Wavelets in High-Capacity DWDM Systems and Modal-Division Multiplexed Transmissions

Advanced Optical Communications II, Technology and Communications Systems Master (ETSIT-UPM), Final Work 2014 Wavelets in High-Capacity DWDM Systems and Modal-Division Multiplexed Transmissions Andrés Macho, Student Member, IEEE Abstract— Optical communications receivers using wavelet signals processing is proposed in this paper for dense wavelength- division multiplexed (DWDM) systems and modal-division multiplexed (MDM) transmissions. The optical signal-to-noise ratio (OSNR) required to demodulate polarization-division multiplexed quadrature phase shift keying (PDM-QPSK) modulation format is alleviated with the wavelet denoising process. This procedure improves the bit error rate (BER) performance and increasing the transmission distance in DWDM systems. Additionally, the wavelet-based design relies on signal decomposition using time-limited basis functions allowing to reduce the computational cost in Digital-Signal-Processing (DSP) module. Attending to MDM systems, a new scheme of encoding data bits based on wavelets is presented to minimize the mode coupling in few-mode (FWF) and multimode fibers (MMF). The Shifted Prolate Wave Spheroidal (SPWS) functions are proposed to reduce the modal interference. Fig. 1. Evolution of single-channel capacity during the last two decades and Index Terms— Digital communications, digital signal aggregate capacities. This figure has been modified for this work from processing (DSP), DWDM, MDM, wavelet transform. reference [60]. figure indicates that, in spite of relatively slow progress on the I. INTRODUCTION single-channel bit rate, a larger WDM capacity, supported by ITH the vast growth of network traffic over the past two the increase in spectral efficiency, had been achieved by W decades it has become increasingly important to realize around 2002. This was, basically, made possible by employing an ultrahigh-speed backbone network. The amount of traffic a closer channel spacing. Consequently, by 2002 high-speed carried on backbone networks has been growing exponentially fiber-optic systems research had started to investigate new at about 30 to 60% per year, and now the increasing number of modulation formats using direct detection and Fourier applications relying on machine-to-machine traffic and cloud approach based in harmonic functions decomposition [56]. services is accelerating the growth of bandwidth requirements However, with the view focused in 100-Gb/s transmissions near to 90% per year [7], [37], [62]. and beyond [63], it became clear that additional techniques Over the past twenty years, the demand for communication were needed to use 100-Gb/s channels in the widely bandwidth has been economically met by wavelength-division established 50-GHz WDM infrastructure [57]. In this context, multiplexed (WDM) optical transmission systems. A series of polarization-division multiplexed (PDM) allowed for a technological breakthroughs have allowed the capacity-per- reduction of symbol rates, which brought 40 and 100-Gb/s fiber to increase around 10x every four years, as illustrated in optical signals. At this point coherent optical receivers Figure 1. Transmission technology has therefore thus far been emerged as the main area of interest in broadband optical able to keep up with the relentless, exponential growth of communications due to the high receiver sensitivities which capacity demand. The cost of transmitting exponentially more they can achieve. They have since been combined with digital data was also manageable, in large part because more data was post-processing to perform all-electronic CD and PMD transmitted over the same fiber by upgrading equipment at the compensation, frequency- and phase locking, and polarization fiber ends. Fig. 1 illustrates the evolution of single-channel bit demultiplexing [43]. The use of coherent optical receivers rates (single-carrier, single-polarization, electronically with the combination of advanced modulation formats has led multiplexed; circles), as well as of aggregate per-fiber to increase the long transmission distances and high spectral capacities using wavelength-, polarization-, and most recently efficiencies required for the current commercial core DWDM space-division multiplexing (SDM) [57]. networks. As can be seen from Fig. 1, WDM was able to boost the But in the coming decade, as optical transmission systems relatively slow progress in single-channel bit rates [62]. The operating over single strands of legacy single-mode fiber Advanced Optical Communications II, Technology and Communications Systems Master (ETSIT-UPM), Final Work 2014 (SMF) are reaching their fundamental capacity limits [17], for the original function. The more important differences SDM based in modal-division multiplexing (MDM) and between the wavelet transform and the Fourier transform are multicore fibers (MCF) is likely to become a critical two: technology to meet future traffic demands. However, attending exclusively to MDM systems, the inability to individually Wavelet transform is based in non-harmonic excite and detect each propagation mode lets modal dispersion functions. In this case, the basic functions of the within each mode group still limit the bandwidth-distance vector space ℱ(ℝ, ℂ) are localized in both time and product of such systems, which makes them unattractive for frequency, in contrast with harmonic functions of long-haul transmission [35, Ch. 10]. Fourier transform, which are completely delocalized As we have mentioned above, DWDM and SDM systems in time. are based in Fourier signal processing, which decomposes To describe finer details in time, wavelet expansion signals in harmonic functions. In this paper, we propose the uses scaled basis functions, contrary to Fourier wavelet approach based in time-limited basis functions for analysis, which uses higher frequencies. This is the DWDM and MDM systems. In the former, the use of wavelets key to reduce in Section IV the DSP complexity to let us to reduce the DSP complexity and increase the compensate the signal degradation due to fiber Kerr sensitivity of the coherent receiver. In the latter, the wavelet nonlinearities. Therefore, we could decompose a approach is an opportunity to increase the bandwidth-distance factor due to the reduction of the mode coupling and the temporal event that occurs briefly only once using OSNR required for QPSK modulation format limited by ASE only a few wavelet basis functions rather than noise [24]. hundreds of Fourier components. The paper is organized as follows. Section II reviews the Any square integrable function 휓(푡) (that is, finite energy) mathematic fundamentals of the wavelet transform, Section III can be a wavelet if it satisfies (in addition to a few other focuses in wavelet applications for DWDM and MDM conditions) the admissibility condition: systems, Section IV describes a wavelet-based FIR filter design used for backward-propagation to nonlinear 2 +∞ |훹̂(휔)| compensation in the DSP of the coherent receivers and Section 퐶휓 = ∫ 푑휔 < ∞ (3) V presents the future perspectives, the open issues and it 0 휔 contains an implementation of the continuous wavelet where Ψ̂(휔) is the Fourier transform of 휓(푡). If the function transform in the optical domain. Finally, in Section VI the 2 paper concludes with a discussion of the main results obtained 휓 satisfies (3) we say that 휓 휖 퐿 (ℝ, ℂ), which is the vector in this work. space of square integrable functions. A (much) less mathematically stringent way of saying the same thing is to describe a wavelet as a few oscillations on a finite interval. If II. WAVELET PACKET TRANSFORM FUNDAMENTALS 휓(푡) is the mother wavelet, we can create a whole family of It is instructive to start our discussion with the Fourier wavelets by scaling and shifting the original mother wavelet: transform that is more well-known in optical communications than the wavelet transform. The Fourier transform is an 1 푡 − 푏 operation that transforms one complex-valued function into 휓푎푏(푡) ≝ 휓 ( ) (4) √2 푎 another. Since the original function is typically of time- domain and the new function is of frequency-domain, the where 푎 is the scaling parameter and 푏 is the shift parameter, Fourier transform is often called the frequency-domain both real parameters varying continuously from −∞ to +∞. representation of the original function. Discrete Fourier The family 휓푎푏(푡) can generate the whole vector space transform (DFT) is a specific kind of Fourier transform that 퐿2(ℝ, ℂ), so an arbitrary signal 푓(푡) 휖 퐿2(ℝ, ℂ) can be written requires the input function be discrete, which means its non- as a function of the basic functions as follow [11], [31]: zero values have a finite duration. Mathematically, forward and inverse DFTs are defined as (N-order): +∞ 1 푡 − 푏 1 푁−1 푓(푡) = ∬ 퐶푊푇푓(푎, 푏)휓 ( ) 2 푑푎푑푏 (5) 퐶휓 푎 푎 푋(푘) = 퐷퐹푇푁{푥[푛]} = ∑ 푥[푛]푒−푗2휋푛푘/푁 (1) −∞ 푛=0 푁−1 where the scalar numbers of this linear combination 1 푥[푛] = 퐼퐷퐹푇푁{푋(푘)} = ∑ 푋(푘)푒+푗2휋푛푘/푁 (2) 퐶푊푇푓(푎, 푏) are known as the continuous wavelet 푁 transformation (CWT), corresponding to the module of the 푘=0 projection of 푓(푡) over the vector sub-space generated by the It is known that DFT only has frequency localization, and its axis 휓푎푏(푡): basis functions, the sinusoids are infinitely long in time domain. In comparison, the wavelet transform is the ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 퐶푊푇 (푎, 푏) = ‖푝푟표푦 (푓⃗⃗⃗⃗(⃗⃗푡⃗⃗) ) ‖ = 푓 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ representation of a function by an orthogonal set called 퐿({휓푎푏(푡)}) “wavelets”. It is another form of time-frequency representation 2 Advanced Optical Communications II, Technology and Communications Systems Master (ETSIT-UPM), Final Work 2014 ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ +∞ practical example: the Haar wavelet [41]. Haar wavelet ⟨푓(푡), 휓푎푏(푡)⟩ 1 ∗ function 휓(푡) is: = = ∫ 푓(푡)휓푎푏(푡)푑푡 (6) ⟨휓⃗⃗⃗⃗⃗⃗⃗⃗⃗(⃗⃗푡⃗⃗) , 휓⃗⃗⃗⃗⃗⃗⃗⃗⃗(⃗⃗푡⃗⃗) ⟩ √|푎| 푎푏 푎푏 −∞ 1 0 ≤ 푡 < 1/2 휓(푡) = {−1 1/2 ≤ 푡 < 1 (10) To perform the WT in a DSP we need to work with digital 0 표푡ℎ푒푟푤푖푠푒 sequences 푓[푛], so it is necessary to create the discrete version of the CWT: the discrete wavelet transformation (DWT) [1].

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