PREPRINT (ACCEPTED VERSION) FOR PUBLICATION IN JOURNAL OF LIGHTWAVE TECHNOLOGY (DOI: 10.1109/JLT.2022.3166754)

1

Fast-Tracking Optical Coherent Receiver Tolerating
Transmitter Component Distortion
Etsushi Yamazaki, Member, IEEE and Shinya Sugiura, Senior Member, IEEE

Abstract—In this paper, we propose a receiver architecture
capable of compensating for a time-varying signal distortion induced by the transmitter analog components. In the proposed architecture, the novel transmitter distortion compensating (TDC)
block is implemented outside of the feedback control loop of theadaptive equalizer to suppress the feedback delay by optimizing
the parameter used in the least mean square (LMS) algorithm.
The proposed TDC block is designed for compensating for the
DC offset, the IQ amplitude imbalance, the IQ orthogonality
error, and the IQ timing skew. Our simulation results show thatthe proposed scheme significantly reduces the required signal-tonoise ratio penalty from the theoretical limit, which is imposeddue to the transmitter components distortion. Furthermore,
our theoretical analysis confirms that the delay induced in the
feedback loop of the adaptive equalizer determines the upperbound of the LMS step size under the stable condition, hence
allowing us to maximize the tracking speed of our receiver.
Index Terms—Adaptive equalization, analog distortion, fasttracking, least mean square, optical fiber communication, theoretical bound.

I. I NTRODUCTION
N optical fiber network acts as a lifeline platform, whichis essential for economic activities. The capacity demandis further driven due to the expansions of inter/intra-data center
communications and the evolution of wireless back-haul networks with a variety of applications, such as on-demand video
streaming and cloud computing. Recent experiment reported
that high capacity per wavelength, exceeding 1 Tbps, wasachieved [1]–[7], where both the symbol rate and modulation
size significantly increased. For example, the symbol rateincreased from 32 Gbaud to more than 96 Gbaud [8]–[10].
This was achieved as the explicit benefits of the improved
analog-to-digital (AD)/digital-to-analog (DA) converters [11]–[13], of with frequency-doubling techniques [2], [6], [14],[15], and of the improved analog components, such as optical modulators [16], [17], [10], drivers, photodiodes, andtransimpedance amplifier (TIA) [18]. The modulation size
was increased to more than 64-point quadrature amplitudeAc 2022
Preprint (Accepted Version). DOI: 10.1109/JLT.2022.3166754. ⃝IEEE. Personal use of this material is permitted. Permission from IEEE must
be obtained for all other uses, in any current or future media, including
reprinting/republishing this material for advertising or promotional purposes,
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E. Yamazaki is with the Network Innovation Laboratories, NTT Corporation, Hikarino-oka, Yokosuka, Kanagawa, 239-0847, Japan, and is also with
the Institute of Industrial Science, The University of Tokyo, Meguro, 1538505, Japan (e-mail: etsushi.yamazaki.wk@hco.ntt.co.jp).
S. Sugiura is with the Institute of Industrial Science, The University ofTokyo, Meguro, 153-8505, Japan (e-mail: sugiura@iis.u-tokyo.ac.jp).

modulation (QAM) constellations with the aid of capacityapproaching probabilistic constellation shaping [19]–[21].
In high-rate optical fiber communication, the signal distortion is induced by analog components at the transmitter and thereceiver. To be more specific, upon increasing the modulation
size, the detrimental effects of analog component distortions
increase since the higher symbol rate causes more severe analog distortions. Several digital-domain equalization techniques
at the receiver have been developed to compensate for the
distortion imposed by the receiver-side components [22]–[24].
The transmitter-side analog components, such as the modulator, the driver, and electrical wiring on the printed circuitboard, also distort signal because of their limited bandwidth,
I/Q crosstalk, IQ skew, DC offsets, and nonlinearity. The
sophisticated digital pre-distortion methods are proposed andtransmitter-side distortions, including nonlinearity can be compensated [25]–[28]. Typically, the distortion on the transmitterside is estimated by the calibration process, and the optimum
coefficients are set in a digital pre-distortion block on thetransmitter side and post-compensation block on the receiverside. However, this calibration process generally needs special
equipment or back-to-back configuration, which is challenging
to be performed during the operation condition over the fiber
channel. Thus, time-varying distortions caused by a variationin physical conditions, such as operating temperature or component aging, may not be tracked without relying on any
additional feedback channel. Also, the digital pre-distortion
method cannot compensate for DC offset because of the AC
coupling of analog devices at the transmitter. Thus, the receiver
has to tolerate residual transmitter-side distortion.
To this end, several receiver architectures were proposed
to compensate for the time-varying transmitter-side distortion
with the aid of receiver-side processing [29]–[31]. The compensating for the transmitter-side distortion at the receiver has
advantages and disadvantages. More specifically, the digitalpre-distortion tends to increase the peak-to-average power
ratio in the DAC and to increase the quantization errors of
the transmitter [27]. Thus, the partial compensation at thereceiver benefits from suppressing the quantization noise of
the transmitting. However, the receiver-side compensation for
the transmitter nonlinearity and bandwidth limitation increases
the associated noise components [32]. The paper shows thatthe receiver-side nonlinearity equalization using the Volterra
filter remains a considerable amount of the SNR penalty. To
reduce the effects, a more sophisticated technique, such as
equalization in the likelihood ratio domain, is required. Thus,
this paper assumes that most nonlinearity is compensated forat the transmitter. This allows us only to consider a small

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amount of residual nonlinearity of a received signal.
Furthermore, in optical fiber transmission, a receiver has to
track the state of polarization (SOP) and the polarization mode
dispersion (PMD) even in a rapidly time-varying channel [33]–[35]. In the extreme case, it was reported that significantly
fast SOP variation is caused by lightning strikes [36]–[38].
Typically, the receiver has to decompose the multiplexed polarization signals while eliminating intersymbol interference (ISI)
caused by PMD and residual chromatic dispersion (CD), in an
adaptive manner. The adaptive algorithms, such as the leastmean squares (LMS) algorithm and constant modulus algorithm (CMA), are used for tracking the channel coefficientsand optimizing the parameters of an equalizer.
An upper limit of the step size of the LMS algorithm or
CMA under the operation of stable convergence depends onthe delay in the feedback control loop [39], [40], while a large
step size allows us to increase the tracking speed. This implies
that the delay in the LMS feedback control loop may be thelimiting factor of an adaptive equalizer. Thus, to compensate
for the signal distortion induced by the transmitter-side analog
components at the receiver, the delay in the LMS feedback
control loop has to be maintained sufficiently low.
Against the above-mentioned background, the novel contributions of this paper are as follows.
• We propose a novel fast-tracking adaptive optical receivercapable of compensating for a time-varying signal distortion induced by the transmitter analog components inthe operation condition for transmission over the fiber
channel. More specifically, we introduce a transmitter
distortion compensating (TDC) block, which is capableof suppressing the feedback delay by optimizing theparameters used in the LMS-based adaptive equalizer.
The proposed TDC block is capable of compensating
for the DC offset, the IQ amplitude imbalance, theIQ orthogonality error, and the IQ timing skew. Morespecifically, the replica signal used in the LMS-based
adaptive equalizer is modified, depending on the TDC
coefficients, which allows our TDC block to be outsideof the adaptive equalizer feedback loop, and hence iscapable of suppressing the feedback delay.
• Moreover, we propose a theoretical analysis framework,
which is used for optimizing the LMS step size inthe presence of the feedback loop delay of an adaptive
equalizer. Our simulation results show that our proposed
scheme achieves a high bit-error-ratio (BER) performance close the theoretical bound while eliminating the
detrimental effects of signal distortion. Furthermore, it
is found that the proposed scheme’s penalty steeply
increases upon increasing the LMS step size beyond
the derived upper limit, which verifies our theoreticalframework.
The remainder of this paper is organized as follows. In
Section II, the system model of our optical-fiber transceiver ispresented. In Section III, the fast-tracking adaptive equalizer
architecture compensating for the transmitter distortion is presented. In Section IV, the theoretical upper limit of the LMS
step size of the adaptive equalizer is derived. In Section V,
we provide performance results on penalty evaluation of the2

tracking speed of the proposed adaptive equalizer. Finally,
Section VI concludes this paper.
II. S YSTEM M ODEL
Fig. 1 shows the system model for the optical-fibertransceiver considered in this paper. We assume the two
orthogonal polarization dimensions of the single-mode opticalfiber, each employing 2M -sized QAM. In the IQ mapping
block of each polarization, M bits are modulated onto 2M QAM symbols to generate u(n) or v(n), where n is the symbol
index. Then, u(n) and v(n) are twice up-sampled with theaid of a root raised cosine (RRC) shaping filter to generatexT (n) and yT (n), respectively. Finally, the up-sampled signals
are fed into the nested Mach-Zehnder (MZ) modulator, which
consists of two inner Mach-Zehnder interferometers (MZIs)
and a single outer MZI. More specifically, modulated baseband signal of the MZI xM is modeled by
(
)
√
π
π
aO
xM = j √aI ej 2 Re{xT } − 1−a2I e−j 2 Re{xT }
2
√
)
(
√
1−a2O
jπ
Im{xT }−j π
Im{xT }22
2
− 1−aI e· ejθe ,
−
aI e
2
(1)
where the variables aI and aO are associated with the extinction ratios of the inner and outer MZIs, respectively. The
extinction ratios are derived as the ratio of the maximum
output power to the minimum power as [41]√(ai + 1 − a2i )2
√
REi =
(2)
(i ∈ {O, I}),(ai − 1 − a2i )2Then, (2) is rewritten as√
1 REi + 1√√
ai =(i ∈ {O, I}),(3)
2 REi + 1
Furthermore, θe denotes the 90-degree phase errors of theouter MZI, and Re{xT } and Im{xT } correspond to I- and Qcomponents of the signal xT (n) output from the RRC shaping
block.
A low extinction ratio of the inner MZI REI causes the DC
offsets in the Re{xT } and Im{xT }. Also, a low extinction ratio of the outer MZI REO induces imbalance between Re{xT }and Im{xT }, which is also caused by the gain differencesof drivers amplifiers or passive path loss differences. The
90-degree phase errors originating from the control error of
the outer MZI DC biases cause signal cross-talk between
Re{xT } and Im{xT }. Additionally, there typically exists theresidual or dynamic skew between Re{xT } and Im{xT },even after the transmitter-side digital pre-equalization.These
distortion factors cannot be perfectly compensated for only by
the transmitter-side optimization, and combating such limitations are the targets of the proposed scheme. In this paper,
we exclude compensation of nonlinear distortions, which istypically caused by the driver and the inner MZI, while the
MZI nonlinearity distortion is included in our simulation.
As mentioned above, we assume that most nonlinear distortion is compensated for with the digital pre-distortion of
the transmitter. Hence, the nonlinear distortion compensation

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Fig. 1. System model of our optical-fiber transceiver.

is not our target at the receiver. Hence, only the nonlinearity
induced by the inner MZI modulator, rather than the driver,is considered. To be specific, the RMS of the signal input tothe inner MZI was set at 0.20 Vπ. The SNR after the inner
MZI was 34.5 dB, which is determined only by nonlinearity
distortion. These are the reasonable conditions for the scenario
where digital pre-distortion is used, and only the residual
nonlinearity is present.
Transmit signals in each polarization are contaminatedby laser phase noises (PNs), and the polarization scrambler
rotates the polarization state. Then, the signal experiences the
additive white Gaussian noises (AWGNs). Then, the AWGNcontaminated signals are fed into the 90-degree hybrid blockto decompose I- and Q-phase orthogonal polarization signals.
Note that the I- and Q-phase orthogonal polarization symbols
at the receiver x(n) and y(n) are different from the Iand Q-lanes of each polarization at the transmitter xT (n)and yT (n) since the PNs rotate the carrier phases and thepolarization scrambler mixes the two polarization symbols.
The two complex-valued signals output from the 90-degreehybrid block are fed into the 2 × 2 complex-valued adaptive
equalizer to de-multiplex two polarization symbols. The tap
coefficients of the equalizer are calculated for minimizing thesquare error loss of the signal output from pilot-based feedforward (PFF) carrier phase recovery (CPR), as defined by(11) and (12) in the next section, and the effects of the delay
are included in the feedback control loop. The proposed TDC
block is placed after the PFF-CPR block.
III. P ROPOSED C OMPENSATION FOR T RANSMITTER -S IDEC OMPONENT D ISTORTION
In this section, we propose a novel receiver architecturethat is capable of tracking the rapidly time-varying channel
and compensating for the signal distortion induced by thetransmitter-side analog components. Our TDC block of Fig. 1
is designed for compensating for the DC offset, the amplitude
imbalance, the 90-degree phase error, and the skew between
I- and Q-lanes, as mentioned in Section II. The effects ofnonlinear distortions are not compensated for in our scheme
since it typically involves the noise enhancement issue. For
detail, refer to [32].
Fig. 2 shows the proposed receiver architecture designed forsingle-mode optical fiber transmission.
A. FIR Block
In the 2×2 complex-valued finite impulse response (FIR)
block of Fig. 2, the polarization cross-talk is canceled out,
and the PMD distortion and ISI are compensated for torecover polarization-demultiplexed signals. Let us introduce
two polarization-diversity input signals ofx(n) = [x(n − L), ..., x(n), ..., x(n + L)]T ∈ C2L+1 (4)y(n) = [y(n − L), ..., y(n), ..., y(n + L)]T ∈ C2L+1 . (5)where n is the time index of symbols, and the filter length
is given by 2L + 1. Consider four complex-valued coefficient vectors of the 2 × 2 complex-valued FIR filters ofhxx (n) ∈ C2L+1 , hxy (n) ∈ C2L+1 , hyx (n) ∈ C2L+1 , andhyy (n) ∈ C2L+1 . Then, we obtain the two de-multiplexed
output symbols as follows:x̂(n) = x(n)T hxx (n) + y(n)T hxy (n)TTŷ(n) = x(n) hyx (n) + y(n) hyy (n),

(6)
(7)

which are fed back into the PFF-CPR block of Fig. 2.
B. PFF-CPR Block
In the PFF-CPR block, a carrier phase error is calculated
by averaging the received and transmitted pilot symbols. Morespecifically, the estimated phase error of the x-polarization is
given by()
NPA∑
x̂(Ip (n, k))1
ϕ(n) = arg, (8)2NPA +1
fDx (u(Ip (n, k)))k=−NPA

where fD (u) is the distortion function (DF) representing theinverse of TDC, which is formulated in the later part ofthis section using TDC coefficients. The DF contributes toreproducing the transmitter distortion. Also, 2NPA + 1 is thewindow size. Furthermore, Ip (n, k) denotes the symbol index
of the kth nearest pilot symbol to the symbol index n. Whenk is a natural number, Ip (n, k) is larger than or equal to n.
Similarly, when k is a negative number, Ip (n, k) is smaller thann. For example, suppose that the pilot symbol is inserted with aunique interval NPI and that the time index of the pilot symbol
is given as NPI × k, then we have Ip (n, k) = n + kNPI . Hence,(8) is the ratio of x̂(n) and fDx (u(n)), averaged over a certain
window size of 2NPA + 1. The notation u(Ip (n, k)) denotes
for the transmitted pilot symbol corresponding to the equalizer
output x̂(Ip (n, k)). Note that the estimated carrier phase error
ϕ(n) of (8) remains unchanged when the index n is between
the two closest pilot symbols. This implies that the estimatedcarrier phase error ϕ(n) is updated every pilot interval. Similar
to (8), the carrier phase error of the y-polarization ψ(n) can be
estimated. A symbol sequence typically has a frame structure
where a synchronization sequence is inserted at the beginningof each frame. With the differential detection algorithm, thesynchronization sequence is identified before the CPR block.

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Fig. 2. Configuration of the proposed receiver architecture with transmitter-side distortion compensation.

This enables frame synchronization and the pilot positionIp (n, k) is identified.
Moreover, in the PFF-CPR block, the carrier phase rotation
between two pilot symbols is output for each polarization
symbol, according to the linear interpolation of two carrier
phase errors estimated at the two pilot symbol positions asfollows:()
ûP (n) = x̂(n) exp [−jFI n, ϕ(n), ϕ̃(n) ]()
v̂P (n) = ŷ(n) exp [−jFI n, ψ(n), ψ̃(n) ],

(9)
(10)

(
)
where FI n, ϕ(n), ϕ̃(n) denotes the carrier phase error at
symbol index n based on interpolation, and ϕ̃(n) and ψ̃(n)denote the phase errors of the x- and y-polarizations, respectively, which are estimated in the previous iteration.1 The
above-mentioned method does not depend on the modulation
format for pilot symbols. Then, the signal output from the
PFF-CPR block is sent to the TDC block of Fig. 2.

employing the steepest gradient descent method [42], [43] asfollows:{}hxx (n+1) = hxx (n)+µ fDx (u(n−ND ))−ûP (n−ND )()× exp [jFI n−ND , ϕ(n−ND ), ϕ̃(n−ND ) ]x∗ (n−ND ) (13){}hxy (n+1) = hxy (n)+µ fDx (u(n−ND ))−ûP (n−ND )()× exp [jFI n−ND , ϕ(n−ND ), ϕ̃(n−ND ) ]y∗ (n−ND ) (14){}hyx (n+1) = hyx (n)+µ fDy (v(n−ND ))−v̂P (n−ND )() ∗× exp [jFI n−ND , ψ(n−ND ), ψ̃(n−ND ) ]x (n−ND ) (15){}hyy (n+1) = hyy (n)+µ fDy (v(n−ND ))−v̂P (n−ND )()× exp [jFI n−ND , ψ(n−ND ), ψ̃(n−ND ) ]y∗ (n−ND ) (16)where µ and ND denote the step size of the LMS algorithm
and the number of delay taps in the feedback control loop,
respectively. Note that (13), (14) are derived from (11) while(15), (16) are derived from (12).
D. TDC Filter Block

C. Adaptive Tap Control Block
In the adaptive tap control block of Fig. 2, the FIR coefficients of our adaptive equalizer are calculated by the decisiondirected LMS algorithm with the recovered data symbols whenfast-tracking is required. Let us introduce square error loss
functions of x- and y-polarizations, respectively, as follows:∑2
Jx =|fDx (u(n))−ûP (n)|(11)

The TDC filter block consists of the 2 × 2 real-valued FIRfilter and the DC offset adder. The 2 × 2 real-valued FIR filter
allows us to mitigate the IQ skew, the IQ amplitude imbalance,
the 90-degree error, and the cross-talk between I- and Q-lanes.
Let us introduce a complex-valued vector with length 2LT +1 asûP (n) = [ûP (n−LT ), ..., ûP (n), ..., ûP (n+LT )]T ∈ C2LT +1
(17)

n

Jy =

∑

2

|fDy (v(n))−v̂P (n)| ,

(12)

n

where u(n) and v(n) denote the transmitted pilot symbols
and the generated replica of data symbols. Then, the FIR coefficients of hxx (n), hxy (n), hyx (n), and hyy (n) are updated
1 Note that while in this paper, we employ a simple linear interpolation
function, it may be readily
possible toone. For example,() rely on a higher-ordern′n′
we may employ FI n, ϕ(n), ϕ̃(n) = (1 − N)ϕ̃(n) + N
ϕ(n), wherePIPI′
n = mod(n, NPI ).

Then, the symbols output from the TDC filter are given byûT (n) = Re{ûP (n)T }{wII + jwQI }+ Im{ûP (n)T }{wIQ + jwQQ } + wI0 + jwQ0 (18)where wII , wIQ , wQI , and wQQ are the coefficient vectors of
2 × 2 real-valued FIR filter having the length of 2LTA + 1,
while wI0 and wQ0 are the DC offset coefficients of I- and Qlanes. Also, (18) corresponds to the process of x-polarization,and that of y-polarization is given by the symmetric copy of(18), hence omitted in this paper.

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The distortion function fD (u) is formulated as follows:fD (u) = Re{u}(w̃QQ −j w̃QI )+ Im{u}(−w̃IQ +j w̃II ) − wI0 − jwQ0 ,5u(n-N)
equalizer(19)

where w̃II , w̃IQ , w̃QI , and w̃QQ denote the middle taps of 2 × 2
real-valued FIR filter of the TDC block, and amplitude scaling
is omitted. The distortion function is formulated as an inverse
function of the TDC filter, where only the center taps areused. The signals output from the TDC block, i.e., ûT (n) andv̂T (n), are input into the decision-directed feed-forward (DFF)
CPR block to compensate for the residual carrier phase errors.
The LMS algorithm is used for adapting the TDC coefficients
based on the output from the DFF-CPR block.

x(n)
H(z)

Z-1

Z-N

-

Z-N

()*

ʹ

ND: delayߤ, stepsize

Fig. 3. The LMS equalizer model with the feedback loop delay.

E. TDC DFF-CPR Block
In the TDC DFF-CPR block, symbol replicas of data
symbols ûR (n) are generated based on the hard decision ofûT (n). Then, the residual carrier phase error at the index n isestimated as
(
)
NDA
∑
1
ûT (n − k), (20)ϕD (n) = arg
2NDA +1ûR (n − k)k=−NDA

which is averaged over 2NDA + 1 symbol intervals to exclude
the effects of the white noise, similar to (8). Then, the symbol
output from the TDC filters ûT (n) is compensated for by theaveraged carrier phase error ϕD (n). Finally, the symbols output
from the DFF-CPR block are used for optimizing the FIR
coefficients of the TDC block. The four FIR coefficient vectors
and the two DC offset coefficients are updated as follows:{}wII ← wII + µRe (u(n)−ûD (n))e−jϕD (n) Re{ûP (n)} (21){}wIQ ← wIQ + µRe (u(n)−ûD (n))e−jϕD (n) Im{ûP (n)} (22){}wQI ← wQI + µIm (u(n)−ûD (n))e−jϕD (n) Re{ûP (n)} (23){}wQQ ← wQQ + µIm (u(n)−ûD (n))e−jϕD (n) Im{ûP (n)}(24){}wI0 ← wI0 + µRe (u(n)−ûD (n))e−jϕD (n)(25){}−jϕD (n)wQ0 ← wQ0 + µIm (u(n)−ûD (n))e(26)
Typically, the distortion condition of the transmitter-side
components changes over time at least a couple of ordersof magnitude slower than a symbol rate, and hence therequirement to suppress the feedback loop delay in the LMS
algorithm for the FIR coefficients of the TDC block is lower
than that imposed on the adaptive channel equalizer.
To elaborate a little further, one of our main contributions
is on the feedback from the TDC block to the 2 × 2 complexvalued FIR equalizer and the PFF-CPR, which is characterized
by the distortion function fD (u) of (19). More specifically,fD (u) reproduces the distortion of the transmitter components
for the pilot symbols or the replica of data symbols, andmodifies it from the ideal mapping position. This allows
us to dispense with the explicit distortion compensation at
the channel equalizer and the PFF-CPR blocks. Instead, thedistortion is compensated for at the TDC block.
A further note is that as shown in (19), the effects of
ISI caused by transmitter-side distortion are ignored, and the
distortion-reproduced symbol fD (u) is obtained without using

the information of adjacent symbols. This feedback specificto our scheme significantly increases the resilience against thetransmitter-side component distortion, which is evaluated in
our numerical simulations of Section V.
IV. S TABLE C ONDITION OF LMS E QUALIZER IN THE
P RESENCE OF F EEDBACK D ELAY
In this section, we derive the stability condition of the
LMS algorithm under the presence of the feedback loop delay.
Note that the LMS step size that satisfies the stable condition
decreases upon increasing the feedback loop delay. Fig. 3
shows the LMS equalizer model to consider the stability
condition of the FIR filter coefficients. The input signal to
the LMS equalizer x(n) is represented in a vector form asx(n) = [x(n − L), ..., x(n), ..., x(n + L)]T ∈ C2L+1 , (27)where n is the time index in a symbol interval, and the filter
length is given by 2L+1. 2 The input signal is convolved by the
FIR filter coefficients h(n) ∈ C2L+1 to output the equalized
symbol xT (n)h(n). Then, the equalized symbol is used for
updating the FIR filter coefficients with the aid of the LMS
algorithm after adding ND -tap delay, where ND represents thetotal delay in the feedback control loop.
The carrier phase recovery (CPR) has two main delay
factors, i.e., the signal path imposed by averaging a sufficientnumber of symbols to estimate a stable carrier phase andthe excess delay caused due to parallel-processing of multiple
symbols per internal clock period. These delay factors causedby the CPR are taken into account in our scheme by the delay
tap ND .
In the LMS algorithm, the FIR-filter coefficients h(n) are
updated as follows:{h(n + 1) = h(n) + µ u(n − ND )}− xT (n − ND )h(n − ND ) x∗ (n − ND ) (28){= h(n) + µ u(n − ND )x∗ (n − ND )}− x∗ (n − ND )xT (n − ND )h(n − ND ) , (29)2 Here, we assume a discrete time frame of a symbol interval for simplicity.
However, our analysis is readily applicable to arbitrary over-sampling scenarios, as mentioned later in this paper.

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where µ is the step size, while u(n) is the nth information
symbol that corresponds to the equalizer output xT (n)h(n).
In our decision-directed LMS algorithm, the recovered information symbol û(n) is used as u(n) of (29).
Since the time-varying term x∗ (n)xT (n) of (29) is hard toexactly obtain, we approximate it by a covariance matrix of
Rx =En [x∗ (n)xT (n)]. This approximation is reasonable when
the step size µ is significantly low, i.e. µ ≪ 1, where thetap coefficient h(n) evolves slowly enough that the termx∗ (n)xT (n) can be approximated by its ensemble on n. Hence,(29) is further modified to{}h(n+1) = h(n) + µ u(n−ND )x∗ (n−ND ) − Rx h(n−ND ) .
(30)
Let us introduce the z-transform of a correlation vector
u(n)x∗ (n) and that of the FIR coefficient vector h(n), respectively, as
C(z) =
H(z) =

∞
∑
n=−∞
∞
∑

z −n u(n)x∗ (n)(31)

z −n h(n)

(32)

n=−∞

With (31), (32), the z-transform of (30) is represented byzH(z) = H(z) + µ{z −ND C(z) − z −ND Rx H(z)}. (33)Then, (33) is rewritten by
H(z) = [(1 − z −1 )I + z −ND −1 µRx ]−1 [µz −ND −1 C(z)].(34)
Recalling that Rx is an Hermitian matrix, it is diagonalizedwith the aid of eigenvalue decomposition as follows: Rx =Q† Λx Q, where Q is a unitary matrix, having the relationshipof Q† Q=QQ† =I, and Λx is a diagonal matrix composed of
eigenvalues. Furthermore, I is the identity matrix. By applying
this diagonalization to (34), we obtain
H(z) = [(1−z −1 )I + z −ND −1 µQ† Λx Q]−1 [µz −ND −1 C(z)]= [Q† {(1−z −1 )I + z −ND −1 µΛx }Q]−1 [µz −ND −1 C(z)]= Q† [(1−z −1 )I + z −ND −1 µΛx ]−1 Q[µz −ND −1 C(z)].
(35)
Since the z-transform H(z) has to satisfy that all the poles
are inside a unit circle, we arrive at the stable condition for
feedback loop control of FIR coefficients h(n) as follows:(1−z −1 )+z −ND −1 µλx (k) = 0(k = 0, ..., 2L + 1), (36)where λx (k) denotes the kth diagonal component of Λx .
Finally, the condition that solution of (36) exists within a unit
circle is equivalently given by [44]
µ<

2
π
sin
.
max λx (k)2(2ND + 1)

(37)

k

Here, the highest eigenvalue of the covariance matrix Rx andthe delay in the feedback control loop ND determine the upper
limit of the LMS step size µ for stable operation.

6Note that the derivation of (37) may be generalized foran arbitrary oversampling rate by modifying the input signal
vector x(n) to that supporting all input samples that are
convoluted by the FIR coefficient vector.
V. P ERFORMANCE R ESULTS
In this section, we provide our performance results toevaluate the adaptive equalizer with the proposed TDC blockand investigate the effects of the delay induced in the feedback
control loop. In the simulations, we employed 64-QAM, i.e.,
M = 6 bits/symbol. The pilot symbol was mapped at oneof four points (±3/8, ±3/8) in 64QAM constellation, which
has the lower average power as the data symbol, and the pilot
interval was set to NPI = 32.
Fig. 4 shows the Q-factor of the proposed and benchmark
schemes, which is calculated as a normal distribution variable
that gives a complementary cumulative probability equal totheBER, where we have the relationship of BER=
∫ ∞ observed
2√1 exp(− x ) dx. The step size of the decision-directed
2Q
2π
LMS equalizer was set to µ = 10−3 . The delay parameter was
set at ND = 256 symbols. The two green curves represent
the scheme that deactivates our TDC block while employing
the PFF-CPR in the presence of the PN. The two red curves
correspond to the proposed schemes activating the TDC andPFF-CPR blocks. For the green and red dotted curves, thenested MZ modulator is activated, while the transmitter distortion is set negligible, i.e. REI = 99 dB, REO = 99 dB, andthe 90-degree phase error of θe = 0 deg. For the green and redsolid curves, the transmitter distortion conditions of REI = 25
dB, REO = 30 dB, and θe = 5 deg were considered. We alsoplotted the theoretical upper bound, the MMSE bound, andthe ideal LMS bound. In the ideal LMS bound, the effectsof the PN contamination were not considered, and the nested
MZ modulator was deactivated. Note that while the MMSE
equalizer exhibits theory-achieving performance in the static
condition, it is not suitable in a rapidly time-varying channel.
The proposed scheme is evaluated in terms of the required
SNR (RSNR) gap from the theoretical bound, where the RSNR
is defined by a minimum SNR to achieve the BER of 1.0 ×
10−2 . In the rest of this paper, the gap from the theoreticalbound is referred to as the RSNR gap. Observe in Fig. 4 thatthe theoretical and the MMSE bounds nearly coincided. The
RSNR gap of the LMS bound was less than 0.1 dB. When the
PNs and the nested MZ modulator were considered, the RSNR
gap increased to 0.9 dB. The laser PN linewidth was set at
200 kHz in the rest of this paper. This penalty includes theeffects of the nonlinear characteristic of a sinusoidal transfer
function in the inner MZI, despite the idealistic scenario of
REI = REO = 99 dB and θe = 0 deg., corresponding to thenegligible imperfection condition. When the TDC block was
activated in the proposed scheme with the optimum parameter
NPA = 7, the RSNR gap was reduced to 0.6 dB, which isachieved as the explicit benefits of the DFF-CPR in our TDC
block.
When the transmitter distortion was loaded with REI =25 dB, REO = 30 dB, and θe = 5 deg. in the benchmark
scheme, the RSNR was not attained since the BER was higher

PREPRINT (ACCEPTED VERSION) FOR PUBLICATION IN JOURNAL OF LIGHTWAVE TECHNOLOGY (DOI: 10.1109/JLT.2022.3166754)ϭϯϮdŚĞŽƌǇdŚĞŽƌǇ
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Fig. 5. The effects of window size 2NPA + 1 of the proposed scheme on the
RSNR gap, where the four scenarios were considered, i.e., those with/without
our TDC block in the presence/absence of the PN.

than 10−2 . By contrast, in the proposed scheme activating
the TDC block with the distortion function, the BER was
sufficiently low, and the RSNR gap was as low as 0.9 dB,
even in the presence of the high transmitter distortion.
Fig. 5 investigates the effects of window size in the PFFCPR NPA on the RSNR gap of the proposed and benchmark
schemes, where the nested MZ modulator was deactivatedfor simplicity, i.e., any transmitter-side impairments are notconsidered. We considered the four scenarios, i.e., 1) the
deactivated TDC block without PN, 2) the deactivated TDC
block with PN, 3) the activated TDC block without PN, and 4)
the activated TDC block with PN. When the effects of PN wereignored in the scheme with and without our TDC block (i.e.,
the yellow and the blue curves), the RSNR gap monotonicallydecreased upon increasing NPA and converged to the RSNR
gap of 0 dB. This is because the PFF-CPR suffers from theestimated carrier phase error induced due to the AWGN when
NPA is small. For a high NPA , such as NPA > 5, the estimatedPN error was maintained to be sufficiently low for the ideal
scenario of no PN. By contrast, when the effects of the PN
were included in the benchmark scheme without our TDC
block (corresponding to the green curve), the PFF-CPR, having
significantly high NPA , did not track the PN variation, where
the optimum NPA was NPA = 9, and the associated RSNR gap
was as high as 0.76 dB. Furthermore, in the proposed scheme,
including our TDC block (corresponding to the red curve),
sufficiently suppressed the PN, where the RSNR gap was as
low as 0.36 dB for the optimal window size of NPA = 7.
Fig. 6 shows the effects of the 90-degree phase error on the
RSNR penalty, where the imperfection of the inner MZI was
given by REI = 99 dB, 30 dB, and 25 dB. For comparisons,we considered the three schemes, such as the proposed schemewith the TDC block and the DF, the proposed scheme with the
TDC block without the DF, and the benchmark without either
our TDC block or DF. The inset of Fig. 6 shows the Q-factorof the three schemes with REI = 25 dB and the 90-degree
phase error of 0 deg.

When REI and the 90-degree phase error were set at 99 dB
and 0 deg., corresponding to no imperfection case, the RSNR
gap was as low as 0.45 dB when the TDC block and the
DF were deactivated. In the proposed scheme with the TDC
block and the DF, the RSNR gap remains similarly low as
0.30 dB. In the benchmark without either our TDC block or
DF, when the REI was set at 30 dB, the RSNR gap increased
to more than 7 dB. When REI was further decreased to 25 dB,
the BER floor was higher than 1.0 × 10−2 , as shown in theinset, and hence the associated RSNR cannot be calculated
in our definition. By contrast, the RSNR gap of the schemewith the TDC blocks and the DF decreased to 0.35 dB and
0.45 dB, when the REI was set at 30 and 25 dB, respectively.
When the 90-degree phase error was increased up to 5 deg., the
RSNR gap increase was below 0.05 dB, which was common
for REI = 99, 30, and 25 dB. Furthermore, by activating the
DF in the proposed scheme with our TDC block, the RSNR
gap decreased by more than 1.0 dB for REI = 30 dB. For
REI = 25 dB, the DF has the benefits of reducing the BER
floor and improving the Q factor from 5.2 dB to 10.6 dB at the
SNR of 24 dB, as shown in the inset of Fig. 6. This indicatesthat the gain achievable by the DF is significantly high.
We evaluated the achievable performance of the conventional scheme where the TDC FIR is inside the feedback control loop of the adaptive equalizer, and the tap coefficients arecontrolled by using ûT (n). The RSNR gap of the conventional
scheme for REI = 30 dB and 25 dB were 0.32 dB and 0.43 dB,
respectively. The difference in the RSNR gap was as low as
0.02–0.03 dB.
Fig. 7 shows the RSNR gap from the theoretical bound fortwo cases, with and without the proposed scheme, where the
IQ skew at the transmitter was varied. When the IQ skew was
set at 0.08 per symbol, the RSNR gap penalty decreased from
2.7 dB to 0.5 dB. The proposed scheme improves tolerance
to IQ skew significantly. The RSNR penalty increase of 0.3
dB was observed with the TDC case at the IQ skew of 0.1
per symbol, which corresponds to 1 ps for the symbol rate

PREPRINT (ACCEPTED VERSION) FOR PUBLICATION IN JOURNAL OF LIGHTWAVE TECHNOLOGY (DOI: 10.1109/JLT.2022.3166754)ϭ͘ϴϭϰZ/сϮϱθĞсϬϭϮ

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Fig. 6. RSNR gap of the proposed scheme, where imperfection of theextinction ratio of the inner MZ REI and the 90-degree phase error θe wereconsidered. When REI was set at 25 dB, the BER floor was higher than
1.0 × 10−2 for the benchmark scenario without TDC and that without DF,
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2.8
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of 100 GBaud. Since we assume that most of the transmitter
IQ skew is compensated in the digital pre-distortion, a 10%
range of symbol period is sufficient to tolerate the residual IQ
skew caused by the variation in time and calibration error ofthe digital pre-distortion scheme.
Fig. 8 shows the proposed scheme’s RSNR gap from thetheoretical bound for BER of 10−2 , where we varied the speedof polarization scrambling normalized by the symbol rate from
2π × 10−8 to 4π × 10−5 rad/symbol. For example, when thenormalized speed of polarization was set at π × 10−5 , theϭ͘ͲϬϳ
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Fig. 8. The effects of the speed of polarization scramble on the RSNR gap
for the different LMS step sizes.

state of signal polarization is rotated such that the x- and ypolarizations are gradually rotated and completely exchanged
after 105 symbols, which corresponds to the rotation in the
S1 –S2 plane of Pointcare sphere [45]. Here, we considered
the effects of the PN while ignoring those of the nested MZ
modulator and transmitter imperfection. Also, both of the PFFCPR and the TDC blocks were activated, while setting NPA =7. The feedback delay in the adaptive equalizer was set toeight pilot-symbol intervals, which corresponded to the total
delay of ND =8×32 symbol intervals. Observe in Fig. 8 that ineach step size, the RSNR gap significantly increased when thenormalized speed of the polarization scramble was higher thana specific threshold. Assuming that the RSNR penalty increaseof 0.5 dB due to the polarization scramble is acceptable, themaximum tracking speed is given by a crossing point with the
RSNR gap of 0.9 dB, as shown by the broken line in Fig. 8.
For µ = 10−4 , the maximum tracking speed was as low as
2.0 × 10−6 rad/symbol. When the step size was increased toµ = 10−3 and 10−2 , the maximum tracking speed were morethan 1.5×10−6 and 1.8×10−5 , respectively. However, for the
step size as high as µ = 10−2 , the RSNR gap increased by
0.3 dB from the case of µ = 10−3 . Hence, there is a tradeoff
between the maximum tracking speed and the RSNR gap. Notethat the step size of 10−3 was employed in our evaluations
except for Figs. 8, 9, and 10, since the step size of 10−3is the optimum value to track the scramble speed of 10−5
rad/symbol, maintaining the RSNR penalty sufficiently low inthe stable condition.
Fig 9 shows the maximum tracking speed as a function
ND , when the RSNR gap increase of 0.5 dB is accepted.
Upon increasing ND , the maximum tracking speed decreased.
Additional blocks in the feedback control loop tend to increase
the feedback latency. Assuming the symbol rate of 100 GBaud,the operating clock frequency of the CMOS digital logic
circuit is less than 1GHz, and the parallel expansion width ismore than 100 symbols. Thus, for example, when the feedback
control loop delay increases from 5 to 10 in the operating clockPREPRINT (ACCEPTED VERSION) FOR PUBLICATION IN JOURNAL OF LIGHTWAVE TECHNOLOGY (DOI: 10.1109/JLT.2022.3166754)

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Fig. 10. Relationship between µλmax and the delay in the feedback loop.

Fig. 9. The normalized maximum tracking speed for different feedback loop
delay ND .

cycle, then ND increases from 500 symbols to 1000 symbols,
which degrades the maximum tracking speed from 6 × 10−6
to 4.5 × 10−6 rad/symbol based on Fig. 9. This corresponds
to the decrease in the maximum tracking speed from 6 to
4.5 Mrad/second in the case of the 100 GBaud symbolrate. Consequently, our proposed architecture excluding the
TDC block from the feedback control loop benefits frommaintaining a fast-tracking feature of the adaptive equalizer.
Fig. 10 shows the contour plot of the RSNR gap, where
the delay in the feedback control loop and the step size µwere varied. According to (37), the maximum step size thatmaintains the stable condition is determined by the delay in the
feedback control loop,3 while it is also the essential limiting
factor of the maximum tracking speed. The vertical axis ofFig. 10 represents the product of the step size µ and the
maximum eigenvalue λmax of the covariance matrix Rx of
the input signal vector x(n) as defined in Section IV. The red
curve indicates the theoretical upper bound of (37). As shownin Fig. 10, the RSNR gap steeply increased when µλmax
became higher than the theoretical upper bound. Thus, it was
confirmed that the theoretical bound is beneficial for striking
the relationship between the delay and the step size.
VI. C ONCLUSION
We proposed the optical coherent receiver architecture capable of fast-tracking the signals in a rapidly time-varying fiber
channel while compensating for the residual signal distortion
induced by the transmitter analog components. The TDC
block in the proposed architecture has the explicit benefits bymodifying the replica symbols used in the LMS tap optimizer
while dispensing the increase in the feedback loop delay of
the adaptive channel equalizer. Our simulation results show
3 The maximum step size also depends on the maximum eigenvalue of thecovariance matrix of the input signal. In Fig. 10, only the step size µ andthe feedback loop delay ND were varied and it was found that µ and ND
unaffected the input signal of the 2 × 2 complex-valued FIR. Hence, in thispaper, the eigenvalues of the covariance matrix of the input signal are assumed
to be fixed.

that the proposed scheme significantly reduces the RSNR
gap from the theoretical bound. The RSNR gap caused by
the low extinction ratio of the inner MZI was reduced bymore than 6.5 dB, exhibiting the RSNR gap of as low as
0.4 dB. Furthermore, our theoretical analysis allows us to
strike the balance of the feedback loop delay in the adaptive
equalizer and the stable LMS step size, hence maximizing the
tracking speed. Importantly, our theoretical analysis confirms
the explicit advantage of implementing our TDC block outsideof the feedback control loop for the sake of suppressing the
feedback delay.
ACKNOWLEDGEMENT
We would like to thank Mitsuteru Yoshida, and Kyo
Minoguchi at NTT Network Innovation Labs for the valuable
discussion.
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