## Abstract

A low-complexity soft-decision 12.5 Gb/s optical receiver for decoding low-density parity-check codes reusing conventional hard-decision receivers is investigated. The simulated and experimental results show that the proposed optical receiver exhibits coding gains of 2.75 and 6.73 dB for post-forward-error-correction bit error rates of 10${}^{-4}$ and 10${}^{-9}$, respectively, over a hard-decision uncoded system for a 60 km single-mode optical fiber transmission. The soft-decision circuitry consumes approximately 5 W and is optimized to provide two confidence levels, one above and one below the hard-decision levels.

©2011 Optical Society of America

## I. Introduction

In recent years, the rapid growth of data-centric services and the general deployment of broadband access networks have been driving the upgrade of dense wavelength division multiplexing (DWDM) networks from 10 Gb/s per channel to more spectrally efficient 40 Gb/s or 100 Gb/s. The major concern at such high data rates over thousands of km of optical fiber is the degradation in the signal quality due to linear and non-linear impairments, in particular, accumulated optical amplifier noise, polarization mode dispersion, and intrachannel nonlinearities [1–4]. Error control coding (ECC) and signal processing play a key role in meeting the ever increasing demand for higher data rate communications and for better quality of service at low cost [1–4]. The extra coding gain provided by ECC can for instance be used to improve the line parameters and maximum span length, relax the optical component and line fiber specifications, or improve the overall quality of communication against degraded operation conditions. Current optical networks employ forward error correction (FEC) based on classical error-correcting codes such as Reed–Solomon (RS) or Bose–Chaudhuri–Hocquenghem (BCH) codes [4,5]. Both RS and BCH codes currently use hard-decision-based receivers that have limited coding gain.

Since the rediscovery of iteratively decodable error-correcting codes performing very close to the Shanon’s capacity limit, wireless communications have been realigned to use a new powerful class of codes including turbo codes and low-density parity-check (LDPC) codes [6–9]. Although LDPC codes were invented by Gallager in 1962 [6,7], they were widely overlooked until the 1990s [8–11]. With the recent increase in computing power, LDPC codes have generated great interest in the wireless community [12–14]. Following this new paradigm, LDPC codes that use a soft decision have been proposed for optical communication systems to mitigate the challenging optical channel impairments in next-generation optical communication systems [15,16]. The performance of different classes of LDPC codes has been assessed extensively through simulation, taking into account certain major transmission impairments such as inter-channel and intra-channel nonlinearities, stimulated Raman scattering, group-velocity dispersion, optical amplifier noise, filtering effect, and channel cross-talk [16]. Results showed that LDPC codes can be an extremely effective solution for high-speed optical systems achieving coding gains of as high as 11 and 5 dB over uncoded and RS(255,239)-based optical systems, respectively [3].

LDPC codes can be very powerful, but their practical implementation for high-data-rate optical communications remains a challenge due to the complex structure of the decoder. Promising works are looking into a simplified structure enabling the same performance [12,13]. Research activities related to LDPC algorithms for optical communication are mostly focused on simulation studies with a few experimental demonstrations [17–22]. In [17–20] a novel soft-decision all-electrical front end is demonstrated where the soft-decision bits are provided by a 3-to-2 encoder with a power consumption of 14 W at 32 GS/s by the soft-decision circuit itself.

In this paper, an alternative low-complexity soft-decision front-end is presented with an experimental investigation with deployable LDPC codes in next-generation optical transmission systems. The proposed optical receiver design can be implemented in deployed hard-decision-based optical systems by tapping the incoming optical signal prior to the photodetector. The implementation is also simplified by replacing the 3-to-2 encoder with a single gate design.

The paper is organized in five sections. In Section II an overview of the soft-decision circuit and the proposed architecture is
described. In Section III the experimental setup
and details of the hardware implementation are presented. In Section IV, the optimum threshold conditions are analyzed. The
performance of the proposed scheme is then evaluated through both simulation and
experimental results for the decoding of an LDPC(32000,29759). Results are presented in
terms of the post-FEC bit error rate (BER) versus the pre-FEC *Q*-factor.
A 12.5 Gb/s non-return-to-zero (NRZ) on–off-keying (OOK) optical system consisting of 60
km of single-mode fiber (SMF-28) with inline chromatic dispersion compensation is used
to experimentally correlate the simulation results. The optimum optical coupling ratio
is studied through optical power penalty measurements and post-FEC BER performance. The
power budget is calculated, and a conceptual implementation at 50 GS/s is proposed.
Finally, we draw our conclusion in Section V.

## II. Soft-Decision Receiver Implementation

In this section we first present the proposed architecture for implementing LDPC decoders within a conventional optical system. We will then discuss the design of a low-complexity 2-bit soft-decision circuit. Finally, the delay matching of the hard-decision and soft-decision branches will be explained.

Conventional direct-detection receivers provide 1 bit of information corresponding to a
hard decision at the optimum decision threshold. The receiver consists of a photodiode
followed by a transimpedance amplifier and a limiting amplifier. The limiting amplifier
digitizes the analog signal. Most of the advanced error correction codes such as LDPC
require soft-decision decoding that involves more than 1 bit of information. The
additional bits provide a confidence level of the digitized hard decision. One approach
is to maintain the analog converted signal and replace the limiting amplifier with a 2–3
bit soft-decision circuit that will provide the confidence levels. This solution has
been proposed in [1,2]. For an *n*-bit soft decision, ${2}^{n}-1$ decision thresholds are required. Figure 1 shows an example of a 2-bit soft-decision where two additional confidence
thresholds are placed on both sides of the optimum hard-decision threshold. At point A
the received signal amplitude is above the hard-decision level as well as above both
confidence threshold levels. As a result, the hard-decision achieved is 1 with a
confidence level of 1. On the contrary, at point B although the hard-decision threshold
is still 1 the upper confidence threshold is 0 and overall confidence will be 0. The
truth table formulated for this approach is presented in Table I where the confidence bit is 1 if the two soft decisions are the
same and 0 if they are not the same [18].

Kobayashi *et al.* demonstrated that after
electrically fanning out the output of the photodetector to four drivers, a 3-to-2
encoder can be used to implement the truth table (Table I) to provide a soft decision for LDPC decoders [18]. In this paper, we propose to extract the soft decision prior to
the optical receiver in a conventional optical system and use it to determine the
confidence levels (Fig. 2). A simple optical
coupler is used in determining the hard decision and confidence levels. However, the
optimum coupling ratio and threshold conditions must be studied carefully because there
is a trade-off between coding gain and power penalty.

From Table I, the output of the encoder that provides the confidence of the received bit can be given by

where ${V}_{Conf}$, ${V}_{Th1}$, and ${V}_{Th0}$ are the output of the soft-decision circuit and upper and lower comparators, respectively. ⊕ and $\overline{\left(\right)}$ denote binary exclusive-or (XOR) and inverse operation, respectively. Therefore, it is quite clear from Eq. (1) that the 3-to-2 encoder functionality can be simplified and implemented using a single high-speed exclusive-nor (XNOR) gate.The soft-decision circuit is implemented for operation at 13 GS/s. The signal from the optical system is optically divided between the soft-decision and hard-decision segments. As shown in Fig. 2, the main building blocks include RF amplifiers, a 1:2 fan-out buffer (HMC744), two high-speed comparators (ADCMP572), and one XNOR gate (HMC745). The optically divided signal is photodetected and amplified using the RF low-noise amplifier with a noise figure (NF) of 2.5 dB and a gain of 26 dB. The photodetector from ${u}^{2}t$ Photonics has an RF bandwidth of 100 GHz and a responsivity of 0.6 A/W. A tunable attenuator is inserted after the RF amplifier to maintain the voltage constant at 320 mVp-p at the input of the 1:2 fan-out buffer. The signal from the fan-out buffer is fed into two comparators. The comparators provide the upper and lower confidence levels. The threshold values ${V}_{Th1}$ and ${V}_{Th0}$ are tuned with precision power supplies with a tuning resolution of the order of 1 mV.

The outputs of the comparators in Fig. 2 are fed into the XNOR gate, which provides the confidence level of the received bit. The output eye of the soft-decision circuit at 12.5 Gb/s is presented in Fig. 3. While reflection can be seen in the rise and fall time of the signal, the eye opening is good (312 mVp-p).

The hard-decision bit and the soft-decision bit traverse through different physical lengths. As a result there will be a delay difference between the hard-decision bit and the confidence bit at the processing device. In order to make sure that during the decoding process the confidence corresponds to the exact hard-decision bit, it is important to make sure that the propagation delay is equal in both branches. We compensated for the measured delay difference of 211 ps with a custom-designed microstrip delay line fabricated in Rogers material [23]. The measured delay responses and bandwidth meet the requirements of the circuit and are presented in Appendix A.

Finally, the hard decision and soft decision are combined using a high-speed 2:1 multiplexer (HMC854). The data is then processed by a high-speed oscilloscope that will be described in detail in Section III.

## III. Experimental Setup

In this section, the experimental setup for performance evaluation of the low-complexity soft-decision circuit is described. For performance analysis, the LDPC has an input data length of 29,759 bits with 7% overhead for an output block length of 32,000 [24]. We choose this code for several reasons: i) longer block-length LDPC codes perform very close to the Shannon limit under iterative decoding algorithms [25,26]; ii) the code has a very low error floor due to the absence of 4-cycles in its graph [24]; and iii) the code length is similar to the LDPC codes proposed in the ITU recommendation, whereas the 7% is the same as currently used Reed–Solomon RS(255,239) codes [5]. The LDPC code is decoded by a sum–product algorithm (SPA) with 32,000 degree-4 variable nodes and 2241 check nodes with 57/58 degrees [24,27]. The number of iterations is set to 50. The code has prospects of future implementation on field-programmable gate arrays (FPGAs) based on reduced-complexity algorithms [28]. Since we want to study the performance of the hardware rather than the code performance, using the SPA satisfies our requirement. The initial log likelihood ratios (LLRs) are calculated accurately using the actual signal amplitude of the received signal [16]. The conditional probability of the observed bit ${x}_{j}$ for the received sample ${y}_{j}$ is given by [16]

*a priori*probabilities and $P\left({y}_{j}\right|{x}_{j})$ are the transition probabilities calculated from the received signal after the photodetector [16]. The initial LLR is calculated by

The initial LLRs calculated by Eq. (2)
are used for decoding the received signal using an iterative soft-decision decoding
algorithm [24,27]. The *n*-bit soft decision is implemented by quantizing
the pre-calculated LLR according to a lookup table for each *Q* value.
The hard-decision decoding is performed by an iterative message-passing algorithm,
commonly known as a Gallager-B algorithm [6,7].

#### A. Optical Experimental Test Bed

The considered system setup for the performance evaluation of LDPC(32000,29759) with optical fiber transmission is shown in Fig. 4. In Fig. 4 the tunable multichannel WDM distributed feedback (DFB) laser source (Thorlabs PRO8000) emits continuous-wave (CW) light at wavelengths of 1537.4, 1538.19, and 1538.98 nm and with an output power of ∼11.5 dBm/channel (point A in Fig. 4). The channels are separated by 0.79 nm to match the 100 GHz ITU grid. Polarization controllers (PCs) are inserted for each of the channels to make sure that all the channels are tuned for equal peak power and maximum extinction at the output of the electro-optic modulator. The channels are multiplexed onto a single fiber using a 4:1 coupler from JDS Uniphase with 6.3 dB of insertion loss. The total optical power at the output of the multiplexer was approximately 10 or 5 dBm per channel (point B in Fig. 4). The CW light is then injected into a single-drive x-cut Mach–Zehnder modulator (MZM) from JDSU, driven by the baseband signal from the output of a programmable pulse pattern generator (PPG), Anritsu MP1800.

The MZM has an insertion loss of 7 dB and a dc extinction ratio of approximately 20
dB. The voltage required to induce a *π* phase shift at the MZM is ${V}_{\pi}=8\text{V}$. We first generated a NRZ OOK signal by biasing the MZM at half the
power of the transmission curve and driven with a ${V}_{\pi}$ amplitude signal. The dc bias of the MZM is tuned at 3.95 V ($\sim {V}_{\pi}/2$) to achieve maximum eye opening at point C before the fiber
transmission. The average total optical power after modulation before being launched
into the fiber was −0.5 or
−5.2 dBm per channel (point C in
Fig. 4).

For an NRZ–OOK signal over fiber transmission, the actual binary bits of the LDPC code are first loaded into the PPG. We insert 225 zeros at the end of each LDPC frame. These bits are used for frame synchronization and identification of the starting of the frames in our decoder simulator. The peak-to-peak voltage of the PPG was set to 250 mV. The baseband signal from the PPG is sent to an optical modulator driver from JDSU (H301-1110) that amplifies input signals to the driving voltage levels of the MZM.

The modulated NRZ–OOK optical channels are first sent through 35.01 km of SMF, with a fiber loss of $\alpha =0.22\text{dB/km}$ and chromatic dispersion of 17 ps/(nm km) at 1550 nm. After the fiber transmission, the signal is optically amplified by an erbium-doped fiber amplifier (EDFA) from INO (FAW CL) with an NF of 5 dB. This EDFA roughly compensates all the fiber and connector losses for 35.01 km, and the output power at point E in Fig. 4 is −0.3 dBm in both cases (gain of ∼7.5 dB).

The optical signal after the first EDFA is then transmitted through 24.23 km of SMF and a spool of dispersion-compensating fiber (DCF). The DCF from Suhner has a total loss of ∼1.2 dB and return loss of 52 dB at 1550 nm. The DCF has a chromatic dispersion of −1035 ps/nm and is designed to perform dispersion compensation for 60 km of SMF-28. The total length of optical fiber we have in our setup is 59.24 km. After the DCF (point F in Fig. 4), the optical signal is amplified by another EDFA from INO (FAD C, NF: 5 dB) to compensate for the losses incurred in the fiber and DCF.

We use a variable optical attenuator to control the amount of optical power launched
into a 50/50 optical coupler. The
other input of the 50/50 coupler is
connected to an ASE source (PriTel FA22) to degrade the optical signal-to-noise ratio
(OSNR) and the eye opening of the received signal, i.e., degrade the pre-FEC
*Q* factor. Then a JDSU tunable grating filter TB45B with bandwidth
of 0.4 nm and insertion loss of 6 dB is used to filter out the middle optical channel
(centered at 1358.19 nm) and remove out-of-band amplified spontaneous emission (ASE)
noise. The output of the filter is eventually fed to the soft-decision circuit of
Fig. 2. We evaluated the eye diagram after
photodetection and the pre-FEC *Q* using a high-speed time-sampling
oscilloscope DCA 86100C from Agilent Technologies with 80 GHz of RF bandwidth. The
received optical spectra for NRZ OOK signals without the ASE noise after the MZM
(point C) (received optical power −0.5
dBm) and before the photodetector but after the optical bandpass filter (point G)
(received optical power −4 dBm) are
shown in Figs. 5(a) and 5(b), respectively. The captured frames and frame intersections
after separating the hard- and soft-decision bits are presented in Fig. 6. There is a time gap of 18 ns (225 bits at 12.5
Gb/s) between each frame.

#### B. Timing and Frame Synchronization

The sampling oscilloscope is programmed to capture 4 samples per bit of the received signal to construct the signal successfully. The oscilloscope is controlled through a GPIB interface and a minimum of 3300 frames (each frame contains 32,000 bits) are captured for each measurement. In total, the number of captured bits is close to the order of 10${}^{8}$ bits. Capturing more frames is not currently feasible due to the requirement of very large disk space and long record times.

After capturing the frames, they were first processed in MATLAB as follows. We first removed 225 zeros that we initially inserted at the end of each LDPC frame. These bits were used for frame synchronization and identification of the start of the frame. Then we removed the dc offset added by the oscilloscope to the signal (40 mV) and downsampled the signal to 1 sample per bit. The optimum sample on the bits was found by performing a hard-decision error counting on the received bits for different sample positions. From the samples of the signal, we decoded the signals by use of a soft-decision algorithm for LDPC decoding.

## IV. Results and Discussion

Using the setup as shown in Fig. 4, we experimentally characterize the impact of optical modulation and fiber transmission using the measured post-FEC BER. The hard-decision and soft-decision thresholds are first analyzed through simulation and experimental studies. We optimize the optical coupling ratio for the best achievable performance and study the possible trade-offs.

For error correction coding analysis, the *Q*-factor is used as a figure
of merit. The *Q*-factor is defined as $Q=20log[({\stackrel{\u0304}{I}}_{1}-{\stackrel{\u0304}{I}}_{0})/({\sigma}_{1}-{\sigma}_{0})]$, where ${\stackrel{\u0304}{I}}_{j}$ and ${\sigma}_{j}$ are the mean and standard deviation of the received mark-bit ($j=1$) and space-bit ($j=0$) [3]. The advantage of using the
*Q*-factor is that it can be measured directly with an oscilloscope
and is easier to correlate with simulation results in the narrow waterfall region of the
LDPC codes. A plot of pre-FEC BER versus *Q*-factor is provided in
Appendix B with experimental points in good
agreement. It is assumed throughout this paper that the optical amplifier noise is the
most significant source of noise and will be the factor degrading the
*Q*-factor. The photodetector noise can be neglected as long as the
received signal at the photodetector is above the sensitivity level of the
photodetector, which is −12 dBm.
Low-noise RF amplifiers are used for amplification after the photodetector and are
assumed not to degrade the *Q*-factor.

#### A. Optimization of the Soft-Decision Threshold

For any soft-decision circuit the best possible error correction performance is obtained only in the case where the decision thresholds are placed precisely [29,30]. In practical systems, all thresholds are set by adaptive decision threshold tracking circuits that continuously tune the levels for optimum decoding performance.

In NRZ OOK systems where the optical amplifier is the primary source of noise the
probability distribution or the amplitude histogram is given by a non-central
chi-squared distribution [31,32]. Although the *Q*-factor
intrinsically assumes a Gaussian distribution, it gives an accurate estimation of the
system performance for non-Gaussian distribution when measured at the optimum
decision threshold [29]. The
*Q*-factor measurement is done with an optimized threshold level
for maximum eye opening on both the signal captured by the sampling oscilloscope and
in simulation. Hence, the measurement methodology is equivalent to finding the
minimum BER of a non-central chi-squared distribution correlating simulation and
experimental results. To generalize the results, we normalize the signal amplitude to
the mean of the mark or the logic 1 [31].
Figure 7 shows the normalized amplitude
histogram of the captured signal for a *Q*-factor of 8.6 dB
(BER = 3.6 × 10^{−3}) from experimental results. The signal shows an asymmetric
distribution for mark (logic 1) and space (logic 0). The hard-decision threshold is
the cross point between the two asymmetric distributions. The optimum normalized
voltage as a function of the *Q*-factor in the waterfall region of the
code is presented in Fig. 8. The normalized
threshold voltage is tuned for a minimum hard-decision BER at each *Q*
point for both the simulation and the experiments. This can be achieved in
experiments by observing the eye diagram on the oscilloscope and selecting the
threshold at the eye crossing. Simulated results in Fig. 8 show that the optimum normalized hard-decision threshold for a
*Q* of 8.6 dB is ∼0.42, which is in good agreement with [31]. The experimental optimum threshold for a hard decision is
around 0.47. The experimental results are in good agreement with the simulated
results. The experimental results are slightly higher in value because the amplitude
distribution broadens to some extent due to the presence of residual chromatic
dispersion in the fiber link.

Now we will consider the confidence thresholds. Due to the chi-squared nature of the
optical channel, a wider spacing for mark and a narrower spacing for space were
chosen for the proposed thresholds [30]. We
follow a similar approach as that for the hard decision and tune the threshold until
the minimum soft-decision post-FEC BER is obtained. The simulated results show that
at the *Q* value of 8.6 dB the simulated upper and lower confidence
levels are at 0.32 and 0.54, respectively. The experimental values are 0.36 and 0.61.
The results show that for optimum confidence levels the ratio of the threshold
spacing between mark-confidence/hard-decision-bit and
space-confidence/hard-decision-bit is ∼1.3. The measured difference between two confidence levels is
approximately 45 mV for an input peak-to-peak voltage level of 320 mV. It is found
that following these ratios provides the optimum BER for a wide range of
*Q* values in the waterfall region of the
code.

#### B. Optimization of the Optical Coupling Ratio

Figure 9 shows the simulated and experimental
results for the decoder. The post-FEC BER is studied as a function of
*Q*-factor. To investigate the performance of the alternate
approach and possible trade-offs, the performance is compared with the electrical
fan-out case where a 1:3 electrical fan-out is used.

Results in Fig. 9 show that if 20% of the
optical power is coupled for a soft decision, there is a degradation of 0.35 dB in
the *Q*-factor with respect to the electrical fan-out case (2-bit
all-electrical soft decision). When compared with the ideal unquantized case, this
penalty is approximately 0.85 dB. The coding gains achieved by extrapolation in
Fig. 9 are 2.75, 6.73, and 9.4 dB for
post-FEC BERs of 10${}^{-4}$, 10${}^{-9}$, and 10${}^{-15}$, respectively. In addition, the coding gain over a hard decision is
1.0 and 1.4 dB for post-FEC BERs of 10${}^{-4}$ and 10${}^{-15}$ (extrapolated), respectively. It should be noted that due to the
structure of its parity check matrix, LDPC codes may exhibit an “error floor”
phenomenon in the experimental post-decoder BER leading to no further improvement in
post-FEC BER after a certain *Q*-factor [1]. The hardware implementation of the LDPC code used in this work
does not exhibit an error floor for a BER as low as 10${}^{-16}$ [5]. The potential existence
of an error floor at a lower BER is not an issue for the system applications
discussed.

Figure 10 shows the optical power penalty for
different coupling ratios for error-free transmission (BER < 10^{−8}). The minimum received optical power for error-free transmission is
first measured when the photodetected signal is electrically fanned out to the soft-
and hard-decision segments. We then observe that for a 20% coupling ratio the optical
power penalty is about 0.9 dB (Fig. 10).

In Fig. 9 the *Q*-factor
degradation increases to 0.54 dB if 10% of the optical power is used. Hence as
expected for a 10% coupling ratio the optical power penalty is also higher,
i.e., 1.6 dB. The performance degrades due to a low signal-to-noise ratio (SNR) for
the soft-decision bit and peak power below the sensitivity of the photodetector.

For a 30% coupling ratio and above, the performance degrades due to lower performance of the hard-decision receiver. For coupling ratios of 40% and 50%, the optical power penalty reaches about 3.1 and 4.2 dB, respectively.

Hence, a coupling ratio of 20% is optimum for our proposed architecture. Although the performance in the optical coupling case is slightly more degraded than the all-electrical fan-out case for the same received optical power, its simplicity makes it an interesting solution. The overall power dissipation is also low, which is analyzed in the following subsection.

#### C. Power Dissipation by the Soft-Decision Circuit

The power consumed by the components is given in Table II. The overall power consumption is approximately 5 W for the soft-decision and 3.8 W for the hard-decision circuitry. It should be noted that more than 70% of the power is consumed by the RF amplifiers. The power consumption can be reduced significantly by using more power efficient RF amplifiers such as the HMC460 from Hittite Microwave Corporation, which consumes 2 W instead of 3.6 W [33]. In such a case, the total power consumed by the hard- and soft-decision circuits would be 5.4 W.

The soft-decision circuit is scalable to 50 GS/s by simply replacing the 13 GS/s components with higher-bandwidth ones. The power consumed by the off-the-shelf components at 50 GS/s are also shown in Table II [33,34]. The overall power consumption at 50 GS/s is 14.3 W (63% power consumed by the MUX). The extra coding gain provided by the soft-decision circuit over the hard-decision circuit can be used with optical differential quadrature phase-shift-keying (DQPSK) modulation for a 100 Gb/s implementation without using a coherent receiver. In addition, the soft-decision circuit can be used as a 50 Gb/s 2-bit analog-to-digital converter (ADC) with a dual-polarization (DP) QPSK-based coherent optical receiver yielding an aggregate bit rate of 200 Gb/s.

## V. Conclusion

In this paper, a low-complexity implementation of a soft-decision circuit has been
proposed for the decoding of LDPC codes in conventional direct detection optical
systems. A certain amount of the optical power prior to the optical receiver was used
for determining the confidence levels of the soft-decision receiver. The proposed scheme
requires no modification of the current optical infrastructure while benefiting from the
soft-decision decoding approach. A low-complexity soft-decision circuit from low-cost
off-the-shelf components was experimentally demonstrated and performance was evaluated
in terms of the post-FEC BER versus the pre-FEC *Q*-factor. The optimized
threshold levels were initially investigated. It was found that the performance of the
hardware is optimum if the difference between two confidence levels is 45 mV for an
input peak-to-peak voltage level of 320 mV where the threshold spacing ratio between
mark-confidence/hard-decision-bit and space-confidence/hard-decision-bit was
∼1.3. The optimum optical coupling
ratio for the soft-decision circuit was also studied. It was shown that, for the
coupling ratio of 80:20, a coding gain of 2.75 and 6.73 dB at an output BER of 10${}^{-4}$ and 10${}^{-9}$ was achieved, respectively, when compared with hard-decision uncoded
systems. The system suffers a *Q*-factor penalty of 0.35 dB. In the
proposed soft-decision receiver for LDPC decoding, fewer components are required and the
implemented hardware consumes relatively lower power.

## Appendix A

The delay length is designed using ANSOFT HFSS software for Rogers RT Duroid 5880 high-frequency laminate [23]. The design requires an electrical length of 360°and Frequency = 1/Delay. The laminate has a dielectric constant of 2.2 and a thickness of 0.508 mm. The simulated microstrip length for 211 ps delay is ∼4.6 cm. The simulated and experimental amplitude and delay responses are presented in Figs. 11(a) and 11(b). The magnitude response tends to decay after 10 GHz due to the bandwidth-limited SMA connectors used for connection. However, the response satisfies our requirements quite well.

## Appendix B

The relationship between the BER and the *Q*-factors is given
by [35]

*Q*-factor at the optimum decision threshold are in good agreement with the calculated BER.

## Acknowledgments

This work was supported in part by Le Fonds québécois de la recherche sur la nature et les technologies (FQRNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to thank Dr. Tho Le-Ngoc (Canada Research Chair in Broadband Communications, McGill University, Montreal, Canada) and Dr. Xiupu Zhang (Advanced Photonic Systems Laboratory, Concordia University, Montreal, Canada) for the use of their lab facilities. The authors are also grateful to Hittite Microwave Corporation, Rogers Corporation, and CMC Microsystems for their kind contributions to this project.

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** Meer N. Sakib** (S’10) received the bachelor’s degree in electrical engineering (with
distinction) from Bangladesh University of Engineering & Technology, Dhaka,
Bangladesh, and the master’s degree in electrical and computer engineering from
Concordia University, Montreal, Canada, in 2006 and 2009, respectively. He is currently
working toward the Ph.D. degree in electrical engineering at the Photonic Systems Group
of McGill University, Montreal, Canada. His research interests include coherent optical
OFDM systems, high-speed optical receivers, and advanced error correction coding for
ultra-high-speed optical communication systems. He currently holds a Vanier Canada
Graduate Scholarship. Mr. Sakib is a student member of the IEEE Photonics Society and
the Society for Photographic Instrumentation Engineers
(SPIE).

** Venkatanarayanan Mahalingam** received the B.Tech degree in electronics and communication engineering from
Shanmugha Arts, Science, Technology, and Research Academy (SASTRA), Thanjavur, India. He
is presently working toward his M.Eng. degree in electrical engineering at the Photonics
System Group, McGill University, Montreal, Canada. His research interests include
advanced error correction coding for optical communications and soft-decision-based LDPC
systems.

**Warren J. Gross** (S’92-M’04-SM’10) received the B.A.Sc. degree in electrical
engineering from the University of Waterloo, Waterloo, Ontario, Canada, in 1996, and the
M.A.Sc. and Ph.D. degrees from the University of Toronto, Toronto, Ontario, Canada, in
1999 and 2003, respectively. Currently, he is an Associate Professor with the Department
of Electrical and Computer Engineering, McGill University, Montreal, Quebec, Canada. His
research interests are in the design and implementation of signal processing systems and
custom computer architectures. Dr. Gross currently serves as an Associate Editor for the
*IEEE Transactions on Signal Processing*. He serves as Secretary of
the Data Storage Technical Committee of the IEEE Communications Society. He is a member
of the Design and Implementation of Signal Processing Systems Technical Committee of the
IEEE Signal Processing Society. He served on the Program Committees of IEEE Globecom,
the IEEE Workshop on Signal Processing Systems, the IEEE Symposium on Field-Programmable
Custom Computing Machines, and the International Conference on Field-Programmable Logic
and Applications and as the General Chair of the 6th Annual Analog Decoding Workshop.
Dr. Gross is a Senior Member of the IEEE and a licensed Professional Engineer in the
Province of Ontario.

**Odile Liboiron-Ladouceur** (M’95) received the B. Eng. degree in electrical
engineering from McGill University, Montréal, QC, Canada, in 1999 and the M.S. and Ph.D.
degrees in electrical engineering from Columbia University, New York, in 2003 and 2007,
respectively. Her doctoral research work focused on the physical layer of optical
interconnection networks for high performance computing. In 2007, Dr. Liboiron-Ladouceur
received a postdoctoral fellowship from the Natural Sciences and Engineering Research
Council of Canada (NSERC). She is currently an Assistant Professor in the electrical and
computer engineering department at McGill university. Her research interests include
energy efficient photonic interconnects for high-performance optical systems. Dr.
Liboiron-Ladouceur currently serves as an Associate Editor for *IEEE Photonics
Technology Letters* and is the Chair of the IEEE Photonics Society Montreal
Chapter. She is the author or coauthor of over 40 papers in peer-reviewed journals and
conferences.