 9 , where the twice iterative equalization is applied after 100-km SSMF and the other operation conditions are identical to those in Fig. 8(c). Because the residual dispersion, i.e. L and/or β₂, may also be measured inaccurately, the effect of inaccurate estimation of residual dispersion is also investigated by providing the dispersion errors of –20%, –10%, 0, + 10% and + 20% to Figs. 9(a)-9(e), respectively. Moreover, the green and red boxes in Fig. 9 indicate the boundaries of ± 10% and ± 20% estimation errors of the chirp parameters, respectively. According to Figs. 9(b)-9(d), if the estimation errors of these three parameters are all within ± 10%, the corresponding penalty would be approximately less than 0.1 dB. Even if the estimation errors are up to ± 20%, the penalty will never exceed 0.5 dB, actually, mostly below 0.3 dB, as shown in Fig. 9. The tolerance to the estimation errors is mainly because the SSII weighting in Fig. 5 would statistically adjust the calculated SSII with the deviated parameters to meet the real distortion.

Fig. 9 The sensitivity penalty as a function of the estimation errors of the chirp parameters with the fix dispersion errors of (a) –20%, (b) –10%, (c) 0, (d) + 10% and (e) + 20%.

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5. Conclusions

In this work, a theoretical model is provided to analyze the effects of both transient and adiabatic chirps on the DML-based IMDD OFDM transmission system for the first time. Using small-signal approximation, the SSM can give a comprehensive insight of how the transient and adiabatic chirps interact with dispersion to result in the power fading of each subcarrier and the distortion caused by intermixing among subcarriers, or the so-called SSII. Notably, the presence of the adiabatic chirp would ease power fading but increase the SSII, and therefore, minor adiabatic chirp would make the subcarriers have higher SIRs, except those suffering from strong fading. Furthermore, based on the laser rate equations, the LSS is also carried out to examine the feasibility of the SSM. In the LSS, the effect of the adiabatic chirp is investigated by adjusting bias currents. If the laser is operated at higher bias current such that the laser resonance relaxation do not affect the OFDM signal much, the SSM can estimate the transmission performance of the LSS well. Furthermore, since the SSM can deterministically estimate the transmission distortion by calculating the SSII, this work proposes an iterative equalization to mitigate the distortion without the knowledge of the transmitted data. In the proposed equalization scheme, the demodulated data at the receiver are used to calculate the SSII and to mitigate the dispersion-induced distortion, and the iterative process would further improve the accuracy of the mitigation. Actually, for most of the cases discussed in this work, twice iterative equalization could perform as well as the ideal equalization with the complete knowledge of the transmitted data. Moreover, since the distortion is mostly eliminated by the proposed equalization, the maximum transmission distance of the DML-based OFDM signals could be significantly improved. When the DML is biased at 30 mA with 30-mA peak-to-peak drive current, for example, the maximum SSMF transmission distance of the 30-Gbps OFDM signal can be extended from 10 km to more than 100 km. In addition, since the distortion could be mitigated by the proposed equalization, the residual effect of the adiabatic chirp is to ease power fading or even to provide gain. Thus, the iteratively equalized OFDM signals could exhibit negative power penalties at some distances.

Appendix

Let $U = \sum_{m = - M}^{M} u_{m} e^{j m ω t}$ and $V = \sum_{m = - M}^{M} v_{m} e^{j m ω t}$ be DC-free complex signals, i.e. $u_{0} = 0$ and $v_{0} = 0$ . Then, the product of them will have twice bandwidth, $Y = U \times V =$ _{$\sum_{m = - 2 M}^{2 M} y_{m} e^{j m ω t}$}, and its frequency component can be obtained by discrete convolution,

y_{m > 0} = \sum_{k = m - M}^{M} u_{k} v_{m - k} = u_{m / 2} v_{m / 2} + \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} u_{k} v_{m - k} + u_{m - k} v_{k}

To derive

D_{R} = S_{1} \times S_{1}^{*}

in Eq. (8), let

u_{m} = s_{m}

and

v_{m} = s_{- m}^{*}

in Eq. (14), and it follows,

d_{R, m > 0} = s_{m / 2} s_{- m / 2}^{*} + \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} s_{k} s_{k - m}^{*} + s_{m - k} s_{- k}^{*}

Inserting Eq. (7) into Eq. (15) and using the relation of

x_{- k}^{*} = x_{k}

and

x_{k - m}^{*} = x_{m - k}

owing to the reality of X, Eqs. (10) and (11a) can be derived. On the other hand, S₂ without dispersion is the linear combination of

X^{2}

,

{\tilde{X}}^{2}

and

X \tilde{X}

as shown in Eq. (6), and they can be represented as

\sum_{m = - 2 M}^{2 M} {\bar{\bar{x}}}_{m} e^{j m ω t}

,

\sum_{m = - 2 M}^{2 M} {\tilde{\tilde{x}}}_{m} e^{j m ω t}

and

\sum_{m = - 2 M}^{2 M} {\bar{\tilde{x}}}_{m} e^{j m ω t}

, respectively. Using Eq. (14) and

{\tilde{x}}_{m \neq 0} =

- j x_{m} / m

, these frequency components are,

{\bar{\bar{x}}}_{m > 0} = x_{m / 2}^{2} + \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} 2 x_{k} x_{m - k}

{\tilde{\tilde{x}}}_{m > 0} = - \frac{4}{m^{2}} x_{m / 2}^{2} - \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} \frac{2}{k (m - k)} x_{k} x_{m - k}

{\bar{\tilde{x}}}_{m > 0} = - j \frac{2}{m} x_{m / 2}^{2} - j \sum_{k = ⌊ \frac{m}{2} + 1 ⌋}^{M} \frac{m}{k (m - k)} x_{k} x_{m - k}

Consequently, if S₂ with dispersion is represented as

\sum_{m = - 2 M}^{2 M} {\hat{s}}_{m} e^{j m ω t}

, then the positive frequency component is,

{\hat{s}}_{m > 0} = [- \frac{1 + α^{2}}{8} {\bar{\bar{x}}}_{m} - \frac{δ^{2}}{8} {\tilde{\tilde{x}}}_{m} - j \frac{δ (1 - j α)}{4} {\bar{\tilde{x}}}_{m}] e^{j m^{2} θ_{D}}

Because

X^{2}

,

{\tilde{X}}^{2}

and

X \tilde{X}

are all real, their frequency components must be Hermitian symmetric, and

{\hat{s}}_{m < 0} = [- \frac{1 + α^{2}}{8} {\bar{\bar{x}}}_{- m}^{*} - \frac{δ^{2}}{8} {\tilde{\tilde{x}}}_{- m}^{*} - j \frac{δ (1 - j α)}{4} {\bar{\tilde{x}}}_{- m}^{*}] e^{j m^{2} θ_{D}}

Finally, due to

D_{T} = S_{2} + S_{2}^{*}

and

d_{T, m > 0} = {\hat{s}}_{m} + {\hat{s}}_{- m}^{*}

, Eqs. (10) and (11b) are obtained via Eqs. (16)-(18).

Acknowledgment

The author would like to thank the National Science Council, Republic of China, Taiwan for financially supporting this research under Contract No. NSC 100-2221-E-110-089-MY3 and NSC 101-2628-E-110-006-MY3.

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Symbol	Value	Units	Symbol	Value	Units	Symbol	Value	Units
N_tr	1.23 × 10²⁴	1/m³	V	3 × 10⁻¹⁷	m³	Γ	0.2
v_g	8.33 × 10⁷	m/s	σ	9.9 × 10⁻²⁰	m²	α	3
A	1 × 10⁸	1/s	τ_p	1.69 × 10⁻¹²	s	ε	2.77 × 10⁻²³	m³
B	1 × 10⁻¹⁶	m³/s	β_sp	1 × 10⁻⁵		ν	1.94 × 10¹⁴	1/s
C	3 × 10⁻⁴¹	m⁶/s	η	0.24

Analysis and iterative equalization of transient and adiabatic chirp effects in DML-based OFDM transmission systems

Abstract

1. Introduction

2. Analysis of the chirp effects

3. Numerical simulation

4. Iterative equalization

5. Conclusions

Appendix

Acknowledgment

References and links

Cited By

Figures (9)

Tables (1)

Equations (23)

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