Advanced Communication Systems course, Technology and Communications Systems Master (ETSIT-UPM), Final Work 2014
The received signal defined by (3) represents the ideal
coherently received optical field, which is: no direct-detection
terms, infinite common-mode rejection between differential
pairs and perfectly matched optical path lengths in the hybrid
resulting in an exact 90º difference between quadratures.
Ideally, the I and the Q channels of a quadrature
communication system are orthogonal to each other. However,
implementation imperfections (e.g., incorrect bias points
settings for the I, Q, and phase ports, imperfect splitting ratio
of couplers, photodiodes responsivity mismatch, and
misadjustment of the polarization controllers) can create
amplitude and phase imbalances that destroy the orthogonality
between the two received channels and degrade performance
of the quadrature system. Also, in practice, phase diversity
receivers are vulnerable to imperfections in the optical 90º
hybrid, which result in dc offsets and both amplitude and
phase errors in the received signals. These effects give rise to
quadrature imbalance in a communication system.
All of these imperfections, can in principle, be compensated
digitally, thereby relaxing the requirements on the optical
components. While blind source separation techniques, such
as independent component analysis could be used [25], it is
often possible to simplify this problem, reducing it to finding
two pairs of mutually orthogonal components
.
In the field of DSP, there are numerous techniques to achieve
this for creating orthogonal components, such as the Gram–
Schmidt orthogonalization algorithm (see subsection II.C).
B. Analog-to-Digital Conversion
Having mapped the signal from the optical domain into the
electrical domain, the next stage is to convert the analog
signals into a set of digital signals. From a functional view, we
can consider the ADC to be made up of two subsystems, a
sampler, which samples the signal in time, converting the
continuous time analog signal into a discrete-time analog
signal, followed by a quantizer, which converts the discrete-
time analog signal into a finite set of values determined by the
bits of resolution in the ADC.
For a digital communication system, which transmits
symbols at a rate of R
s
symbols per second, the minimum
sampling rate is R
s
(in hertz). In general, however, for
asynchronous sampling a sampling rate of 2·R
s
Hz, is
advantageous, giving rise to two samples per symbol, thereby
enabling digital timing recovery. While in practice, there may
be a slight difference between clocks of the transmitter and the
receiver, the signal may be resampled by interpolating of the
digital signal, which will be discussed in subsection II.F.
The receive-side ADC, usually specified in terms of its
effective number of bits (ENoB). As shown in [65], the
required ADC resolution at a 1-dB receiver sensitivity penalty
and at a pre-FEC bit error ratio (BER) typical of coded
systems (e.g., is approximately 3 bits more than what would
be needed to recover the
amplitude levels of the two
signal quadratures if the constellation were received without
any distortions and phase rotations. The technological trade-
off between ADC resolution and bandwidth is analyzed in a
series of papers by Walden [56], [57], and reveals a reduction
of about 3.3 ENoB per decade of analog input bandwidth.
C. Orthogonalization: front-end imperfection compensation
The functionality of this subsystem is to recover the
orthogonality between the I and Q components in each
polarization transmitted. Ideally, the I and the Q channels of a
quadrature communication system are orthogonal to each
other.
However, as we can see in Fig. 4 (constellations simulated
with OptiSystem) the main optical impairments (chromatic
dispersion, polarization-mode dispersion and fiber Kerr
nonlinearities) and hardware imperfections (e.g., incorrect bias
points settings for the I, Q, and phase ports, imperfect splitting
ratio of couplers, photodiodes responsivity mismatch, and
misadjustment of the polarization controllers) can create
amplitude and phase imbalances that destroy the orthogonality
between the two received channels and degrade performance
of the multilevel modulation formats. Also, in practice, phase
diversity receivers are vulnerable to imperfections in the
optical 90º hybrid, which result in dc offsets and both
amplitude and phase errors in the received signals. These
effects give rise to quadrature imbalance in a digital
communication system.
The Gram–Schmidt orthogonalization procedure (GSOP)
enables a set of nonorthogonal samples to be transformed into
a set of orthogonal samples. Given two nonorthogonal
components of the received signal, denoted by
and
,
the GSOP results in a new pair of orthonormal signals,
denoted by
and
, as follows:
Fig. 4. (a) QPSK constellation imbalanced due to the photodiodes
responsivity mismatch at the quadrature receiver structure. (b) I and Q
channels imbalanced in a homodyne receiver due to misalignment between
the axes of the two PBS inside of the quadrature receiver. (c) I and Q
channels imbalanced in a heterodyne receiver due to misalignment between
the axes of the two PBS inside of the quadrature receiver. (d) Correction of
nonorthogonality in the above case using GSOP in OptiSystem.
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