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Multidimensional Vector Quantization-Based Signal Constellation Design Enabling Beyond 1 Pb/s Serial Optical Transport Networks Volume 5, Number 4, August 2013 Ivan B. Djordjevic Aleksandra Z. Jovanovic Milorad Cvijetic Zoran Peric

DOI: 10.1109/JPHOT.2013.2269678 1943-0655/$31.00 Ó2013 IEEE

IEEE Photonics Journal

VQ-SCD Enabling Beyond 1Pb/s Serial OTNs

Multidimensional Vector Quantization-Based Signal Constellation Design Enabling Beyond 1 Pb/s Serial Optical Transport Networks Ivan B. Djordjevic, 1;2;3 Aleksandra Z. Jovanovic, 4 Milorad Cvijetic, 2 and Zoran Peric 4 1

Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721 USA 2 University of Arizona, College of Optical Sciences, Tucson, AZ 85721 USA 3 Department Electrical Engineering & Information Technology, Technische Universita¨t Darmstadt, 64283 Darmstadt, Germany 4 University of Nisˇ, Faculty of Electronic Engineering, 18000 Nisˇ, Serbia DOI: 10.1109/JPHOT.2013.2269678 1943-0655/$31.00 Ó2013 IEEE

Manuscript received May 29, 2013; revised June 7, 2013; accepted June 13, 2013. Date of publication June 18, 2013; date of current version June 27, 2013. This paper was supported in part by the National Science Foundation (NSF) CAREER under Grant CCF-0952711. Corresponding author: I. B. Djordjevic (e-mail: [email protected]).

Abstract: In this paper, we propose a hybrid multidimensional coded-modulation (CM) scheme based on a new multidimensional signal constellation design as a solution to limited bandwidth and high energy consumption of information infrastructure. This multidimensional signal constellation, herewith called vector-quantization-based signal constellation design (VQ-SCD), is based on the vector quantization theory. The proposed scheme employs both the electrical basis functions (in the form of the prolate spheroidal wave functions) and the optical basis functions (in the form of polarization and spatial mode states) as optical basis functions. The proposed coded VQ-SCD scheme is the enabling technology for optical serial transport with bit rates exceeding 1000 Tb/s (1 Pb/s). In addition, the CM scheme we proposed allows for the adaptive, elastic, and dynamic allocation of the bandwidth. This fine granularity bandwidth manipulation is envisioned as a part of future software-defined optical networking. Index Terms: Microwave photonics signal processing, optical Ethernet, advanced FEC, LDPC coding, coded modulation, multidimensional signaling, elastic optical networks.

1. Introduction The exponential Internet traffic growth trends have placed enormous transmission rate demands on currently existing information infrastructure. As the response to these never ending demands for higher data rates, the 100 Gb/s Ethernet (100 GbE) standard has been adopted by IEEE 802.3ba [1] with a massive deployment currently under way, while the research focus has moved to beyond 1 Tb/s Ethernet (1 TbE) technologies [2]. It has become evident that optical Ethernet technologies will be affected by three main problems: limited bandwidth of the information-infrastructure, high energy consumption, and heterogeneity of next generation optical networks. The key enabling technologies for the next generation of optical transport networks (OTNs) [2], [3] are the most advanced coded modulation (CM) schemes, such as LDPC based ones. In this paper, we advocate the use of hybrid multidimensional quasi-cyclic (QC)-LDPC CM as a solution to the limited bandwidth and high energy consumption problems. This multidimensional

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signaling scheme employs simultaneously both electrical and optical degrees of freedom. The optical degrees of freedom include the polarization and spatial modes in few-mode fibers (FMFs), while the electrical degrees of freedom include orthogonal prolate spheroidal wave (OPSW) functions [4]. The use of this multidimensional signaling brings at least two important advantages as compared to conventional polarization-division multiplexed (PDM) QAM scheme: (i) for the same symbol energy, the Euclidean distance among signal constellation points in multidimensional signaling can be increased as compared to distance in conventional 2-D constellations; and (ii) the nonlinear interaction among spatial modes in FMFs (as well as the nonlinear PMD effects in SMF applications) can be compensated for by LDPC-coded turbo equalization. In order to enable beyond 1 Pb/s serial optical transport with commercially available electronics, the novel multidimensional signal constellations have been developed as described in following sections. These multidimensional signal constellations are obtained by the proposed vector quantization-based signal constellation design (VQ-SCD) algorithm. The paper is organized as follows. The proposed VQ-SCD algorithm is described in Section 2. In Section 3, we introduce novel hybrid multidimensional CM scheme, employing both electrical and optical basis functions and developed signal constellations. In Section 4, the proposed softwaredefined multiband spectral-spatial MIMO scheme enabling dynamic allocation of bandwidth and fine granularity, together with the tree-level spectral-spatial hierarchy suitable for beyond 1 Pb/s serial OTNs, is described. Performance analysis of the proposed hybrid CM scheme is done in Section 5, while some important concluding remarks are provided in Section 6.

2. VQ-SCD The proposed VQ-SCD algorithm establishes the analogy between the signal constellation design for Gaussian-like channels, such as ASE noise dominated scenario in fiber-optics communications, and the vector quantization of a Gaussian source. Notice that in links with no in-line chromatic dispersion compensation, the distribution of samples upon compensation of chromatic dispersion and nonlinearity phase compensation is still Gaussian-like as shown in the experimental study [5], which justifies the use of the ASE noise dominated scenario. This analogy allows us to employ the VQ theory to design a signal constellation to come closer to the channel capacity limit. Moreover, the method is suitable for multidimensional signal constellation design, which is of high importance to both the SMF applications (where the signaling space is 4-D) and the FMF applications (where the signaling space is larger than 4). As a compromise solution to an excellent BER performance and low complexity of this implementation, we opt for an optimum vector quantizer design for discretized source probability density function, which gives a near-optimum vector quantizer of a memoryless Gaussian source [6]. Namely, it is known that the simplest quantization model (uniform quantizer) is the optimal solution for quantization of a source with uniform probability density function, i.e., a piecewise-uniform quantizer is the optimal one for a piecewise-uniform probability density function [7], [8]. It has been shown in some previous works that the distortion of a piecewise-uniform vector quantizer, designed for the probability density function approximation, approaches the distortion minimum of VQ for any source type. The main problem appears to be the lack of a method for the vector space partition into the regions [8]. This problem can be overcome with introduction of the geometric principle in quantizer design [6]. This means that the surfaces of a constant input-vector probability density function should be taken for the boundaries among the regions. In the case of geometric VQ of a memoryless Gaussian source that we consider, those surfaces are concentric n-dimensional spheres. Since the lattice VQ is highly structured, we also assume that each region is further uniformly partitioned using Z n lattice. The aforementioned geometric VQ can be applied on the constellation design. The signal constellation must be selected in accordance with the non-uniform probability density function of the channel, so that the geometric VQ of corresponding source can be applied. In this paper, we are interested in the signal constellation design for the ASE noise dominated optical channels.


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Accordingly, the corresponding design is recognized as BVQ-SCD[. The details of this signal constellation design are provided below. The input to the K -point VQ-SCD is n-dimensional vector x ¼ ½x1 x2 . . . xn with coordinates being independent and identically distributed Gaussian variables xi , i ¼ 1; 2; . . . ; n with zero mean and unit variance. The probability density function of the input is given as ! n n Y 1X n2 2 pðxÞ ¼ f ðxi Þ ¼ ð2Þ exp x : 2 i¼1 i i¼1 The vector space partition into Kq regions is guided by geometric principle, which means that region boundaries are surfaces of a constant probability density function. For a memoryless Gaussian source, the surface of constant probability density function is given by n X

xi2 ¼ ln pc2 ð2Þn r02 ;

r0 0;

i¼1

where pc is the probability density function value corresponding to x . This is an P expression for the n-dimensional sphere of radius r0 , where the radius is defined as r ¼ ð ni¼1 xi2 Þ1=2 . The regions are quantized by using uniform Z n lattice quantizers with specified cell numbers as follows [6]: 2

Ki ¼ K

n

Vinþ2 Pinþ2 Kq P

2 nþ2

k ¼1

;

i ¼ 1; . . . ; Kq

(1)

n nþ2

Vk Pk

where Vi is the volume of the i-th region whose radii of boundaries are ri and riþ1 ðr1 ¼ 0Þ n

n

2 r n 2 r n Vi ¼ n iþ1 n i ; 2þ1 2þ1

(2)

while Pi is the probability that vector x , which has a radius r with the following probability density function pn ðr Þ ¼

1 1 2 r n1 e2r ; n 1 ðn=2Þ22

r 0

(3)

is in the i-th region Pi ¼

Zriþ1

pn ðr Þdr :

(4)

ri

(The ðÞ denotes the well-known gamma function.) Finally, the Z n lattice cell length for the i-th region is i ¼ ðVi =Ki Þ1=n ;

(5)

where Vi and Ki are determined with (2) and (1), respectively. The analysis conducted in [6] is asymptotic one as it provides the valid results for higher dimensions. In [9], the iterative polar quantization is applied to a constellation design for dimension 2, so that in this paper we are focused on the constellation design for dimensions higher than 2. What now remains is to utilize the established analogy between the VQ and the signal constellation and formulate the corresponding VQ-SCD algorithm. For the given number of regions and pre-specified values of the boundary radii, we calculate the cell length for all regions by using


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VQ-SCD Enabling Beyond 1Pb/s Serial OTNs TABLE 1

The specifications for 512-ary 3-D VQ-SCD

TABLE 2 The specifications for 512-ary 4-D VQ-SCD

^j ¼ ½x^j;1 Eq. (5). Then, we determine the constellation points x the following

x^j;2

x^j;n , j ¼ 1; 2; . . . ; K by

VQ-SCD algorithm: 1. For all i , i ¼ 1; 2; . . . Kq calculate the corresponding n-dimensional vectors of the Z n lattice as follows: xii1 ;i2 ;...in ¼ ½ ði1 1=2Þi

ði2 1=2Þi

ðin 1=2Þi ;

(6)

where i1 ¼ K =2n þ 1; . . . ; 0; 1; 2; . . . ; K =2n i2 ¼ K =2n þ 1; . . . ; 0; 1; 2; . . . ; K =2n ... in ¼ K =2n þ 1; . . . ; 0; 1; 2; . . . K =2n : 2. If the point xii1 ;i2 ;...in belongs to the i-th region, i.e. if its radius h i1=2 rii1 ;i2 ;...in ¼ ði1 1=2Þ2 þ þ ðin 1=2Þ2 i

(7)

satisfies the condition ri rii1 ;i2 ;...in G riþ1 , select the point xii1 ;i2 ;...in as the signal constellation point ^ , 2 ½1; K in VQ-SCD. x The proposed algorithm for multidimensional signal constellation design is quite simple and requires only the specification of Kq , ri ði ¼ 1; . . . ; Kq þ 1Þ and i ði ¼ 1; 2; . . . Kq Þ. Moreover, this algorithm can further be simplified by exploiting the symmetry in the point distribution thanks to the regularity in the structure of Z n lattice. Based on this observation, the algorithm can be even reduced down to the determination of the constellation points inside one quadrant. To illustrate this method for the proposed signal constellation design algorithm, Tables 1–3 summarize the details for the 512-point 3-D VQ-SCD, 512-point 4-D VQ-SCD, and 512-point 8-D VQ-SCD, respectively. In Fig. 1 we present the 256-ary and 512-ary 3-D constellations obtained by VQ-OSCD algorithm and with specifications from Table 1. It should be also noted that this method neglects the boundary effects so that sometimes it is required to adjust the number of points inside the region or the lattice cell size with respect to the values obtained from Eqs. (1) and (5), which is always true for any asymptotic analysis. The number of points in each region, computed by means


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VQ-SCD Enabling Beyond 1Pb/s Serial OTNs TABLE 3

The specifications for 512-ary 8-D VQ-SCD

Fig. 1. Illustration of 256-point (left) and 512-point (right) 3-D signal constellations obtained by VQ-SCD algorithm.

of Eq. (1), should be rounded to the integer value [8]. Further, the summation of point numbers from all the regions must be K .

3. Hybrid CM Employing OPSW Functions as Electrical Basis Functions and Spatial Modes as Optical Basis Functions The use of multidimensional signaling brings several important advantages as compared to conventional PDM QAM. The most important advantages of multidimensional signal constellations are: (i) for the same energy, the Euclidean distance among signal constellation points can be increased versus distances in 2-D constellations, which leads to energy-efficient multidimensional signaling; and (ii) the nonlinear interaction among spatial modes in FMF/FCF (as well as the nonlinear PMD effects in SMF) applications can be compensated for by LDPC-coded turbo equalization. The overall system configuration of proposed hybrid D-dimensional discrete-time (DT) CM scheme is shown in Fig. 2. The D-dimensional modulator, shown in Fig. 2(a), generates the signal constellation points as follows: si ¼

D X

i;d d :

(8)

d ¼1

In (8), i;d denotes the d -th coordinate ðd ¼ 1; 2; . . . ; DÞ of the i-th signal-constellation point, while the set f1 ; . . . ; D g denotes the set of basis functions: M electrical basis functions corresponding to in-phase channel and M electrical basis functions corresponding to quadrature channel. There are also two polarization states and N spatial modes. Therefore, the corresponding signal space is 4MN-dimensional. Alternatively, the set of M complex basis functions, such as complex orthogonal polynomials [11], can be used instead of set of 2M real basis functions. As it can be seen in [12], the modified orthogonal polynomials offer a limited flexibility in terms of time bandwidth product. The prolate spheroidal wave functions [13] appear to be better candidates [4]. Namely, the OPSW


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Fig. 2. The proposed hybrid 4MN-dimensional VQ-SCD coded-modulation scheme: (a) generic hybrid D-dimensional modulator, (b) generic hybrid D-dimensional demodulator, (c) 4MN-dimensional modulator, (d) 4M-dimensional modulator, (e) 2M-dimensional modulator, (f) 4MN-dimensional demodulator, and (g) 2M-dimensional demodulator.

functions are simultaneously time-limited to symbol duration Ts and bandwidth-limited to band and can be obtained as solutions of the following integral equation [13]: Ts =2 Z

n ðuÞ

sinðt uÞ du ¼ n n ðt Þ; ðt uÞ

n 2 ð0; 1

(9)

Ts =2

where the coefficient n is related to the energy concentration in the interval ½Ts =2; Ts =2. The OSPWs satisfy double-orthogonality principle: Ts =2 Z

n ðuÞm ðuÞdu ¼ n nm ;

Ts =2

Z1

n ðuÞm ðuÞdu ¼ nm :

(10)

1

As such, these functions are an ideal match for the optical communication applications. The configurations of mode-multiplexers/demultiplexers were described in [2], and are provided in Fig. 3, for the completeness of presentation. In the rest of this section, we describe the remaining blocks.


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Fig. 3. Configurations of mode-multiplexer (right) and mode-demultiplexer (left).

The generic hybrid D-dimensional modulators and demodulators are shown in Fig. 2(a) and (b), respectively. The B binary data streams are encoded by using B ðn; k Þ LDPC codes. The codewords generated by LDPC encoders are written in row-wise fashion into corresponding blockinterleaver. In code-rate adaptive applications, the code rates of individual LDPC codes could be different for the same codeword length. The B bits at the time instance i are taken from blockinterleaver column-wise fashion and used as the input of the corresponding D-dimensional mapper, implemented as a look-up table (LUT), to select a constellation point for the corresponding D-dimensional mapper. The D-coordinates of corresponding VQ-SCD from mapper are used as input to the D-dimensional modulator. The D-dimensional signal constellation point, transmitted over an FMF communication system of interest, is reconstructed in the hybrid D-dimensional demodulator shown in Fig. 2(b), which provides the estimates of projections along the basis functions. The reconstructed coordinates are used as input to the D-dimensional a posteriori probability (APP) demapper, which calculates symbol log-likelihood ratios (LLRs). In bit-LLR calculation block, we calculate the bit likelihoods needed for LDPC decoding. After LDPC decoding, the extrinsic information is passed back to the APP demapper. We iterate the extrinsic information between LDPC decoders and APP demapper until convergence or until pre-determined number of iterations has been reached, in a similar fashion to that we described in [2]. The configuration of 4MN-dimensional modulator is depicted in Fig. 2(c). It is composed of one 1 : N power splitter, N 4M-dimensional E/O modulators, and one N : 1 mode-multiplexer. The 4M-dimensional modulator, shown in Fig. 2(d), is composed of one polarization beam splitter (PBS), two 2M-dimensional modulators, and one polarization beam combiner (PBC). The configuration of 2M-dimensional modulator is shown in Fig. 2(e). The even (odd) coordinates of 2M-dimensional signal-constellation after up-sampling are passed through corresponding DT pulse-shaping filters of impulse responses hm ðnÞ ¼ m ðnT Þ, whose outputs are combined together into a single real (imaginary) data stream representing the inphase (quadrature) signal. After digital-to-analog conversion (DAC), the corresponding in-phase and quadrature signals are used as inputs to I/Q modulator. The 4MN-dimensional demodulator is shown in Fig. 2(f). After mode-demultiplexing, every mode projection is forwarded to the conventional polarization-diversity receiver, which provides the projections along the basis functions in both polarizations (and in-phase/quadrature channels). Each projection represents M-dimensional electrical signal. Two M-dimensional projections (corresponding to x-/y-polarization) are after analog-to-digital conversion (ADC) used as inputs to corresponding matched filters with impulse responses hm ðnÞ ¼ m ðnT Þ, as indicated in Fig. 2(g). Finally, the outputs after re-sampling represent projections along the corresponding basis functions, and these projections are used as inputs to the D-dimensional APP demapper, as shown in Fig. 2(b). The proposed scheme is flexible and can be used in various configurations ranging from multiplexing of 2N multidimensional signals to fully 4MN-dimensional signaling. The spectral efficiency of proposed multidimensional CM scheme is by factor SE =SEPDMQAM ¼

log2 ð24MN ÞRs ¼ 2MN=log2 K 2log2 ðK ÞRs

(11)

times better than that of the conventional SMF-based PDM-QAM scheme for the same symbol rate Rs .


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Fig. 4. Conceptual scheme of spectral-spatial processing enabling up to 1 Pb/s serial optical transport networking.

In (11), K denotes QAM signal-constellation size. For instance, the 4 Tb/s data rate can be achieved by using the commercially available electronics operating at 25 Giga symbols/s (GS/s) for M ¼ 5 and N ¼ 8. The conventional single-carrier PDM-QAM systems operating at 25 GS/s to achieve 4 Tb/s would require enormous signal constellation size ðK ¼ 2160 Þ, which is impossible to implement in practice. It is possible to achieve 1 Tb/s traffic by employing only fundamental fiber mode LP01 by setting M to 10 (and N ¼ 1) in previous example. Clearly, with this approach multiTb/s serial optical transport can be achieved with a single-carrier only, without even introducing the concept of space-division multiplexing.

4. Software-Defined Coded Multiband Spectral-Spatial-MIMO Scheme Enabling Beyond 1 Pb/s Serial OTNs To facilitate the explanation of a software-defined QC-LDPC-coded multiband spectral-spatialMIMO scheme suitable for 1 Pb/s serial optical transport and beyond, we will utilize illustration presented in Fig. 4. The signal frame is flexible and envisioned to support bit rates of 1 Pb/s. It is organized into 10 band-groups with center frequencies being orthogonal to each other. Each spectral component caries 1 Tb/s Ethernet (1 TbE), while each spectral band group carries 10 TbE traffic. We can employ a three-step hierarchical architecture with a building block being 1 Tb/s signal originating from either 10 100 GbE channels, 25 40 GbE channels, 2 400 GbE þ 2 100 GbE channels, or one TbE channel, respectively, which gives us the maximum flexibility on terms of bandwidth grooming and manipulation. The 1 TbE traffic can be generated as described in previous section. Also, several optical subcarriers of all-optical OFDM scheme can be used to create a supper channel structure. Next, 1 TbE spectral slots are arranged in spectral band-groups to enable up to 10 TbE. By combining two (four) spectral band-groups, the scheme can enable 20 TbE (40 TbE). We assume that 10 spectral band groups can be aligned along the optical spectrum as a content of the spatial mode. The second layer is related to spectral-division multiplexing, resulting in 100 Tb/s aggregate data rate per spatial mode, corresponding to 100 TbE. By combining two (four) spatial modes, the scheme is compatible with 200 TbE (400 TbE). Finally, the fiber link layer is implemented by combining the signals from spatial modes to achieve 1 Pb/s serial optical transport. The proposed scheme represents one of the possible scenarios; the spectral-spatial arrangement can also be done in a different way. Notice that, in this consideration, we assumed that 25% of the line bit rate is occupied by advanced FEC schemes, such as LDPC codes. We now describe the proposed software-defined QC-LDPC-coded multiband spectral-spatial MIMO scheme, which is shown in Fig. 5(a). (Only a single polarization state is shown to better facilitate explanations.) The mi independent data streams are used as inputs of mi QC-LDPC codes [see Fig. 5(b)]. The mi bits are used from block-interleaver to select the coordinates of 2M-dimensional signal constellation written into an LUT (designed as described in Section 2). The configuration of corresponding modulator is provided in Fig. 2(e). The band selection within the band group is chosen by complex multiplication by expðj2fn t Þ term, as shown in Fig. 5(b), where fn is the center frequency of the n-th band in the band-group. Such obtained signals are first


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Fig. 5. (a) Block diagram of proposed software-defined coded multiband spectral-spatial-MIMO scheme. (b) Details of LDPC-coded 2M-dimensional modulator.

spectrally-multiplexed to create spectral band group. The all-optical OFDM approach is used for spectral-multiplexing. Alternatively, the spectral multiplexing can be achieved by the complex multiplication of corresponding 2M-dimensional signals by exp½j2ðfc þ fn Þt, where fc is the central frequency of the c-th spectral band group and a power coupler. The corresponding spectral bandgroup signals are then coupled into FMF by mode-multiplexer, as shown in Fig. 5(a). To facilitate the demodulation process, the central frequencies of bands within the band-group, as well as among the band-groups, are properly chosen so that orthogonality principle is satisfied. The code-rate-adaptation is an important component of the proposed scheme, since in opticallyrouted networks different lightwave paths can experience different penalties due to deployment of ROADMs and wavelength cross-connects, so that their optical OSNRs at destination sides could be quite different. In order to provide seamless integrated transport platforms, which can support elastic and heterogeneous networking, we need to have an opportunity to change the error correction strength depending on channel conditions. The code-rate adaptation can be performed by either selecting different number of block-rows in corresponding parity-check matrix (H-matrix) as described in [2] or by changing permutation matrix size. In particular, the code rate adaptation performed by partial reconfiguration of decoder by varying the number of employed block-rows in H-matrix, while keeping codeword length intact, is quite suitable for hardware implementation. The aggregate data rate of the proposed scheme is given by RD ¼ 2mRRs N1 N2 N3 ;

(12)

where the factor two comes from two polarizations, R denotes the code rate, Rs denotes the symbol rate, N1 denotes the number of bands within the spectral group, N2 denotes the number of spectral groups, and N3 denotes the number of spatial bands. In (12), m denotes number of bits per symbol in the 2M-dimensional VQ-SCD. For instance, the aggregate data rate of 256-ary 3-D constellation is given by 2bRRs N1 N2 N3 ¼ 2 log2 256 0:8 31:25 GS/s 10 10 10 ¼ 400 Tb/s. On the other hand, by using 1024-ary 8-D constellation and 40 Gb/s equipment (and other parameters as in previous example), the aggregate data rate of 800 Tb/s is achieved. Further on, by using 512-ary 4-D constellations with additional electrical basis functions for multiplexing and 25 GS/s based equipment, we can generate Pb/s serial data rate. The multi-Tb/s optical transport can be achieved with single carrier only. For 400 GbE, by setting in the first example N1 ¼ N2 ¼ N3 ¼ 1, we achieve the desired aggregate data rate. Therefore, the proposed scheme is quite flexible ranging anywhere


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Fig. 6. The BER performance of spectral-spatial multiplexed ðN1 ¼ N2 ¼ 14; N3 ¼ 12Þ QC-LDPC (16935,13550,0.8)-coded 3-D modulations. BER performances are obtained by averaging over all bands and spatial modes.

Fig. 7. The BER performance of spectral-spatial multiplexed ðN1 ¼ N2 ¼ 14; N3 ¼ 12Þ QC-LDPC (16935,13550,0.8)-coded 4-D and 8-D modulations. BER performances are obtained by averaging over all bands and spatial modes.

from 400 Gb/s to several Pb/s. Notice that proposed QC-LDPC-coded multidimensional VQ-SCD spectral-spatial MIMO scheme is very well aligned with current optical communication trends [14]–[25].

5. Performance Analysis To demonstrate high potential of the proposed VQ-SCD hybrid CM scheme, we perform Monte Carlo C++ simulations for ASE noise dominated scenario. It is done for information symbol rate per single-band of 25 GS/s and by setting N1 ¼ N2 ¼ 14, N3 ¼ 12. The simulation results, summarized in Figs. 6 and 7, are obtained by averaging over all bands and spatial modes. To compensate for mode-coupling effects, the MIMO-OFDM signal processing has been used, as described in [24]. In simulations, natural mapping rule is used, as the determination of the optimum mapping rule for such large constellations is challenging optimization problem, as discussed in [25], where the optimization for small constellations and 2-D signaling is observed. The aggregate data rate for 512-ary constellation-based spectral-spatial multiplexing is 1.06 Pb/s. A QC-LDPC(16935, 13550,0.8) code used in our simulation has a column-weight 3, and it is designed as described


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Fig. 8. Information capacities of multidimensional VQ-SCDs (per single spatial mode and single polarization state).

in [2], while the multidimensional signal constellations are obtained by VQ-SCD algorithm described in Section 2. (For the OSNR per bit definition, an interested reader is referred to ref. [2].) For illustrative purposes, the corresponding scheme based on 64-QAM has been shown as a reference. As we see, the LDPC-coded 64-ary 4-D scheme based on VQ-SCD algorithm outperforms the LDPC-coded 64-QAM by 6 dB. On the other hand, the improvements of 256-ary 8-D scheme over corresponding 4-D and 3-D schemes are 3.9 dB and 6.5 dB, respectively. Given that the QC-LDPC-coded VQ-SCD-based schemes show an excellent BER performance, naturally raises the question how far away we are from the information capacity limits. To address this question, in Fig. 8, we report the information capacities for various VQ-SCD-modulation schemes of different dimensionalities. We conclude that the QC-LDPC-coded 256-ary VQ-SCD-8-D scheme is 1.97 dB away from 256-ary 8-D information capacity limit. This indicates that even though excellent performance of QC-LDPC-coded VQ-SCD schemes can be obtained, there is still some space for improvement. To come close to information capacity curves, we need to use stronger, nonbinary LDPC codes, and optimum signal constellation designs obtained from maximization of mutual information, which are quite challenging to obtain for large constellation sizes considered in this paper.

6. Summary The adaptive coded-VQ-SCD spectral-spatial-MIMO scheme based on regular QC-LDPC coding suitable for next generation software-defined networking with petabit transport bit rates has been proposed. The proposed hybrid CM based on combination of the OPSW functions as electrical basis functions and spatial modes as optical basis functions, represents beyond 1 Pb/s serial optical transport enabling technology, while employing the commercially available electronics. Accordingly, the proposed QC-LDPC-coded multidimensional VQ-SCD scheme, which is far superior to schemes based on QAM and well aligned with current optical communication trends, will directly address the Boptical networks capacity crunch[ problem.

References [1] P. Winzer, BBeyond 100G Ethernet,[ IEEE Comm. Mag., vol. 48, pp. 26–30, Jul. 2010. [2] I. B. Djordjevic, BSpatial-domain-based hybrid multidimensional coded-modulation schemes enabling multi-Tb/s optical transport,[ J. Lightw. Technol., vol. 30, no. 14, pp. 2315–2328, Jul. 2012. [3] Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuochi, BA tripleconcatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,[ presented at the Proc. OFC/NFOEC, San Diego, CA, USA, 2010, Paper OThL3.


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[4] I. B. Djordjevic and M. Cvijetic, BOptical transport exceeding 10 Tb/s based on adaptive LDPC-coded multidimensional spatial-spectral scheme and orthogonal prolate spheroidal wave functions,[ presented at the Proc. IEEE ICTON, Cartagena, Spain, Jun. 23–27, 2013, (Invited paper). [5] Y. Zhao, J. Qi, F. N. Hauske, C. Xie, D. Pflueger, and G. Bauch, BBeyond 100G optical channel noise modeling for optimized soft-decision FEC performance,[ presented at the Proc. OFC/NFOEC, Los Angeles, CA, USA, Mar. 4–8, 2012, Paper OW1H.3. [6] A. Zˇ. Jovanovic and Z. H. Peric, BGeometric piecewise uniform lattice vector quantization of the memoryless Gaussian source,[ Inf. Sci., vol. 181, no. 14, pp. 3043–3053, Jul. 2011. [7] A. Gersho and R. Gray, Vector Quantization and Signal Compression. Norwell, MA, USA: Kluwer, 2003. [8] F. Kuhlmann and J. A. Bucklew, BPiecewise uniform vector quantizers,[ IEEE Trans. Inf. Theory, vol. 34, no. 5, pp. 1259– 1263, Sep. 1988. [9] H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, BIterative polar quantization-based modulation to achieve channel capacity in ultrahigh-speed optical communication systems,[ IEEE Photon. J., vol. 2, no. 4, pp. 593–599, Aug. 2010. [10] P. F. Swaszek and T. W. Ku, BAsymptotic performances of unrestricted polar quantizer,[ IEEE Trans. Inf. Theory, vol. 32, no. 2, pp. 330–333, Mar. 1986. [11] V. I. Smirnov, BOn the theory of orthogonal polynomials of a complex variable,[ Zh. Leningrad. Fiz.-Mat. Obshch., vol. 2, no. 1, pp. 155–179, 1928. [12] I. B. Djordjevic and T. Wang, BOrthogonal polynomials based hybrid coded-modulation for multi-Tb/s optical transport,[ presented at the Proc. CLEO-PR&OECC/PS, Kyoto, Japan, Jun. 30–Jul. 4, 2013, Paper ThR2-5. [13] D. Slepian, BProlate spheroidal wave functions, Fourier analysis and uncertainty V: The discrete case,[ Bell Syst. Tech. J., vol. 57, no. 5, pp. 1371–1430, May/Jun. 1978. [14] T. Liu and I. B. Djordjevic, BOn the optimum signal constellation design for high-speed optical transport networks,[ Opt. Exp., vol. 20, no. 18, pp. 20 396–20 406, Aug. 27, 2012. [15] I. B. Djordjevic and T. Wang, BRate-adaptive irregular QC-LDPC codes from pairwise balanced designs for ultra-highspeed optical transport,[ presented at the Proc. CLEO, San Jose, CA, USA, Jun. 9–14, 2013, Paper CM1G.8. [16] M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks. Boston, MA, USA: Artech House, Jan. 2013. [17] P. J. Winzer, BOptical networking beyond WDM,[ IEEE Photon. J., vol. 4, no. 2, pp. 647–651, Apr. 2012. [18] M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, BPolarization-multiplexed rate-adaptive non-binary-LDPCcoded multilevel modulation with coherent detection for optical transport networks,[ Opt. Exp., vol. 18, no. 3, pp. 1820– 1832, Feb. 2010. [19] R. Ryf, S. Randel, N. K. Fontaine, M. Montoliu, E. Burrows, S. Chandrasekhar, A. H. Gnauck, C. Xie, R.-J. Essiambre, P. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, L. Gruner-Nielsen, R.V. Jensen, and R. Lingle, B32-bit/s/Hz spectral efficiency WDM transmission over 177-km few-mode fiber,[ presented at the Proc. OFC/NFOEC, Anaheim, CA, USA, 2013, Paper PDP5A.1. [20] E. Ip, M.-J. Li, E. Ip, Y.-K. M.-j. Li, A. Huang, E. Tanaka, W. Mateo, J. Wood, Y. Hu, and K. Yano, B146 6 19-Gbaud wavelength- and mode-division multiplexed transmission over 10 50-km spans of few-mode fiber with a gainequalized few-mode EDFA,[ presented at the Proc. OFC/NFOEC, Anaheim, CA, USA, 2013, Paper PDP5A.2. [21] X. Chen, J. Ye, Y. Xiao, A. Li, J. He, Q. Hu, and W. Shieh, BEqualization of two-mode fiber based MIMO signals with larger receiver sets,[ Opt. Exp., vol. 20, no. 26, pp. B413–B418, Dec. 2012. [22] X. Chen A. Li et al., BReception of mode-division multiplexed superchannel via few-mode compatible optical add/drop multiplexer,[ Opt. Exp., vol. 20, no. 13, pp. 14 302–14 307, Jun. 2012. [23] Z. Chen, L. Yan, W. Pan, B. Luo, Y. Guo, H. Jiang, A. Yi, Y. Sun, and X. Wu, BTransmission of multi-polarizationmultiplexed signals: Another freedom to explore?[ Opt. Exp., vol. 21, no. 9, pp. 11 590–11 605, May 2013. [24] C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, BNon-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,[ IEEE Photon. J., vol. 4, no. 5, pp. 1922–1929, Oct. 2012. [25] T. Liu and I. B. Djordjevic, BEXIT chart analysis of optimal signal constellation sets and symbol mappings for blockinterleaved coded-modulation enabling ultra-high-speed optical transport,[ in Proc. IEEE ICTON, Cartagena, Spain, Jun. 23–27, 2013.


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