## Abstract

We propose a new way to structure the digital signal processing for *reduced guard-interval* (RGI) OFDM optical receivers. The idea is to digitally parallelize the processing over multiple parallel virtual sub-channels, occupying disjoint spectral sub-bands. This concept is well known in the optical or analog sub-carrier domains, but it turns out that it can also be performed* efficiently in the digital domain.* Here we apply critically sampled uniform analysis and synthesis DFT filter bank signal processing techniques in order to realize a novel hardware efficient variant of RGI OFDM, referred to as *Multi-Sub-Band OFDM* (MSB-OFDM), reducing by 10% receiver computational complexity, relative to a single-polarization version of the CD pre-equalizer. In addition to being more computationally efficient than a conventional RGI OFDM system, the signal flow architecture of our scheme is amenable to being more readily realized over multiple FPGAs, for experimental demonstrations or flexible prototyping.

© 2011 OSA

## 1. Introduction

*Reduced-Guard-Interval* (RGI) coherent optical *Orthogonal Frequency Division Multiplexing* (OFDM) [1] is a leading method leveraging the spectral efficiency advantages of OFDM while mitigating the excessive penalty of the Cyclic Prefix (CP) overhead.

In this paper we propose a new way to structure the *digital signal processing* (DSP) for RGI OFDM optical receivers: *Multi-Sub-Band OFDM* (MSB-OFDM), expanding on our brief introduction [2]. The idea (Fig. 1
) is to digitally parallelize the transmitter receiver processing into multiple, *M*, parallel virtual sub-channels occupying disjoint spectral sub-bands, each of bandwidth *B*/*M*, where *B* is the total channel bandwidth (the channel may be one of multiple WDM channels, i.e. the sub-banding provides a lower multiplexing tier under the WDM level). Our optical communication community is well-used to such (de)multiplexing concepts in the photonic or analog sub-carrier domains, but it turns out that the assembly and partitioning of sub-bands may also be performed* efficiently in the digital domain*.

This novel digital sub-banding method attains reduced *cyclic prefix* (CP) overhead while saving computational complexity, as the received samples are digitally partitioned into independent sub-streams, over spectrally disjoint sub-bands which may be simply and accurately processed, generally improving almost all coherent OFDM receiver functions.

The core processing technique is the incorporation of a *critically sampled* (CS) analysis filter bank algorithm into the DSP front-end, breaking the digitized high-speed stream of a single WDM channel into multiple spectral sub-bands, enabling to reduce the CP overhead. This essentially performs the same function as the conventional frequency domain pre-equalizer of RGI OFDM, but the processing is parallelized in frequency rather than in time.

In terms of prior art addressing related multiple-sub-bands concepts, we should credit previous work on OFDM multi-banding, namely [3] which proposed the dual of our receiver based scheme, but at the transmitter for *M* = 2 sub-bands. Unfortunately, complexity gets excessive with lots of bands (or even for the two bands demonstrated in [3]), as *M*-1 extra full-band FFTs are required. Moreover, that scheme necessitates transmitter (Tx) to receiver (Rx)feedback, requiring the transmitter to be cognizant of the channel state information – the *Chromatic Dispersion* (CD) delay spread experienced by the receiver.

A previous paper [4] highlighted the advantage of OFDM multi-banding at the Rx, re CP overhead reduction, but pursued an analog sub-carrier or optically multiplexed version of multi-band OFDM, which requires more complex modulator/receiver and more DACs/ADCs. We pick up where the analog/optical multi-band approach ends up, showing that the CP overhead reduction may be entirely realized digitally, within a single OFDM receiver (no need for extra modulators, DACs, ADCs) and the filter-bank based realization is actually more efficient than that of conventional RGI OFDM, rather than being less efficient.

The paper is structured as follows: In section 2 we present the receiver concept of digital sub-band de-multiplexing for RGI OFDM. Section 3 introduces the synthesis filter bank at the transmitter. In section 4 we review critically sampled filter bank implementations, and apply them to our configuration. Section 5 treats the performance of MSB-OFDM over the coherent optical link. Section 6 addresses orthogonal polarizations processing. The final section 7 discusses the multiple features of filter-bank realizations and outlines future work.

## 2. Digitally partitioning the processing into multiple sub-bands by means of a filter-bank

In this section we introduce the digital sub-band de-multiplexing rationale at the Rx side. We start by reviewing a conventional Coherent OFDM link using *MN*-point (I)FFTs (the motivation for expressing the FFT size as the product of two integers *M*,*N* will be presented shortly). Figure 2
illustrates an OFDM symbol or block at the Tx. The CP-add operation consists of replicating a section of the OFDM symbol tail at the symbol head. Due to the CD, in the fiber link, the OFDM symbol is received with a delay spread,$\Delta {\tau}_{CD}=2\pi \left|{\beta}_{2}\right|\cdot L\cdot B$proportional to the optical bandwidth, *B*, and the fiber length, *L*. The CP duration must be at least as long as the CD delay spread:$\Delta {\tau}_{CP}\ge \Delta {\tau}_{CD}$.

Let us now review the concept of RGI-OFDM. Figure 3
introduces an overlap-save CD pre-equalizer implemented as a *Frequency-Domain-Equalizer* (FDE), ahead of the CP-drop and FFT operations at the Rx. The combined impulse response of the optical channel and the FDE is now shorter than$\Delta {\tau}_{CD}$*,* thus we may now use a tiny CP. Therefore, we have a tradeoff between the large CP overhead of conventional OFDM and the high HW complexity in RGI OFDM, required to reduce the CP overhead.

Another way to reduce the CD delay spread, and hence the CP, would be to cut down the bandwidth, but along with it the bit rate would be reduced as well. Thus, to attain our high-speed target bit rate, we might use multiple narrowband sub-channels in parallel, as indicated in Fig. 1, which illustrates our MSB OFDM Rx top-level structure. The key element is a bank of band pass filters in the digital domain, say *M* = 4 band pass filters in parallel, each handling 1/*M* (here a quarter) of the channel bandwidth, assumed here *B* = 25 GHz. Therefore, as each sub-channel at each band pass filter is narrowband, its CD-induced delay spread is very small. In addition, as different sub-bands have different center frequencies, they propagate with different group velocities, thus the CD induces a successive time staggering of the individual sub-channels. Thus, each sub-band experiences little delay spread internally, however different sub-bands arrive at different times at the Rx. Notice that at this point each filter-bank output still runs at the high sampling rate of the ADC, which is *M* times faster than a rate commensurate with its reduced *B*/*M* bandwidth. Therefore, we may place down-samplers (as described by arrow-down-*M* blocks) at the band pass filter outputs, retaining every *M*-th sample and discarding the samples in between, in effect reducing the sampling rate by a factor of *M*. The resulting Rx digital front-end is referred to in the DSP literature as *critically-sampled uniform-DFT-filter-bank*. The multiple outputs of the filter-bank are taken at the decimator outputs. Notice that as the sub-bands of the band pass filters are all equal in spectral widths, and their spectral supports are contiguous and non-overlapping, this type of filter bank alludes to an efficient implementation, based on uniformly frequency shifting a single low-pass *prototype filter* by means of a DFT (section 4). Here, the *uniform* term refers to all the contiguous sub-bands having the same spectral width, while the term *critically sampled* (CS) refers to the decimation rate, *K*, coinciding with the number of paths, *M*, in the filter bank: *K* = *M*. In contrast, a filter bank with *K*<*M*, as treated in [5] is referred to as *oversampled* (OS). In this paper we solely treat CS uniform-DFT-filter-banks, which will be henceforth simply referred to, for brevity, as *filter banks*.

As indicated in Fig. 4
, each of the *M* filter bank outputs feeds a sub-band Rx. Thus, we have an array of *M* low-speed sub-band receivers following the filter bank. The rationale is that while we have invested some computational overhead in partitioning the incoming spectrum into *M* sub-bands, each sub-band receiver is now quite slow, therefore will be considerably simpler to realize, requiring much less than 1/*M*-th of the complexity of a full-band conventional OFDM receiver (including the FDE pre-equalizer), therefore we win in overall complexity, even when accounting for the filter-bank extra overhead. Beyond complexity reduction, additional key complexity and performance advantages of partitioning into sub-bands will be outlined in section7. Here we highlight the tree structure of the DSP structure of Fig. 4, with the filter-bank at the stem of the tree being the high-speed bottleneck, whereas the tree branches are terminated into slow rate sub-band receivers, wherein not only is the processing simplified, but programmable hardware such as FPGA or even software based DSP may be used to provide flexibility of realization and prototyping. Evidently, in order for this structure to make sense, it is crucial to devise an efficient implementation for the high-speed filter bank (else all the advantages gained in the slow sub-band Rx array would be offset by the added filter-bank complexity). This challenge will be addressed in section 4, wherein a detailed block diagram for the implementation of each slow sub-band receiver will be shown. Actually, each sub-band Rx is considerably simpler than a conventional full-rate OFDM receiver. As per Fig. 4, the time-staggering of the various sub-bands is partially corrected by discrete-time delays of integer number of slow-rate samples applied to each sub-band stream. These delays perform FFT window synchronizations. Simple and accurate Schmidl-Cox [6] based algorithms are applicable for estimating the required delay in each sub-band, but this is outside the scope of this paper. Next, each sub-band Rx performs an *N*-point FFT, followed by an array of 1-tap *equalizers* (EQZ) (complex scaling of each of the FFT sub-carrier outputs). For sufficiently large number *M* of sub-bands (e.g. *M* = 16, for our 25 GHz channel), it turns out that the residual CD delay spread is less than the duration Ts of a single sample (0.644∙Ts for 2000km fiber), thus there is no need for a FDE pre-equalizer in each sub-band receiver, which simply contains an integer-delay for coarse FFT window synchronization, the *N*-point FFT and the 1-tap EQZ, terminated in QAM-slicing of each of the 1-tap EQZ outputs, corresponding to the *N tones* (OFDM subcarriers) within each OFDM band. The 1-tap EQZ may correct any residual CD over the small sub-band (though in practice for say, *M* = 16 sub-bands and *B* = 25 GHz aggregate bandwidth the CD induced quadratic phase-shift over each sub-band is negligible), as well as compensate for the fine (fractional) timing. Notice that, in principle, the frequency domain 1-tap EQZ is able to provide timing correction with arbitrary resolution, provided the time shift is within the CP span. To improve spectral efficiency, we use the shortest CP possible, just a single sample per sub-band, hence the 1-tap EQZ provides perfect fractional time delay correction, while the integer pre-delay in the FFT window synchronization addresses the required integer delay correction. This indicates that the multi-sub-band approach also enables simplified and robust OFDM timing recovery, and the overall sub-band receiver structure is indeed very simple.

Thus, the overall scheme adopts a *divide&conquer* strategy, with the *divide* occurring in the filter bank, while the *conquer* is completed in the individual sub-band receivers. In particular, the sub-band Rx operations are performed at the *M* times slower rate, thus the *N*-points FFTs are manageable, although we have *M* of them. The overall processing is now equivalent to performing larger MN-points FFT, which would have been quite computationally demanding at the full rate of the aggregate channel. Those versed in FFT complexity may concur that *M* decoupled FFTs each of *N* points, are simpler than a single *MN*-points FFT, even if efficiently realized by the Cooley-Tuckey algorithm. Thus, if high spectral and temporal (low CP overhead) efficiency is our objective, the number of OFDM tones should be large, and so should the FFT size, which would be challenging at high speed (see [7] for state-of-the-art FFT sizes at high speed). The proposed filter bank approach effectively enables a very large FFT size, coupled with the other heavy operations required in the FDE equalization. The savings are not only in the number of multipliers, but also in the elimination of a considerable amount of data shuffling involved in large FFT size generation – e.g. if the full *MN*-points FFT is realized as a radix-*N* structure, we have *M*FFT-s of size *N*, followed by twiddle factor multiplications then followed by *M*FFTs of size *N*. However, the data needs to be re-organized after the first array of *M* sub-FFT-s of size *N,* such that each of these FFTs feeds each of the *N*FFTs of size *M* of the second array; it is this data shuffling as well as the second array of sub-FFTs that get eliminated in our filter-bank based approach, which may be viewed as an efficient way of organizing the large FFT, coupled with the dramatic impact of eliminating of the FDE pre-equalizer, which simplifies the overall processing. The FDE pre-equalization is not required at all ahead of the OFDM DFT in each of the sub-band receivers, as just 1-tap equalization suffices after the sub-band FFT – this is the case provided that a sufficient number of sub-bands is used, such that each sub-band is effectively frequency flat, seeing very little CD. To account for the FDE elimination, notice that it is well-known that the complexity of a CD equalizer, realized in the time domain as an FIR filter, is quadratic rather than linear in the bandwidth *B* (more bandwidth implies more CD delay spread in seconds, but in addition, the sampling rate also gets proportionately higher with *B*, thus the number of samples to be processed in the CD equalizer goes as $\propto {B}^{2}$(note that even if the CD equalizer is realized in the frequency domain, the effective temporal window to address, measured in samples, is still$\propto {B}^{2}$- it is just that in a frequency-domain realization the number of multiplications required to implement the FDE is less than the number of FIR taps). Conversely, if the bandwidth is reduced by a factor of *M*, (upon moving from the full band, *B*, to a sub-band, *B/M*) the quadratic dependence,$\propto {B}^{2}$, implies that the impulse response window of the CD, as measured in full-rate samples, is reduced by a factor $\propto 1/{M}^{2}$. For ultra-long haul 2000km transmission of a 25 GHz channel over standard SMF, the CD delay spread duration is164 full-rate samples (here full-rate means a sampling rate slightly exceeding the Nyquist rate of 25 GS/s). When the bandwidth is reduced by *M* = 16, i.e., for a single sub-band, we must scale down the sampling rate by the 1/*M*^{2} = 1/16^{2} factor, obtaining the result that the CD impulse response duration is less than a single sample. Accordingly, as already mentioned, it suffices to use a reduced *CP of a single sample per sub-band* (i.e., CP overhead, of just 1/*N* for the *N*-point FFT assumed per sub-band). This indicates that we may realize RGI OFDM, with low residual overhead, e.g. 1/128 = 0.78% CP OH for *N* = 128 points FFT in an *M* = 16 sub-bands system.

The general principle at work here is that the overall optical receiver complexity is generally super-linear (increases faster than linear) in *B*, thus when the bandwidth is reduced by a factor of *M*, the receiver complexity reduces by factor larger than 1/*M*, and as we have *M* sub-band receivers, the overall sub-band complexity of the sub-band receiver array is much reduced relative to the full-rate conventional receiver. This principle will apply to every receiver function or block, with some functions of the conventional full-rate receiver being completely eliminated (such as the FDE), while other functions (e.g. the timing recovery), will be seen to require less complexity overall, when summed up across all sub-band receivers. These savings are partially offset by the overhead incurred in partitioning the signal into sub-bands by means of the filter-bank. In this introductory presentation of the filter bank concept we shall not be able to address all receiver DSP functions, nor carry out a full comparison of the “full-rate” vs. filter-bank implementations, but in the remainder of this paper we shall focus on a thorough comparative analysis of the key functionality of *CD equalization* which weighs heavily on the complexity of reduced guard band interval OFDM realizations. In section 6 we shall also briefly outline the filter-bank based receiver structure required to handle polarization de-multiplexing.

## 3. Synthesis filter bank at the transmitter

Heretofore, we have addressed the receiver digital FDM de-multiplexing – the partitioning of the overall channel into multiple sub-channels, each transmitted over a sub-band, by means of a filter-bank structure which may be characterized as an analysis filter bank (as it analyzes or separates the overall spectrum into sub-bands). Notice that in the absence of nonlinearities, the linear impairments experienced by each of the sub-band Rx-s are decoupled -the signal received by each sub-band Rx is processed independently from the processing occurring in the other sub-band receivers. This implies that each sub-band signal has also been generated at the transmitter independently from other sub-band signals. Thus, each sub-band receiver detects a single sub-channel, correspondingly generated at the transmitter, decoupled from the other sub-channels. At the Rx we have the analysis filter bank - the dual concept at the transmitter is that of FDM multiplexing of multiple sub-channels, digitally forming the overall channel spectrum by means of a synthesis filter bank, juxtaposing multiple sub-bands each carrying a sub-channel to be addressed to a corresponding sub-band receiver (Fig. 5 ). As is well known in filter-bank DSP theory, the analysis and synthesis filter banks are duals of each other in the critically sampled case. Efficient implementation of synthesis and analysis CS filter banks is addressed in section 4. Low filter-bank complexity is a key to achieving low overhead for the filter-bank operations, enabling to take advantage of the complexity savings in the sub-band receivers.

The dual *analysis-synthesis* filter bank description implies that any modulation format, besides OFDM, may be supported for the sub-channel streams injected at the transmitter in each of the sub-bands, by means of the synthesis filter bank. This is indeed the case – e.g. each sub-channel may consist of single-carrier QAM – however this case will not be further pursued here. Instead, this paper focuses on the case wherein each sub-channel, as transmitted over a sub-band, consists of a tributary OFDM signal with *N tones* (subcarriers). It is evident that the frequency-domain juxtaposition of *M* such OFDM tributaries, each containing *N* OFDM tones actually forms a single aggregate OFDM signal with *MN* tones. A necessary condition for it is that the sub-channels be properly synchronized so that all the tones from all sub-bands fall onto a common frequency grid (this synchronization condition is readily achieved digitally, provided the center frequencies of the sub-bands are made to fall onto the same grid). This further indicates, that in the OFDM case, with each of the sub-band receivers detecting independent OFDM signal, we do not actually have to use a synthesis filter bank at the transmitter, but we may simply synthesize the overall transmitted signal by means of a conventional OFDM transmitter with *MN-*points FFT size. This is actually the approach adopted in [5], nevertheless in this paper we do pursue a filter bank (a synthesis one) at the transmitter as well, as this may be advantageous for future joint processing of polarizations at the transmitter (e.g. in order to realize polarization-time coding).

## 4. Critically Sampled analysis and synthesis Filter Bank implementation

In this section we pursue the computationally efficient digital implementation of the filter banks. Figure 6
shows the block-diagram of a communication system employing filter-bank modulation and demodulation concepts [8]. A set of *M* modulation symbols${A}_{k}^{(\beta )},\beta =0,1,\mathrm{...},M-1$, (with *k* discrete-time at the rate of 1/*T*), is input in parallel into a set of *M* discrete-time filters with transfer functions${H}_{\beta}(f)$. This set of *M* filters with common additive output represents a so-called *synthesis* filter-bank. At the receiver, demodulation is achieved by an *analysis* filter bank (a set of filters with common input) comprising *M* filters ${G}_{\beta}(f)$ followed by *K*-fold down-samplers. When *M* = *K* (i.e. the number of filter-bank paths equals the down-sampling factor), a critically sampled filter-bank structure is obtained.

In practice, filter-bank modulation systems are almost never directly implemented as shown in Fig. 6 (as band pass filters), the reason being that in this configuration the filters must operate at a rate that is *K* times faster than the symbol rate 1/*T*. If the band-pass frequency responses are appropriately selected, it is possible to achieve quite efficient realizations. For example in the critically sampled case, if the *M* transmit (receive) filters are selected as frequency-shifted versions of a single baseband filter H(f) (G(f)), the so-called *prototype filter*, the system of Fig. 6 becomes equivalent to that shown in Fig. 7
and Fig. 8
. The next step is to realize the discrete-time modulations with the complex exponentials, by means of an inverse discrete Fourier Transform (IDFT) applying LTI filtering operations on the *M* branches, inserting *M* filters corresponding to the so-called *polyphase components* of the prototype filter [8]. These structures are quite standard in DSP theory but have not yet been applied in the digital domain for optical communication, to the best of our knowledge. The complexity of the resulting DFT + polyphase filters (Fig. 9
) structure is very low, and will be evaluated in the next section.

## 5. Performance of Multi-Sub-Band (MSB) OFDM for Optical Link

#### 5.1 MSB OFDM system

Figure 10
presents a detailed MSB-OFDM block diagram, spectrally partitioning the channel in the Rx by means of a critically sampled filter bank, into multiple (*M* = 16) decoupled sub-channels, each of which is pulse-shaped in the Tx for tight spectral confinement, requiring *M-*fold digital interpolation. Matched filters are used in the Rx filter bank, with *M*-fold decimation. To generate MSB-OFDM, a single *Root Raised Cosine* (RRC) *prototype filter* (PF) pulse shaped with tight α = 0.015 roll-off(and truncated to a finite number *N*_{PF}*M* of taps, i.e. *N*_{PF} taps per polyphase) is frequency shifted by means of discrete-time modulations with complex exponentials, corresponding to uniformly frequency multiplexing the multiple sub-bands to form each of the Quasi-Nyquist WDM [1,5] aggregate channels. Efficient implementation is based on a *uniform DFT filter bank* as described earlier, wherein the filtering operations on the *M* = 16 (I)DFT branches correspond to the so-called *polyphase* components of the RRC prototype filter. Each polyphase filter with *N*_{PF} = 30 taps is in turn implemented in the frequency domain based on *L*_{PF} *= 128* (I)FFTs, with overlap equal to *N*_{PF} = 30.Note that polyphase*-*based uniform DFT filter bank are modern multi-rate DSP structures, first applied to optical transmission in our recent work [2], reducing the conceptual filter bank of Fig. 4 to the very low complexity implementation of Fig. 9.

Hardware-wise, in addition to the top-tier DFT with *M* = 16 points, used in the filter-bank, we also require a processing bottom tier of slow *N* = 128 points (I)DFTs, implementing, for each sub-channel, OFDM transmission with *N* subcarriers, with a minimal single-sample cyclic prefix. The shifted RRC spectral profiles are cleverly designed to slightly spectrally overlap (Fig. 11a
), and in addition, three OFDM subcarriers per sub-channel, namely those located at the RRC filters roll-off transitions, are turned off (or used as pilots). The net effect is to generate spectral guard bands wasting as little as 3%, robustly decoupling the individual sub-channels, allowing them to be processed independently at the bottom tier using slow *N* = 128 (I)FFTs for the OFDM modulations per sub-channel. At the top tier we have the *M* = 16 points fast (I)FFT required for the filter bank realization, but fortunately these fast (I)FFTs are kept short (a 16 points (I)FFT requires 8 fast multipliers). Our two-tiered FFT processing effectively realizes the equivalent of a very long *MN* = 2048 points FFT in an equivalent RGI OFDM system. It is this tiered FFT structure that enables efficient *real-time* FPGA/ASIC parallelization. As for our system performance, the received QPSK constellation for the worst OFDM subcarrier (at the band edge) is shown in Fig. 12a
; the Modulation Error Ratio (MER) vs. OFDM subcarrier index (identical for all sub-channels) is shown in Fig. 12b). Figure 13
plots average *Bit Error Ratio* (BER) vs. OSNR, over all sub-channels of the aggregate Nyquist WDM channel are shown in Fig. 13.

We next itemize the *complex multiplier* (CM) counts of the various stages of the proposed algorithm, yielding complexity formula for our scheme, with *M*,*N*, *N*_{PF}, *L*_{PF} as defined above. We assume for complexity calculations:

- - Polyphase filtering is performed in frequency domain using the
*OverLap-Save*(OLS) method for each polyphase channel. - - A Filter bank is used at RX only. The Tx comprises a conventional FFT based OFDM realization.
- - Sub-band processing includes an
*MN*-points IFFT at Tx and an*N*-points FFT at RX followed by 1-tap equalizer.

The complexity of the filter banks based MSB-OFDM scheme is expressed as follows:

The complexity due to sub-band processing is:

The total complexity is therefore:

Figure 14 compares our complexity, as per the last formula, with that of a reference RGI OFDM system using the OS-FDE, adjusted to provide the same spectral efficiency, the complexity of which was already modeled in [9]:

## 6. Orthogonal Polarizations processing

Heretofore, we have treated a single polarization. The full treatment of polarization multiplexing and de-multiplexing is outside the scope of this paper, however, it is important to get a sense of the “polarization processing roadmap”. Hence, in this section we briefly preview the MSB-OFDM receiver architecture modifications required in order to accommodate polarization de-multiplexing.

Figure 15
illustrates the proposed receiver architecture. The analysis filter bank is doubled up, with one instance of a filter bank for each of the X and Y received orthogonal polarizations. The X and Y sub-channels associated with each of the sub-bands are routed in pairs to corresponding 2x2 MIMO polarization sub-band receivers. Assuming *M* = 16 sub-bands, the 25 GHz channel, is partitioned into 25/15 = 1.666 GHzsub-bands. Each of the 16 MIMO sub-band receivers then takes two 1.66 GS/s X and Y inputs from the two filter banks and generates two 1.66GBd outputs, corresponding to the de-mixed X and Y polarizations (estimates of the original X and Y polarizations at the Tx).

The detailed analysis of the dual polarization sub-band receiver processing and resulting performance are outside the scope of this paper, but let us just mention that similarly to the CD handling reducing to memoriless single-tap processing per each narrow sub-band, the polarization transformation also becomes memoriless, as each sub-band sees a fixed (frequency flat) birefringence. Accordingly, there is no longer a need for a complex ‘butter-fly’ MIMO 2x2 polarization equalizer with memory with $4{\mu}_{DGD}$taps (where ${\mu}_{DGD}$is the DGD between the two principal polarizations, expressed in sample units), but rather 4 complex taps per OFDM tone suffice to realize a “scalar” inverse Jones 2x2 matrix inversion, following the sub-band IFFTs of the X and Y polarization signals.

## 7. Discussion and Conclusions

In this paper we introduced a “divide&conquer” digital sub-band (de)multiplexing strategy, digitally partitioning the wideband spectrum of each WDM channel into *M* narrow sub-bands, to be separately processed. This technique improves almost every aspect of receiver signal processing. Some of the advantages were already surveyed above, whereas the remaining ones are briefly outlined in this section.

One remaining deficiency of our scheme is 3% loss in spectral efficiency due to turning off three sub-carriers in the roll-off transitions of the band-pass filters of the filter bank. Thus our spectral efficiency is 97%, whereas our CP-overhead temporal efficiency is 1-1/*N* = 1-1/128 = 99.2% .In a follow-up publication expanding on our brief introduction [5], we shall show how to replace the critically sampled filter banks treated in this paper, by *oversampled* (OS) filter banks, which are 100% spectrally efficient and are even more hardware efficient than our first-generation CS filter-bank based systems described here. Nevertheless, the CS filter banks treated here are easier to understand than their OS counterparts, therefore the current CS approach, serves as the best introductory approach to get acquainted with the FB method, the advantages of which are briefly outlined in the following:

- 1. Here we have only established modest savings of 10% due to the elimination of the FDE (accounting for the filter-bank overhead), however, the HW complexity (as measured in terms of the number of complex multipliers) of the overall sub-band MSB-OFDM receiver may be improved even further by introducing oversampled filter banks. E.g. the FDE improvement evaluated in [5] attained 19% savings in the complex multiplier count relative to the FDE of conventional RGI OFDM.
- 2. The filter-bank Tx and Rx are highly amenable to FPGA parallelization, e.g. for the purpose of demonstrations and flexible prototyping. Partitioning of the overall processing task over multiple FPGAs is facilitated, and so is efficient parallelized processing in ASIC realizations, taking advantage of the tree structure of the filter banks, with the full data stream being split into (combined from) multiple independent slower parallel paths which do not exchange any information, and which directly interface to the ADC/DAC in parallel form. In contrast, in a conventional realization the multiple FPGAs must communicate among them at full rate.
- 3. Filter-banks effectively provide a novel method to generate arbitrarily large FFT sizes (e.g. 512-4096 points) and the effective large FFT is readily partitioned over multiple (few) FPGA(s), requiring far less inter-FPGA communication. Thus, FFT algorithms of arbitrarily large sizes can now be readily parallelized and spread across multiple processors (which might have significance for other processing areas as well).
- 4. Each sub-channel is quite narrowband (
*M*times narrower in bandwidth than the overall channel), hence sees an almost frequency-flat end-to-end transmission environment: Each sub-band experiences negligible CD and PMD. This points to extremely simple sub-band receivers, Pol-Demux and PMD equalization are substantially easier per sub-band, requiring just 2x2 MIMO memoriless processing. - 5. OFDM Rx synchronization (timing recovery, coarse and fine) are substantially simpler and more accurate per sub-band. In particular, wireless window synchronization algorithms [6], which do not work well full band due to the CD of the overall channel (which is not known prior to timing synchronization –“chicken&egg problem”) may now be made to work “by-the-textbook” for each frequency-flat sub-band.
- 6. Channel estimation becomes much simpler for each sub-band; Moreover, it may be further improved by joint sub-bands processing. Very simple and accurate monitoring of the channel CD is possible.
- 7. Carrier Recovery advantages:
*Equalization*- or*Dispersion-Enhanced Phase-Noise*(EEPN/DEPN) [10,11] is cut down by a factor of*M*, practically eliminated. EEPN is the effect whereby the phase noise of the LO laser is enhanced through the CD equalizer which has long impulse response (large delay). With the filter-bank method, as each sub-band is narrowband, its CD impulse response duration is*M*times shorter, therefore EEPN is reduced by a factor of*M*. - 8. Adaptive parameters adjustment algorithms (for CD, PMD, CR, etc.) converge faster and more accurately, due to the sub-banding, as is well-known in adaptive signal processing. Not only is the number of coefficients in each sub-band smaller, but also each sub-band is considerably flatter in its frequency response, which implies much smaller eigenvalue spread, hence faster adaptive algorithm convergence. This convergence speed-up will be manifested in every adaptive DSP algorithm, e.g. CMA for polarization de-multiplexing.
- 9. IQ imbalance correction algorithms may be more effectively formulated in the filter-bank context. It will be seen that pairs of sub-bands (with center frequencies symmetric vs. the mid-band frequency) will be coupled in pairs in order to generate simple and rapidly converging IQ imbalance correction.
- 10. Nonlinear compensation (NLC) is facilitated and improved. NLC may be applied per individual sub-band, and may be further improved by joint sub-bands processing- a recent study highlighting the NLC advantage upon processing by sub-bands may be found in [12].
- 11. The proposed sub-band based algorithms do not require special excessive allocation of bits, relative to the full channel conventional implementations.

Two remaining disadvantages of the proposed technique are: (i) the realization of the sub-band partitioning by means of the critically sampled filter bank is still somewhat computationally intensive (due to the requirement to realize spectrally sharp filters) and partially offsets the savings in the sub-band receivers complexity (though we still win overall, attaining a total 10% complexity reduction relative to conventional RGI OFDM FDE). (ii): There is some spectral inefficiency penalty incurred in the filter transitions at the sub-band boundaries. In fact, this penalty trades off with the complexity of the polyphase filters, i.e. we have a tradeoff between (i) and (ii), which we would like to further improve. In a follow-up publication expanding on [5], we shall introduce and develop the concept of *oversampled filter banks* to address the two remaining deficiencies mentioned above – this approach will remarkably attain 100% spectral efficiency while even further improving the computational efficiency.

A final disadvantage to be mentioned re the filter bank concept, is that it is hard to explain to skeptics not versed in multirate DSP.

Additional future work on the filter-bank based MSB-OFDM scheme, which is already underway with promising interim results, will further explore all the remaining points mentioned in 1-11above, which were not addressed in this paper: channel estimation, FFT window synchronization, carrier recovery (phase and frequency estimation and compensation), polarization de-multiplexing, IQ imbalance compensation and non-linear compensation, all of which functionalities are likely to be improved, once pursued in the context the decoupled narrowband sub-channels. A detailed treatment of these multiple issues is outside the scope of this introductory paper on the filter-bank approach, however, initial studies show advantages of the filter-bank based architecture in all these aspects.

## References and links

**1. **X. Liu, S. Chandrasekhar, B. Zhu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “448-Gb/s Reduced-Guard-Interval CO-OFDM Transmission Over 2000 km of Ultra-Large-Area Fiber and Five 80-GHz-Grid ROADMs,” J. Lightwave Technol. **29**(4), 483–490 (2011). [CrossRef]

**2. **A. Tolmachev and M. Nazarathy, “Real-time-realizable Filtered-Multi-Tone (FMT) Modulation for Layered-FFT Nyquist WDM Spectral Shaping - paper SPMB3,” in *European Conference of Optical Communication (ECOC)**(2011).*

**3. **L. B. Du and A. J. Lowery, “Mitigation of dispersion penalty for short-cyclic-prefix coherent optical OFDM systems,” in *European Conference of Optical Communication (ECOC)**(2011).*

**4. **S. L. Jansen and T. Schenk, “Optical OFDM for Long-Haul Transport Networks - Tutorial MH1,” in *LEOS - IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings**(2008).*

**5. **A. Tolmachev and M. Nazarathy, “Low-Complexity Multi-Band Polyphase Filter Bank for Reduced-Guard-Interval Coherent Optical OFDM - paper SPMB3,” in *Signal Processing in Photonic Communications (SPPCom), Advanced Photonics OSA Conference**(2011).*

**6. **T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. **45**(12), 1613–1621 (1997). [CrossRef]

**7. **R. I. Killey, Y. Benlachtar, R. Bouziane, P. A. Milder, R. J. Koutsoyannis, C. R. Berger, J. C. Hoe, M. Püschel, P. M. Watts, and M. Glick, “Recent Progress on Real-Time DSP for Direct Detection Optical OFDM Transceivers - paper OMS1,” in *Optical Fiber Communication Conference (OFC/NFOEC)**(2011).*

**8. **F. J. Harris, *Multirate Signal Processing for Communication Systems* (Prentice Hall, 2004).

**9. **J. Leibrich and W. Rosenkranz, “Frequency Domain Equalization with Minimum Complexity in Coherent Optical Transmission Systems,” in *Optical Fiber Communication Conference (OFC/NFOEC)**(2010).*

**10. **W. Shieh and K. P. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express **16**(20), 15718–15727 (2008). [CrossRef] [PubMed]

**11. **Q. Zhuge, B. Châtelain, C. Chen, and D. V. Plant, “Mitigation of Equalization-Enhanced Phase Noise Using Reduced-Guard-Interval CO-OFDM,” in *Optical Fiber Communication Conference (OFC/NFOEC)**(2011).*

**12. **E. Ip, N. Bai, and T. Wang, “Complexity versus Performance Tradeoff for Fiber Nonlinearity Compensation Using Frequency-Shaped, Multi-Sub band Back propagation - paper OThF4,” in *Optical Fiber Communication Conference (OFC/NFOEC)**(2010).*