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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 23 — Nov. 5, 2012
  • pp: 25884–25901
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An analytical study of the improved nonlinear tolerance of DFT-spread OFDM and its unitary-spread OFDM generalization

Gal Shulkind and Moshe Nazarathy  »View Author Affiliations


Optics Express, Vol. 20, Issue 23, pp. 25884-25901 (2012)
http://dx.doi.org/10.1364/OE.20.025884


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Abstract

DFT-spread (DFT-S) coherent optical OFDM was numerically and experimentally shown to provide improved nonlinear tolerance over an optically amplified dispersion uncompensated fiber link, relative to both conventional coherent OFDM and single-carrier transmission. Here we provide an analytic model rigorously accounting for this numerical result and precisely predicting the optimal bandwidth per DFT-S sub-band (or equivalently the optimal number of sub-bands per optical channel) required in order to maximize the link non-linear tolerance (NLT). The NLT advantage of DFT-S OFDM is traced to the particular statistical dependency introduced among the OFDM sub-carriers by means of the DFT spreading operation. We further extend DFT-S to a unitary-spread generalized modulation format which includes as special cases the DFT-S scheme as well as a new format which we refer to as wavelet-spread (WAV-S) OFDM, replacing the spreading DFTs by Hadamard matrices which have elements +/−1 hence are multiplier-free. The extra complexity incurred in the spreading operation is almost negligible, however the performance improvement with WAV-S relative to plain OFDM is more modest than that achieved by DFT-S, which remains the preferred format for nonlinear tolerance improvement, outperforming both plain OFDM and single-carrier schemes.

© 2012 OSA

1. Introduction

DFT-spread (DFT-S) OFDM is a variant of OFDM transmission ported into optical communication by Shieh et al [1

1. W. Shieh, Y. Tang, and B. S. Krongold, “DFT-Spread OFDM for Optical Communications,” in 9th International Conference on Optical Internet (COIN), (2010).

], simulated in [2

2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. 2(3), 276–283 (2010). [CrossRef]

6

6. C. Li, Q. Yang, T. Jiang, Z. He, M. Luo, C. Li, X. Xiao, D. Xue, and X. Yi, “Investigation of Coherent Optical Multi-band DFT-S OFDM in Long Haul Transmission,” IEEE Photon. Technol. Lett. 24, 1704–1707 (2012).

] and experimentally demonstrated [7

7. Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, A. A. Amin, and W. Shieh, “Coherent optical DFT-Spread OFDM in Band-Multiplexed Transmissions, We.8.A.6,” in European Conference of Optical Communication (ECOC), (2011).

10

10. A. Li and G. Chen, Xi, A. Guanjun, A. Amin, W. Shieh, William, B. S. Krongold, “Transmission of 1. 63-Tb/s PDM-16QAM Unique-word DFT-Spread OFDM Signal over 1, 010-km SSMF,” in OFC/NFOEC, paper OW4C.1, (2012).

]. In the wireless literature this method is referred to as Single-Carrier Frequency Division Multiple Access (SC-FDMA) [11

11. C. Ciochina and H. Sari, “A review of OFDMA and single-carrier FDMA,” in Wireless Conference (EW), 706– 710, (2010).

] and this scheme has been adopted in the fourth generation wireless cellular networks.

The DFT (de)spreading idea consists of applying additional (I)DFT based array pre(post)-processing ahead (after) of the main (DFT) IDFT in the OFDM transmitter(receiver). This typically results in a reduction of the Peak to Average Power Ratio (PAPR) of the OFDM signal. The DFT-S OFDM format has been gaining traction [12

12. X. Liu and S. Chandrasekhar, “High Spectral-Efficiency Transmission Techniques for Beyond 100-Gb/s Systems, SPMA1,” in SPPCom - Signal Processing in Photonic Communications - OSA Technical Digest, 1–36, (2011).

] as it is a multicarrier format [2

2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. 2(3), 276–283 (2010). [CrossRef]

] more tolerant to fiber nonlinearity than either plain OFDM or single carrier transmission are. An intuitive explanation advanced in [2

2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. 2(3), 276–283 (2010). [CrossRef]

] accounts for the nonlinear tolerance (NLT) advantage in terms of the reduced PAPR, the rationale being that each DFT-S sub-band amounts to a conventional single carrier of relatively low PAPR. As there are far fewer sub-bands per channel than there are OFDM sub-carriers, the PAPR is reduced.

We further introduce and analyze a generalization of DFT-S OFDM, referred to here as unitary-spread OFDM (U-S OFDM), which consists of replacing the (de)spreading (I)DFTs by more general unitary transformations. The statistics of the U-S OFDM signals are theoretically analyzed and simulated and some subtle effects are clarified – in particular the unitary spread signals (DFT-S in particular) are marginally yet not jointly Gaussian and are statistically correlated but not independent. This provides insight into the mechanism of FWM nonlinear interference buildup in U-S OFDM, ultimately enabling a full analytic model of the NL performance of the generic U-S OFDM format as well as that of special cases such as DFT-S OFDM.

One interesting special case of U-S OFDM is obtained by having the generalized (de)spreading performed by means of N-pnt Hadamard matrices. We refer to the resulting system as Wavelet Spread (WAV-S) OFDM. This OFDM variant was previously proposed in wireless transmission under the name Walsh-Hadamard OFDM [15

15. M. P. H. Jun and J. Cho, “PAPR Reduction in OFDM transmission using Hadamard Transform,” in IEEE International Conference on Communications1, 430–433, (2000).

,16

16. Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in Proceedings IEEE 56th Vehicular Technology Conference4, 2096–2100, (2002).

] but for a single sub-band. Both WAV-S and DFT-S OFDM are viewed here as special cases of our newly introduced generic U-S OFDM transmission scheme. The advantage of the WAV-S OFDM is its reduced complexity relative to the DFT-S OFDM, as the Hadamard unitary spreading matrix has elements +/−1, hence matrix multiplication reduces to signed addition, much simpler than DFT-spreading; the WAV-S pre/post processing is then multiplier-free and its complexity just marginally exceeds that of plain OFDM. However, the WAV-S nonlinear tolerance turns out to be inferior to that of DFT-S OFDM but still exceeds that of plain OFDM. Thus, WAV-S OFDM practically provides a new complexity-penalty-free option for slightly improving OFDM NL performance.

2. Review of DFT-S OFDM and its interpretation as FDM of sub-single-carriers

In the DFT-S OFDM receiver the inverse processes occur. The output of the main OFDM DFT is partitioned into M equal sub-blocks, each of which is passed through an N-pnt IDFT. The sub-band IDFTs in the Rx form the de-spreading IDFT array. The cascade of the main MN-pnt DFT and the de-spreading array acts to demultiplex and decimate each of the sub-bands, undoing the action of the DFT-S Tx. The de-spreading DFT-S post-processing corresponds to inverting Eq. (1) by performing the exchanges A(r)B(r)andDFTNIDFTN. The associated Rx transformation matrix is not reproduced here but it is similar to that of Eq. (1), having a block-diagonal formdiag{IDFTN,IDFTN,...,IDFTN},with M identical IDFT blocks on the main diagonal.

Throughout the paper we refer to the OFDM sub-carriers as tones, and we also make the convention of referring to the DFT-S sub-bands as sub-single-carriers (SSC), as the signal within each sub-band is effectively indistinguishable from a time-domain single-carrier signal, albeit of reduced bandwidth B/M where B is the overall channel bandwidth, as shown next.

2.1 DFT-S OFDM as frequency division (de)multiplexer with interpolation (decimation)

We now show that the overall DFT-S signal is equivalent to a frequency-division-multiplex (FDM) carrying M SSCs side-by-side in frequency. To this end consider exciting a single SSC input of the DFT-S OFDM Tx (turning off the other M-1 sub-band signals). The block diagram now reduces to a single N-pnt DFT feeding an MN-pnt IDFT (Fig. 1(b)) with zero-padded frequency inputs on either side of the active N inputs fed by the single N-pnt DFT singled out of the DFT-S spreading array. It is well-known in DSP theory [17

17. B. Porat, A Course in Digital Signal Processing (John Wiley and Sons, 1997).

] that zero-padding input of an IDFT corresponds to time-interpolation of the IDFT output. It follows that the DFT-IDFT cascade generates a time→frequency→time map composition, compounding to a time→time mapping generating TD interpolation. Thus, the DFT-S transmitter time-domain interpolates its center SSC. Moreover, whenever the SSC is shifted off center, as the inputs into the main IDFT are effectively in the frequency-domain (FD), then this is a frequency shift amounting to modulation (multiplication by an harmonic signal) in the TD. We conclude that each of the SSCs may be viewed as a TD single-carrier signal which is both interpolated and modulated onto a sub-carrier at a center frequency corresponding to the sub-band center position. When all the SSCs are excited together by simultaneously applying M TD inputs to the DFT spreading array, we then obtain multiple interpolated SSCs modulated onto a regular grid of center frequencies. A composite FDM signal is then formed. The DFT-S transmitter provides sub-carrier multiplexing of multiple (M) narrowband single carrier QAM transmissions. This intuitive account of DFT-S Tx operation is mathematically formalized in Appendix A, rigorously proving that the DFT-S Tx indeed functions as FDM + interpolator.

2.2 OFDM and cyclically extended SC as special cases of DFT-S OFDM

Notice that in the special case N = 1 (but M1) we retrieve a conventional OFDM system, as each SSC now coincides with a single OFDM tone.

At the other extreme, for M = 1 (but N1) a DFT-S system with a single sub-band (Fig. 2(a)
Fig. 2 Cyclically extended Single-Carrier Tx (a) obtained as a special case of DFT-S with M = 1 SSC sub-band. (b) resulting CE-SC block diagram. The Tx is just a single-carrier one with added CP; the Rx drops the CP and performs DFT-based FD one-tap per tone equalization.
, degenerates to Single Carrier (SC) transmission (with cyclic extension) as shown in Fig. 2(b), referred to in the wireless literature as SC-FDMA. A more appropriate name adopted here in the optical context is Cyclically Extended Single Carrier (CE-SC), where CE alludes to equipping the conventional SC transmission with a cyclic prefix (CP). The CE-SC Tx (Fig. 2(b)) is akin to a single carrier conventional QAM Tx, except that each block of N consecutive SC symbols is prepended or appended a cyclic extension (cyclic prefix or suffix), e.g., a CP consisting of the last μsymbols of the N-pnt block is replicated ahead of the N-pnt block to form an N+μpnt extended block of samples transmitted on the line. This differs from an OFDM Tx in that the CE-SC Tx lacks the main IDFT, which is now cancelled out by the single spreading DFT of the DFT-S array (which degenerates to a single DFT for M = 1).

The corresponding CE-SC Rx comprises a N-pnt DFT followed by an N-pnt IDFT (which corresponds to a degenerate DFT-S array with a single de-spreading IDFT transformation). However, in the Rx, the DFT and IDFT no longer cancel out, as in between them there are inserted one-tap complex multiplications used to equalize the cyclically extended fiber channel.

3. Numeric simulations of the nonlinear tolerance of DFT-S OFDM

In this section we present our own numeric simulation of the NLC of DFT-S OFDM as plotted in Fig. 3
Fig. 3 Numeric simulation of the received FWM NLI MER over a 16-QAM DFT-S OFDM optically amplified fiber link, for an SSF transmission experiment with the following parameters:MN=128;BW=25GHz;α=0.2dB/Km;γ=1.3(WKm)1;D=17ps/(nmKm);L=6×100Km;P=0dBm.38400 data-symbols were used to gather the statistics (and we have verified that the MER converged to steady values).
, corroborating the key conclusion of [2

2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. 2(3), 276–283 (2010). [CrossRef]

6

6. C. Li, Q. Yang, T. Jiang, Z. He, M. Luo, C. Li, X. Xiao, D. Xue, and X. Yi, “Investigation of Coherent Optical Multi-band DFT-S OFDM in Long Haul Transmission,” IEEE Photon. Technol. Lett. 24, 1704–1707 (2012).

] that there is an optimal (>1) number of SSCs per DFT-S channel whereat the NL performance improvement attains its maximum, exceeding the performance of plain OFDM (by more than 2 dB for the specific simulated case) and single-carrier systems (by 1 dB), and setting the stage for additional analytic study of the NL behavior of DFT-S systems.

4. PAPR is not an accurate predictor of the nonlinear tolerance of DFT-S OFDM

To summarize the conclusions of the last section in operational terms, the DFT-S transmission scheme is superior in its NLI MER to both OFDM and to single carrier transmission which are the extreme points of the peaked plot of Fig. 3.

There have been several publications, e.g [19

19. B. Goebel, S. Hellerbrand, N. Haufe, and N. Hanik, “PAPR reduction techniques for coherent optical OFDM transmission,” in 2009 11th International Conference on Transparent Optical Networks1, 1–4, (2009).

,20

20. C. R. Berger, Y. Benlachtar, R. I. Killey, and P. A. Milder, “Theoretical and experimental evaluation of clipping and quantization noise for optical OFDM,” Opt. Express 19(18), 17713–17728 (2011). [CrossRef] [PubMed]

], suggesting to adopt PAPR [18

18. H. Myung, J. Lim, and D. Godman, “Peak-To-Average Power Ratio of Single Carrier FDMA Signals with Pulse Shapingý,” in IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications, (2006)ý, pp. 1–5.

] as a meaningful measure of nonlinear behavior in fiber optic transmission. In this section we consider how the numerically predicted NL performance seen in the last section correlates with the PAPR properties of the DFT-S Tx.

These numerical simulations confirm the known property that PAPR is a monotonically decreasing function of number of SSCs – the fewer carriers are used, the lower the PAPR. However, as illustrated in Fig. 4, the FWM NLI MER exhibits quite a different behavior - it is not a monotonically decreasing function of the number of SSCs but it rather peaks up at a certain SSC bandwidth, becoming worse for either higher or lower values. We conclude that Tx-side PAPR is a highly inaccurate predictor of the nonlinear tolerance of DFT-S OFDM. Indeed, if NLT were well-correlated with PAPR, then the NLT would peak for SC transmission whereat the PAPR peaks up, yet we have seen that it is DFT-S with a certain M>1 number of sub-bands that attains the optimal NLT, exceeding the CE-SC performance.

5. Monte Carlo Volterra series NL simulation of the DFT-S link

In the sequel we resort to the analytic frequency-domain FWM NLI generation model over optically amplified fiber links, as developed in [13

13. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

] and reviewed in [14

14. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

], based on Volterra Series Transfer Function (VSTF) formalism, e.g [21

21. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997). [CrossRef]

,22

22. L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightwave Technol. 30(3), 310–316 (2012). [CrossRef]

], evaluating the complex amplitude of the FWM intermodulation tone resulting from each FD triplet of OFDM tones complex amplitudes by coherently superposing all the FWM contributions falling on the i-th tone.

6. Unitary-Spread OFDM as generalization of DFT-S and its FWM generation model

The analytical derivation of nonlinear performance may be directly carried out for DFT-S OFDM, but it is more convenient to first generalize DFT-S to a more general modulation format with wider applicability, to which we refer to as Unitary-Spread OFDM. We proceed to derive the nonlinear tolerance of this more generic format, finally restricting the general U-S NLT results to the special DFT-S case, as well as to other special cases.

6.1 Extending DFT-S OFDM to the generalized U-S OFDM modulation format

The DFT-S Tx pre-processor was specified by a linear transformation via the block DFT matrix of Eq. (1), operating onto a data block of MN samples. DFT-S may be viewed as a special case of a more general unitary transformation operating on the MN–pnt block according to,
Ai=t=1MNWi,tBtA=WB
(4)
which is referred to as spreading transform (here Wi,tdenotes the i,t-element of W). The associatedW-matrix will be assumed unitary, i.e., satisfyingW1=W,wheredenotes the conjugate-transpose. Evidently, the particular block-diagonal transforming matrix of Eq. (1) is unitary, as its sub-blocks (the N-pnt DFT matrices) are unitary. A more general linear spreading transform with non-unitary W might be introduced. However, imposing a unitarity constraint on the spreading transform ensures desirable qualities in transmission over noisy channels, namely per-tone energy conservation and lack of statistical correlation between the spread OFDM tones. Indeed, as a consequence of the unitarity of W (implying that the rows of W are orthonormal) when the transmitted zero-mean samples are uncorrelated, Bt1Bt2*=σ2δt1,t2, the resulting spread samples are also uncorrelated:
Ai1Ai2*=t1t2Wi1,t1Wi2,t2*Bt1Bt2*=σ2t1Wi1,t1Wi2,t1*=σ2δi1,i2
(5)
In addition to imposing unitarity onto the spreading transform, it is useful to have its -matrix assume the following block-diagonal regular form
WMN×MN=diag{UN×N,UN×N,...,UN×N}
(6)
where the M identical sub-blocks, UN×N,are all positioned along the main diagonal of the matrix (with zeros elsewhere) and the matrix UN×Nis an arbitrary unitary transformation. The particular form of Eq. (6) for the spreading transform results in structuring the hardware signal processing which is all performed with identical building blocks, UN×N.

In the U-S OFDM generalization, except for replacing the array of identical (de)spreading (I)DFTs at the Tx and Rx by an array of alternative unitary transformations, the rest of the transceiver processing remains the same, namely the main IDFT at the Tx and the main DFT at the Rx are still performed but they are now fed from / feed into the alternative (de)spreading transformation of Eq. (6) at the Tx and by its Rx dual, WMN×MN=diag{UN×N,UN×N,...,UN×N},for various UN×Nselections which now differ from N-pnt (I)DFTs. Evidently, taking UN×N=DFTNas the N-pnt DFT matrix (with its (m,n) element given by exp(j2πNmn)) reduces the generalized U-S transform to the conventional DFT-S transform of Eq. (1).

6.2 Wavelet-Spread OFDM

A useful special case of U-S OFDM is obtained by adoptingUN×N=HADN,with HADN the N-pnt Hadamard matrix. The resulting transmission system will be referred to as WAV-S OFDM. We mention that this technique is unrelated to that of [23

23. A. Li, W. Shieh, R. S. Tucker, and A. Wavelet, “Wavelet Packet Transform-Based OFDM for Optical Communications,” J. Lightwave Technol. 28, 3519–3528 (2010).

] which superficially bears a similar name. Our WAV-S spreading transform is described by:
WMN×MNWAV-S=diag{HADN,HADN,...,HADN}
(7)
where the M identical sub-blocks consist of the N-pnt Hadamard matrix, recursively defined as follows:
HAD2N=(HADNHADNHADNHADN);HAD2=(1111).
(8)
A key advantage of WAV-S OFDM, compared to DFT-S OFDM, is that the HAD matrix contains only + 1's or −1's rendering trivial the evaluation of the WAV-S spreading transform, which becomes multiplier-free just requiring signed additions.

6.3 Exploring additional U-S OFDM special cases

A large variety of additional U-S matrices may be further investigated beyond the DFT-S and WAV-S ones. One important consideration is whether the spreading matrix elements are real-valued or complex-valued. WAV-S (DFT-S) belongs to the real- (complex-) valued U-S family respectively. Intuitively, having complex-valued elements might yield an advantage, as exemplified by the relative performance of WAV-S vs. DFT-S, however this conjecture should be further explored. In terms of computational complexity, efficient complex-valued U-S matrices would be those with elements {1,j,-1,-j} which are trivial to multiply by. In [24

24. M. H. Lee, S. Member, B. S. Rajan, and J. Y. Park, “A Generalized Reverse Jacket Transform,” IEEE Trans. Circ. Syst. II 48(7), 684–690 (2001). [CrossRef]

,25

25. A. Aung, B. P. Ng, and S. Rahardja, “Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications,” IEEE Trans. Sig. Process. 56(8), 3562–3571 (2008). [CrossRef]

] complex-valued generalizations of the Hadamard transform matrices have been suggested – those might be suitable for our efficient U-S OFDM as some of the unitary matrices presented in both references do have elements in the {1,j,-1,-j} set. Follow-up work should further explore of the wealth of unitary matrices which might potentially provide improved NL tolerance relative to WAV-S while retaining very low hardware complexity.

7. Statistics of the unitary-spread samples

In this section we reveal some non-intuitive statistical properties of U-S OFDM resolving some apparent paradoxes. Surprisingly, while the spread MN samples,{Ai},at the output of the U-S transform are statistically uncorrelated, they are nevertheless statistically dependent, as opposed to the input samples, {Bt} as generated by the m-QAM or m-PSK mapper, which are independent identically distributed (IID), all drawn from the same m-pnt discrete constellation distribution. The oddity of this statement stands out once we observe that the unitary-spread MN samples,{Ai},are asymptotically circular Gaussian distributed for sufficiently large N. This follows from the Lyapunov form of the Central Limit Theorem (CLT), stating that under certain conditions (satisfied here) a sum of independent equal mean and equal variance (but not necessarily IID) random variables tends to a normal distribution. The Gaussian distribution is numerically verified in Fig. 6
Fig. 6 Numeric Monte Carlo simulation of the PDF of the modulus (magnitude) and angle of the output samples of a 16-QAM DFT-S transmitter, parameterized by the number N (64,128,256) of tones per sub-band. The empirical histograms track the respective theoretical PDFs which are Rayleigh for the modulus and uniform over 2π for the phase. The plot pertains to the tone indexed i = 1.
, which plots the modulus and phase (arg) of the complex samples {Ai} (for i = 1, but the same applies for the other i-indexes), parameterized by the number of tones. As the complex variables are circular Gaussian, the distributions of modulus and phase are supposed to be respectively Rayleigh and uniform. The empirical PDFs are seen to plausibly track the expected analytic profiles, verifying that at least forN64the asymptotic approximation seems accurate.

Applying the plain CLT (no need for the more specialized Lyapunov version) we may also readily establish that the WAV-S spreading transform introduced in section 6.2 yields {Ai} samples which should be asymptotically Gaussian distributed as MN becomes sufficiently large. The asymptotically Gaussian distribution was also verified by simulation, though the results are omitted.

Now, to reason that the spread samples,{Ai}, are statistically dependent yet uncorrelated, recall that the inputs, {Bt}, are IID and zero-mean, hence are uncorrelated (form a white sequence). Therefore, the output of the unitary spreading transform should also be white, as well-known and also proven in Eq. (5). To summarize, the U-S samples {Ai}are Gaussian, uncorrelated, dependent. This might seem odd at first sight, as it is a knee-jerk reaction of those trained in elementary random signals theory to state that lack of correlation for Gaussian stochastic variables leads to their independence. There is nevertheless no contradiction here, as the spread {Ai}samples are individually Gaussian, as random variables, however the A-vector is not a Gaussian vector, i.e. the{Ai}are just not jointly Gaussian. In other words the distribution of the A-vector is not multi-variate normal, just the marginal distributions of the{Ai}variables are individually normal. The statistical dependency of the spread {Ai}may also be alternatively verified as follows: Imagine the U-S Tx being cascaded back-to-back with a corresponding U-S Rx. After traversing the main IDFT in the Tx, the main DFT and the de-spreading transform in the Rx, we should retrieve the original samples {Bt}which are drawn from the m-pnt discrete constellation distribution. The fact that Gaussian variables are linearly combined to yield an m-QAM or m-PSK distribution attests to their statistical dependency and to their not being jointly Gaussian. Indeed, by negation, if these individually Gaussian variables were actually statistically independent, then they would be jointly Gaussian, and any linear transformation of theirs would still yield jointly Gaussian variables rather than m-QAM or m-PSK distributed variables.

8. Nonlinear performance of U-S OFDM

In this section we derive a third analytic model (further to the SSF and tri-linear numerical models) for the FWM generation. This is actually a novel analytic version of the trilinear model - it predicts the FWM NLI of DFT-S by means of a closed-form formula (further generalized to additional OFDM variants based on alternative spreading transforms). This third model still requires numerical evaluation of a very large analytic formula involving summations, but it no longer resorts to Monte-Carlo simulation. This new trilinear analytic model precisely assesses the numerical FWM tolerance of DFT-S and more generalized OFDM variants.

In the case of DFT-S OFDM, the TD QAM/PSK samples,Bt, are first spread as per Eq. (4), generating the FD samples Ai, triple products of which appear in Eq. (3), As per subsection 5, the post-spread samples, Ai, are uncorrelated yet they are not statistically independent as they would be in plain OFDM generation. This makes it challenging to evaluate the statistics of the output of the third-order nonlinear (trilinear) transformation of Eq. (3) for spread OFDM. The solution is to retrace Eq. (3) to the underlying IID TD samples,Bk, by substituting the spreading transform of Eq. (4) and simplifying the resulting expression. This yields the following compact formula for the FWM NLI detected in the DFT-S OFDM Rx, directly expressed in terms of transmitted TD Bksamples:
riFWM=(j,k)S[i]Hi;jkt1t2t3Bt1Wj,t1Bt2Wk,t2(Bt3W(j+ki),t3)*=t1t2t3Bt1Bt2Bt3*(j,k)S[i]Hi;jkWj,t1Wk,t2W(j+ki),t3*=t1t2t3hi;t1t2t3Bt1Bt2Bt3*
(10)
where we have introduced the TD nonlinear Volterra Series Kernel (VSK):
hi;t1t2t3(j,k)S[i]Hi;jkWj,t1Wk,t2W(j+ki),t3*
(11)
In the special case of DFT-S transmission, the Bt samples are effectively in the time-domain, in which case Eq. (10) is interpreted as a TD Volterra series.

Using the unitarity of the spreading W-matrix (irrespective of the specific nature of the spreading transform), a useful Parseval-like formula may be readily derived between the VSTF and the VSK:
t1t2t3|hi;t1t2t3|2=(j,k)S[i]|Hi;j,k|2
(12)
An important symmetry property of the VSK readily stems from Eq. (11) which turns out to be symmetric in its first and second indices (the proof that t1,t2 may be exchanged resorts to the indices symmetry property(j,k)S[i](k,j)S[i]):

hi;t1t2t3=hi;t2t1t3
(13)

8.1 Evaluating the VSK in the case of cyclically-extended single-carrier transmission

For CE-SC transmission with block size N (to which CP is added) the spreading DFT and main IDFT of DFT-S now both have the same size, N, hence cancel out, yet it is still useful to think of the {Bt} samples emitted by the CE-SC Tx as having been generated by a back-to-back cascade of N-pnt DFT-S and N-pnt IDFT. The generic unitary spreading W-matrix now reduces to the DFT matrix with elementsWj,t=exp(-j2πNjt).Eq. (11) then takes the following form:
hi;t1t2t3(j,k)S[i]Hi;jkWj,t1Wk,t2W(j+ki),t3*=(j,k)S[i]Hi;jkej2πNjt1ej2πNkt2ej2πN(j+ki)t3=ej2πNit3(j,k)S[i]Hi;jkej2πN[j(t1t3)+k(t2t3)]=ej2πNit3(j,k)1S[i][j,k]Hi;jkej2πN[j(t1t3)+k(t2t3)]==(u,v)DFTj,k{1S[i][j,k]Hi;jk}|(u,v)=(t1t3,t2t3)ej2πNit3
(14)
The VSK for CE-SC transmission is then seen to be given (for a fixed observation index, i) by the 2-D DFT-like transform acting on the 2-D VSTF sequence Hi;jk, which is hard-windowed in the (j,k) plane to the set of points S[i].

8.2 NLI power of U-S OFDM

Back to the NL modeling of the generic U-S OFDM format, in order to obtain the expected NLI power we first write an expression for the instantaneous NLI power (based on Eq. (10)):
|riFWM|2=t1,t2,t3,t1,t2,t3Bt1Bt1*Bt2Bt2*Bt3*Bt3hi;t1t2t3hi;t1t2t3*
(15)
Taking the expectation of both sides yields for a PSK constellation the following result, the proof of which is omitted for brevity:
|riFWM|2=2P3t3t1t2t3|hi;t1t2t3|2+P3t1t2|hi;t1t1t2|2+P3t1|hi;t1t1t1|2=P3[2t1,t2,t3|hi;t1t2t3|2t1t2|hi;t1t1t2|24t1t2|hi;t1t2t2|2t1|hi;t1t1t1|2]
(16)
where P is the per-carrier power (1/MN of the total power PB|Bk|2) . Notice that the first term in the second line is seen not to depend on the spreading transformation W(Eq. (12)). It follows that minimizing the NLI power amounts to maximizing the last three terms overW, thus a new U-S transformation with NL performance which exceed DFT-S might be found, but this will not be further pursued here. The last two Eqs. may also be cast in yet another form wherein the Wdependency is made explicit:
|riFWM|2=2P3(j,k)S[i]|Hi;j,k|2P3(j1,k1)S[i](j2,k2)S[i]Hi;j1,k1Hi;j2,k2*Q[j1,j2,k1,k2]
(17)
whereQ[j1,j2,k1,k2]tWj1,tWj2,t*Wk1,tWk2,t*δj1+k1,j2+k2++4Wk1,tWk2,t*Wj1+k1i,t*Wj2+k2i,tδj1,j24Wj1,tWj2,t*Wk1,tWk2,t*Wj1+k1i,t*Wj2+k2i,t
(18)
This last expression comprises a sum of quadruple and sextuple products of certain rows of the spreading matrix. We omit the tedious algebraic derivation of this last expression, however Eq. (17) along with the definition of Eq. (18) have been numerically verified to come out equal to Eq. (16).

Equation (16) or (17) analytically predicting the expected NLI power in closed form (in terms of the VSK of Eq. (11) which is also analytically computable) are a key result of this paper, applicable to the U-S OFDM generic format. This result is evaluated in Appendix B for the special cases of OFDM and CE-SC modulation formats. In the OFDM case a known result [13

13. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

,14

14. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

] is retrieved, whereas for CE-SC an interesting new formulation is obtained.

8.3 DFT-S OFDM NL performance

8.4 WAV-S OFDM NL performance

Finally, let us consider the NL performance of WAV-S OFDM. An OFDM system based on this particular spreading transform has already been considered in the wireless literature but evidently only in the context of its PAPR properties [15

15. M. P. H. Jun and J. Cho, “PAPR Reduction in OFDM transmission using Hadamard Transform,” in IEEE International Conference on Communications1, 430–433, (2000).

,16

16. Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in Proceedings IEEE 56th Vehicular Technology Conference4, 2096–2100, (2002).

]. However, in the photonic transmission context, PAPR is not a faithful predictor of NL performance, as discussed in section 4. Here we investigate for the first time the NL tolerance of WAV-S OFDM in fiber transmission.

Notwithstanding the WAV-S low hardware complexity (multiplier-free implementation) as mentioned in section 6.2, unfortunately, as indicated in Fig. 8, the WAV-S NL performance while exceeding that of plain OFDM, it still does not match that of DFT-S OFDM, which retains its status as the highest performance format at this point. Nevertheless we now have two alternative OFDM variants both exceeding plain OFDM in NL performance, with these two variants mutually trading complexity vs. the amount of NLT improvement. Practically marginally higher complexity is required in WAV-S but its NL performance improvement is modest, whereas DFT-S OFDM provides more improvement at the expense of extra complexity.

9. Conclusions

An interesting open question relegated to future work is finding a spreading transformation which ‘beats’ DFT-S in its NLT performance, and whether such transformation would further fall in the class of hardware efficient linear transforms with O[N log(N)]complexity in par with that of the DFT. Our treatment should be further extended to address dual polarization operation. Other types of unitary spreading matrixes (e.g. complex hadamard generalizations) should be further explored.

Appendix A. DFT-S as a superposition of single carrier sub-bands

Conventional OFDM transmission essentially amounts to generating an IDFT sn=i=0MN1Aie+j2πMNni. The DFT-S Tx precedes the IDFT by a blockwise DFT spreading transformation (Eq. (1)) onto the MN data symbolsB={Bt}k=0NM1, generating the vector:
A={DFT{Bt}t=0N1,DFT{Bt}t=N2N1,...,DFT{Bt}t=(M1)NMN1}
(19)
It is this signal which is input into the MN-pnt IDFT yielding the Tx output. Due to the linearity of the IDFT operation it is possible to analyze its output as a superposition of responses sn(m),m=1,2,...,Mto each of the SSC sub-band blocks. Let us inspect the m-th sub-band. The N-pnt data segment {Bt}t=mN(m+1)N1 is applied to the spreading DFT, then the DFT output is appropriately zero-padded over the index intervals [0,mN-1] and [(m + 1)N,MN-1], then the MN-pnt IDFT is applied:
sn(m)=ν=mN(m+1)N1(t=0N1BmN+tej2πN(νmN)t)e+j2πMNνn=t=0N1BmN+tν=mN(m+1)N1ej2πNνte+j2πMNνn=t=0N1BmN+tν=mN(m+1)N1e+j2πNν(nMt)=ν=νmNt=0N1BmN+tν=0N1e+j2πN(ν+mN)(nMt)=t=0N1BmN+tν=0N1e+j2πNν(nMt)e+j2πMnm=Ne+j2πMnmt=0N1BmN+tejπ(nMt)N1NdincN[n/Mt]wheredincN[u]sin(πu)Nsin(πu/N)
(20)
and we used the identity
ν=0N1e+j2πNν(nMt)=ejπ(n/Mt)(N1)/Nsin[π(n/Mt)]sin[πN(n/Mt)]=Nejπ(n/Mt)(N1)/NdincN[n/Mt]
(21)
Equation (20) indicates that, apart from a constant phase factor, the m-th SSC is given by an interpolated single-carrier signal, frequency shifted (in the digital domain) by modulation with an harmonic phase factor given bye+j2πMmn. The overall output signal is given by a superposition sn=m=1Msn(m)of all sub-band (SSC) signals.

Appendix B. Analytic NLT evaluation for U-S special cases: OFDM, SC, DFT-S

Conventional OFDM

In this case the U-S matrix coincides with the unity matrix -W=I.
Q[j1,j2,k1,k2]tWj1,tWj2,t*Wk1,tWk2,t*δj1+k1,j2+k2++4Wk1,tWk2,t*Wj1+k1i,t*Wj2+k2i,tδj1,j24Wj1,tWj2,t*Wk1,tWk2,t*Wj1+k1i,t*Wj2+k2i,t=1{(j1,j2,k1,k2)|j1=j2=k1=k2}(j1,j2,k1,k2)+41{(j1,j2,k1,k2)|k1=k2=j1+k1i=j2+k2i,j1=j2}(j1,j2,k1,k2)41{(j1,j2,k1,k2)|j1=j2=k1=k2=j1+k1i=j2+k2i}(j1,j2,k1,k2)
(22)
The last two terms don't contribute to the NLI calculation Eq. (17) as they require either j1=ior k1=iand these are not proper FWM terms (they are not included inS[i]). Thus, finally
|riNL|2=P3[2(j,k)S[i]|Hi;j,k|2(j,j)S[i]|Hi;j,j|2]
(23)
This recovers a formula for the OFDM distortion power previously established in [13

13. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

,14

14. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

].

Cyclically extended Single Carrier (CE-SC) transmission

In this caseUN×N=DFTN,Wm,n=ej2πNmn, therefore:
Q[j1,j2,k1,k2]=t[ej2πNt((j1+k1)(j2+k2))δj1+k1,j2+k2+4ej2πNt((k1+j2+k2i)(k2+j1+k1i))δj1,j22ej2πNt((j1+k1+j2+k2i)(j2+k2+j1+k1i))]=t[1{(j1,j2,k1,k2)|j1+k1=j2+k2}(j1,j2,k1,k2)+41{(j1,j2,k1,k2)|j1=j2}(j1,j2,k1,k2)4]
(24)
finally yielding the following expression for the FWM NLI:

|riNL|2=2P3(j,k)S[i]|Hi;j,k|2+4NP3(j1,k1)S[i](j2j1,k2)S[i]Hi;j1,k1Hi;j2,k2*NP3(j1,k1)S[i](j2,k2)S[i]j1+k1=j2+k2Hi;j1,k1Hi;j2,k2*.

Appendix C - Glossary

ADC = Analog to Digital Converter
MC = Monte Carlo
Rx = Receiver
CE-SC = Cyclically Extended Single Carrier
MER = Modulation Error Rate
RRC = Root Raised Cosine
CCDF = Complementary Cumulative Distribution Function
NL = Nonlinear
SSC = Sub Single Carrier
CLT = Central Limit Theorem
NLC = Nonlinear Compensator
SPM = Self Phase Modulation
CP = Cyclic Prefix
NLT = Nonlinear Tolerance
SSF = Split Step Fourier
CD = Chromatic Dispersion
NLI = Nonlinear Interference
Tx = Transmitter
DAC = Digital to Analog Converter
PAPR = Peak to Average Power Ratio
TD = Time Domain
FWM = Four Wave Mixing
PDF = Probability Distribution Function
VSK = Volterra Series Kernel
FDM = Frequency Domain Multiplexer
PSK = Phase Shift Keying
VSTF = Volterra Series Transfer Function
IID = Independent Identically DistributedQAM = Quadrature Amplitude ModulationXPM = Cross Phase Modulation

Acknowledgments

This work was supported by the Chief Scientist Office of the Israeli Ministry of Industry, Trade and Labor within the ‘Tera Santa’ consortium.

References and links

1.

W. Shieh, Y. Tang, and B. S. Krongold, “DFT-Spread OFDM for Optical Communications,” in 9th International Conference on Optical Internet (COIN), (2010).

2.

W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J. 2(3), 276–283 (2010). [CrossRef]

3.

F. Wang and X. Wang, “Coherent Optical DFT-Spread OFDM,” in Advances in Optical Technologies (Hindawi Publishing Corporation, 2011).

4.

Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett. 22(16), 1250–1252 (2010). [CrossRef]

5.

Y. Tang, W. Shieh, and B. S. Krongold, “Fiber Nonlinearity Mitigation in 428-Gb / s Multiband Coherent Optical OFDM Systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2010).

6.

C. Li, Q. Yang, T. Jiang, Z. He, M. Luo, C. Li, X. Xiao, D. Xue, and X. Yi, “Investigation of Coherent Optical Multi-band DFT-S OFDM in Long Haul Transmission,” IEEE Photon. Technol. Lett. 24, 1704–1707 (2012).

7.

Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, A. A. Amin, and W. Shieh, “Coherent optical DFT-Spread OFDM in Band-Multiplexed Transmissions, We.8.A.6,” in European Conference of Optical Communication (ECOC), (2011).

8.

Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, and W. Shieh, “Coherent optical DFT-spread OFDM transmission using orthogonal band multiplexing,” Opt. Express 20(3), 2379–2385 (2012). [CrossRef] [PubMed]

9.

X. Chen, A. Li, G. Gao, and W. Shieh, “Experimental demonstration of improved fiber nonlinearity tolerance for unique-word DFT-spread OFDM systems,” Opt. Express 19(27), 26198–26207 (2011). [CrossRef] [PubMed]

10.

A. Li and G. Chen, Xi, A. Guanjun, A. Amin, W. Shieh, William, B. S. Krongold, “Transmission of 1. 63-Tb/s PDM-16QAM Unique-word DFT-Spread OFDM Signal over 1, 010-km SSMF,” in OFC/NFOEC, paper OW4C.1, (2012).

11.

C. Ciochina and H. Sari, “A review of OFDMA and single-carrier FDMA,” in Wireless Conference (EW), 706– 710, (2010).

12.

X. Liu and S. Chandrasekhar, “High Spectral-Efficiency Transmission Techniques for Beyond 100-Gb/s Systems, SPMA1,” in SPPCom - Signal Processing in Photonic Communications - OSA Technical Digest, 1–36, (2011).

13.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

14.

S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

15.

M. P. H. Jun and J. Cho, “PAPR Reduction in OFDM transmission using Hadamard Transform,” in IEEE International Conference on Communications1, 430–433, (2000).

16.

Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in Proceedings IEEE 56th Vehicular Technology Conference4, 2096–2100, (2002).

17.

B. Porat, A Course in Digital Signal Processing (John Wiley and Sons, 1997).

18.

H. Myung, J. Lim, and D. Godman, “Peak-To-Average Power Ratio of Single Carrier FDMA Signals with Pulse Shapingý,” in IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications, (2006)ý, pp. 1–5.

19.

B. Goebel, S. Hellerbrand, N. Haufe, and N. Hanik, “PAPR reduction techniques for coherent optical OFDM transmission,” in 2009 11th International Conference on Transparent Optical Networks1, 1–4, (2009).

20.

C. R. Berger, Y. Benlachtar, R. I. Killey, and P. A. Milder, “Theoretical and experimental evaluation of clipping and quantization noise for optical OFDM,” Opt. Express 19(18), 17713–17728 (2011). [CrossRef] [PubMed]

21.

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997). [CrossRef]

22.

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightwave Technol. 30(3), 310–316 (2012). [CrossRef]

23.

A. Li, W. Shieh, R. S. Tucker, and A. Wavelet, “Wavelet Packet Transform-Based OFDM for Optical Communications,” J. Lightwave Technol. 28, 3519–3528 (2010).

24.

M. H. Lee, S. Member, B. S. Rajan, and J. Y. Park, “A Generalized Reverse Jacket Transform,” IEEE Trans. Circ. Syst. II 48(7), 684–690 (2001). [CrossRef]

25.

A. Aung, B. P. Ng, and S. Rahardja, “Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications,” IEEE Trans. Sig. Process. 56(8), 3562–3571 (2008). [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: September 4, 2012
Revised Manuscript: October 19, 2012
Manuscript Accepted: October 22, 2012
Published: November 1, 2012

Citation
Gal Shulkind and Moshe Nazarathy, "An analytical study of the improved nonlinear tolerance of DFT-spread OFDM and its unitary-spread OFDM generalization," Opt. Express 20, 25884-25901 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25884


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References

  1. W. Shieh, Y. Tang, and B. S. Krongold, “DFT-Spread OFDM for Optical Communications,” in 9th International Conference on Optical Internet (COIN), (2010).
  2. W. Shieh and Yan Tang, “Ultrahigh-Speed Signal Transmission Over Nonlinear and Dispersive Fiber Optic Channel: The Multicarrier Advantage,” IEEE Photonics J.2(3), 276–283 (2010). [CrossRef]
  3. F. Wang and X. Wang, “Coherent Optical DFT-Spread OFDM,” in Advances in Optical Technologies (Hindawi Publishing Corporation, 2011).
  4. Y. Tang, W. Shieh, and B. S. Krongold, “DFT-Spread OFDM for Fiber Nonlinearity Mitigation,” IEEE Photon. Technol. Lett.22(16), 1250–1252 (2010). [CrossRef]
  5. Y. Tang, W. Shieh, and B. S. Krongold, “Fiber Nonlinearity Mitigation in 428-Gb / s Multiband Coherent Optical OFDM Systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2010).
  6. C. Li, Q. Yang, T. Jiang, Z. He, M. Luo, C. Li, X. Xiao, D. Xue, and X. Yi, “Investigation of Coherent Optical Multi-band DFT-S OFDM in Long Haul Transmission,” IEEE Photon. Technol. Lett.24, 1704–1707 (2012).
  7. Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, A. A. Amin, and W. Shieh, “Coherent optical DFT-Spread OFDM in Band-Multiplexed Transmissions, We.8.A.6,” in European Conference of Optical Communication (ECOC), (2011).
  8. Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, and W. Shieh, “Coherent optical DFT-spread OFDM transmission using orthogonal band multiplexing,” Opt. Express20(3), 2379–2385 (2012). [CrossRef] [PubMed]
  9. X. Chen, A. Li, G. Gao, and W. Shieh, “Experimental demonstration of improved fiber nonlinearity tolerance for unique-word DFT-spread OFDM systems,” Opt. Express19(27), 26198–26207 (2011). [CrossRef] [PubMed]
  10. A. Li and G. Chen, Xi, A. Guanjun, A. Amin, W. Shieh, William, B. S. Krongold, “Transmission of 1. 63-Tb/s PDM-16QAM Unique-word DFT-Spread OFDM Signal over 1, 010-km SSMF,” in OFC/NFOEC, paper OW4C.1, (2012).
  11. C. Ciochina and H. Sari, “A review of OFDMA and single-carrier FDMA,” in Wireless Conference (EW), 706– 710, (2010).
  12. X. Liu and S. Chandrasekhar, “High Spectral-Efficiency Transmission Techniques for Beyond 100-Gb/s Systems, SPMA1,” in SPPCom - Signal Processing in Photonic Communications - OSA Technical Digest, 1–36, (2011).
  13. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008). [CrossRef] [PubMed]
  14. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).
  15. M. P. H. Jun and J. Cho, “PAPR Reduction in OFDM transmission using Hadamard Transform,” in IEEE International Conference on Communications1, 430–433, (2000).
  16. Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in Proceedings IEEE 56th Vehicular Technology Conference4, 2096–2100, (2002).
  17. B. Porat, A Course in Digital Signal Processing (John Wiley and Sons, 1997).
  18. H. Myung, J. Lim, and D. Godman, “Peak-To-Average Power Ratio of Single Carrier FDMA Signals with Pulse Shapingý,” in IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications, (2006)ý, pp. 1–5.
  19. B. Goebel, S. Hellerbrand, N. Haufe, and N. Hanik, “PAPR reduction techniques for coherent optical OFDM transmission,” in 2009 11th International Conference on Transparent Optical Networks1, 1–4, (2009).
  20. C. R. Berger, Y. Benlachtar, R. I. Killey, and P. A. Milder, “Theoretical and experimental evaluation of clipping and quantization noise for optical OFDM,” Opt. Express19(18), 17713–17728 (2011). [CrossRef] [PubMed]
  21. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997). [CrossRef]
  22. L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightwave Technol.30(3), 310–316 (2012). [CrossRef]
  23. A. Li, W. Shieh, R. S. Tucker, and A. Wavelet, “Wavelet Packet Transform-Based OFDM for Optical Communications,” J. Lightwave Technol.28, 3519–3528 (2010).
  24. M. H. Lee, S. Member, B. S. Rajan, and J. Y. Park, “A Generalized Reverse Jacket Transform,” IEEE Trans. Circ. Syst. II48(7), 684–690 (2001). [CrossRef]
  25. A. Aung, B. P. Ng, and S. Rahardja, “Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications,” IEEE Trans. Sig. Process.56(8), 3562–3571 (2008). [CrossRef]

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