Erwan Pincemin
email: erwan.pincemin@orange.com

Mengdi Song
email: mengdi.song@orange.com

Julie Karaki
email: julie.karaki@telecom-bretagne.eu

Omid Zia-Chahabi
email: omid.ziachahabi@orange.com

Thierry Guillossou
email: thierry.guillossou@orange.com

Didier Grot
email: didier.grot@orange.com

Gilles Thouenon
email: gilles.thouenon@orange.com

Christophe Betoule
email: christophe.betoule@orange.com

Rémi Clavier
email: remi.clavier@orange.com

A Poudoulec
email: alain.poudoulec@yenista.com

M Van Der Keur

Y Jaouën

R Le Bidan

T Le Gall

P Gravey

M Morvan

B Dumas-Feris
email: barbara.dumasferis@telecom-bretagne.eu

M L Moulinard

G Froc
email: gfroc@merce.fr

Multi-band OFDM transmission at 100 Gbps with sub-band optical switching

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Index Terms-CAPEX cost, DP-QPSK, MB-OFDM, subwavelength optical switching, 100 Gbps WDM transmission.

I. INTRODUCTION

I N the next generations of optical transport networks, flexibil- ity will be carried out to a high degree of efficiency, in order to increase significantly the capacity transported by the fiber bandwidth but also to optimize as much as possible the aggregation process through the various network layers (from optical to IP layer). Indeed, the filling of the communication connections into a network is dimensioned for the peak of traffic and not for the average data-rate, involving a traffic over-provisioning of ß45% [START_REF] Morea | Data-rate adaptation for night-time energy savings in core networks[END_REF]. The idea is thus to perform an optimization of the multi-layer transport network architecture in order to deploy only the equipments that are necessary. That starts by introducing flexibility into the optical layer of the network [START_REF] Zami | What is the benefit of elastic superchannel for WDM network?[END_REF], and in particular by performing sub-wavelength optical switching over the up-coming super-channels. The proposed scheme consists in operating an optical separation-aggregation of the independent sub-bands which constitute the super-channels thanks to the use of ultra-selective optical pass-band and stop-band filters [START_REF] Dischler | Demonstration of bit rate variable ROADM functionality on an optical OFDM super-channel[END_REF], [START_REF] Pincemin | Multi-band OFDM transmission with sub-band optical switching[END_REF]. To push farther the flexibility, the idea would be also to adapt the capacity carried by the sub-bands to the services that are transported. For instance and as illustrated in the lower part of the Fig. 1 below, the data-rates carried by each sub-wavelength could take the following values: 12.5, 25, 33.3, 50, 66.6 and 100 Gbps. By using the MB-OFDM technique, this bit-rate elasticity would be obtained by playing both over the modulation format transported by the sub-carriers constituting the OFDM sub-bands (i.e., BPSK, QPSK, 8-QAM, 16-QAM), and over the number of sub-carriers constituting each OFDM sub-band (or equivalently over their bandwidth).

According to us and as demonstrated in this manuscript, MB-OFDM is the preferred candidate to achieve such a high degree of flexibility. Indeed, thanks to its rectangular spectrum and the possibility to easily pre-distort electronically its profile (to take into account for instance the distortion induced by the OADM cascade), OFDM is naturally robust to ultra-narrow optical filtering. From its side, Nyquist-WDM super-channel at 32 Gbauds seems particularly sensitive to 37.5-GHz optical filtering [START_REF] Morea | Impact of reducing channel spacing from 50 GHz to 37.5 GHz in fully transparent meshed networks[END_REF], [START_REF] Zami | Interest of mix-grid setup for elastic spectral efficiency[END_REF]. Furthermore, the opportunity to play both on the modulation format and bandwidth (or equivalently on the number of sub-carriers) of each OFDM sub-band gives the means to better manage the fiber bandwidth and to limit as much as possible the gaps which are created inside the spectrum. Note that this fiber bandwidth management is less easy to achieve with the Nyquist-WDM technology operating at a constant baud-rate, as proposed today by the equipment suppliers [START_REF]Corestream agility optical transport system brochure[END_REF]- [START_REF] Laperle | Flexible transceivers[END_REF]. Fig. 1 below compares the short-term approach of flexibility proposed today by the system vendors, based on the combination of Nyquist-WDM super-channel at 32 Gbauds and mixed 37.5/50-GHz spectral grid [START_REF] Morea | Impact of reducing channel spacing from 50 GHz to 37.5 GHz in fully transparent meshed networks[END_REF]- [START_REF] Gringeri | Technical considerations for supporting data rates beyond 100Gbps[END_REF], and our MB-OFDM approach, based on the implementation of MB-OFDM technique and gridless management of the fiber bandwidth. In the short-term approach, the sub-bands inside the Nyquist-WDM super-channel are spaced by 37.5 GHz, the guard-band between the WDM channels is typically of ß18 GHz, and the capability to optically switch the sub-bands inside the super-channel is not authorized. The data-rate carried by the WDM channels are either 50 Gbps (with BPSK), 100 Gbps (with QPSK), 200 Gbps (with 16-QAM), or 400 Gbps for the super-channel grouping two 200 Gbps 16-QAM sub-bands [START_REF]Corestream agility optical transport system brochure[END_REF]- [START_REF] Laperle | Flexible transceivers[END_REF]. In our MB-OFDM approach, the spectral granularity for the OFDM sub-bands is 10 GHz, the OFDM sub-bands occupy a spectral slot of 10 or 20 GHz and can be grouped together to constitute a superchannel transporting 50, 100, 200, or 400 Gbps (as shown in Fig. 1 below). Note that we have deliberately chosen a spectral granularity (10 GHz) which is different from the one defined in the standards (12.5 GHz), in order to illustrate more concretely what is the gridless approach. Thanks to the use of ultra-narrow optical pass-band and stop-band filters, the optical add-drop of OFDM sub-bands is allowed. The guard-band between the OFDM sub-bands is typically of ß2-4 GHz (as shown later in this paper). With this scheme, the adaptability of the optical layer to the services transported is very high and the management of spectral gaps in the fiber bandwidth is greatly eased by the flexibility provided by the OFDM format. Note however that a certain complexity is associated with this proposed flexibility scheme, as the performance of Nyquist-WDM or MB-OFDM sub-bands in terms of maximum transmission distance depends on the modulation format implemented within each sub-band.

In this paper, we present an original work on sub-band optical switching performed over a coherent MB-OFDM superchannel operating at 100 Gbps. In Section I, we detail how the coherent MB-OFDM signal is generated and detected. In particular, we describe the main steps of the digital signal processing (DSP) operated at the transmitter and receiver side. In Section II, we compare the transmission performance of our 100 Gbps MB-OFDM super-channel constituted of four polarization-multiplexed sub-bands with that of single-carrier 100 Gbps DP-QPSK over an uncompensated 10 × 100-km G.652 fiber-based transmission line. In Section III, the design and realization of the ultra-narrow optical filters required for the "add-drop" process is discussed, and the obtained profiles for both the pass-band and stop-band filters are presented. In Section IV, the assessment of the efficiency of our FOADM to add-drop OFDM sub-bands as narrow as 8 GHz inside the 100 Gbps MB-OFDM super-channel is carried out both in backto-back and in the previously introduced 1000-km transmission line [START_REF] Pincemin | Multi-band OFDM transmission with sub-band optical switching[END_REF]. In particular, the bandwidths of the "drop" and "notch" filters used into the FOADM are optimized as a function of the guard-band separating two consecutive sub-channels. In Section V, the question of the cascadability of the FOADM is studied thanks to the insertion of the add-drop multiplexer into a 2 × 100-km G.652 fiber-based recirculating loop. The impact of the cumulated filtering transfer function, as well as the crosstalk inside the FOADM, is investigated. Finally, in the last part of the paper, a typical use case for the FOADM introduction into a long-haul transport network is discussed, and the capital expenditure (CAPEX) cost advantage of MB-OFDM and sub-band optical switching techniques for multi-layer transport networks is highlighted.

II. COHERENT OFDM: GENERATION AND DETECTION

In this section, we first describe the transmitter and receiver architecture for the generation and detection of a single-band coherent optical OFDM signal. We then introduce the experimental setup used to build a 100 Gbps MB-OFDM signal from four independent 25 Gbps OFDM sub-bands.

A. Single-Band Coherent OFDM Transmitter Architecture

The basic idea behind OFDM is the sharing of a high bitrate data stream between a high number (in general, several hundreds) of low bit-rate tributaries transported each by one sub-carrier [START_REF] Jansen | Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF[END_REF]- [START_REF] Yang | Experimental demonstration and numerical simulation of 107-Gb/s high spectral efficiency coherent optical OFDM[END_REF]. The essential operation of OFDM is the inverse fast Fourier transform (IFFT) which permits to superimpose temporally into one OFDM symbol the symbols transported by the previously introduced sub-carriers, at the transmitter side. The number of time-domain samples into the created OFDM symbol is then equal to the number of sub-carriers that feed the IFFT. Fig. 2 presents the block diagram of a typical OFDM transmitter and receiver. A high bit-rate binary data stream is first split into a high number of low bit-rate binary data tributaries by a serial-to-parallel conversion. After conversion of these binary data streams into quadrature amplitude modulation (QAM) symbols, some training symbols and sub-carriers (named pilot tones) are introduced for channel and phase noise estimation/correction (performed at the receiver side). As already explained, the IFFT permits to superimpose temporally into one OFDM symbol the QAM symbols carried by the low symbol-rate tributaries, while their orthogonality is ensured by spacing the sub-carriers of a value inversely proportional to the symbol duration. Orthogonality can be declined into the spectral or temporal domain. In the spectral domain, orthogonality is guaranteed by the sub-carrier spacing equal to a multiple of the inverse of the symbol duration, as well as by the sinc spectrum of the OFDM sub-carriers. This sinc profile ensures that sub-carriers will not interfere after demodulation and will not be affected by inter-carrier interference (ICI). Orthogonality in the time domain is guaranteed by the presence of an integer number of cycles into the OFDM symbol for each of the considered OFDM sub-carriers. A cyclic prefix (or guard interval) is then added to the beginning of each OFDM symbol in order to prevent inter-symbol interference (ISI) over time-dispersive channels. The cyclic prefix is formed by the copy of the last samples of the OFDM symbol, the number of samples being determined by the maximum channel delay spread. Clipping is then performed in order to suppress the peaks of amplitude, which affect the OFDM signal and could result into an inefficient use of the dynamic range of the digital-to-analogue converters (DAC). Spectral pre-emphasis is also carried out to compensate for the non-flat frequency response of DAC, low-pass frequency filter (LPF) and RF driver. A parallel-to-serial conversion is then applied to obtain the in-phase (I) and quadrature (Q) components of the OFDM signal. The I/Q tributaries feed a pair of DAC embedded into an arbitrary waveform generator (AWG), whose 3-dB bandwidth and sampling speed are equal to ß10 GHz and 12 GSamples/s, respectively. LPF are applied at the DAC outputs to remove the aliasing products produced by the sampling process. RF drivers operating in their linear regime feed a complex Mach-Zehnder modulator (MZM), which optically mixes together the I and Q data streams. An external cavity laser (ECL) with 100-kHz linewidth feeds the complex MZM, which is itself connected to a polarization-division-multiplexing (PDM) unit. The role of this PDM unit is to generate the dual-polarization OFDM signal at a reasonable cost. It consists in replicating the OFDM signal and in delaying the two copies by one symbol delay. A variable optical attenuator (VOA) inserted into the lower arm of the PDM unit equilibrates the power of each of the two OFDM replicas, which are coupled by the means of a polarization beam combiner (PBC). The polarization extinction ratio of each OFDM replica is increased by polarizers embedded into the lower and upper arms of the PDM unit.

B. Single-Band Coherent Optical OFDM Receiver Architecture

At the receiver side, as shown in the lower part of Fig. 2, a polarization beam splitter (PBS) separates the polarizationmultiplexed OFDM signal in its two components, carried respectively by the X and Y polarizations. For each of these two polarization components, a 90°hybrid firstly mixes together the local oscillator (LO) and the OFDM signal, and secondly separates the I and Q tributaries. Four balanced photodiodes detect the I X , Q X , I Y and Q Y data streams, which are sent over four analog-to-digital converters (ADC) embedded into a real-time storage oscilloscope working at 50 GSamples/s.

The DSP at the receiver side is then performed in order to compensate for system as well as propagation impairments and recover the binary data sent over the X and Y polarizations. The expression of the two time-domain samples r nj received over the X and Y polarizations at relative time index n within OFDM symbol j can be written in vector form as:

r nj = e i 2π∆ CFO • (j (N SC + N g )+n) 1 f S + ϕ nj × L m -1 l=0 h (l) * x (n -l-p)j + n nj .
∆ CFO is the carrier frequency offset (CFO) between the LO and the laser used into the transmitter, N sc is the number of OFDM sub-carriers, N g is the cyclic prefix length, f s is the DAC/ADC sampling frequency, φ nj is the laser phase noise, h is the channel impulse response having at most L m discrete paths, x nj are the transmitted time-domain samples at time index n within OFDM symbol j, p is an integer timing offset, and n nj is the amplified spontaneous emission noise.

Timing synchronization is first implemented in order to determine the start of the OFDM symbol. Timing errors will cause the FFT window to be applied at an incorrect location in the received signal, resulting in both ISI and ICI, and leading to a severe degradation of the system performance. Hereafter, we use synchronization algorithms based on the insertion of a training symbol constituted by the L-fold repetition of a pre-determined basic pattern B, possibly with alternation of signs. Schmidl and Cox [START_REF] Schmidl | Robust frequency and timing synchronization for OFDM[END_REF], Minn and Barghava [START_REF] Minn | A simple and efficient timing offset estimation for OFDM systems[END_REF] and Shi and Serpedin [START_REF] Shi | Coarse frame and carrier synchronization of OFDM systems: A new metric and comparison[END_REF] where K is an appropriate auto-correlation metric, E is an energy normalization metric, and p ranges over the admissible time index. In the following, the Minn and Barghava algorithm has been chosen for its accuracy.

Compensation of CFO between the LO and the laser used into the transmitter is the second DSP step. Generally, CFO results in a phase offset common to all the sub-carriers of a given symbol, in a complex attenuation common to all the sub-carriers and symbols, and in ICI. While the two first effects can be corrected, ICI is very detrimental and cannot be compensated. In order to constrain CFO to a relatively low value and avoid too much ICI, the wavelength of the LO has to be tuned to match (at least approximately) the wavelength of the ECL used at the transmitter side. The technique used to coarsely tune the LO can be decomposed in two steps: firstly, the spectrum of the data sampled by the ADC is generated by Fourier transform; secondly, the base-band spectrum of the OFDM signal originally emitted is recovered by shifting the wavelength of the LO accordingly. The usual approach for CFO compensation is then to decompose CFO into its fractional and integer part. is then obtained for the fractional CFO part:

∆ CFO = L 2 • f s N SC • arg ⎛ ⎝ L N SC N SC L -1 n =0 r * nj r n +1. N SC L j ⎞ ⎠ .
The estimation of the integer CFO part is carried out by recovering the spectrum of the data over the X and Y polarizations through a discrete Fourier transform (DFT). By comparing the recovered spectrum at the receiver side with the original spectrum generated at the transmitter side, it is straightforward to determine the integer CFO [START_REF] Karaki | Frequency offset estimation robustness in a polarization-multiplexed coherent OFDM system stressed by chromatic dispersion and polarization mode dispersion[END_REF]. Fig. 3 below illustrates this method. In this example, the integer CFO estimated by comparing the spectrum obtained after DFT at the transmitter and receiver side is equal to -21∆f( ∆f denoting the sub-carrier spacing).

After removal of the cyclic prefix, the OFDM signal is converted back to the frequency domain by fast Fourier transform (FFT). Channel estimation and equalization is then performed. Its role is to compensate for the gain/phase distortion experienced by the sub-carriers, as well as to carry out polarization separation. The channel estimate is computed by using the training symbols and by supposing that the data symbols are disturbed by the same impairments than the training symbols. 2 × 2 multipleinput-multiple-output (MIMO) least square channel estimation is implemented. We reuse here a special structure for the training symbols, which has been proposed in [START_REF] Yang | Experimental demonstration and numerical simulation of 107-Gb/s high spectral efficiency coherent optical OFDM[END_REF], [START_REF] Jansen | Long-haul transmission of 16 × 52.5 Gbits/s polarization-division multiplexed OFDM enabled by MIMO processing[END_REF] and is depicted in Fig. 4 above. It consists in transmitting two training symbols on each polarization, for instance twice in the duration of the OFDM frame (as illustrated in Fig. 4). One arbitrary OFDM symbol (in blue over Fig. 4) followed by an empty symbol is sent over the X polarization, while an empty symbol followed by another arbitrary OFDM symbol (in green over Fig. 4) is launched over the Y polarization. In a dual-polarization OFDM system, the signal received on each polarization is a linear combination of the two polarization-multiplexed transmitted signals. Specifically, for a given sub-carrier k, H xx (k) and H yx (k) represent the energy transferred from the transmitted polarization X towards the received polarizations X and Y, respectively. Similarly, H xy (k) and H yy (k) represent the energy transferred from the transmitted polarization Y towards the received polarizations X and Y, respectively. For our two previously defined training symbols [c kj 0] over the X polarization and [0, c k (j +1) ] over the Y polarization, the received symbols [R kj R k (j +1) ] in the frequency domain on each sub-carrier k are given by ⎡

⎣ R kj,x R k (j +1),x R kj,y R k (j +1),y ⎤ ⎦ = ⎡ ⎣ H xx (k) H xy (k) H yx (k) H yy (k) ⎤ ⎦ • ⎡ ⎣ c kj,x 0 0 c k (j +1),y ⎤ ⎦ .
The four elements of the H matrix are then easily obtained by

H xx (k)= R kj,x c kj,x H xy (k)= R k ( j +1),x c k ( j +1),y H yx (k)= R kjy c kj,x H yy (k)= R k ( j +1),y c k ( j +1),y .
Zero-forcing (ZF) equalization is used. It has the double advantage of not requiring any other knowledge than the channel frequency response, and more importantly of being optimal for optical channels whose frequency response has unit magnitude ([H k | =1), thereby avoiding the classical problem of noise enhancement. The 2 × 2 MIMO ZF equalizer output can then be written as

Rkj = H -1 k × R kj
where H -1 k is the inverse of the 2 × 2 channel frequency response (or, in practice, its estimate) for the sub-carrier k.

The last processing to perform is the compensation of the laser phase noise, which constitutes a major impairment for the performance of OFDM systems. By decomposing the phase noise φ nj into a mean rotation φ j and a random fluctuation ∆φ nj around the mean, the received signal R kj in the frequency domain at the DFT output can be written as

R kj = N SC -1 n =0 e i φ j α j e i φ j H k c kj + N SC -1 n =0 e i∆ φ nj N SC -1 l =0 l = k H l c lj e -i2π. n (1-k ) N SC ICI .
Laser phase noise has two kind of effect on the OFDM signal. First, it results in an attenuation α j and a phase shift φ j , which impairs all the sub-carriers in the symbol j and differs from one symbol to the other. Second, it involves ICI. If we assume that the phase noise trajectory slowly evolves around its mean value along the OFDM symbol, and neglect the small fluctuations ∆φ nj , then α j ß1 and ICI can be neglected. Only the common phase error (CPE) term φ j has to be considered and corrected [START_REF] Petrovic | Effects of phase noise on OFDM systems with and without PLL: Characterization and compensation[END_REF]. CPE estimation is performed here thanks to the pilot tone method. It allows correcting the phase rotation common to all the sub-carriers in an OFDM symbol. After extraction of the pilot tones for each OFDM symbol j, the phase difference between the received pilot sub-carriers and the transmitted ones is evaluated. The CPE term φ j affecting the OFDM symbol j is estimated by averaging the phase difference over the pilot tones

exp (i φ j )= 1 N p k ∈I Rkj ×|c kj | c kj × Rkj
where I is the set of pilot tone indices, N p is the number of pilot tones, Rkj is the equalizer output for the pilot tone of index k, and c kj is the known transmitted symbol on this pilot tone. The CPE is finally corrected by multiplying each OFDM symbol by the conjugate of the previously introduced estimated CPE complex exponential function ĉkj = Rkj . exp (-i. φ j ) .

C. Generation of the 100 Gbps Multi-Band OFDM Signal

We are now describing in detail how the 100 Gbps coherent DP-MB-OFDM signal is generated experimentally. We first explain how one polarization-multiplexed OFDM sub-band carrying 25 Gbps is produced [START_REF] Karaki | First and secondorder PMD impact over 100 Gbps polarization-multiplexed coherent MB-OFDM system under realistic "field" conditions[END_REF]. The two DAC of a Tektronix 7122B arbitrary waveform generator operating at a sampling rate of 12 GSamples/s generate the I and Q tributaries of the OFDM signal, which have both a bandwidth of ß4 GHz. 256 sub-carriers constitutes the OFDM signal. Each OFDM subcarrier is modulated by a QPSK constellation. A first sequence of training symbols [0TS1] is inserted at the frame beginning We compare its performance with that of 100 Gbps DP-QPSK. Our previously described 124.4 Gbps DP-MB-OFDM transmitter operating at 1552.93 nm is combined with a multiplex of 79 wavelengths spaced by 50 GHz and modulated at 112 Gbps by DP-QPSK (to take into account the 7% FEC overhead). The 100 Gbps DP-QPSK channel used here as a reference is fed by a 100-kHz linewidth ECL at 1548.11 nm (to limit the impact of laser phase noise), while the 78 other channels ranging from 1529.16 to 1560.61 nm are fed by cheap standard laser diodes (LD). Decorrelated 2 15 -1 pseudo-random bit sequences (PRBS) are used to drive the I and Q ports of the CMZM which generates the QPSK constellation. A 10-ns timing delay is introduced between the two replica of the QPSK signal inside the PDM unit, in order to generate the DP-QPSK signal. The 112 Gbps DP-QPSK odd and even channels are separately multiplexed, independently modulated, and combined with the 124.4 Gbps DP-MB-OFDM signal, before being injected into the optical link. The optical link is constituted of 10 spans of G.652 standard single-mode fiber (SSMF), separated by Erbium-doped fibre amplifiers (EDFA) with 20 dB gain and 5.5 dB noise figure (NF). In the middle of our transmission line, a dynamic gain equalizer (DGE) is inserted in order to flatten the multiplex power after 1000-km. This uncompensated transmission configuration is realistic of the 100 Gbps WDM systems which are today deployed by operators and which presents the best performance, when compared to legacy dispersion-managed systems [START_REF]Corestream agility optical transport system brochure[END_REF], [8].

At the receiver side, the DP-MB-OFDM and DP-QPSK signals are selected by a square flat-top optical band-pass filter (OBPF) of 0.4 nm bandwidth, and detected by a polarization diversity coherent receiver using a ß100-kHz bandwidth ECL as LO. The signals are converted back to the digital domain thanks to four ADC operating at 50 GSamples/s and embedded into a Tektronix DPO 72004C real-time storage oscilloscope. In the DP-MB-OFDM case, the LO wavelength is tuned to the centre of the OFDM sub-band under measurement, while the DSP is the one described in part II. In the DP-QPSK case, DSP is based on blind equalization, and more particularly on the constant modulus algorithm which carries out polarization separation and residual chromatic dispersion compensation [START_REF] Godard | Self-recovering equalization and carrier tracking in twodimensional data communication systems[END_REF]. The fixed part of chromatic dispersion is compensated thanks to two finite-impulse response filters located on the X and Y polarizations [START_REF] Savory | Digital filters for coherent optical receiver[END_REF], respectively. CFO compensation and carrier phase estimation are carried out by the methods described in [START_REF] Leven | Frequency estimation in intradyne reception[END_REF] and [START_REF] Ly-Gagnon | Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation[END_REF], respectively. To avoid cycle slips, coherent detection followed by differential decoding is used. Bit-error rates (BER) are evaluated by launching several tens of successive acquisitions and by averaging the corresponding BER. In the OFDM case, 400 acquisitions are carried out and 125 OFDM symbols on each polarization are saved at each acquisition. BER is thus calculated by using 25.6 millions of QPSK symbols (125 OFDM symbols x 256 sub-carriers × 2 polarizations × 400 acquisitions). In the DP-QPSK case, 50 acquisitions are performed and 1.12 millions of QPSK symbols on each polarization are saved at each acquisition. BER is thus calculated by using 112 millions of QPSK symbols (1.12 millions symbols × 2 polarizations × 50 acquisitions).

In Fig. 7, the bit-error rate (BER) of DP-MB-OFDM and DP-QPSK is measured as a function of the optical signal-tonoise ratio (OSNR) in 0.1 nm, in the back-to-back configuration. We note that the two curves are nearly superimposed, showing that the quality of our experimental implementation of 100 Gbps DP-MB-OFDM is quite good, and ensuring that the transmission comparison between the two modulation formats will not be impacted by limitations of the DP-MB-OFDM transmitter/receiver. Furthermore, the FEC limit (fixed here at a BER = 2 × 10 -3 ) is reached for an OSNR in 0.1 nm equal to 15.36 dB for both 100 Gbps DP-MB-OFDM and 100 Gbps DP-QPSK. The 10 × 100-km uncompensated transmission line is now inserted between the transmitter and the coherent receiver. The transmission performance is measured in different configurations for the two modulation formats under study. In the singlechannel (SC) configuration, the channels are not modulated any more, except the DP-MB-OFDM channel at 1552.93 nm and the DP-QPSK channel at 1548.11 nm. The WDM configuration corresponds to the experiment depicted at the beginning of the paragraph III. Fig. 8 shows the BER evolution as a function of the span input power per channel (P IN SPAN ) for both the SC and WDM configurations and the two modulation formats under study. We notice first that the SC configuration is better than the WDM one from approximately one BER decade, while ß1 dB more power per channel can be injected into the spans. It clearly means that the cross-phase modulation (XPM) and its related cross-polarization modulation (XPolM) exacerbated by the WDM scheme badly impact the WDM transmission. In the SC configuration, only the self-phase modulation (SPM) excited at the locations of the high peak power impacts the transmission performance. In the WDM configuration, the interplay between the high peak power in the neighboring channels (through XPM and XPolM) adds an extra penalty to the WDM system performance. When we now compare the two modulation formats between them, we notice that in the SC configuration, after the 1000-km transmission, the 100 Gbps DP-QPSK channel (BERß1 × 10 -5 at P IN SPAN ß1 dBm) is a little better than the 100 Gbps DP-MB-OFDM channel (BERß2.4 × 10 -5 at P IN SPAN ß0 dBm) both in terms of BER and optimal span input power. Also, in the WDM configuration, DP-QPSK (BERß6 × 10 -5 at P IN SPAN ß0 dBm) is still slightly better than DP-MB-OFDM (BERß2 × 10 -4 at P IN SPAN ß 0dBm). This indicates a slightly higher sensitivity of 100 Gbps DP-MB-OFDM to nonlinear effects compared to 100 Gbps DP-QPSK. The exacerbation of four-wave mixing (FWM) that occurs between OFDM sub-carriers [START_REF] Lowery | Calculation of power limit due to nonlinearity in optical OFDM systems[END_REF], [START_REF] Nazarathy | The FWM impairment in coherent OFDM compounds on a phased-array basis over dispersive multi-span links[END_REF] and between the OFDM sub-bands is probably the source of this slightly higher sensitivity to nonlinearities. However, in the present case of uncompensated transmission line, this effect is controlled by the continuous accumulation of chromatic dispersion along the line: the phase matching conditions which are mandatory to exacerbate intra and inter-band FWM are rarely met, which limits the FWM impact [START_REF] Nazarathy | Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links[END_REF].

IV. ULTRA-NARROW PASS-BAND AND STOP-BAND OPTICAL FILTERS

A double target is followed for the design and building of optical pass-band and stop-band filters used in the sub-wavelength optical switch. They have to be ultra-selective to extract and suppress OFDM sub-bands as narrow as 8 GHz, while having a transfer function as rectangular as possible in order not to distort too much the neighboring sub-bands. Free-space optical (FSO) technology, based on bulk diffraction grating and filtering slit, is particularly adapted to ultra-narrow optical filtering [START_REF] Lefèvre | Rectangular response optical filter for partitioning a limited spectral interval[END_REF], and permits to obtain typical full-width-half-maximum (FWHM) of 10 GHz and edge roll-off of 1000 dB/nm. The tunability of the filter central wavelength and bandwidth, authorized by the FSO approach, constitutes also an important feature of the developed devices. Note that optical filters based on fibre Bragg gratings can be as selective and steep as FSO-based filters, but with limited tunability capabilities [START_REF]WSS ROADMs product guide[END_REF]. From their side, wavelength selective switch, based on combination of diffraction grating and liquid crystal on silicon matrix, cannot reach the same spectral selectivity than FSO-based filters [START_REF]Ultra-narrow-band tunable optical filter data sheet[END_REF].

The optical pass-band and stop-band filters employ the same basic architecture. The main difference comes from the filtering slit, which is specific either to the pass-band or to the stop-band filter. The filter architecture is shown in Fig. 9 below, and is now detailed. The input divergent beam coming from the In/Out single mode fiber is collimated thanks to a lens, whose focal point is adjusted on the In/Out fiber. The filter architecture uses a polarization-diversity configuration in order to deal with the X and Y polarizations independently. A PBS separates the two polarization components and generates two elementary parallel light beams. A half wave-plate inserted over the lower beam turns its polarization by 90°. The diffraction grating is thus fed by two beams polarized in a direction perpendicular to the grating grooves. An optical expander is inserted before the grating, in order to spread the beams on the whole grating length and to maximize the separation strength of the Littman-Metcalf setup. A dihedron is placed after the grating as a retro-reflector, to increase the resolution of the device by doubling the number of passage into the grating. The beams are then sent to the upper filtering stage, and are finally focused onto a concave mirror, after passing through a slit. The role of this slit is to filter the signal either in the pass-band or stop-band configuration. The slit is fixed over a translation stage and can be moved into the beam: it allows changing the bandwidth of the filters. The filter central frequency is chosen thanks to a rotation of the retro-reflector. In the case of the pass-band filter, the slit shape is triangular. Note that the concave mirror has limited dimensions in the dispersion plane. Thus, the wavelengths corresponding to the beam portion reflected onto the concave mirror and not stopped by the slit are re-coupled on the way back in the In/Out fiber. This last one is connected to a circulator, which enables to decorrelate the input from the output signals. The optical configuration chosen here is symmetrical when considering the input and output beams, that in turn results in very low polarization dependant losses (PDL) for both the optical pass-band and stop-band filters. The typical PDL is equal to 0.2 dB and is mainly due to the circulator. Table I gives the main optical features of the developed filters. Their insertion losses do not exceed 5 dB for the pass-band filter and 5.5 dB for the stop-band filter. Their edge roll-off are typically higher than 800 dB/nm for the pass-band filter and 750 dB/nm for the stop-band filter. The minimum 3-dB bandwidth is equal to 32 pm (or 4 GHz) for the pass-band filter and 65 pm (or and 13 shows the typical transfer function of the stop-band optical filter for 3-dB bandwidths of ß20 GHz and ß10 GHz, respectively. While the "notch" function is nearly rectangular when the 3-dB bandwidth is ß20 GHz, it becomes triangular when the 3-dB bandwidth is decreased up to ß10 GHz. Further reduction of the filter bandwidth is limited by several factors: firstly, the slit size cannot be reduced indefinitely and have to be higher than 20 µm; secondly, limitation over the number of lines of the diffraction gratings limits the grating dispersion strength; and finally, chromatic and spherical aberrations affecting optics degrade the resolution of the filters.

V. OFDM SUB-BAND OPTICAL SWITCHING

We are now introducing our sub-band optical switch into the experiment [START_REF] Pincemin | Multi-band OFDM transmission with sub-band optical switching[END_REF]. Fig. 14 depicts the structure of our FOADM. The OADM is constituted of a first 3-dB coupler associated with an optical pass-band filter (OPBF) over its lower arm and an optical stop-band filter (OSBF) over its upper arm. In our experiment, the OSBF removes the third OFDM sub-band of the 100 Gbps DP-MB-OFDM signal, while the OPBF selects it. Once dropped, the third OFDM sub-band enters into another 3-dB coupler, whose lower arm is connected for instance to a coherent receiver and upper arm feeds a 5-km fiber spool. The role of this fiber spool is to de-correlate the signals once the third sub-band will be reinserted into the remaining truncated OFDM multiplex. The "add" operation is carried out by connecting the de-correlating fiber to the OSBF output, as shown in Fig. 14. The spectrum of the OFDM signals and filter shapes are measured with a high-resolution (20 MHz) optical spectral analyzer [START_REF]High resolution optical spectrum analyzer AP2050[END_REF] and shown in the insets of Fig. 14. In the back-to-back configuration, the OADM is inserted between the DP-MB-OFDM transmitter and the coherent receiver. In the transmission configuration, as shown in Fig. 15, the OADM is introduced in the middle of the previously depicted transmission line after the fifth SSMF spans. A 3-dB coupler located at the transmitter side firstly duplicates the 100 Gbps DP-MB-OFDM signal into two identical copies. The first signal replica is combined with the 79 odd and even 100 Gbps DP-QPSK channels and launched into the first five SSMF spans. After an EDFA, a 3-dB coupler sends the WDM multiplex firstly towards an OPBF in order to proceed to the "drop" of the third OFDM sub-band, and secondly towards the OSBF which removes the third OFDM sub-band from the OFDM multiplex. The second replica of the DP-MB-OFDM signal (generated at the transmitter side) is filtered by a second OPBF, which selects the third OFDM sub-band. Thanks to a third 3-dB coupler, this sub-band is re-injected into the trun- cated OFDM multiplex at the place of the sub-band, which has been removed by the OSBF. Note that here it is not necessary to use a de-correlating fiber before the "add" operation, as 500 km of fiber separate the added sub-band from the truncated OFDM multiplex (which receives this sub-band). After a DGE, the mix of 100 Gbps DP-MB-OFDM and 100 Gbps DP-QPSK channels is sent into the last five SSMF spans of the transmission link.

Our FOADM is firstly evaluated in back-to-back. The BER versus OSNR sensitivity curves are measured by degrading the OSNR thanks to a white noise source, which is coupled with the DP-MB-OFDM signal. Fig. 16 below gives the BER of the dropped sub-band (the third one) versus the OSNR in 0.1 nm for various 3-dB bandwidths (3-dB BW) of the OPBF. From 14 GHz to 6 GHz, the OSNR penalty is acceptable compared to the case where no OADM is inserted, but an error floor at BERß1 × 10 -4 appears for a 3-dB BW of 4 GHz. In the following, we choose the 3-dB BW of 10 GHz for the "drop" OPBF, which results in a very low OSNR penalty with respect to the reference case without OADM.

Then, we plot in Fig. 17 the BER of the sub-bands 2, 3 and 4 as a function of the 3-dB BW of the OSBF after the "add-drop" operation when the guard-band (GB) is 2 GHz and OSNR = 18.5 dB (OSNR required for having a BER = 10 -4 without OADM for the 100 Gbps DP-MB-OFDM signal). The impact of the crosstalk over the sub-band 3, between the added and remaining residual signals, is maximum (respectively, minimum) at low (respectively, at high) 3-dB BW of the OSBF. It is the opposite observation for the "pass-through" sub-bands 2 and 4, for which a large OSBF results in high spectral distortions. The optimum OSBF 3-dB BW results from a trade-off between the distortions affecting sub-band 3 from one side and sub-bands 2 and 4 from the other side, and is obtained at the intersection of these three measured curves. For a GB = 2 GHz, the optimum OSBF 3-dB BW is equal to ß14 GHz, while the BER of sub-bands 2, 3 and 4isß3 × 10 -3 .

The same measurements have been carried out in Fig. 18 for a GB = 4 GHz and OSNR = 18.5 dB. The optimum 3-dB BW is now equal to ß15.7 GHz, while the BER of sub-bands 2, 3 and 4 has been improved up to ß3 × 10 -4 , i.e., one BER decade better than the BER previously obtained for the GB = 2 GHz.

In Fig. 19 is now plotted the BER of the dropped sub-band 3, added sub-band 3 and "pass-through" sub-bands 2 and 4 as a function of the OSNR for the optimum tunings of the OPBF and OSBF, obtained with the GB of 2 and 4 GHz. While the BER of the various sub-bands reaches a ß10 -3 error floor (near from the FEC limit fixed here at BER = 2 × 10 -3 )a th i g h OSNR for a GB = 2GHz, the BER results are satisfying for a GB = 4 GHz. The corresponding OSNR penalty compared to the case "without OADM" is less than 1 dB at BER = 10 -3 . It is the reason why we now evaluate the performance of our FOADM in the transmission line previously depicted with a GB tuned over the 4-GHz value. Results are presented in Fig. 20 by the means of BER versus OSNR sensitivity curves for the subbands 2, 3 and 4. Previously implemented experiments over the 10 × 100-km transmission line without the OADM (see Fig. 8) show that the optimum channel span input power is ß0d B m , while the corresponding BER and OSNR are ß2 × 10 -4 and ß20 dB, respectively. Note that this optimum power per channel is kept hereafter. Firstly, as a reference, the OSNR sensitivity curve of the 100 Gbps DP-MB-OFDM signal transmitted over the 10 × 100-km transmission line without the OADM is plotted. The performance of the sub-band 3, dropped after 500 km, and reinserted in the last 500-km of the link (after having been duplicated at the transmitter side) is measured, as well as the performance of the "pass-through" sub-bands 2 & 4 (which are transmitted over 1000 km on the ten SSMF spans). One can notice first that a negligible OSNR penalty is observed at BER = 10 -3 whatever the configuration tested, and secondly that the BER obtained at the best OSNR (measured when the white noise source is not activated) is below the FEC limit for the dropped, added and "pass-through" sub-bands. Note that the best achievable OSNR on the sensitivity curves differ as a function of the configuration considered, since the transmission distance and accumulated ASE are different.

VI. CASCADABILITY OF THE SUB-BAND OPTICAL SWITCH

The question of the cascadability of the previously defined FOADM is now studied thanks to the insertion of the add-drop multiplexer into a 2 × 100-km G.652 fiber-based recirculating loop. The guard-band between the OFDM sub-bands is still fixed at 4 GHz. The 3-dB bandwidths of the OPBF and OSBF are equal to 10 GHz and 15.7 GHz, respectively, as previously optimized. The impact over the sub-band 3 (the sub-band which is dropped and reinserted) but also over the neighboring subbands 2 and 4 of the cumulated filtering transfer function as well as the crosstalk inside the OADM is investigated. Into the recirculating loop, the architecture of the OADM is identical to the one depicted in Fig. 14. After a 3-dB coupler, the 100 Gbps DP-MB-OFDM signal is filtered by a OPBF (respectively, OSBF) which extracts (respectively, removes) the subband 3 from the OFDM multiplex. After a 5-km de-correlating fiber, the OPBF and OSBF outputs are re-coupled together, and the reconstituted OFDM multiplex with the "pass-through" and "added" sub-bands is sent into the loop for a new round-trip. This OADM configuration allows us evaluating the impact of the repeated action of the OPBF and OSBF over the OFDM signal (more particularly over the sub-band 2, 3 and 4), both in terms of filtering and crosstalk.

Fig. 23 below describes the experimental set-up. The transmitter, depicted in Fig. 6 and in Fig. 15, which mixes the 124 Gbps DP-MB-OFDM and the 79 odd and even 112 Gbps DP-QPSK, is connected to the recirculating loop through an acousto-optical modulator (AOM). The loop is constituted of two 100-km spans of G.652 fiber. An EDFA with 20-dB gain and 4.2-dB NF is located at the span input and output, as shown in Fig. 23 below. The OADM described above is then inserted into the loop. The sub-band 3 is firstly dropped and finally reinserted into the OFDM multiplex. It experiences thus the effect of the repeated action of the OPBF (over the "drop" path) and OSBF (over the "stop" path) transfer function, as well as the crosstalk coming from the interaction between the "drop" and "stop" paths. From their side, the sub-bands 2 and 4 experience the repeated distortions of the OSBF. An EDFA compensates for the introduced OADM losses. A DGE and a synchronous polarization scrambler (SPS) achieve power equalization and polarization rotation of the WDM multiplex after each loop round-trip, respectively. The role of the SPS is to stabilize the loop operation. Note that EDFA used in this Section VI are better than those used in the previous experiments (NF = 4.2 dB here against NF = 5.5 dB before), as the objective here is to cascade the highest possible number of OADM.

The performance of sub-bands 2, 3 and 4 is shown in Fig. 21. The BER is plotted as a function of the number of loop roundtrips. For each transmission distance, the span input power per channel is optimized. Two configurations are studied. The case for which no OADM is inserted into the loop is taken as reference. The FEC limit (fixed here at BER = 2 × 10 -3 ) is reached after 10 loop round-trips or 2000 km. When the OADM is introduced into the loop, the BER of sub-band 3 (BER ß 6 × 10 -4 ) after five round-trips or 1000 km, including the crossing of five OADM, is well below the FEC limit. The BER of sub-bands 2 and 4 (BER ß 1.5 × 10 -3 ) is slightly worse than that of subband 3. Nonetheless, five loop round-trips can be achieved as well. This BER degradation with respect to sub-band 3 comes from the higher distortions which affect the edge sub-carriers of sub-bands 2 and 4 when compared to those of sub-band 3. Indeed, as indicated in the Table I, the edge roll-off of the OSBF (ß750 dB/nm) and its corresponding frequency selectivity are lower than those of the OPBF (ß800 dB/nm). The bandwidths of the OPBF and OSBF would have deserved to be re-optimized after each new loop round-trip. However, the performance improvement (respectively, degradation) for the sub-bands 2 and 4 (respectively, for the sub-band 3) would have been limited to less than ½ BER decade (as Fig. 21 lets it imagine), and no more than five loop round-trips would have been possible. To exhibit the progressive distortion affecting the sub-bands 2, 3 and 4, the spectrum of the MB-OFDM channel after each loop round-trip is also shown in Fig. 24.

The sensitivity to frequency drift of such ultra-narrow optical pass-band and stop-band filters is also a crucial question when considering their deployment in the field. Indeed, the mechanical or temperature stability of these devices is not as well ensured in the field as in the laboratory. Fig. 22 presents the BER evolution of the sub-band 3 (on which the "add-drop" operation is achieved) as a function of the OPBF and OSBF central frequency drift after five loop round-trips. In the OPBF case, the FEC limit is reached for absolute values of frequency drift higher than 0.8 GHz, which represents 20% of the OFDM signal bandwidth (fixed here at ß8 GHz) if we consider the whole acceptance window going from -0.8 to +0.8 GHz. When considering now the OSBF, the acceptance window spreads from -1 GHz up to 0.8 GHz, namely 22.5% of the OFDM signal bandwidth. Consequently, the sensitivity of the OFDM signal (here the sub-band 3) to the central frequency drift of the optical filters is slightly higher in the case of the OPBF than in the case of the OSBF. As expected, these values are relatively small, and will require that the optical filters be immunized against both mechanical vibrations and temperature variations.

These results are very encouraging when considering the introduction of such flexible OADM into meshed optical transport networks, for which several OADM (4 or 5) are often cascaded over a given path.

VII. TYPICAL USE CASE FOR THE INTRODUCTION OF OFDM SUB-BAND OPTICAL SWITCHING IN THE NETWORK

The efficiency of ultra-narrow sub-wavelength optical switching has been extensively demonstrated above. We now consider the applicability of this technology in a multi-layer transport network. More precisely, we aim to evaluate the benefit of the combined introduction of flexibility and multi-band transmission techniques over the capital expenditure (CAPEX) cost of a representative national multi-layer transport network [START_REF] Christodoulopoulos | Elastic bandwidth allocation in flexible OFDM-based optical networks[END_REF]. We propose to compare different scenarios in order to highlight the advantages that flexibility and MB-OFDM provide in terms of capacity transported by the network, in terms of filling of communication connections through the various network layers (from optical to IP layer), and, as a consequence, in terms of CAPEX cost for the overall network. Scenario S0 serves as a reference and corresponds to the currently deployed optical transport networks at 100 Gbps which use coherent single-carrier 100 Gbps DP-QPSK channels and fixed 50-GHz ITU-T grid. Scenarios S1 and S2 correspond to the up-coming use case of flexible transport networks for which the optical channel capacity will be better adapted to the traffic demands in order to overcome as much as possible over-provisioning. From one hand, scenario S1 reflects the upper part of Fig. 1 which describes the Nyquist-WDM super-channel solution employing a mixed 37.5/50 GHz spectral grid. On the other hand, scenario S2 corresponds to what is exhibited in the lower part of Fig. 1 with the achievement of a fully flexible management of the fiber bandwidth thanks to the MB-OFDM and sub-wavelength optical switching technologies.

In order to obtain a holistic view and evaluation of the introduction of optical flexibility in a multi-layer transport network, we first model the network by considering its essential elements. A multi-layer transport network can be divided in two main macroscopic layers composed of different kinds of equipments, as depicted in Fig. 25. A packet layer aggregating IP traffic is located above a transmission layer which aggregates both timedivision-multiplexing (TDM) traffic and IP flows while multiplexing and transporting optical channel inside the optical fiber. From an organic architecture point of view, the packet layer uses IP routers, whereas the transmission layer uses ROADM (if we consider the S0 scenario) and FOADM (if we consider the S1/S2 scenarios). The right hand-side of Fig. 25 represents our functional generic multi-layer network architecture model, using client/server relationships between the various layers [START_REF] Berthold | Optical networking: Past, present, and future[END_REF]. In the scenario S0, the transmission layer is constituted of two sub-layers: the "Optical Channel" sub-layer represents a WDM channel, while the "Colored Fiber" sub-layer stands for the multiplex of 80 WDM channels transmitted in the optical fiber. In the scenarios S1 and S2, the transmission layer is decomposed in three different sub-layers called "Sub-Band," "Super-Channel" and "Slotted Colored Fiber," respectively. The "Sub-Band" sublayer performs the choice of the adequate modulation format and data-rate that can be carried by a sub-band (or equivalently by a sub-wavelength) as a function of the transmission reach of the service demand.

The sub-bands/sub-wavelengths coming from the "Sub-Band" sub-layer are then aggregated inside a super-channel in the "Super-Channel" sub-layer in order to create an adaptive transmission path whose capacity is well-adjusted to the traffic demand. The "Super-Channel" entity is autonomous and transports the traffic from a point A to a point B without undergoing any sub-band optical switching process. While the "Colored Fiber" sub-layer supports up to 80 WDM channels (or equivalently 80 spectral slots of 50 GHz) in the scenario S0, 320 spectral slots of 12.5 GHz and 400 spectral slots of 10 GHz (i.e., 4000 GHz or 32 nm of overall bandwidth) can be transported by the "Colored Slotted Fiber" sub-layer of the scenarios S1 and S2, respectively. In the scenario S1, the subbands which constitute the super-channel occupy three spectral cases a substantial capacity over-provisioning but no CAPEX cost increase (by definition in our model). Fig. 26 below shows the topology of the transport network used hereafter. It is representative of a transport backbone operated by a Tier-1 operator in a European country. The network is constituted of 51 nodes connected together thanks to 75 WDM links. The mean node degree is equal to 2.9. 80% of the links corresponds to transmission distances lower than 300 km. 25% corresponds to intra-city links and are shorter than 50 km. The average link length is equal to 217 km.

The traffic matrix associated with this network topology is composed of 371 demands divided into 185 IP demands and 186 leased-line (LL) demands (corresponding to TDM-based traffic) that represents globally more than 7 Tbps of ingress traffic. Although the number of IP and LL demands is quite equivalent, the part of the IP traffic volume into the global ingress traffic is twice more important than that of the LL demands. Note that the traffic demands in a real transport network are not uniformly spread over the nodes: some nodes are used for pure transit, while other nodes can manage up to 1 Tbps of edge services.

By sake of simplicity, we consider a global traffic matrix growth which does not make any difference between the TDM and IP-based services. In other words, the growing ratio is identical for the IP and LL demands. In particular, complex traffic matrix evolutions integrating the apparition of new service demands along the time are not taken into account here. Instead of that, the variation of the traffic load is based on a yearly and fixed growing ratio of 35% whatever the type of demand. It constitutes a rather realistic assumption applicable to core transport networks where traffic is already largely aggregated. Three traffic volumes V1, V2 and V3 are considered, corresponding to a traffic projection for now, and for the next 4 and 7 years, respectively. As already mentioned, the scenario V1 corresponds to 7 Tbps of ingress traffic.

In this study, the routing of optical paths and the combined dimensioning of the various layers are optimized in order to make the global network cost as low as possible. An internallydeveloped multi-layer network optimization tool, based on a simulated annealing meta-heuristic, finds an adequate combination of the multi-layer paths yielding to a nearly optimal network cost. However, in this multi-layer network architecture environment, some dimensioning rules are required to map a client layer on a server layer. IP traffic is handled by IP routers, and the mapping of IP demands in the transmission layer is performed through the use of 10 and 100 GbE client interfaces. LL demands are managed directly by the transmission layer and are generated/detected by 10 and 100 GbE client interfaces, as well. The threshold to choose a 100 GbE interface rather than N-fold 10 GbE interfaces is fixed to 70 and 100 Gbps for IP and LL demands, respectively. For instance, if an IP path requires a bandwidth of 80 Gbps, a 100 GbE interface is installed in the IP router, rather than eight 10 GbE interfaces. At the opposite, if a LL path transports 80 Gbps, eight 10 GbE interfaces will be activated. Moreover, the dimensioning process does not integrate any resiliency mechanism, and corresponds to a "best fit" strategy that favors aggregation as much as possible. It means that "Optical Channel" sub-layer in scenario S0 can aggregate various IP/LL demands that are routed along the same physical path. In the scenarios S1 and S2, each IP/LL demand is allocated to a dedicated sub-band which is then incorporated into a superchannel. For instance, a traffic demand of 290 Gbps routed over a "Slotted Colored Fiber" of 300 km will use in the scenario S1 a super-channel composed of two sub-bands of 200 Gbps (with the DP-16QAM modulation format) occupying 7 slots of 12.5 GHz, i.e., six slots for the traffic itself and one additional slot for the guard-bands located from each side of the 2 × 200 Gbps super-channel (as shown in the upper part of Fig. 1). In the scenario S2, the same demand will use a super-channel constituted of three sub-bands of 100 Gbps occupying six slots of 10 GHz. Note that in this last case, in accordance with what has been shown in the sections before, the guard-band is integrated into the 10 GHz spectral slot (with for instance, OFDM sub-bands of 8 GHz and guard-bands of 2 GHz from each side). Note as well that we consider here one super-channel entity per demand due to the fact that the super-channel is used as an aggregate of sub-bands, in order to adapt as finely as possible the allocated capacity to the traffic demand. Another way to proceed would be to consider super-channel as a dedicated switching layer in itself, able to groom any sub-band from any traffic demand. This will be studied into the on-going FP7 Eu- ropean project FOX-C [START_REF]FP7 Call 8 FOX-C project[END_REF]. However, if the "Super-Channel" sub-layer may make sense from an optical node implementation point of view, its benefits in terms of networking architecture is not obvious because it could introduce aggregation constraints.

For each iteration of the optimization process, the global network cost is inferred form the addition of the individual cost of the various network elements. The optimization is driven by the filling of the fiber in terms of spectral slot of 50, 37.5 and 10 GHz in the S0 and S1/S2 scenarios, respectively. When the maximum capacity of the fiber is reached, an additional degree is automatically added to the OADM node.

The Table III above presents the costs that are considered in our model for the most essential elements of our multi-layer transport network. The cost of the 12.5/10 GHz spectral slot in the scenarios S1/S2 is established as follows. We take as a reference the price of the flexible coherent WDM interface working at 100 Gbps and implementing both the Nyquist filtering and SD-FEC functionalities. We then make the two following assumptions: firstly we suppose that the cost ratio between this flexible coherent 100 Gbps WDM interface (operating indifferently over a 50 or 37.5 GHz spectral grid) and the 12.5/10 GHz spectral slot considered in the scenarios S1/S2 is equal to 4; secondly we assume that the price of a standard coherent 100 Gbps WDM interface considered in the scenario S0 with the SD-FEC functionality but without the Nyquist filtering is equal to the cost of this flexible coherent WDM interface working at 100 Gbps (at the expense of a slight over-estimation of its cost). Note at last that the cost difference between the flex-rate coherent WDM transceivers (for instance, between the 100 and 200 Gbps version of the flex-rate transceiver) comes uniquely from the embedded client interfaces whose number varies as a function of the data-rate addressed by the WDM part of the transceiver.

The results of our cost optimization for the multi-layer transport network previously described are presented in Fig. 27 for the three scenarios S0, S1, S2 and the three traffic matrix V1, V2, V3 under investigation. First of all, the cost of the packet layer is more or less the same whatever the considered scenario. It is the indication that IP traffic in transit is managed by the transmission layer and not by the packet layer, which is just used for the termination points of IP demands. Actually, the major part of the network cost is related to the transmission layer and optical transport, whatever the scenario (S1, S2, S3) and the traffic matrix (V1, V2, V3) considered. Any technique that permits to use more efficiently the fiber bandwidth is thus relevant for reducing the cost of the optical transport. With the traffic matrix V1, scenario S1 (which combines Nyquist-WDM technique and 37.5/50 GHz spectral grid) does not bring any benefit in terms of CAPEX cost reduction. Its cost is even ß50% larger than that of the classical scenario (S0) using coherent 100 Gbps DP-QPSK channels and 50 GHz ITU-T grid. At the opposite, the scenario S2 combining MB-OFDM and sub-wavelength optical switching is dramatically better with a cost reduction of ß52% when compared to scenario S0 and ß68% when compared to scenario S1. These results prove very clearly all the potential of the MB-OFDM and sub-wavelength optical switching techniques to manage as efficiently as possible the fiber bandwidth. With the traffic matrix V2, scenario S0 is still ß7% better than scenario S1, but ß28% worse than scenario S2. With the highest traffic matrix V3, the flexible scenarios S1 and S2 are better than the scenario S0 with a CAPEX gain of ß5% and ß23%, respectively. The traffic matrix growth makes the use of Nyquist-WDM super-channels and 37.5/50 GHz spectral grid more relevant, because the traffic going from a point A to a point B becomes larger and the need for sub-wavelength optical switching is less stringent. The same explanation is valid for justifying that the cost reduction allowed by the scenario S2 becomes lower and lower when the traffic matrix increases. In consequence, the migration towards scenario S1 is not entirely justified in a short-term perspective, and implementation of the scenario S0 (coherent 100 Gbps DP-QPSK channels over a 50 GHz ITU grid) during the next 4 or 5 years is very pertinent. In all cases studied here, use of the scenario S2 (com-bination of MB-OFDM and sub-wavelength optical switching techniques) is always relevant, and justify the effort made to design and build ultra-selective FOADM (as presented in this paper). Furthermore, in the context of a real transport network operation, the resources have to be regularly up-graded to support the traffic increase. The global network cost follows more or less the traffic growth evolution with the scenario S2, which is well-adapted to "a pay as you grow" investment model. This is due to the fact that the optical transport network dimensioning is finely adjusted to the traffic demands without any capacity over-provisioning. From their side, scenarios S0 and S1, which may allocate higher capacity than that required by the real traffic load, have the capacity to support smooth traffic increase without adding any new resources. It explains why the cost evolutions are more irregular with the scenarios S0 and S1 than with the scenario S2, when the traffic matrix increases. The cost evolution trends that we have highlighted there should be reinforced if we integrate the cost erosion parameters in our study. Note at last that end-to-end protection resiliency mechanisms have still to be integrated in our study to evaluate the total cost of the protected multi-layer transport network. Nonetheless, it could be reasonably assumed that the protected network cost will be slightly higher than two, as resources for service transport and protection are twofold more important, while protected routes (which can be longer than the original ones) could engender extra-cost related to the insertion of new regeneration sites.

VIII. CONCLUSION

In this manuscript, we provide the experimental proof of concept of sub-band optical switching over a 100 Gbps MB-OFDM signal. After having demonstrated that 100 Gbps DP-MB-OFDM presented the same performance than 100 Gbps DP-QPSK on an uncompensated 10 × 100-km G.652 fiberbased transmission line, we successfully perform an optical "add-drop" of an OFDM sub-band as narrow as 8 GHz inside a 100 Gbps DP-MB-OFDM signal constituted of four subbands spaced by 4-GHz guard-bands. The essential elements of this FOADM are composed of two ultra-selective optical passband and stop-band filters whose rectangular shape and narrow bandwidth allowed OFDM sub-band extraction and suppression, respectively. The crucial question of the cascadability of such ultra-selective optical filters is addressed, by inserting our FOADM into a G.652 fiber-based recirculating loop. We prove that five OADM can be cascaded without degrading too much the transmission performance. Such demonstration opens the way to optical meshed transport networks with a high degree of flexibility. Indeed, the combination of MB-OFDM technique and sub-band optical switching allows to increase significantly the fiber throughput and to manage finely the fiber bandwidth, while adapting the optical connections (i.e. the OFDM subbands) to the transported services. As shown in the last part of this paper, another crucial advantage of this scheme is the ability to optimize the aggregation process in its multi-layer dimension (from the optical layer up to the IP layer), and in turn to make significant cost savings on the deployed equipments and, as a consequence, on the energy consumption (even if this point has not been investigated here). Our proposal pushes network flexibility far away of what is today proposed by system vendors, opening new horizons for an optimized use of multi-layer transport networks.

Fig. 1

 1 Fig. 1 Bandwidth management proposed today by the Nyquist-WDM super-channel working at 32 Gbaud and the mixed 37.5/50 GHz spectral grid approach (upper figure part), and by the MB-OFDM technology and the grid-less management of the fiber bandwidth approach (lower figure part).

Fig. 2

 2 Fig. 2 Block diagram of a typical OFDM transmitter (upper figure part) and receiver (lower figure part), with S/P = Serial-to-Parallel Conversion, IFFT = Inverse Fast Fourier Transform, P/S = Parallel-to-Serial Conversion, AWG = Arbitrary Waveform Generator, DAC = Digital-to-Analogue Converter, LPF = Low-Pass Filter, ECL = External Cavity Laser, MZM = Mach-Zehnder modulator, VOA = Variable Optical Attenuator, ODL = Optical Delay Line, PBC = Polarization Beam Combiner, PBS = Polarization Beam Splitter, ADC = Analog-to-Digital Converter, CFO = Carrier Frequency Offset, FFT = Fast Fourier Transform, MIMO = Multiple-Input-Multiple-Output.

  The determination of the fractional part reuses the training symbol previously defined for synchronization. If L denotes the number of repetitions of the basic pattern B into the training symbol, an accurate CFO estimate ∆ CFO can be obtained by averaging ∆ CFO over the N sc /L samples of the basic pattern B. The following expression

Fig. 3

 3 Fig. 3 Power spectral density versus sub-carrier index for the transmitted and received OFDM signal, with an integer CFO measured at -21∆f.

Fig. 4

 4 Fig. 4 Particular structure of the training symbols used for channel estimation and equalization, in the case of a dual-polarization OFDM system.

Fig. 5

 5 Fig. 5 Set-up of the 100 Gbps DP-MB-OFDM transmitter. In inset is shown the spectrum of the 100 Gbps DP-MB-OFDM signal. OSA = Optical Spectral Analyzer, PM = Polarization-Maintaining, PM-EDFA = Polarization-Maintaining Erbium-doped fiber amplifier.

Fig. 6

 6 Fig. 6 Set-up of the 10 × 100-km transmission test-bed, with the 124.4 Gbps DP-MB-OFDM channel, the 79 odd and even 112 Gbps DP-QPSK wavelengths, the 10 × 100-km uncompensated G.652 SSMF transmission line, and the coherent receiver. LD = Laser Diode, PRBS = Pseudo-Random Bit Sequence, SSMF = Standard Single-Mode Fiber, DGE = Dynamic Gain Equalizer, OBPF = Optical Band-Pass Filter.

Fig. 7

 7 Fig. 7 BER versus OSNR (in 0.1 nm) of the 100 Gbps DP-MB-OFDM and 100 Gbps DP-QPSK modulation formats in the back-to-back configuration. The FEC limit is reached for an OSNR (in 0.1 nm) = 15.36 dB.

Fig. 8

 8 Fig. 8 BER versus channel span input power (P IN SPAN ) of the 100 Gbps DP-MB-OFDM and 100 Gbps DP-QPSK modulation formats, in the singlechannel (SC) and WDM configurations.

Fig. 9

 9 Fig. 9 Typical architecture of the ultra-narrow pass-band and stop-band optical filters used in the FOADM.

Fig. 10

 10 Fig. 10 Typical transfer function of the optical pass-band filter when its 3-dB bandwidth is ß14 GHz.

Fig. 11

 11 Fig. 11 Typical transfer function of the optical pass-band filter when its 3-dB bandwidth is ß4 GHz.

Fig. 12

 12 Fig. 12 Typical transfer function of the optical stop-band filter when its 3-dB bandwidth is ß20 GHz.

Fig. 13

 13 Fig. 13 Typical transfer function of the optical stop-band filter when its 3-dB bandwidth is ß10 GHz.

Fig. 14

 14 Fig. 14 Set-up of the OFDM sub-band optical add-drop multiplexer with the spectra of signals and optical filters in insets.

Fig. 15

 15 Fig. 15 Set-up of the 10 × 100-km transmission test-bed, with the 124.4 Gbps DP-MB-OFDM channel, the 79 odd and even 112 Gbps DP-QPSK wavelengths, the 10 × 100-km uncompensated G.652 SSMF transmission line with the sub-wavelength optical switch located in its middle.

Fig. 16

 16 Fig. 16 BER versus OSNR of the sub-band 3 for various 3-dB BW of the OPBF, with the shapes of the OBPF for various 3-dB BW in insets.

Fig. 17 BER

 17 Fig. 17 BER versus 3-dB BW of the OSBF for the sub-bands 2, 3 and 4 and aGB= 2 GHz. In insets, the spectra of the signals at the optimum.

Fig. 18 BER

 18 Fig. 18 BER versus 3-dB BW of the OSBF for the sub-bands 2, 3 & 4 and a GB = 4 GHz. In insets, the spectra of the signals at the optimum.

Fig. 19

 19 Fig. 19 BER versus OSNR of sub-bands 2, 3 and 4 in BtB (OADM inserted between the Tx and Rx) for a GB = 2 GHz and GB = 4 GHz.

Fig. 20

 20 Fig. 20 BER versus OSNR of sub-bands 2, 3 & 4 after transmission (OADM inserted after 500 km) for a GB = 4 GHz.

Fig. 21 BER

 21 Fig. 21 BER versus number of 2 × 100-km loop round-trips for the sub-bands 2, 3 and 4, when the OADM is inserted or not into the loop.

Fig. 22

 22 Fig. 22 BER versus OPBF and OSPF central frequency drifts for the sub-band 3 after 5 loop round-trips (or 1000 km).

Fig. 23

 23 Fig. 23 Set-up of the 2 × 100-km G.652 fiber-based recirculating loop including the OFDM sub-band optical switch. AOM = Acoustic-Optical Modulator, SPS = Synchronous Polarization Scrambler, DGE = Dynamic Gain Equalizer.

Fig. 24 :

 24 Fig. 24: Evolution of the 124.4 Gbps MB-OFDM channel spectrum measured with the high-resolution (20 MHz) OSA after each loop round-trip.

Fig. 25

 25 Fig. 25 Multi-layer transport network architecture with the details of the essential elements of the model. LL = Leased Line.

Fig. 26

 26 Fig. 26 Topology associated with the transport network studied here.

Fig. 27

 27 Fig. 27 Multi-layer transport network cost comparison in the scenarios S0, S1 and S2 for the traffic matrix V1, V2 and V3.

TABLE I TYPICAL

 I FEATURES OF THE PASS-BAND AND STOP-BAND OPTICAL FILTERS

		Optical Pass-Band Filter	Optical Stop-Band Filter
	Insertion Loss	5 dB typical	5.5 dB typical
	Edge roll-off	> 800 dB/nm	> 750 dB/nm
	3-dB Bandwidth	32 pm / 4 GHz	65 pm/8 GHz
	Isolation	60 dB typical	Not Applicable
	Rejection	Not Applicable	60 dB typical

TABLE III CONSIDERED

 III COSTS FOR THE MAIN ELEMENTS OF THE MULTI-LAYER TRANSPORT NETWORK UNDER STUDY.

	Item	Cost
	IP Router
	Router (shelf and switch fabric)	1.62
	10 GbE interface card	5.08
	100 GbE interface card	8.42
	Directionless/Colorless OADM
	OADM degree 1 (shelf and switch fabric)	2
	OADM degree 2	12.36
	OADM degree 3	17.87
	OADM degree 4	23.77
	Additional degree	34
	10 GbE interface	0.04
	100 GbE interface	4.17
	100 Gbps WDM interface (Scenario S0)	8.33
	12.5/10 GHz spectral slot (Scenarios S1/S2)	2.08


Nhat-Quang Nhan
email: nhat-quang.nhan@univ-brest.fr

Telex M N Ngatched
email: tngatched@grenfell.mun.ca

Octavia A Dobre
email: odobre@mun.ca

Philippe Rostaing
email: philippe.rostaing@univ-brest.fr

Karine Amis
email: karine.amis@telecom-bretagne.eu

Emanuel Radoi
email: emanuel.radoi@univ-brest.fr

N.-Q Nhan

Multiple-Votes Parallel Symbol-Flipping Decoding Algorithm for Non-Binary LDPC Codes

Keywords: LDPC codes, low complexity decoding algorithm, symbol flipping

A novel decoding algorithm for non-binary low density parity check (NB-LDPC) codes is proposed. The algorithm builds on the recently designed parallel symbol-flipping decoding (PSFD) algorithm and combines a technique of error estimation and a method of multiple voting levels from each unsatisfied checksum to the corresponding variable nodes. Simulations results, performed on a number of NB-LDPC codes of various lengths and column weights constructed using several methods, show that the new algorithm not only avoids using code-dependent voting threshold but also improves the error rate performance of the PSFD algorithm, particularly for low column weight parity-check matrices.

I. INTRODUCTION

F IRST introduced by Gallager in [1], and subsequently rediscovered by Davey and Mackay in [START_REF] Davey | Low-density parity-check codes over GF(q)[END_REF], non-binary low density parity check (NB-LDPC) codes outperform their binary counterparts, especially for short-to-moderate length codes. Just like with binary LDPC codes, the algorithms for decoding NB-LDPC codes can be classified into three general categories: hard-decision decoding [START_REF] Zhang | Low-complexity reliability-based messagepassing decoder architectures for non-binary LDPC codes[END_REF], soft-decision decoding [START_REF] Wymeersch | Log-domain decoding of LDPC codes over GF(q)[END_REF]- [START_REF] Wang | A simplified min-sum decoding algorithm for non-binary LDPC codes[END_REF], and hybrid decoding (also known as reliability-based decoding) [START_REF] Chen | A symbol-reliability based message-passing decoding algorithm for non-binary LDPC codes over finite fields[END_REF]- [START_REF] Garia-Herrero | Non-binary LDPC decoder based on symbol flipping with multiple votes[END_REF]. From an implementation point of view, hard-decision decoding is the simplest in complexity. However, its simplicity results in a relatively poor performance that can be as far away as a few decibels from that of soft-decision decoding. Softdecision decoding provides the best performance but requires the highest computational complexity. Hybrid decoding is in between the two extremes and provides a good trade-off between performance and complexity.

Among reliability-based message-passing algorithms, the parallel symbol flipping decoding (PSFD) algorithm recently introduced in [START_REF] Huang | Parallel symbolflipping decoding for non-binary LDPC codes[END_REF] offers one of the best trade-off between performance and complexity. Using a flipping function which combines both the weighted check-based message of the normalized check-sums and the variable-based message of the received sequence, the algorithm identifies, in each decoding iteration, the relatively unreliable symbols in the hard-decision symbol sequence and decodes them based on the corresponding flipping symbols. The unreliable symbols are found using a voting system whereby each unsatisfied check node (CN) gives one vote to the variable node (VN) with the largest flipping function checked by it. Variable nodes thus accumulate votes and those with a total number of votes exceeding a predefined threshold are identified as unreliable. By only playing with the reliabilities of hard-decisions, PSFD is a good choice to decode high rate finite field NB-LDPC codes. However, simulation results show that PSFD is suitable for decoding only regular NB-LDPC codes whose parity-check matrices have large column weights, e.g., column weights of at least 8. In addition, PSFD employs a code-dependent voting threshold, which should be optimized through simulation. Furthermore, the computation of its flipping function involves a pair of scaling factors, (η, λ), which also depend on the code and is optimized through simulation.

This letter proposes a new algorithm, referred to as multiplevotes PSFD (MV-PSFD). When compared with PSFD, it introduces a method of error estimation which results in avoiding the use of the voting threshold, and passes multiple votes from the unsatisfied CNs to the corresponding VNs. The proposed algorithm significantly outperforms PSFD for low column weight parity check matrices, as shown by simulation results. Such improvements are achieved with a similar computational complexity.

The organization of this letter is as follows: In Section II, some preliminaries and the PSFD algorithm are briefly reviewed. The proposed algorithm and its complexity analysis are introduced in Section III. Simulation results are presented in Section IV. Finally, Section V concludes the letter.

II. PRELIMINARIES A. Notations and Definitions

A regular NB-LDPC code C of length N and dimension K over the Galois field of order q (GF(q)) is completely described by a row-column (RC)-constrained parity-check matrix of size M × N (M ≥ K) H =[h j,i ] over GF(q), which has a constant column weight d v and a constant row weight d c . For practical purpose, we only consider binary extension fields, where q =2 p .L e tM j = {i :0≤ i<N,h ji =0} be the set of all indices of VNs which connect to the CN c j and N i = {j :0≤ j<M,h ji =0} be the set of all indices of CNs which connect to the VN v i .

Suppose that a regular (N, K) NB-LDPC code is used for error control over a binary-input additive white Gaussian noise (BIAWGN) channel with zero mean and two-sided power spectral density N 0 /2. Assume binary phase-shiftkeying (BPSK) signaling with unit energy. A codeword c = (c 0 ,c 1 ,...,c N -1 ) in C, where c n =(c n,0 ,c n,1 , ••• ,c n,p-1 ) ∈ GF(2 p ) with c n,i ∈ GF [START_REF] Davey | Low-density parity-check codes over GF(q)[END_REF], is mapped into the sequence x =(x 0 ,x 1 ,...,x N -1 ) before its transmission, where x n = (x n,0 ,x n,1 , ••• ,x n,p-1 ) with the mapping rule x n,i =(2c n,i -1) ∈{+1, -1}.L e ty =(y 0 ,y 1 ,...,y N -1 ) be the softdecision received sequence at the output of the receiver matched filter. For 0 ≤ i<N, y n =(y n,0 ,y n,1 ,

••• ,y n,p-1 )=(x n,0 + ν n,0 ,x n,1 + ν n,1 , ••• ,x n,p-1 + ν n,p-1
) in which the ν n,i 's are statistically independent Gaussian random variable with zero mean and variance N 0 /2. The hard-decision symbol sequence at the decoder is denoted as z =(z 0 ,z 1 ,...,z N -1 ).

For any hard-decision symbol sequence z, the syndrome vector s is computed as s = z • H T , where

s j = h j0 z 0 + h j1 z 1 + •••+ h j(N -1) z N -1 with 0 ≤ j<M. s j is known as the j th check-sum of z. z is a valid codeword in C if and only if s = 0.
Let S i (z)={s j : j ∈N i } be a set of check-sums that contain the symbol z i at the VN v i and Si (z)={s ji (z)=h -1 ji s j : s j ∈ S i (z)} be a normalized check-sum set. Finally, let S * i (z) be Si (z) excluding zero elements, i.e., S * i (z)= Si (z) \{0}.

B. PSFD Algorithm [10]

Given the initial hard-decision symbol vector z (0) =(z

(0) 0 ,z (0) 1 ,
...,z (0)
N -1 ), the PSFD algorithm iteratively flips erroneous symbols in z (0) until either a codeword is found or a maximum number of iteration is reached. In other words, the algorithm finds an error symbol vector e =[e 0 ,e 1 ,...,e N -1 ], such that z = z (0) -e is a codeword. The error symbol vector is recursively computed by e (0) = 0 and e (l+1) = e (l) +ε (l) , where each component of ε (l) is chosen so as to maximize a flipping function that is updated at each iteration and which is defined by

F (l) i =m a x α∈ S * (l) i (z) F (l) i (α). (1)
In (1), F

i (α) with α ∈ GF (q) is given by

F (l) i (α)=E (l) i (α) -I i α + e (l) i , (2)
where the weighted check-based message about z i (l), E

i (α), and the variable-based message, I i (α + e (l) i ), are respectively defined by

E (l) i (α)= j ′ ∈N i :s (l) j ′ i (z)=α W j ′ i - j ′ ∈N i :s (l) j ′ i (z)=0 W j ′ i , (3)
and

I i (α)=ηφ i (α), (4)
where η is a scaling factor. W ji in (3) is the weighting coefficient contributed to z i by other z i ′ 's checked by s j and is defined by

W ji = λ min i ′ ∈M j \{i} R i ′ , ( 5
)
where λ is a scaling factor. R i is the reliability of the initial hard-decision symbol z (0) i and is given by

R i =m i n α∈GF (q) φ i (α), (6)
with φ i (α) being a log-likelihood ratio representing the gap of likelihood between z (0) i and z

(0) i α defined by

φ i (α)= N 0 4 ln P v i = z (0) i |y i P v i = z (0) i -α|y i . ( 7
)
To speed up the decoding, it is essential to be able to flip multiple symbols in each decoding iteration. The PSFD algorithm achieves this by adopting a voting system whereby each unsatisfied CN gives one vote to the VN with the largest flipping function checked by it. At the l th iteration, VN v i accumulates the number of votes, V

i , from all the unsatisfied CNs. That is

V (l) i = j∈N i V (l) ji , (8)
where

V (l) ji 0 =1 if s (l) j =0; otherwise V (l) ji =0 with i 0 = arg max i ′ ∈M j F (l) i ′ . The hard-decision z (l) i will be flipped to z (l+1) i = z (l) i -ε (l) i when V (l)
i >V th , where V th is a codedependent threshold.

III. MULTIPLE-VOTES PSFD ALGORITHM A. Algorithm

The voting threshold, V th , in the PSFD is code-dependent and should be optimized through simulation [START_REF] Huang | Parallel symbolflipping decoding for non-binary LDPC codes[END_REF]. In the sequel, we design a new decoding algorithm which mitigates this drawback while maintaining the strong aspects of the PSFD.

First, we introduce multiple voting levels, allowing each unsatisfied CN to pass more than one vote to the VNs checked by it. Without loss of generalization, let us introduce two voting levels ζ 0 >ζ 1 > 0. At each VN v i , the voting function is still given by ( 8) but with the voting principle defined as follows For s

(l) j =0, • V (l) ji 0 = ζ 0 with i 0 =argmax i ′ ∈M j F (l) i ′ . • V (l) ji 1 = ζ 1 with i 1 =argmax i ′ ∈M j \{i 0 } F (l) i ′ . • V (l)
ji =0elsewhere. The unsatisfied CN c j gives ζ 0 to the VN that has the largest flipping function and gives ζ 1 to the VN that has the second largest flipping function. Though it might appear that the two factors ζ 0 and ζ 1 should be optimized, we will show through simulations, in the next section, that their optimal values are not code-dependent.

Secondly, we derive a relationship between the syndrome weight, the number of errors, and the parity-check matrix column weight d v from the following Lemma [START_REF] Bossert | Channel Coding for Telecommunications[END_REF]. Lemma 1: For regular LDPC matrices, the average syndrome weight increases linearly with the number of errors.

Proof: (Heuristic.) For any regular LDPC matrix, a plot of the average syndrome weight in terms of the number of errors yield a straight line with slope d v .

Based on the above Lemma, at iteration l, the number of errors in the hard-decision vector z (l) , and hence the number of symbols to be flipped, can be estimated by

N e (l) = w (l) (s) d v , (9)
where ⌊x⌋ denotes the greatest integer less than or equal to x, and w (l) (s) is the syndrome weight at iteration l defined by

w (l) (s)= j s (l) j =0 . ( 10
)
The use of a voting threshold can therefore be avoided by flipping the first N e (l) symbols with the highest votes V (l)

i . The steps of the new algorithm, which we call Multiple-Votes Parallel Symbol-Flipping Decoding (MV-PSFD), can be summarized as follows.

MV-PSFD algorithm 1: Inputs: Maximum number of iterations l max , harddecision z (0) and the received symbol vector y. 2: Initialization: Set iteration index l =0, maximum number of iterations to l max , and error symbol vector e = 0. Find φ i (α) by ( 7) and the variable-based message I i (α) by ( 4). Compute the coefficients W ji by ( 5) and store them. 3: while l ≤ l max do 4:

Compute the syndrome s (l) 5:

if s (l) = 0 then 6:

Stop the while loop 7:

end if 8:

With 0 ≤ i<N, compute the flipping function

F (l)
i and find its corresponding flipping symbol α = ε i by ( 8) using the multiple-votes principle 10:

Estimate the number of errors N e (l) by ( 9) 11:

With 0 ≤ i<N, sort VNs according to descending order of V (l) i

12:

Flip the first N e (l) VNs by setting new hard-decision symbols z

(l+1) i = z (l) i -ε (l) i
and new error symbols e

(l+1) i = e (l) i + ε (l) i
13: l ← l +1 14: end while 15: Output: z (l) .

B. Complexity Computation

We evaluate the computational complexity of the new algorithm based on the number of integer addition (IA), multiplication (IM) and comparison (IC) operators as in [START_REF] Wymeersch | Log-domain decoding of LDPC codes over GF(q)[END_REF].

At the initialization, the computations of I i (α) and R i (BPSK modulation) need O(Nqlog . The total computational complexity of MV-PSFD and PSFD algorithm for each iteration is shown in Table I. From [START_REF] Huang | Parallel symbolflipping decoding for non-binary LDPC codes[END_REF], it is proved that PSFD is the algorithm that costs the lowest decoding complexity among the reliability-based decoding algorithms. The MV-PSFD requires slightly more comparison operators than PSFD due to the sorting process. However, in practice, the cost of comparators is much cheaper than those of multiplications and additions. Thus, for small to moderate length codes, we can conclude that the complexity of MV-PSFD is close to the original PSFD.

IV. SIMULATION RESULTS

In this section, using Monte Carlo simulations, the error performance in terms of bit-error rate (BER), frame-error rate (FER), and average number of iterations as a function of the rate-normalized signal-to-noise ratio (SNR) per information bit (E b /N 0 ), of the proposed algorithm is compared with that of the PSFD algorithm on a number of NB-LDPC codes of various constructions with short to medium block lengths. These constructions include Euclidean geometry [START_REF] Zeng | Construction of non-binary cyclic, quasi-cyclic and regular LDPC codes: A finite geometry approach[END_REF], progressive edgegrowth [START_REF] Hu | Progressive edgegrowth Tanner graphs[END_REF] and the original method of Gallager. The maximum number of iterations is set to 50 for both algorithms. At each SNR value, at least 100 erroneously received codewords are detected.

The optimal values of η, λ, V th , ζ 1 , and ζ 0 , are selected through simulations for each code. It should be noted that only the ratios η λ and ζ 1 ζ 0 are actually needed. An interesting result is that the optimum ratio ζ 1 ζ 0 is 2 3 for all codes. Moreover, no significant difference in error performance was observed using Due to space considerations, we only present results for two codes, referred to as Code 1 and 2. Code 1 is a (d v =3,d c =6) [START_REF] Mackay | s Code Resource[END_REF] and Code 2 is a (d v =5,d c = 10) [START_REF] Mackay | s Code Resource[END_REF]; both are regular (102, 204) NB-LDPC codes over GF (2 4 ) constructed based on Gallager's method, whose parity-check matrix satisfies the RC-constraint.

Fig. 1 shows the BER and FER performances of Code 1. It can be observed that, unlike PSFD, the performance of the proposed algorithm is not sensitive to the parameters η and λ. Moreover, the proposed algorithm outperforms the optimal PSFD by about 0.5 dB at the BER of 10 -5 . Note that the optimal PSFD is plotted with the optimized η λ and V th . Fig. 2 shows the BER and FER performances of Code 2. Contrary to the previous case, the performance of the proposed algorithm is sensitive to the parameters η and λ. However, using the optimal value of η λ , the MV-PSFD outperforms the PSFD with a gain of about 0.3 dB at the BER of 10 -5 and 0.4 dB at the BER of 10 -6 . It can also be observed that, when MV-PSFD is used without the optimal values of η and λ, the performance is close to that of PSFD. We would like to point out that, although the proposed algorithm could be generalized to more than two voting levels, no significant performance improvement was observed for more than two voting levels for all the codes simulated. Therefore, taking into account the additional complexity, two voting levels appears to be the best choice for the proposed algorithm. Furthermore, for regular NB-LDPC codes whose parity-check matrices have large column weights, i.e., column weights of at least 8, the proposed algorithm only slightly outperforms the standard one. Results are not included here due to the space consideration.

With these performance studies in mind, we move on to the average number of iterations comparison. Fig. 3 depicts the average number of iterations for Codes 1 and 2. We see that, at low to medium SNR, the proposed algorithm with optimized parameters converges faster than the standard one. However, at high SNR, the convergence of the two algorithms is similar. Overall, one can conclude that there is no significant difference between the two algorithms in terms of convergence rate. As such, the excellent performance of the proposed algorithm is not obtained at the expense of the convergence rate.

V. C ONCLUSION

In this letter, a new algorithm based on the PSFD algorithm for regular NB-LDPC codes has been proposed. Simulation results performed on a number of NB-LDPC codes of various column weights show that the proposed algorithm significantly outperforms the standard one for low column weight paritycheck matrices, i.e., column weights less than 8, with a similar complexity.

  2 (q)) IAs, O(Nq) IMs, and O(N (log 2 (q) -1)) ICs. Additionally, O(M (2d c -3)) ICs and O(Md c ) IMs are needed for W ji . At each iteration, check-sums computations require O(M (d c -1)) IAs and O(Md c ) IMs. The normalized checksums S * (l) i need O(Md c ) IMs. The flipping functions and flipping symbols require at most O(2N (d v -1)) IAs for F (l) i (α) and at most O(N (d v -1)) ICs for F (l) i . The error estimation needs at most O(M -1) IAs for d s (l) and O(1) IM for N e (l) .

i

  , it requires at most O(M (d c -1)) ICs and at most O(M ) IAs. If the Quick Sort algorithm[START_REF] Hoare | Quicksort[END_REF] is used, the sorting needs at most O(Nlog 2 (N )) ICs. Note that the Quick Sort is not the best choice in terms of operators saving, but it is the fastest algorithm to achieve high system throughput. Finally, O(2N ) IAs are used for z

ζ 1 ζ 0 = 2 3 and ζ 1 ζ 0

 0 =1. Thus, the parameters ζ 1 and ζ 0 in MV-PSFD are not code-dependent.

Fig. 1 .

 1 Fig. 1. FER (solid) and BER (dashed) performance versus rate-normalized SNR of MV-PSFD and PSFD for Code 1.

Fig. 2 .

 2 Fig. 2. FER (solid) and BER (dashed) performance versus rate-normalized SNR of MV-PSFD and PSFD for Code 2.

Fig. 3 .

 3 Fig. 3. Average number of iterations versus rate-normalized SNR of MV-PSFD and PSFD for Code 1 (solid) and 2 (dashed).

TABLE I COMPLEXITY

 I BY OPERATORS AT EACH ITERATION

OF NB-LDPC DECODING ALGORITHMS

For voting function V

ACKNOWLEDGMENT

We would like to thank the Editor, Prof. Zheng Ma, and the anonymous reviewers for their constructive feedback. We also would like to acknowledge Prof. Emmanuel Boutillon and Dr. Deyuan Chang for their helpful discussions.