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Abstract
We present computationally-efficient implementation techniques of multiple-input multiple-output (MIMO) frequency-domain equalization (FDE) that deals with a modal dispersion effect underlying in long-haul mode-division-multiplexed (MDM) signal transmission. Computational complexity reduction comes from twofold approaches: the first is out-of-band-exclusive (OBE) FDE for removing redundant computations in MIMO-FDE processing by taking a practical signal bandwidth occupation into account. A second appoarch, referred to as overlapping-reduced (OR) FDE, further reduces the complexity by modifying an overlapping ratio in the overlap-and-save FDE method to below 0.5. We show that OBE-FDE reduces complexity for processing with a larger number of spatial channels compared with the conventional FDE, and that the overlapping ratio is minimized up to 0.25 by the OR-FDE approach. Also shown is an experimental demonstration using the results of three-mode MDM transmission with different inter-mode coupling scenarios, verifying that the complexity is decreased by up to 37 % with negligible deterioration in signal performance by using the proposed method.
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1. Introduction
Scaling transmission capacity per optical fiber is attained through increasing the number of information-carrying optical channels. The approach exploiting multiple spatial modes has been intensively studied as promising candidates of newly available spatial channels utilizing transmission media including multi-mode fiber/few-mode fiber (FMF) [1–5], or coupled-core multi-core fiber (CC-MCF) [6–8], known as the family of the mode-division-multiplexing (MDM) transmission strategy. Towards future deployment of MDM transmission technology to ultra-high-capacity optical transport systems, one technical challenge is how to handle the fundamental effect of modal dispersion (MD), arising from the difference of propagation constant in each spatial mode. With the increased transmission reach, MD exhibits an accumulation nature in a linear manner for weakly-coupled MDM transmission links where spatial channels are mutually interact with weak interference [9], or in squared-root manner for strongly-coupled ones [6,7]. While the multiple-input multiple-output (MIMO) equalization technique may provide a solution for undoing MD effects at a coherent MDM transceiver, enormous computational effort will be required for implementation of real-time high-throughput MIMO digital signal processing to deal with an MD-induced wide spread of an impulse response ranging up to several tens of nanoseconds [10–12], likely occurring in MDM transmission links with long distances [4,5]. One common approach to mitigate the computational effort of MIMO equalization is to switch the processing domain from time to frequency [13], referred to as MIMO frequency-domain equalization (FDE). It offers computationally efficient processing especially in the filter output and the weight update by utilizing fast Fourier transform (FFT). Several experimental demonstrations have been conducted on long-distance MDM transmission compensating for MD effects by using MIMO-FDE technique [13–15]. To date, MIMO-FDE algorithm in most of these reports is usually processed on the basis of the overlap-and-save (OS) method with an overlapping ratio of 0.5 in the same manner as other traditional applications of acoustical noise suppression [16]. One remaining task regarding MIMO-FDE for MDM links is to make it simpler and scalable for supporting many spatial channels.
In this paper, especially tailored to MDM transmission links, we introduce two MIMO-FDE techniques contributing to its lower-complexity implementation. The first approach is out-of-band-exclusive (OBE) FDE for omitting the redundant complex multiplications in MIMO-FDE processing with the consideration of signal-bandwidth occupation. Taking the physical property of MD accumulation into account, the second approach, overlapping-reduced (OR)-FDE, attempts to further decrease computational complexity by reducing the overlapping ratio of MIMO-FDE less than 0.5. We show that an effect of complexity reduction is enhanced in cases of MDM links with a larger number of spatial channels. Through an experimental results involving long-haul three-mode-multiplexed transmission links, we also verified the feasibility and equalization performance of the proposed approaches with decreased computational complexity in either weak or strong inter-mode coupling cases.
2. Proposed low-complexity MIMO-FDE algorithms
In this section, techniques for a design of low-complexity MIMO-FDE are introduced. We consider an MDM system where $D$ spatial channels including polarization ones are transmitted over MDM links as information carriers of independent data streams, and received by $D$ coherent receivers. After digitization with the twofold oversampling ratio, MIMO-FDE is carried out with the input signals $\mathbf {u}_i(k)$ reshaped for a block-by-block processing, where $i$ and $k$ denote the $i$-th spatial channel and the $k$-th block, respectively. To simplify discussion, we only consider the radix-2 FFT algorithm in MIMO-FDE processing, hence set a block size of $N$ with a sufficiently large integer of power of two. Since we assume an employmemt of the traditional OS MIMO-FDE algorithm working at the oversampling rate of 2 and the overlapping rate $\rho$ of 0.5 [13,14], each block processing produces $N/4$ output symbols per block per spatial channel. As the adaptation example of our approaches to MIMO-FDE, we employ the unconstrained least squared error (LMS) FDE [15,16] because of its easy-to-understand description. The benefit of the proposed low-complexity MIMO-FDE techniques will be generally expanded to other adaptation algorithms including constrained FDE which ensures that a time-domain filter update is exactly performed by a corresponding frequency-domain processing [17], although it is out of the main scope of this paper.
2.1 Conventional MIMO-FDE
We give a brief review of the conventional MIMO-FDE [13] (Fig. 1(a)). At the $k$-th block time, the $i$-th input blocks $\mathbf {u}_{i}(k)$ $\in \mathbb {C}^{N \times 1}$ is configured as
Each filter weight vector is then updated in the frequency domain as
Fig. 1. Schematics of (a) conventional approach processing with sub-equalizers for odd/even input blocks, and of (b) proposed OBE-FDE preventing out-of-band multiplications in processing of filter output and filter weight update.
As mentioned above, while the conventional MIMO-FDE performs prior down-sampling at its input into sub-equalizers, this may result in the loss of the a priori knowledge for signal occupation in the frequency domain. The fact that commercially available modern coherent transceiver is capable of producing Nyquist-shaped signals realizing densely packed wavelength division multiplexed (WDM) allocations motivates us to modify the algorithm to deal with twofold-oversampled input signals. The detailed description for this new FDE algorithm, OBE-FDE, is provided in Section 2.2. The approach in performing MIMO-FDE that accepts twofold-oversampled inputs for the standard single-mode fiber (SSMF) $2 \times 2$ transmission systems has been reported [18], while our main objective is to improve computational efficiency of MIMO-FDE by removing redundant processing for scalable MDM transmission systems.
Another issue is the discrete increase in the block size satisfying the condition of the power of two, which does not always match the required one in a physical sense. Since an MD-induced spread in the impulse response grows gradually with increased distance, symbols that are additionally available in $\mathbf {v}_{j}(k)$ in Eq. (5) may be discarded in the conventional MIMO-FDE with $\rho$ of 0.5. Our second approach, OR-FDE, increases the number of symbol outputs in Eq. (5), which decreases $\rho$ and improves computational efficiency. The description of OR-FDE is given in Section 2.3. Compared with a similar approach [19], in which the overlap-and-add FDE minimizes overlapping to compensate for chromatic dispersion, the main contribution of the presented work in terms of decreasing $\rho$ is that we introduce an OR-FDE method based on the OS method, and also that the verification of $\rho$ reduction with respect to MD is experimentally presented for the first time (Section 3).
2.2 Out-of-band exclusive (OBE) FDE
This subsection describes the newly-proposed OBE-FDE (Fig. 1(b)). With the a priori knowledge of signal bandwidth occupation (e.g., the roll-off factor), OBE-FDE selectively executes multiplications only for the in-band frequency components of the signals. In the rest of this subsection, OBE-FDE is described under the assumption of using a sufficiently small roll-off factor $\alpha \sim$ 0. OBE-FDE starts processing with the direct conversion of the input blocks $\mathbf {u}_{i}(k)$ into the frequency domain as
The $i$-th filter weight vector for the $j$-th spatial channel $\mathbf {W}_{i,j}(k)$ $\in \mathbb {C}^{N \times 1}$ is exclusively multiplied by $\mathbf {U}_{i}(k)$, for only the components corresponding to the frequency bins of a signal’s presence. This roughly halves the number of complex multiplications from $N$ to $\sim N/2$ for the “rectangular-shaped” signals. For the rest (i.e., out-of-band entries of the size of $\sim N/2$), the multiplication results are set to zeros. Output signals are then obtained by saving only odd (or, even) components at the last half of the output from IFFT. The processing above for obtaining $\mathbf {v}_{j}(k)$ is mathematically written as
Here, $e_j(k)[m]$ is the $m$-th entry of $\mathbf {e}_{j}(k)$. With the comparison of Eqs. (5) and (7), it is apparent that the number of complex multiplications is roughly halved in Eqs. (10) and (11). This complexity mitigation is attributed to the drop in the multiplication operations with respect to the out-of-band frequency components. We provide the detailed analysis on this complexity mitigation in Section 2.4.
2.3 Overlapping-reduced (OR) FDE
We introduce another approach of OR-FDE enabling an improvement of the complexity requirement by decreasing $\rho$, the input-output blocks of which are schematically illustrated in Fig. 2. With the aim of combining OBE-FDE and OR-FDE methods, we describe the algorithm by adding modifications to that of OBE-FDE. By denoting $\Delta$ as an increased number of symbol outputs per block processing which varies in the range of $0 \leq \Delta \leq N/8$ , an $i$-th input block for OR-FDE computation at a $k$-th block time $\mathbf {u}_{i}^\text {OR}$ $\in \mathbb {C}^{N \times 1}$ is configured as
For a tentative use, we define $\mathbf {P}_{j}(k) \triangleq \mathrm {IFFT}\left [ \sum _{i=1}^{D} \mathbf {W}_{i,j}(k) \odot _{\text {in}} \mathbf {U}_{i}^\text {OR}(k)\right ]$. In a filter output computation in the OR-FDE approach, a discarded portion of the IFFT outputs is modified so as to discard its both sides, instead of one side taken in the conventional FDE approach (Fig. 2). If we perform it by saving output symbols in a “symmetrical” manner, Eq. (10) is modified as
Fig. 2. Schematics of FDE processing with consecutive blocks for (a) conventional FDE with $\rho = 0.5$, and for (b) OR-FDE scheme with $\rho < 0.5$.
It is understood that the main modification of OR-FDE is the saved entries in Eqs. (15) and (17). Figure 2 illustrates how the conventional FDE and OR-FDE works at consecutive blocks. Note that each block size $N$ is represented on the two-fold oversampling basis. As shown in Fig. 2 (a), with the conventional FDE, one side of the filter output is picked up as $\mathbf {v}_{j}$. On the contrary, this area is shifted to the center portion of the filter output block with the size of $N/4+\Delta$ with OR-FDE (Fig. 2 (b)). On the design of the OR-FDE, it is not trivial how we can apply proper $\Delta$ to have no (or negligible) effect of inter-block interference in these output symbols. To consider this, we take a scenario of applying OR-FDE at an MDM transmission link where $N = N_1$ to cover MD-induced broadening of the impulse response at distance $L = L_1$. FDE processing produces $N_1/4$ symbol output per block processing per spatial channel. With a slight increase in the transmission distance $dL$, the block size is doubled as $N = 2N_1$ at $L=L_1 + dL$, providing the per-block processing output of $2N_1/4$ plus $\Delta \sim N_1/4$ symbols (equivalently, $N_1/4$ plus $\Delta \sim N_1/8$ symbols when $N = N_1$) without inter-block interference, because of the small increase in the pulse broadening. The number of these additional $\Delta$ will gradually decrease at longer distance for the fixed $N$. In the most optimistic scenario described above, $\rho$ is suppressed from 0.5 (=$N_1/(2N_1)$) to 0.25 (=$(N_1/2)/(2N_1)$).
2.4 Complexity analysis
This subsection discusses achieved computational complexity reduction by two computationally efficient FDE approaches introduced in Sections 2.2 and 2.3. We first discuss reduced complexity effect by using OBE-FDE. Table 1 lists the complexity requirements per block processing for $D$ spatial channels (i.e., $ND/4$ output symbols) between the conventional FDE and OBE-FDE schemes. All FFT/IFFT operations are assumed executed by the radix-2 FFT algorithm, requiring $(N/2) \log _2 N$ complex multiplications for an input block of the size $N$. For the conventional FDE, FFT/IFFT computations with a block size of $N/2$ are required $2D$, $D$, and $D$ times in Eqs. (4), (5), and (8), respectively. In addition, the element-wise product totally requires $ND^2$ and $ND^2 + ND$ complex multiplications in Eqs. (5) and (7), respectively. On the other hand, for the OBE-FDE scheme, $D$ FFT/IFFT computations with a block size of $N$ are performed each in Eqs. (9), (10), and (12). An additional number of complex multiplications in the element-wise product operations is $ND^2/2$ and $ND^2/2 + ND/2$ in Eqs. (10) and (11), respectively. The computational complexity per output symbol per spatial channel $C$ is used as a metric for fair comparisons with different $N$ and $D$, simply obtained by dividing the total number of complex multiplications with $ND/4$:
for the conventional FDE, and for OBE-FDE. We notice from the table that complexity reduction is achieved in the computation of the filter output (Eqs. (5) and (10)) and weight-vector update (Eqs. (7) and (11)), rather than in the FFT/IFFT operations. Clearly, this reduced-complexity effect is due to the selection of the in-band components in performing complex multiplications at these steps. Another important consequence is that complexity mitigation is enhanced at the larger count of spatial channels $D$. In other words, OBE-FDE is suitable for MDM transmission with higher-order spatial modes rather than lower-order ones including SSMF transmission.
Table 1. Complexity requirement comparison between conventional FDE and OBE-FDE.
A simple example is given in Fig. 3 for MDM transmission links with the use of $D = \{2, 6, 8, 12, 20, 24\}$ and $N = \{256, 2048\}$ based on the evaluation of the complexity reduction ratio $\eta _\text {OBE} \triangleq C_\text {OBE}/C_\text {conv}$. Note that $D$ of 2, $\{6, 12, 20\}$, and $\{8, 24\}$ is obtained using SSMFs, FMFs [1–5], and CC-MCFs [6–8], respectively. We see that complexity reduction of $\eta _\text {OBE} < 1$ is achieved for almost all MDM links where $D \geq 6$, while complexity is not unfortunately reduced for $\{D, N\} =\{6, 2048\}$. The corresponding MD-induced pulse broadening reaches 51.2 ns with the use of 10-GBaud signals, which is considered to be too large to handle even with modern advanced DSP application specific integrated circuit (ASIC) technology [20]. It is noteworthy that smaller $\eta _\text {OBE}$ is obtained for larger $D$, indicating that OBE-FDE is suitable for scaling the spatial channels for MDM transmission. In the extreme case in which $D \gg \log _2{N}$, $\eta _\text {OBE}$ converges to 0.5. On the contrary, computational complexity of OBE-FDE processing would unfortunately rather increase as $\eta _\text {OBE} > 1$ for smaller $D$ scenarios including an application to SSMF transmission with $D = 2$. This is because, for both FDE schemes, a majority of the processing in the smaller $D$ regime would be governed by FFT/IFFT operations of which complexity is simply determined by its input FFT block size. Somewhat practical cases considered above give $\eta _\text {OBE}$ as $0.65 \leq \eta _\text {OBE} \leq 0.93$ for $N=256$, and $0.70 \leq \eta _\text {OBE} \leq 1$ for $N=2048$.
Fig. 3. Complexity reduction ratio $\eta _\text {OBE}$ achieved with OBE-FDE for $D = \{2, 6, 8, 12, 20, 24\}$ and $N = \{256, 2048\}$.
To see a complexity reduction by the OR-FDE, recalling that output symbol count per block per spatial channel is enhanced from $N/4$ up to $N/4 + \Delta$ ($0 \leq \Delta \leq N/8$), the computational complexity metric $C$ should be changed accordingly:
which gives a complexity reduction ratio $\eta _\text {OR} \triangleq C_\text {OR}/C_\text {conv}$ ofHence OR-FDE gives $2/3 \leq \eta _\text {OR} \leq 1$ because of the variation of $\Delta$. When both OBE-FDE and OR-FDE are jointly employed, it is not difficult to show that the achieved complexity reduction ratio in the comparison to the conventional FDE, $\eta _{\textrm{OBE}\&\textrm{OR}}$ becomes
indicating that relevant parameters regarding both FDE schemes are independently optimized in a design of them. It is difficult to conduct more comprehensive analysis of reduced complexity because achieved $\eta _\text {OBE}$, $\eta _\text {OR}$, and $\eta _{\textrm{OBE}\&\textrm{OR}}$ is highly depending on system parameters including $D$, $N$, and MD values per unit fiber length. Nevertheless, we try to unveil the complexity reduction effects by both schemes with an experimental verification using three-mode FMFs in Section 3.At a given mode-multiplexed transmission link over a certain distance, an amount of MD-induced pulse broadening behaves static rather than dynamic [12]. In a practial ASIC design scenario, FDE parameters including a FFT block size $N$ and an overlapping ratio $\rho$ would be pre-detemined at an initial stage of a system deployment with a joint consideration for other fundamental design parameters including latency and ASIC power consumption [21].
3. Experimental setup and results
In this section, we experimentally investigate the effects of the reduced complexity together with equalization performance when OBE-FDE and/or OR-FDE are employed in MDM transmission links with decreased overlapping portion in its block-wise processing. To this end, we constructed a setup for an MDM transmission experiment with a 51.2-km-long FMF supporting three modes (i.e., LP$_\text {01}$, LP$_\text {11a}$, and LP$_\text {11b}$), which is almost identical to the one we have previously reported [4]. In the experiment, 11-WDM dual 6-GBaud PDM-QPSK signals shaped with a roll-off factor $\alpha$ of 0.01 were transmitted over three-fold recirculating loop systems, received at the set of coherent receivers, and processed offline with each MIMO-FDE approach employing a step-size parameter $\mu$ of $10^{-4}$. One noteworthy feature of our setup is the switching capability of inter-mode coupling regimes by employing the method of cyclic mode permutation (CMP) [4,5]: without CMP, weak inter-mode coupling occurrs along with signal propagation, producing an almost linear growth in the spread of the impulse response. Alternatively, the introduction of the CMP scheme offers mutual interference for spatial channels during propagation, achieving quasi-strongly-coupling with a squared-root growth in the impulse response with respect to the transmission distance. Hereinafter the transmission without and with CMP is referred to as case-I and case-II, respectively. Figure 4 highlights the transitions of impulse response for (a) case-I and (b) case-II, showing a clear difference in its growth with increased transmission distance. In particular, the impulse response exhibited a “bell-shaped” one in case-II, which is typically observed in strongly coupled MDM transmission using CC-MCFs [6–8]. Figure 5 represents the required equalizer memory length for both transmission regimes, defined as the window covering the pulse power at 99%. As expected, the memory length accumulated in a linear manner for case-I, while this changed in a squared-root growth for case-II. From the evaluation based on the power approximation, the required memory length $M$ in ns for each transmission regime as a function of transmission distance $L$ in km is expressed as
for weakly-coupled transmission in case-I, and for the (quasi) strongly-coupled transmission in case-II. On the basis of the results obtained in Eqs. (23) and (24), Fig. 6 shows the predicted output symbols per block per spatial channel for both schemes at different transmission distances. Letting $k$ be an integer satisfying $k = \lceil \log _2 (4BM) \rceil$ where $B$ and $\lceil \cdot \rceil$ are a symbol rate in GBaud and the ceil function which outputs a nearest integer larger than its input, respectively, FFT block size $N$ is designed as $N = 2^k$.Fig. 4. Impulse response evolution with increased distance observed in three-mode multiplexed transmission for (a) weakly-coupled scenario (case-I) and (b) strongly-coupled scenario (case-II).
Fig. 5. Required equalizer memory length as function of distance for (a) weakly-coupled scenario (case-I) and (b) strongly-coupled scenario (case-II).
Fig. 6. Number of output symbols per block processing per spatial channel for the conventional FDE and OR-FDE in (a) case-I, and (b) case-II. Solid lines show the predicted ones based on Eqs. (23) and (24), while squares are ones obtained by OR-based MIMO equalization with negligible penalty in signal performance.
We start the experimental analysis on the equalization performance of OBE-FDE. Denoting $N_\text {OBE}$ as the number of frequency-bin entries used in the OBE processing which is designed as $N_\text {OBE} = \lceil NB(1+\alpha )/2 \rceil$, we changed it in the range from 0 to $N$, whose results are depicted in Fig. 7 using the normalized general mutual information (NGMI) penalty in case-II at a distance of 1536 km. Note that the NGMI penalty in this figure is defined using the baseline when $N_\text {OBE} = N$. In contrast to a steep decrease in $N_\text {OBE}/N < 0.5$, negligible penalty was observed in $N_\text {OBE}/N \geq 0.5$, verifying that out-of-band multiplications can be avoided for the signals with the roll-off factor $\alpha$ of 0.01. These results assures the feasibility of reducing $N_\text {OBE}$ up to $N/2$, giving $\eta _\text {OBE}$ of 0.95 for the case of $\{D, N\} = \{6, 128\}$.
Fig. 7. NGMI penalty in OBE-FDE processing for various frequency-bin entries when MDM signals are transmitted over 1536 km in case-II.
To further verify the effect of the lower complexity achieved with OR-FDE, the predicted output symbols per block per spatial channel for both schemes at different transmission distance are shown in Fig. 6. At each transition point changing the order of power of two for the FFT block size $N$, the largest $\Delta$ (and also, smallest $\eta _\text {OR}$) was obtained. Then $\Delta$ reached zero towards the next transition point, which is repeated in every region of a fixed $N$. To evaluate the signal performance, we swept number of output symbols $\Delta$ in the region with the fixed $N = 512$, corresponding to the distance ranging from 250 to 460 km for case-I, and one ranging from 560 to 1640 km for case-II. Figure 8 summarized the NGMI penalty for both transmission cases, defined as the NGMI decrease from the baseline when $\Delta = 0$. From the figure, we see that the substantial increase in $\Delta$ is possible for both cases. Interestingly, the NGMI penalty for case-II traced gentler negative curves compared with that for case-I, originating from the wider tail of the bell-shaped impulse response brought by CMP. We added obtained $\Delta$ to Fig. 6 (red squares) based on the NGMI results in Figs. 7 and 8 with an acceptable NGMI penalty of 0.01, confirming the excellent agreement with the predicted one. Signal constellations obtained by OR-FDE are displayed in Fig. 9 at a distance of 614 km with case-II, showing that inter-block interference appears with $\Delta = 64$. Figure 10 depicts complexity reduction ratios $\eta$ in each reduced-complexity FDE schemes, achieving $0.89 \leq \eta _\text {OBE} \leq 0.98$, $2/3 \leq \eta _\text {OR} \leq 1$, and $0.63 \leq \eta _{\textrm{OBE}\&\textrm{OR}} \leq 0.95$ at distances over 100 km for the case-II scenario with reduced pulse broadening presented in our three-mode-transmission setup. The complexity reduction obtained here seems relatively small in this low-$D$ configuration (see discussions in Section 2.4). We will extend the application of the schemes for MDM links supporting many spatial channels for future work.
Fig. 8. NGMI penalty in OR-FDE processing with fixed block size $N=512$ for (a) case-I at distances $L =\{256, 307, \ldots, 460\}$ km, and (b) case-II at distances $L =\{563, 614, \ldots, 1638\}$ km. Note that each curve is drawn with distance increase of 51 km.
Fig. 10. Achieved complexity reduction ratio $\eta$ in the conducted three-mode transmission experiment with the strongly-coupled scenario (case-II).
4. Conclusions
In this paper, novel schemes are presented to realize a computationally-effcient implementation of MIMO-FDE for a design of scalable MDM transmission systems. The OBE-FDE selectively performs complex multiplications only for in-band entries of frequency-domain signals, directly contributing to lower complexity especially for MDM links with an employment of a larger count of spatial channels. In addition, the OR-FDE approach modifies a traditional overlap-and-save FDE to have reduced overlapping ratio to less than 0.5, substantially improving computation effciency. The effects of complexity reduction together with equalization performance were investigated through three-mode MDM transmission experiments over 1000 km with two different inter-mode coupling scenarios, demonstrating that the complexity effort was decreased by up to 37 % with the combined use of OBE-FDE and OR-FDE. We believe that these FDE schemes will contribute to a realization of scalable long-haul MDM transmission systems in the near future.
Funding
National Institute of Information and Communications Technology (NICT).
Acknowledgments
Part of this research utilized the results of the project "Research and Development of Spatial-Mode-Controllable Optical Transmission System" (No. 01001) of the National Institute of Information and Communications Technology (NICT).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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