SOP short=SOP, long= state of polarization, AIR short=AIR, long= achievable
information rate, DSP short=DSP, long= digital signal processing, PMD
short=PMD, long= polarization-mode dispersion, DOF short=DOF, long= degrees of
freedom, CMA short=CMA, long= constant modulus algorithm, MMA short=MMA, long=
multi-modulus algorithm, RDE short=RDE, long= radially directed equalizer, QAM
short=QAM, long= quadrature amplitude modulation, LMS short=LMS, long= least
mean square, LS short=LS, long= least-square error, DP short=DP, long= dual-
polarization, PN short=PN, long= phase noise, GN short=GN, long= Gaussian
Noise, NLI short=NLI, long= nonlinear impairments, MIMO short=MIMO, long=
multiple-input multiple-output, SNR short=SNR, long= signal-to-noise ratio, MI
short=MI, long= mutual information, pdf short=pdf, long= probability density
function, AWGN short=AWGN, long= additive white Gaussian noise, ASE short=ASE,
long= amplified spontaneous emission, QPSK short=QPSK, long= quadrature phase-
shift keying,

#  Capacity Bounds under Imperfect Polarization Tracking

Mohammad Farsi, Magnus Karlsson, and Erik Agrell M. Farsi and E. Agrell are
with the Department of Electrical Engineering, Chalmers University of
Technology, SE-41296 Gothenburg, Sweden (e-mail<EMAIL_ADDRESS>agrell@chalmers.se)M. Karlsson is with the Department of Microtechnology and
Nanoscience, Chalmers University of Technology, SE-41296 Gothenburg, Sweden
(e-mail: magnus.karlsson@chalmers.se).This work was supported by the Knut and
Alice Wallenberg Foundation under grant 2018.0090.

###### Abstract

In optical fiber communication, due to the random variation of the
environment, the state of polarization (SOP) fluctuates randomly with time
leading to distortion and performance degradation. The memory-less SOP
fluctuations can be regarded as a two-by-two random unitary matrix. In this
paper, for what we believe to be the first time, the capacity of the
polarization drift channel under an average power constraint with imperfect
channel knowledge is characterized. An achievable information rate (AIR) is
derived when imperfect channel knowledge is available and is shown to be
highly dependent on the channel estimation technique. It is also shown that a
tighter lower bound can be achieved when a unitary estimation of the channel
is available. However, the conventional estimation algorithms do not guarantee
a unitary channel estimation. Therefore, by considering the unitary constraint
of the channel, a data-aided channel estimator based on the Kabsch algorithm
is proposed, and its performance is numerically evaluated in terms of AIR.
Monte Carlo simulations show that Kabsch outperforms the least-square error
algorithm. In particular, with complex, Gaussian inputs and eight pilot
symbols per block, Kabsch improves the AIR by $0.2$ to $0.35$ bits/symbol
throughout the range of studied signal-to-noise ratios.

###### Index Terms:

Achievable information rate, constant modulus algorithm, capacity, channel
estimation, decision-directed least mean square, Kabsch algorithm, lower
bound, least square error, mutual information, polarization-mode dispersion,
state of polarization.

##  I Introduction

The growing demand for reliable long-distance communication makes it essential
to determine the capacity limits of optical links [1, 2, 3]. The emergence of
coherent optical communication systems enabled DSP and polarization-division
multiplexing to achieve higher spectral efficiencies. However, higher data
rates come with increased sensitivity to impairments such as PMD and SOP
fluctuations, which must be tracked dynamically in the receiver [4]. Due to
the random variation of both internal and environmental impairments, the SOP
fluctuates randomly with time [5]. Previous long-term measurements showed that
the SOP drift might vary from days and hours in buried fibers [6, 7] to
microseconds in aerial fibers [8, 9, 10].

The conventional DSP solutions for SOP tracking are replete with both blind
and data-aided algorithms. The CMA [11], thanks to its low complexity and
tolerance to PN, has been widely considered for blind polarization tracking
[12, 13, 14]. Many studies have been conducted to overcome the so-called
“singularity” problem of CMA [15, 16, 17], where the two estimated channels
converge twice to the same polarization. The RDE and its variants were
proposed to account for higher-order QAM in [18, 19, 20]. The MMA [21] is also
applicable to higher-order modulations. More reliable convergence can be
obtained using data-aided algorithms such as LMS, which was adopted for SOP
estimation by [13, 22], or standard LS, which has been extensively studied for
both wireless and optical applications [23, 24, 25, 26]. While the LMS
algorithm performs real-time continuous equalization, LS applies to block-
based estimation.

While the DP channel can be represented by a unitary matrix, the majority of
the available estimation techniques including LS and LMS have no unitary
constraint on the estimated channel, leading to sub-optimal solutions. This
makes room for algorithms that provide a unitary estimation of the DP channel.
Louchet et al. [27] proposed the Kabsch algorithm [28] as a joint blind PN and
PMD estimation method. In [29], a blind modulation-format-independent joint PN
and polarization tracking algorithm was proposed, and its performance was
compared with the blind Kabsch algorithm.

Depending on the estimation algorithm, the speed of the fluctuations, or the
additive noise, the polarization tracking might be imperfect. This makes it
relevant to ask how the capacity is affected by such an estimation error?
While there are no capacity studies regarding the polarization drift channels,
to the best of our knowledge, the literature is replete with fiber capacity
results. The first finite capacity limit for the single-wavelength optical
channel based on a sort of GN model was introduced in [30, 31], showing a
lower bound that increases with the power until it reaches a maximum, and then
it decreases due to the NLI. Although different nonlinear models were applied,
similar results were reported in [32, 33]. Since the GN model cannot truly
describe the NLI [34], some modeled the NLI as a linear time-variant
distortion highly correlated over time [35, 36]. This made it possible to
counteract NLI using the techniques that were conventionally used for linear
impairments (e.g., PN, chromatic dispersion, PMD, and SOP [37, 38, 39]). For
example, in [40], a tighter lower bound for single-polarization, dispersion-
unmanaged systems was achieved by considering the PN.

In this paper, for the first time, we study the capacity of the polarization
drift channel in the presence of imperfect knowledge of the channel (e.g.,
imperfect polarization tracking). Using the mismatched decoding method, we
derive an AIR, which is a lower bound on the capacity, in the presence of a
channel estimation error. We show that a unitary estimation of the channel
leads to a tighter lower bound, which makes it reasonable to seek unitary
estimators. A data-aided version of the Kabsch algorithm is proposed for DP
channel estimation, for the first time to the best of our knowledge. We
compare Kabsch with an LS algorithm in terms of AIR and show that Kabsch
outperforms LS throughout the range of considered SNR. For instance, with only
eight pilot symbols per channel and block, Kabsch improves the AIR by at least
$0.2$ bits per symbol compared to LS.

Notation: Column vectors are denoted by underlined letters $\underline{x}$ and
matrices by uppercase roman letters $\mathrm{X}$. We use bold-face letters
$\mathbf{x}$ for random quantities and the corresponding nonbold letters $x$
for their realizations. pdf are denoted as $p_{\mathbf{x}}(x)$ and conditional
pdf as $p_{\mathbf{y|x}}(y|x)$, where the subscripts will sometimes be omitted
if they are clear from the context. Expectation over random variables is
denoted by $\mathbb{E}[\cdot]$. Sets are indicated by uppercase calligraphic
letters $\mathcal{X}$. The complex zero-mean circularly symmetric Gaussian
distribution of a vector is denoted by
$\mathbf{\underline{x}}\sim\mathcal{CN}(0,\Lambda_{\underline{x}})$, where
$\Lambda_{\underline{x}}=\mathbb{E}[\mathbf{x}\mathbf{x}^{\dagger}]$ is the
covariance matrix. All logarithms are in base two. In the context of matrix
operations, $|\cdot|$, $(\cdot)^{T}$, $(\cdot)^{\dagger}$, and
$\left\lVert.\right\rVert$ represent the determinant, transpose, conjugate
transpose, and Frobenius norm operators, respectively.

##  II Channel Model and Mutual Information

###  II-A Channel Model

We consider transmission over $n$ channels in the presence of ASE noise at the
receiver. The channel is assumed to be constant during a transmission block of
length $N$, and changes randomly and independently between the blocks. The
assumption that the channel does not change within a block is consistent with
the fact that the SOP drifts at a much slower rate than typical transmission
rates in optical links [6, 7] and is well established in optical
communications literature [38, 40]. The block length $N$ is chosen based on
the application and the drift speed of the channel. In most optical
communication systems, there is no feedback channel between the transmitter
and the receiver. Therefore, the problem of input distribution optimization,
corresponding to the case that the channel is known at the transmitter, is not
investigated. It is assumed that PMD is negligible and all channel
impairments, including nonlinearities and chromatic dispersion are ideally
compensated, with the exception of polarization fluctuation and ASE noise,
which are modeled by a unitary matrix and AWGN, respectively.

Since different blocks are independent, we just model the symbols in one
transmission block in the following. The transmitted signal in each channel at
time $\smash{k=0,\ldots,N-1}$ is an $n$-dimensional random vector
$\mathbf{\underline{s}}_{k}$ that takes on values from a set $\mathcal{S}$ of
zero-mean constellation points. After filtering and resampling the received
signals into one sample per symbol, the vector of received samples
$\mathbf{\underline{x}}_{k}$ can be expressed as

$\mathbf{\underline{x}}_{k}=\mathrm{H}\mathbf{\underline{s}}_{k}+\mathbf{\underline{z}}_{k}$
(1)

where the $n\times n$ matrix $\mathrm{H}$ represents a MIMO channel and
$\mathbf{\underline{z}}_{k}$ denotes the complex ASE noise samples at time
$k$, which is assumed to be $\mathcal{CN}(0,\sigma_{z}^{2}\mathrm{I}_{n})$ and
independent of $\mathbf{\underline{s}}_{k}$. In the remainder of this paper,
we will omit the time index $k$ explicitly for notational convenience. This is
possible because the input and noise are independent and identically
distributed (i.i.d.) over time. We define the covariance matrix of the input
vector as

$\Lambda_{\underline{s}}=\mathbb{E}_{\mathbf{\underline{s}}}[\mathbf{\underline{s}}\mathbf{\underline{s}}^{\dagger}].$
(2)

In order to maintain the generality of the results, they are given for an
arbitrary number of channels $n$ whenever possible. Thus, the MIMO-AWGN
channel (1) can describe a wide range of applications and impairments.
However, for the purpose of DP optical channel modeling, we are particularly
interested in the special cases of $\smash{n=2}$ and $\mathrm{H}$ being
unitary, denoted by $\mathrm{H_{u}}$.

###  II-B Mutual Information with Perfect Knowledge of Channel at the Receiver

The conditional MI between two random vectors $\mathbf{\underline{s}}$,
$\mathbf{\underline{x}}$, when the channel $\smash{\mathbf{H}=\mathrm{H}}$ is
given, is defined as [41, Eq. 2.61]

$\displaystyle
I(\mathbf{\underline{s}};\mathbf{\underline{x}})=\mathbb{E}_{\mathbf{\underline{s}},\mathbf{\underline{x}}}\left[\log\left(\frac{p(\mathbf{\underline{x}}|\mathbf{\underline{s}},\mathrm{H})}{p(\mathbf{\underline{x}}|\mathrm{H})}\right)\right].$
(3)

The capacity of this channel under an average power constraint is [42]

$\displaystyle
C=\max\limits_{p(\mathbf{\underline{s}})}{I}(\mathbf{\underline{s}};\mathbf{\underline{x}})\quad\text{s.t.}\quad\operatorname{tr}(\Lambda_{\underline{s}})\leq
P,$ (4)

where $P$ is the transmission power constraint. For the MIMO-AWGN channel in
(1), the channel law given $\mathrm{H}$ and $\underline{s}$ is characterized
by the pdf

$p(\underline{x}|\underline{s},\mathrm{H})=\dfrac{1}{\pi^{n}\sigma_{z}^{2n}}\exp\left(-\frac{\left\lVert\underline{x}-\mathrm{H}\underline{s}\right\rVert^{2}}{\sigma_{z}^{2}}\right).$
(5)

Then, the pdf of the channel output can be calculated as

$\displaystyle
p(\underline{x}|\mathrm{H})=\mathbb{E}_{\mathbf{\underline{s}}}\left[p\right(\underline{x}|\mathbf{\underline{s}},\mathrm{H})].$
(6)

Given $\mathrm{H}$, the capacity-achieving distribution of the MIMO-AWGN
channel law (5) is $\mathcal{CN}$ [43]. Therefore, assuming
$\mathcal{CN}(0,\Lambda_{\underline{s}})$ inputs,

$\displaystyle
p(\underline{x}|\mathrm{H})=\dfrac{1}{\pi^{n}|\Lambda_{\underline{x}}|}\exp\left(-\underline{x}^{{\dagger}}\Lambda_{\underline{x}}^{-1}\underline{x}\right),$
(7)

where

$\displaystyle\Lambda_{\underline{x}}$
$\displaystyle=\mathbb{E}_{\mathbf{\underline{x}}}\left[\mathbf{\underline{x}}\mathbf{\underline{x}}^{\dagger}\right]$
$\displaystyle=\mathrm{H}\Lambda_{\underline{s}}\mathrm{H}^{\dagger}+\sigma_{z}^{2}\mathrm{I}_{n}$
(8)

is the covariance matrix of the received samples when the channel is given.
The MI of a general MIMO system for a given channel is [43]

$\centering
I(\mathbf{\underline{s}};\mathbf{\underline{x}})=\log\left|\mathrm{I}_{n}+\mathrm{H}^{{\dagger}}\mathrm{H}\mathrm{Q}\right|,\@add@centering$
(9)

where

$\displaystyle\mathrm{Q}=\Lambda_{\underline{s}}/\sigma_{z}^{2},$ (10)

and hence $\smash{\operatorname{tr}(\mathrm{Q})\leq n\eta}$, where
$\smash{\eta=P/(n\sigma_{z}^{2})}$ is the SNR of each channel. If the channel
matrix $\mathrm{H}$ is confined to the set of unitary matrices (i.e.,
$\mathrm{H}=\mathrm{H_{u}}$), (9) gives

$I(\mathbf{\underline{s}};\mathbf{\underline{x}})=\log\left|\mathrm{I}_{n}+\mathrm{Q}\right|.$
(11)

The capacity of a unitary MIMO-AWGN channel is the supremum of (11) for all
possible $\mathrm{Q}$. In general, $\mathrm{Q}$ needs to be optimized for each
realization of the channel $\mathrm{H}$, if it is known at the transmitter;
however, based on (11), the MI is independent of $\mathrm{H}$. Since
maximizing (11) is equivalent to maximizing
$|\smash{\mathrm{I}_{n}+\mathrm{Q}}|$ and $\mathrm{Q}$ is positive definite,
the nondiagonal elements of $\mathrm{Q}$ must be zero, yielding $\mathrm{Q}$
to be a diagonal positive definite matrix. From the well-known theorem that
the geometric mean is always upper-bounded by the arithmetic mean, it is
straight-forward to show that uniform power distribution at the transmitter
(i.e., $\smash{\mathrm{Q}=\eta\mathrm{I}_{n}}$) maximizes (11). Thus, the
capacity of an $n$-dimensional unitary MIMO-AWGN channel, still assuming that
the channel is perfectly known at the receiver, is

$\centering C=n\log\left(1+\eta\right).\@add@centering$ (12)

The capacity of the DP channel is given by simply setting $\smash{n=2}$ in
(12).

For a uniform input distribution over a given constellation $\mathcal{S}$, the
integral in (6) must be replaced by a summation over all the constellation
points, i.e.,

$\displaystyle
p(\underline{x}|\mathrm{H})=\sum_{\underline{s}\in\mathcal{S}}{P(\underline{s})p(\underline{x}|\underline{s},\mathrm{H})},$
(13)

where $P(\underline{s})$ denotes the probability mass function of the input
vector and $p(\underline{x}|\underline{s},\mathrm{H})$ is defined in (5).
Using (3) and (13), the MI of the channel for a uniformly distributed discrete
input can be expressed as

$\displaystyle{I}(\mathbf{\underline{s}};\mathbf{\underline{x}})=\mathbb{E}_{\mathbf{\underline{s}},\mathbf{\underline{x}}}\left[\log\left(\frac{|\mathcal{S}|\exp(-\frac{\left\lVert\mathbf{\underline{x}}-\mathrm{H}\mathbf{\underline{s}}\right\rVert^{2}}{\sigma_{z}^{2}})}{\underset{\underline{s}^{\prime}\in\mathcal{S}}{\sum}\exp(-\frac{\left\lVert\mathbf{\underline{x}}-\mathrm{H}\underline{s}^{\prime}\right\rVert^{2}}{\sigma_{z}^{2}})}\right)\right],$
(14)

where the expectations can be estimated numerically.

##  III the Mutual information in presence of Channel Estimation Error

We derived the capacity of the DP channel with the assumption of perfect
channel knowledge at the receiver in (12). In this section, we derive a lower
bound on the MI of the channel in the presence of an estimation error. As
already shown in [43], when the channel is known, the capacity-achieving
distribution is a zero-mean $\mathcal{CN}$ input with a power constraint.
Thus, keeping the capacity-achieving distribution, i.e.,
$\smash{\mathbf{\underline{s}}\sim\mathcal{CN}(0,\Lambda_{\underline{s}})}$
seems reasonable for the imperfectly estimated channel as well. To derive the
AIR $I_{q}$, which is a lower bound on the MI (3) between
$\mathbf{\underline{s}}$ and $\mathbf{\underline{x}}$, the mismatched decoding
inequality [44]

$I(\mathbf{\underline{s}};\mathbf{\underline{x}})\geq
I_{q}=\mathbb{E}_{\mathbf{\underline{x}},\mathbf{\underline{s}}}\left[\log\left(\frac{q(\mathbf{\underline{x}}|\mathbf{\underline{s}})}{q(\mathbf{\underline{x}})}\right)\right]$
(15)

is used, where $q(\cdot)$ stands for the pdf of an auxiliary channel. Note
that this inequality holds for an arbitrary distribution of $q(\cdot)$. The
mismatched channel law is here assumed to be

$q(\underline{x}|\underline{s})=p(\underline{x}|\underline{s},\mathrm{\hat{H}}),$
(16)

which is obtained by replacing $\mathrm{H}$ in (5) with the estimated channel
$\mathrm{\hat{H}}$. This leads to

$\displaystyle q(\underline{x})=p(\underline{x}|\mathrm{\hat{H}}).$ (17)

Also, the covariance matrix of the auxiliary channel’s output is obtained by
replacing $\mathrm{H}$ with $\mathrm{\hat{H}}$ in (II-B), i.e.,

$\displaystyle\hat{\Lambda}_{\underline{x}}=\hat{\mathrm{H}}\Lambda_{\underline{s}}\hat{\mathrm{H}}^{\dagger}+\sigma_{z}^{2}\mathrm{I}_{n}.$
(18)

The average AIR when the estimated channel is random can be written as

$\displaystyle\bar{I}_{q}=\mathbb{E}_{\mathbf{\hat{H}}}[I_{q}]$ (19)

In the following, we first derive an AIR for the MIMO-AWGN channel model with
a fixed channel $\mathrm{H}$ and estimated channel $\mathrm{\hat{H}}$. Then,
we extend the derived AIR to $n$-dimensional unitary channels. By assuming a
unitary estimate of the channel (i.e.,
$\smash{\mathrm{\hat{H}}\mathrm{\hat{H}}^{\dagger}=\mathrm{I}_{n}}$), a
tighter AIR is derived. Finally, we use (19) to consider a random estimated
channel $\mathbf{\hat{H}}$.

###### Theorem 1

Th:Lower_Bound Consider an arbitrary complex MIMO channel matrix $\mathrm{H}$
and a fixed estimated channel matrix $\mathrm{\hat{H}}$. Defining
$\smash{\mathrm{E}=\mathrm{H}-\mathrm{\hat{H}}}$, the AIR is

$\displaystyle I_{q}$
$\displaystyle=\log\left|\mathrm{I}_{n}+\mathrm{\hat{H}}\mathrm{Q}\mathrm{\hat{H}}^{\dagger}\right|-\frac{1}{\ln{2}}\operatorname{tr}\left(\mathrm{Q}\mathrm{E}^{\dagger}\mathrm{E}\right)$
$\displaystyle-\frac{1}{\ln{2}}\operatorname{tr}\left(\mathrm{I}_{n}-\Lambda_{\underline{x}}\hat{\Lambda}_{\underline{x}}^{-1}\right),$
(20)

where the definitions of $\Lambda_{\underline{x}}$, $Q$, and
$\hat{\Lambda}_{\underline{x}}$ can be found in (II-B), (10), and (18),
respectively.

###### Proof:

One may rewrite (15) as

$\displaystyle I_{q}$
$\displaystyle=\overbrace{-\mathbb{E}_{\mathbf{\underline{x}}}\left[\log
q(\mathbf{\underline{x}})\right]}^{A}+\overbrace{\mathbb{E}_{\mathbf{\underline{s}},\mathbf{\underline{x}}}\left[\log
q(\mathbf{\underline{x}}|\mathbf{\underline{s}})\right]}^{B}.$ (21)

Using (7) and (17), given $\mathrm{H}$, the first term $A$ can be calculated
as

$\displaystyle A$
$\displaystyle=-\mathbb{E}_{\mathbf{\underline{x}}}\left[\log\left(\dfrac{1}{\pi^{n}|\hat{\Lambda}_{\underline{x}}|}\exp\left(-\mathbf{\underline{x}}^{{\dagger}}\hat{\Lambda}_{\underline{x}}^{-1}\mathbf{\underline{x}}\right)\right)\right]$
$\displaystyle=\log\left(\pi^{n}|\hat{\Lambda}_{\underline{x}}|\right)+\frac{1}{\ln{2}}\mathbb{E}_{\mathbf{\underline{x}}}\left[\mathbf{\underline{x}}^{\dagger}\hat{\Lambda}_{\underline{x}}^{-1}\mathbf{\underline{x}}\right]$
$\displaystyle=\log\left(\pi^{n}|\hat{\Lambda}_{\underline{x}}|\right)+\frac{1}{\ln{2}}\operatorname{tr}\left(\Lambda_{\underline{x}}\hat{\Lambda}_{\underline{x}}^{-1}\right),$
(22)

where the last step follows the cyclic permutation rule of the trace. For the
second term, $B$, when $\mathrm{H}$ is given, we use (5) and (16) to obtain

$\displaystyle B=$
$\displaystyle~{}\mathbb{E}_{\mathbf{\underline{s}},\mathbf{\underline{x}}}\left[\log\left(\dfrac{1}{\pi^{n}\sigma_{z}^{2n}}\exp\left(-\frac{\left\lVert\mathbf{\underline{x}}-\mathrm{\hat{H}}\mathbf{\underline{s}}\right\rVert^{2}}{\sigma_{z}^{2}}\right)\right)\right]$
$\displaystyle=$
$\displaystyle-\log\left(\pi^{n}\sigma_{z}^{2n}\right)-\frac{\mathbb{E}_{\mathbf{\underline{z}},\mathbf{\underline{s}}}\left[\left\lVert(\mathrm{H}-\mathrm{\hat{H}})\mathbf{\underline{s}}+\mathbf{\underline{z}}\right\rVert^{2}\right]}{\sigma_{z}^{2}\ln{2}}$
$\displaystyle=$
$\displaystyle-\log\left(\pi^{n}\sigma_{z}^{2n}\right)-\frac{\mathbb{E}_{\mathbf{\underline{s}}}\left[\left\lVert\mathrm{E}\mathbf{\underline{s}}\right\rVert^{2}\right]+\mathbb{E}_{\mathbf{\underline{z}}}\left[\left\lVert\mathbf{\underline{z}}\right\rVert^{2}\right]}{\sigma_{z}^{2}\ln{2}}$
(23) $\displaystyle=$
$\displaystyle-\log\left(\pi^{n}\sigma_{z}^{2n}\right)-\frac{\operatorname{tr}\left(\sigma_{z}^{2}\mathrm{I}_{n}+\mathrm{E}\Lambda_{\underline{s}}\mathrm{E}^{\dagger}\right)}{\sigma_{z}^{2}\ln{2}}$
(24) $\displaystyle=$
$\displaystyle-\log\left(\pi^{n}\sigma_{z}^{2n}\right)-\frac{1}{\ln{2}}\operatorname{tr}\left(\mathrm{I}_{n}+\mathrm{Q}\mathrm{E}^{\dagger}\mathrm{E}\right),$
(25)

where (23) follows from the fact that $\mathbf{\underline{z}}$ and
$\mathbf{\underline{s}}$ are independent, and the cyclic permutation rule of
the trace is used in (24). Finally, substituting (22) and (25) into (21)
completes the proof. ∎

For the sake of keeping Theorem LABEL:Th:Lower_Bound as general as possible,
no assumption is made about the transmitter knowledge of the estimated
channel. However, in the context of the optical communications, we are more
interested in the case that the transmitter has no channel knowledge. When the
channel is not known at the transmitter, the only reasonable choice for
$\Lambda_{\underline{s}}$ is to apply uniform power distribution between the
channels.

###### Corollary 1

corollary:1 Assume that the channel is unitary (i.e.,
$\smash{\mathrm{H}=\mathrm{H_{u}}}$) and that uniform power distribution takes
place at the transmitter. Then the AIR of a unitary channel can be written as

$\displaystyle I_{q}$
$\displaystyle=\log\left|\mathrm{I}_{n}+\eta\mathrm{\hat{H}}\mathrm{\hat{H}}^{\dagger}\right|-\frac{1}{\ln{2}}\operatorname{tr}\left(\eta\mathrm{E}^{\dagger}\mathrm{E}\right)$
$\displaystyle-\frac{1}{\ln{2}}\operatorname{tr}\left(\mathrm{I}_{n}-(1+\eta)(\mathrm{I}_{n}+\eta\mathrm{\hat{H}}\mathrm{\hat{H}}^{\dagger})^{-1}\right).$
(26)

###### Proof:

The uniform power distribution
$\smash{\Lambda_{\underline{s}}=P\mathrm{I}_{n}/n}$ yields
$\smash{\mathrm{Q}=\eta\mathrm{I}_{n}}$. Knowing the channel is unitary (i.e.,
$\smash{\mathrm{H_{u}}\mathrm{H_{u}}^{\dagger}=\mathrm{I}_{n}}$) and using
(II-B) and (18), we can write
$\smash{\Lambda_{\underline{x}}=\sigma_{z}^{2}(1+\eta)\mathrm{I}_{n}}$ and
$\smash{\hat{\Lambda}_{\underline{x}}^{-1}=\sigma_{z}^{-2}(\mathrm{I}_{n}+\eta\mathrm{\hat{H}}\mathrm{\hat{H}}^{\dagger})^{-1}}$.
Finally, substituting $\mathrm{Q}$, $\Lambda_{\underline{x}}$, and
$\hat{\Lambda}_{x}^{-1}$ in (20) completes the proof. ∎

###### Corollary 2

corollary:2 Assume that the channel is a fixed unitary matrix $\mathrm{H_{u}}$
and that the estimated channel is an arbitrary random matrix
$\smash{\mathbf{\hat{H}}=\mathrm{H_{u}}-\mathbf{E}}$. Then with a uniform
power distribution, the average AIR is

$\displaystyle\bar{I}_{q}$
$\displaystyle=\mathbb{E}_{\mathbf{\hat{H}}}\left[\log\left|\mathrm{I}_{n}+\eta\mathbf{\hat{H}}\mathbf{\hat{H}}^{\dagger}\right|\right]-\frac{1}{\ln{2}}\operatorname{tr}\left(\eta\mathrm{R}_{\mathrm{E}}\right)$
$\displaystyle-\frac{1}{\ln{2}}\operatorname{tr}\left(\mathrm{I}_{n}-(1+\eta)\mathbb{E}_{\mathbf{\hat{H}}}\left[(\mathrm{I}_{n}+\eta\mathbf{\hat{H}}\mathbf{\hat{H}}^{\dagger})^{-1}\right]\right),$
(27)

where

$\displaystyle\mathrm{R}_{\mathrm{E}}$
$\displaystyle=\mathbb{E}_{\mathbf{E}}[\mathbf{E}^{\dagger}\mathbf{E}].$ (28)

###### Proof:

By applying (19) to (1) the proof is complete. ∎

###### Corollary 3

corollary:3 If $\mathbf{E}$ has a spherically symmetric distribution, then (2)
gives the same $\bar{I}_{q}$ for any $\mathrm{H_{u}}$.

###### Proof:

Since $\mathrm{H_{u}}\mathrm{H_{u}}^{\dagger}=\mathrm{I}_{n}$, we can write

$\displaystyle\mathbf{\hat{H}}\mathbf{\hat{H}}^{\dagger}$
$\displaystyle=(\mathrm{H_{u}}-\mathbf{E})(\mathrm{H_{u}}-\mathbf{E})^{\dagger}$
$\displaystyle=(\mathrm{I}_{n}-\mathbf{E}\mathrm{H_{u}}^{\dagger})(\mathrm{I}_{n}-\mathbf{E}\mathrm{H_{u}}^{\dagger})^{\dagger}$
(29)

and since $\mathbf{E}$ has a spherically symmetric distribution, it is
invariant to rotation. Therefore, for any $\mathrm{H_{u}}$,
$\mathbf{E}\mathrm{H_{u}}^{\dagger}$ has the same distribution as $\mathbf{E}$
and $\mathbf{\hat{H}}\mathbf{\hat{H}}^{\dagger}$ has the same distribution as
$(\mathrm{I}_{n}-\mathbf{E})(\mathrm{I}_{n}-\mathbf{E})^{\dagger}$. Thus, (2)
yields the same AIR independently of $\mathrm{H_{u}}$. ∎

We have not made any assumption on the estimation technique, so the derived
lower bounds hold for an arbitrary estimator. It can be seen that (2) highly
depends on the choice of estimation technique, so one can tighten the bound by
choosing a suitable estimator.

###### Corollary 4

corollary:4 For a complex unitary channel, a unitary estimated channel, and a
uniform power distribution, the AIR is

$\displaystyle\bar{I}_{q}$
$\displaystyle=n\log(1+\eta)-\frac{\eta}{\ln{2}}\operatorname{tr}(\mathrm{R}_{\mathrm{E}}).$
(30)

###### Proof:

By applying
$\smash{\mathbf{\hat{H}}\mathbf{\hat{H}}^{\dagger}=\mathrm{I}_{n}}$ to (2) the
proof is complete. ∎

Note that (30) is independent of $\mathrm{H}$, meaning that the AIR is the
same for any unitary channel. Interestingly, the unitary estimation of the
channel removes two terms on the right-hand side of (2), leading to a simpler
bound.

For uniformly distributed discrete inputs, the pdf of the output of the
auxiliary channel is

$\displaystyle
q(\underline{x})=\sum_{\underline{s}\in\mathcal{S}}{P(\underline{s})q(\underline{x}|\underline{s})},$
(31)

where $q(\underline{x}|\underline{s})$ is defined in (16). Using (15), (16),
and (31), the average AIR for a uniformly distributed discrete input can be
expressed as

$\displaystyle\bar{I}_{q}=\mathbb{E}\left[\log\left(\frac{|\mathcal{S}|\exp(-\frac{\left\lVert\mathbf{\underline{x}}-\mathbf{\hat{H}}\mathbf{\underline{s}}\right\rVert^{2}}{\sigma_{z}^{2}})}{\underset{\underline{s}^{\prime}\in\mathcal{S}}{\sum}\exp(-\frac{\left\lVert\mathbf{\underline{x}}-\mathbf{\hat{H}}\underline{s}^{\prime}\right\rVert^{2}}{\sigma_{z}^{2}})}\right)\right],$
(32)

where the expectation is over $\mathbf{\underline{s}}$,
$\mathbf{\underline{x}}$, and $\mathbf{\hat{H}}$, which can be estimated
numerically.

##  IV Channel Estimation

Figure 1: Transmission block model

In this section, first, the well-established LS estimation algorithm is
presented. Then, a unitary estimation method is proposed for unitary channels.
As illustrated in Fig. 1, to make data-aided channel estimation possible, the
first $L$ symbols of the transmission block are pilot symbols
$\smash{\mathrm{D}=[\underline{s}_{0},...,\underline{s}_{L-1}]}$, which are
assumed to be known at both transmitter and receiver. The optimal pilot
assignment $\mathrm{D}$ should have the following properties [45]:

  * •

$\smash{L\geq n}$

  * •

$\smash{\mathrm{D}\mathrm{D}^{\dagger}=\frac{PL}{n}\mathrm{I}_{n}}$

The complex matrix
$\smash{\mathbf{X}=[\mathbf{\underline{x}}_{0},...,\mathbf{\underline{x}}_{L-1}]}$
of received symbols is

$\displaystyle\mathbf{X}=\mathbf{H}_{\mathrm{u}}\mathrm{D}+\mathbf{Z},$ (33)

where $\smash{\mathbf{Z}\sim\mathcal{CN}(0,\sigma_{z}^{2}\mathrm{I}_{n})}$ is
an $\smash{n\times L}$ matrix of i.i.d. noise samples and
$\mathbf{H}_{\mathrm{u}}$ is an $\smash{n\times n}$ random unitary channel
matrix, which is assumed to remain constant during a transmission block.

Note that the channel estimation problem can be translated to finding a
certain number of independent real values, often regarded as the DOF and
denoted by $\nu$.

###  IV-A LS Algorithm

Conventional optical transmission systems often use LMS for a real-time
estimate of the channel, because LMS tracks the channel change with each
received symbol. However, in this paper it is assumed that the channel is
constant during a transmission block and the LS method is well adapted to
block based transmissions. The LS estimator estimates the channel by
minimizing the squared error between the desired and received signal. For a
MIMO-AWGN channel, the LS optimization problem can be expressed as [46]

$\displaystyle\min\limits_{\mathbf{\hat{H}}}\left\lVert\mathbf{X}-\hat{\mathbf{H}}\mathrm{D}\right\rVert^{2}$
(34)

Knowing $\mathrm{D}$ and $\mathbf{X}$, the solution of (34) is [46],[47, Ch.
8]

$\mathbf{\hat{H}}_{\text{LS}}=\mathbf{X}\mathrm{D}^{\dagger}(\mathrm{D}\mathrm{D}^{\dagger})^{-1}$
(35)

However, the pilots are chosen in such a way that
$\smash{\mathrm{D}\mathrm{D}^{\dagger}=\frac{PL}{n}\mathrm{I}_{n}}$.
Therefore, we can write

$\mathbf{\hat{H}}_{\text{LS}}=\frac{n}{PL}\mathbf{X}\mathrm{D}^{\dagger}.$
(36)

It can be shown that $\mathrm{R_{E}}$ for the LS algorithm is [46]

$\displaystyle\mathrm{R_{E}}=\mathbb{E}_{\mathbf{E}}\left[\mathrm{\mathbf{E}}^{\dagger}\mathrm{\mathbf{E}}\right]=\frac{n\sigma_{z}^{2}}{PL}\mathrm{I}_{n}=\frac{1}{\eta
L}\mathrm{I}_{n}$ (37)

showing that the estimation error of the LS algorithm is inversely
proportional to the SNR and the pilot length $L$.

It is worth mentioning that the LS algorithm has to find $\nu=2n^{2}$
independent real values to make an estimate.

###  IV-B Kabsch Algorithm

The problem with both blind estimation algorithms (e.g., CMA, RDE, and MMA)
and pilot-aided estimation algorithms (e.g., LS and LMS) is that their
optimization problems are not adopted for unitary channel estimation, making
their solutions suboptimal if $\mathbf{H}$ is known to be unitary. Thus, in
this part, we apply the unitary constraint of the channel to the estimation
problem of (34) and write

$\displaystyle\min\limits_{\mathbf{\hat{H}}}\left\lVert\mathbf{X}-\mathbf{\hat{H}}\mathrm{D}\right\rVert^{2}\quad\text{s.t.}\quad\mathbf{\hat{H}}\mathbf{\hat{H}}^{\dagger}=\mathrm{I}_{n}.$
(38)

The optimal solution to this problem is given by the Kabsch algorithm [28] as

$\displaystyle\mathbf{\hat{H}}_{\text{Kabsch}}=\mathbf{UV}^{\dagger},$ (39)

where $\smash{\mathbf{U}\Sigma\mathbf{V}^{\dagger}}$ is the singular value
decomposition function of $\mathbf{X}\mathrm{D}^{\dagger}$. As can be seen,
unlike the blind conventional estimators, Kabsch is independent of the
modulation format of the transmission. The Kabsch algorithm was proposed for
optical communication by Louchet et al. [27] as a blind polarization tracking
algorithm, where decision-directed symbols were used instead of pilots.

In contrast to LS, no analytical result is known for $\mathrm{R_{E}}$ of the
Kabsch algorithm. Although we cannot analytically prove it, we can make an
intuitive prediction by considering that $\smash{\nu=n^{2}}$ for a unitary
estimation of the channel while for a general estimation of the channel
$\smash{\nu=2n^{2}}$. Since the channel estimation problem is equivalent to
finding $\nu$ independent real-valued quantities, one can predict that the
estimation error of Kabsch would be half of LS. More interestingly, for
special unitary channels where $\smash{|\mathrm{H_{u}}|=1}$, the DOF is
$\smash{n^{2}-1}$ and the gain by the unitary algorithm can be even higher;
however, this gain vanishes for large $n$.

Note that in the case of the DP channel, the singular value decomposition is
deployed only on a two-by-two matrix, making it less computationally complex
than for higher $n$.

##  V Numerical Results

In this section, through Monte Carlo trials, the AIR of the DP channel (i.e.,
$\smash{n=2}$) for the estimation algorithms detailed in Section IV is
computed. While AIRs are derived for a fixed channel matrix $\mathrm{H}$, the
estimated channel $\mathbf{\hat{H}}$ is dependent on each realization of the
channel. A deterministic sequence of $L$ QPSK symbols is selected to satisfy
the pilot conditions detailed in Section IV.

$\eta$(dB)

Information Rate (bits/symbol)

Figure 2: The average AIR $\bar{I}_{q}$ of the DP channel as a function of SNR
$\eta$ and estimation error per DOF $\mathcal{E}^{2}$.

Numerical results verify that the estimation error of the Kabsch algorithm
completely follows our prediction in Sec. IV-B. Thus, to perform a fair
comparison between the unitary and nonunitary estimators, we define a new
parameter called estimation error per DOF as
$\mathcal{E}^{2}=\operatorname{tr}{\mathrm{R}_{\mathrm{E}}}/(n\nu)$. Note that
for a general nonunitary estimator $\nu=2n^{2}$ and for a general unitary
estimator $\nu=n^{2}$.

Using $\mathcal{CN}$ inputs, Fig. 2 shows the Monte-Carlo averaged AIR
$\bar{I}_{q}$ of the DP channel according to (2) and (30) for $\smash{n=2}$.
Note that based on corollary:3, for an estimation error $\mathbf{E}$ with a
spherically symmetric distribution, (2) is independent of $\mathrm{H_{u}}$.
For the blue curves, we have assumed that $\mathbf{E}$ is distributed
according to $\mathcal{CN}(0,\mathrm{R_{E}})$ with
$\smash{\mathrm{R}_{\mathrm{E}}=2n^{2}\mathcal{E}^{2}\mathrm{I}_{2}}$. For the
dashed magenta curves, $\mathbf{\hat{H}}$ is unitary and
$\smash{\mathrm{R}_{\mathrm{E}}=n^{2}\mathcal{E}^{2}}\mathrm{I}_{n}$. The
results illustrate that when we have the same $\mathcal{E}^{2}$, the unitary
estimate of the channel leads to a higher AIR. Besides, it can be seen that
for a constant estimation error per DOF $\mathcal{E}^{2}$, the average AIR
reaches a maximum for a specific SNR and then decreases. This can be justified
according to the right-hand side of (2) and (30), where the first term
increases in a logarithmic manner with respect to SNR, but the second term
decreases linearly with SNR. Therefore there is an optimum SNR that maximizes
the AIR. This behavior comes from the assumption of a constant
$\mathcal{E}^{2}$ regardless of the SNR, which, as we shall see next, is not
fully realistic.

(a)$\eta$ (dB)

Information Rate (bits/symbol)

$\mathbf{0.35}$

$\mathbf{0.20}$

(b)$\eta$ (dB)

Information Rate (bits/symbol)

Figure 3: The average AIR $\bar{I}_{q}$ when $\smash{L=8}$ DP pilot symbols
are used. (a) Using $\mathcal{CN}$ inputs where for LS and Kabsch,
$\bar{I}_{q}$ is according to (2) and (30), respectively. (b) Using uniformly
distributed DP-$16$-QAM inputs where $\bar{I}_{q}$ for both LS and Kabsch is
according to (32).

Fig. 3(a) presents the AIR of LS and Kabsch when $\mathcal{CN}$ inputs are
used. The solid red line indicates the receiver perfectly knows the channel.
It can be concluded that with $\smash{L=8}$, Kabsch surpasses LS throughout
the range of considered SNRs. More specifically, at $4$ and $14$ dB SNRs (see
insets), Kabsch has at least $0.2$ and $0.35$ bits per symbol higher AIR,
respectively. Additionally, it is clear from the results that LS is upper-
bounded by Kabsch, and increasing the SNR cannot fill the gap. This behavior
can be justified because, unlike LS, Kabsch guarantees a unitary estimation of
the channel leading to a lower estimation error. Moreover, as the SNR
increases, the gap between Kabsch and LS and the actual channel MI is almost
constant. Since the theoretical gap is
$(\eta/\ln{2})\operatorname{tr}(\mathrm{R_{E}})$ according to corollary:4, we
conclude that the error covariance $\mathrm{R}_{\mathrm{E}}$ of Kabsch is
inversely proportional to the SNR, i.e.,
$\smash{\mathrm{R}_{\mathrm{E}}\propto{1}/{\eta}}$. Unlike Fig. 2, the AIR
bounds are monotonically increasing with SNR which is due to the fact that the
estimation error is decreasing with SNR.

A comparison between LS and Kabsch with DP-$16$-QAM inputs is provided in Fig.
3(b). The MI when the channel is perfectly known at the receiver is marked by
the solid red line. Evidently, Kabsch outperforms LS throughout the considered
range of SNRs. The results also support the fact that Kabsch upper-bounds LS
for various inputs, which completely agrees with Fig. 2, where the unitary
estimation of the channel leads to a higher AIR.

(a)$L$

Information Gap (bits/symbol)

(b)$L$

Information Gap (bits/symbol)

Figure 4: The average information gap $I-\bar{I}_{q}$ for a range of pilot
lengths. (a) $\mathcal{CN}$ inputs where $\bar{I}_{q}$ for LS and Kabsch
algorithms is according to (2) and (30), respectively. (b) Uniformly
distributed DP-$16$-QAM inputs where $\bar{I}_{q}$ for both LS and Kabsch is
according to (32).

Fig. 4 displays the information gap between the AIRs and the capacity of the
DP channel, when (a) $\mathcal{CN}$ inputs and (b) DP-$16$-QAM inputs are
used. The overall dominance of Kabsch throughout the range of considered SNRs
can be easily verified. Evidently, it is beneficial to use higher pilot
numbers at low SNRs. Given the fact that
$\smash{\mathrm{R}_{\mathrm{E}}\propto{1}/({L\eta})}$ for LS (37) and that
Kabsch in Fig. 4 follows the same trend with respect to $L$, we can
empirically conclude that $\mathrm{R_{E}}$ of Kabsch is also inversely
proportional to the pilot length and the SNR.

It is beneficial to use Kabsch to limit the rate loss due to the pilots. For
example, a reasonable performance is achieved by only a block of
$\smash{L=16}$ pilot symbols, which is relatively small compared to the
transmission block size in optical communication systems, implying that the
rate loss is negligible. For instance, for a system operating at a rate of
$28$ Gbaud, even if the channel remains constant for one microsecond (i.e.,
the SOP drift time is one microsecond), it corresponds to a block length $N$
of at least $28,000$ symbols and the rate loss due to $16$ pilots is
negligible.

##  VI Conclusion

The capacity of unitary MIMO-AWGN channels and specifically DP was
investigated. With perfect channel knowledge, the capacity of unitary channels
is the same as for the regular AWGN channel. An AIR with imperfect channel
knowledge was derived and showed that the AIR is highly dependent on the
estimation algorithm. In the case of unitary channels, higher AIR are obtained
with a unitary estimation of the channel. The bounds are derived for any
$\smash{n\geq 2}$ dimensions, meaning that the results can be applied to other
optical channels. In particular, Th:Lower_Bound can be directly applied to
space-division multiplexed channels which are impaired with polarization- and
mode-dependent loss. In contrast to the conventional estimation algorithms,
the proposed data-aided Kabsch algorithm ensures a unitary estimate of the
channel. Numerical results showed that for various input distributions, Kabsch
outperforms LS in terms of AIR. Also, like LS, Kabsch can perform very well
with only a few pilot symbols, making the transmission rate loss due to the
pilots negligible.

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