1. Introduction

DFT-spread (DFT-S) OFDM is a variant of OFDM transmission ported into optical communication by Shieh et al [1], simulated in [2–6] and experimentally demonstrated [7–10]. In the wireless literature this method is referred to as Single-Carrier Frequency Division Multiple Access (SC-FDMA) [11] 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] as it is a multicarrier format [2] more tolerant to fiber nonlinearity than either plain OFDM or single carrier transmission are. An intuitive explanation advanced in [2] 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.

Going beyond this PAPR-based intuitive explanation which is unable to quantify the nonlinear (NL) performance of DFT-S OFDM, we develop here for the first time to our knowledge a rigorous theoretical framework accurately accounting for the mechanism of nonlinear tolerance improvement in DFT-S OFDM. Remarkably, our novel NL analytical model accurately predicts the number of DFT-S sub-bands (or equivalently the optimal sub-band bandwidth) required for maximizing the nonlinear tolerance. Our Volterra-series [13,14] based closed-form analytic predictions and related Monte-Carlo simulations for the NL performance of DFT-S OFDM well agree with Monte-Carlo Split-Step-Fourier (MC-SSF) numeric simulations over the same non-linear fiber link, and precisely predict the peaking of NL tolerance at a certain sub-band bandwidth)

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 [15,16] 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.

The paper is structured as follows: Section 2 reviews the DFT-S OFDM modulation format and section 3 presents numerical simulations of its NL performance. In section 4 it is shown that PAPR is not an accurate predictor of NL performance. Section 5 analytically models and numerically simulates the NL performance of the DFT-S link by means of a third-order Volterra series. Section 6 extends DFT-S to a more general unitary-spread OFDM formats and identifies the WAV-S format as a special case. In section 7 we elaborate on the NonLinear Interference (NLI) statistical properties of U-S and DFT-S OFDM. Section 8 presents a key result of the paper, deriving the compact analytic model for NL performance of U-S OFDM and its DFT-S and WAV-S special cases. Section 9 concludes the paper. Two appendices detail various mathematical derivations. The abbreviations used in this paper are stated in Appendix C.

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

The pre/post-processing in the DFT-S OFDM Tx/Rx [1–6] is reviewed in Fig. 1(a) . This paper treats a single polarization scalar channel, thus the DFT-S DSP structure reviewed here applies to a single polarization path. In the DFT-S Tx (Rx), the main OFDM (I)DFT of size MN is preceded (followed) by an array of (I)DFT modules each of size N. Here M is also the number of sub-bands, which is also the number of (I)DFTs forming the pre/post-processing arrays while N is the number of OFDM tones per sub-band. A mathematical description of the DFT-S transmitter pre-processing expresses the vector $A$of DFT-S outputs in terms of a block matrix transformation:

 

Fig. 1 (a) DFT-S OFDM link with (de)spreading pre(post)-processing. M sub-single-carriers are transmitted over M FDM sub-bands per channel (b). Tx signal flow for a single sub-band.

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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 $A^{(r)}\leftrightarrow B^{(r)}$and${\text{DFT}}_N\leftrightarrow {\text{IDFT}}_N$. The associated Rx transformation matrix is not reproduced here but it is similar to that of Eq. (1), having a block-diagonal form$diag\left\{IDF{T}_N,IDF{T}_N,\mathrm{...},IDF{T}_N\right\},$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] 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 $M\ne 1$) we retrieve a conventional OFDM system, as each SSC now coincides with a single OFDM tone.

At the other extreme, for M = 1 (but $N\ne 1$) a DFT-S system with a single sub-band (Fig. 2(a) , 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 $\mu$symbols of the N-pnt block is replicated ahead of the N-pnt block to form an $N+\mu -$pnt 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).

 

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.

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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 the wireless transmission context, the DFT spreading enhancement of OFDM has been motivated by the desire to improve the poor PAPR performance of conventional OFDM [10,18]. Shieh et al [1] ported DFT-S to photonic transmission guided by the consideration that the improved PAPR of DFT-S should also benefit nonlinear tolerance of the optical signal in fiber transmission. The NLT of a DFT-S OFDM link was investigated in [2–6] where it was found that there is an optimal SSC bandwidth maximizing the NLT. Remarkably, as established in those works by numeric simulation, at the optimal bandwidth (or equivalently at the optimal number of SSCs per channel) the NLC performance is superior to that of either plain OFDM or CE-SC transmission.

In this section we present our own numeric simulation of the NLC of DFT-S OFDM as plotted in Fig. 3 , corroborating the key conclusion of [2–6] 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.

 

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;\alpha =0.2dB/Km;\gamma =1.3{(W\cdot Km)}^{-1};D=17ps/(nm\cdot Km);L=6\times 100Km;P=0dBm.$38400 data-symbols were used to gather the statistics (and we have verified that the MER converged to steady values).

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We evaluate the NL performance of the DFT-S OFDM link by means of a Monte Carlo simulation, with the optically amplified fiber link numerically modeled by the Split-Step-Fourier (SSF) method for a single polarization scalar channel and DFT-S transceivers as described in Fig. 1(a).

The link NL performance is represented here by the Modulation Error Ratio (MER) of the NLI accompanying the received signal due to the Four-Wave-Mixing (FWM) between the tones, after compensating the Chromatic Dispersion (CD) as well as compensating the Self-Phase-Modulation (SPM) of each tone as well as the Cross-Phase-Modulation (XPM) among the tones [13,14].The MER (or the equivalent EVM measure) is mathematically defined as an ensemble (or time average over k for an ergodic process) of the following ‘instantaneous MER’ expression (here i is the SSC index, k is discrete-time, and $P_B\equiv 〈{\left|B_k\right|}^2〉$is the output power per tone ($1/(MN)$ of the total output power) and $R_i$are the elements of the R-vector at the Rx de-spreading array output as defined in Fig. 1(a)).

The simulated NLI MER (Fig. 3) vs. the number of SSC sub-bands, clearly indicates an optimum at 4 SSCs per 25 GHz channel, corresponding to a SSC sub-band bandwidth of 6.25 GHz for the particular assumed link parameters which are listed in the Fig. caption. Evidently, the optimal SSC bandwidth would change for a different set of link parameters (e.g. constellation type and size (PSK/QAM), number and lengths of spans, physical parameters, number of OFDM subcarriers and bandwidth), however the general principle holds that there is an optimal bandwidth maximizing NL performance. Remarkably, with optimal sub-banding, DFT-S may attain NL tolerance exceeding both single carrier and plain OFDM transmission.

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 [19,20], suggesting to adopt PAPR [18] 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.

The digital domain PAPR is defined (in the context of a DFT-S OFDM Tx) as ${\text{PAPR}}_{\text{digi}}\equiv \max {\left|s_k\right|}^2/〈{\left|s_k\right|}^2〉$where $s_k$are the samples in a single OFDM symbol (e.g. this is per-frame-PAPR). Analog domain PAPR is similarly defined at the DAC output as ${\text{PAPR}}_{\text{ana}}\equiv \max {\left|s(t)\right|}^2/〈{\left|s(t)\right|}^2〉$. We numerically simulated the PAPR of the transmitted DFT-S OFDM signal, varying the number of SSC sub-bands. The PAPR Complementary Cumulative Distribution Function (CCDF) of the DFT-S Tx signal launched into the fiber link is plotted in Fig. 4 , showing both the digital-domain and analog-domain PAPR distributions per OFDM symbol for a Root-Raised-Cosine (RRC) Tx filter and 16-QAM transmission. The plots are parameterized by the number M of SSC sub-bands. The PAPR of conventional OFDM is retrieved for M = MN (when each sub-band is one-tone wide, and the link simplifies to conventional OFDM, as previously discussed), whereas the single-carrier CE-SC case corresponds to DFT-S with M = 1. It is apparent that OFDM exhibits the worst PAPR. The PAPR monotonically improves with decreasing M, finally reaching a minimum for the CE-SC case (M = 1).

 

Fig. 4 PAPR of the transmitted DFT-S OFDM signal, parameterized by the number of SSC sub-bands. Other parameters: RRC Tx filter with parameter alpha = 0.1; 4x upsampling; QAM16 constellation; main FFT size MN = 128. The abbreviations anlg = analog and dig = digital refer to the PAPR type.

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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.

Shieh et al realized that as the OFDM signal propagates along the fiber link the PAPR evolves along the fiber due to the dispersive propagation, rendering it an inadequate parameter for determining the Rx-side NLT. Those authors attempted to salvage the PAPR model by advancing the following intuitive explanation why the DFT-S performance peaks at a certain M value, invoking the interaction with chromatic dispersion (CD) [1]: “On one hand, if the subband bandwidth is too broad the PAPR reduction will not be effective due to the fiber dispersion. On the other hand, if the sub-bands are too narrow, the neighbouring bands interact just as narrowly-spaced OFDM subcarriers, generating large inter-band crosstalk due to narrow subband spacing and incurring a large penalty.”

While this explanation is intuitively plausible, the problem is that it does not seem amenable to closed-form modeling. As PAPR and NLT performances correlate poorly, we advocate to abandon the PAPR measure altogether when assessing nonlinear tolerance of optically amplified fiber links with various modulation formats. PAPR remains a useful measure for quantifying DAC/ADC clipping and quantization noise but in our view it is an inaccurate predictor of fiber NL performance.

In the current case of DFT-S transmission as well as for related OFDM variants, we propose to analytically evaluate the NLI power generated (with plain OFDM and CE-SC as extreme cases) directly from first principles. Thus, we shall replace the intuitive imprecise PAPR description above by a novel rigorous analytic FWM generation model, precisely predicting the NLT peaking mechanism for DFT-S and related OFDM variants.

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] and reviewed in [14], based on Volterra Series Transfer Function (VSTF) formalism, e.g [21,22], 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.

In the context of conventional OFDM, the FWM NLI received at the OFDM DFT output of the NL dispersive noiseless optically amplified fiber link (referenced to the fiber input) is expressible as follows [13,14]:

We set up a Volterra Series (VS) based simulation of the DFT-S based link starting from the spreading transformation of Eq. (1) acting on the MN-pnt input vector, $B$of the DFT-S Tx. We draw the MN-elements of $B$ independently from some transmission constellation (we consider 16-PSK and 16-QAM constellations in this paper) then apply the M spreading DFTs, yielding the A-vector at the IDFT input, the elements of which are substituted into the trilinear transform of Eq. (3), the summation of which is numerically evaluated to yield the NLI realization for each of the i-indexes (OFDM tones) at the main DFT output in the Rx. We then partition the MN resulting samples into M subsets of N points, and apply the de-spreading IDFTs, yielding MN outputs. We then numerically evaluate the MER (reflecting averaging over all i-indexed tones) at the DFT-S Rx output. The resulting MER curve for this VS-based simulation is now shown in Fig. 5 , superposed on the SSF method MER curve of Fig. 3 and the match is seen to be accurate, validating the VS model of Eq. (3), along with the particular expression for the VSTF,$H_{i;jk},$as derived in [13,14] which is not reproduced here. The small discrepancies between the MC-trilinear and MC-SSF curves are attributed to several factors differing between the two cases: (i) CP insertion in the SSF vs. lack thereof in trilinear model. (ii): the finite x4 upsampling in the SSF simulation (whereas the trilinear model is not affected by upsampling related aliasing effects). (iii): finite OFDM symbol-length and Root Raised Cosine shaping at the Tx are accounted for in the SSF simulation but are not included in the Volterra model of Eq. (3) in [12]. Equipped with this analytic NL model we aim to rationalize the NL behavior of the DFT-S OFDM link, as empirically investigated heretofore. The result will be a fully analytic model for the mechanism of NL impairment generation in DFT-S OFDM and related OFDM variants.

 

Fig. 5 Numeric simulation of the received FWM NLI MER over a 16-QAM DFT-S OFDM optically amplified fiber link. The two curves present Monte Carlo simulations respectively based on the SSF and third-order Volterra series models with the following parameters: $MN=128;BW=25GHz;\alpha =0.2dB/Km;\gamma =1.3{(W\cdot Km)}^{-1};D=17ps/(nm\cdot Km);L=6\times 100Km;P=0dBm.$Notice that the worst case deviation of the Volterra-based trilinear model is bounded by 0.4 dB. 38400 data-symbols were used to gather the statistics (and we have verified that the MER converged to steady values).

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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,

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, $W_{MN\times MN}^{�}=diag\left\{U_{N\times N}^{�},U_{N\times N}^{�},\mathrm{...},U_{N\times N}^{�}\right\},$for various $U_{N\times N}$selections which now differ from N-pnt (I)DFTs. Evidently, taking $U_{N\times N}=DF{T}_N$as the N-pnt DFT matrix (with its (m,n) element given by $\exp \left(-j\frac{2\pi }{N}mn\right)$) 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 adopting$U_{N\times N}={\text{HAD}}_N,$with ${\text{HAD}}_N$ 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] which superficially bears a similar name. Our WAV-S spreading transform is described by:

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 [24,25] 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,$\left\{A_i\right\},$at the output of the U-S transform are statistically uncorrelated, they are nevertheless statistically dependent, as opposed to the input samples, $\left\{B_t\right\}$ 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,$\left\{A_i\right\},$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 , which plots the modulus and phase (arg) of the complex samples $\left\{A_i\right\}$ (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 for$N\ge 64$the asymptotic approximation seems accurate.

 

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 $2\pi$ for the phase. The plot pertains to the tone indexed i = 1.

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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 $\left\{A_i\right\}$ 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,$\left\{A_i\right\},$ are statistically dependent yet uncorrelated, recall that the inputs, $\left\{B_t\right\},$ 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 $\left\{A_i\right\}$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 $\left\{A_i\right\}$samples are individually Gaussian, as random variables, however the A-vector is not a Gaussian vector, i.e. the$\left\{A_i\right\}$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$\left\{A_i\right\}$variables are individually normal. The statistical dependency of the spread $\left\{A_i\right\}$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 $\left\{B_t\right\}$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.

Having clarified this subtle point, we further note that while we do not know the particular joint distribution of $\left\{A_i\right\}$ (we just know that it results in Gaussian marginal distributions for its components, and that it does not factor out into the product of these Gaussian marginal distributions), the resulting joint distribution is dependent on the nature of the spreading transform ($U_{N\times N}^{}$matrices). Thus, DFT-S, WAV-S or other specific U-S transforms yield at their output various joint distributions and varying degrees of NL tolerance as further explored in section 8. This may be seen by writing an expression for the average FWM NLI power, obtained by squaring the trilinear transform of Eq. (3) and taking its expectation:

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.

We now proceed to analytically evaluate the FWM NLI generated after propagation of the U-S OFDM signal through an optically amplified fiber link (assumed noiseless).

In the case of DFT-S OFDM, the TD QAM/PSK samples,$B_t,$ are first spread as per Eq. (4), generating the FD samples $A_i,$ triple products of which appear in Eq. (3), As per subsection 5, the post-spread samples, $A_i$, 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,$B_k,$ 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 $B_k$samples:

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:

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 $\left\{B_t\right\}$ 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 elements$W_{j,t}=\exp \left(\text{-j}\frac{2\pi }{N}jt\right).$Eq. (11) then takes the following form:

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)):

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 [13,14] is retrieved, whereas for CE-SC an interesting new formulation is obtained.

8.3 DFT-S OFDM NL performance

We next verify the validity of the key analytic result of Eq. (17) for the special case of DFT-S OFDM, by calculating the average power in closed form and comparing with the corresponding numerical expression already simulated in section 3. The average received FWM NLI MER is plotted in Fig. 7(a) , superposed onto the two numerical simulations of NLI MER namely Monte-Carlo SSF and Monte-Carlo trilinear Volterra series transform (as discussed in section 5). Here the simulations are restricted to MN = 32 and M = 1,2,4,8,16,32, due to the heavy numerical combinatorial complexity of the analytic evaluation averaging over all i-s. It is apparent that the analytic version tracks the Monte-Carlo trilinear Volterra based version very well, however the SSF simulations do deviate from the two tracking (analytic and MC-trilinear) curves. It turns out that the match is much better for higher MN values (e.g. as shown in Fig. 3), but for those values the evaluation of the two agreeing trilinear and analytic Volterra based expressions would just take too long due to the large number of combinations. In Fig. 7(b) we further compare the nonlinear performance of DFT-S for 16-QAM and 16-PSK constellations. It is apparent that in both cases there is a similar trend, however 16-QAM generally incurs worse NLI MER.

 

Fig. 7 (a) Averaged (over all tones) NLI MER vs. the number of sub-bands for a 16-PSK DFT-S OFDM channel over an optically amplified fiber link with the following parameters: Three curves are shown. The analytic curve accurately tracks the Monte-Carlo trilinear Volterra series simulation while the SSF slightly deviates away for the high plotted range of M values. (b) Averaged NLI MER vs. sub-bands, for 16-QAM vs. 16-PSK, over a link with:

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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 [15,16]. 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.

Figure 8 presents the FWM NLI MER attainable with WAV-S OFDM, as compared with that of DFT-S OFDM using various simulation methods: Monte-Carlo SSF, Volterra trilinear transform and analytic performance as per Eq. (17).

 

Fig. 8 NLI MER averaged (over all tones) vs. the number of unitary-spread sub-bands over an optically amplified fiber link with the following parameters: $BW=25GHz;\alpha =0.2dB/Km;\gamma =1.3{(W\cdot Km)}^{-1};D=17ps/(nm\cdot Km);P=0dBm.$(a) 16-QAM MC-SSF simulation over $L=10\times 100Km$ comparing WAV-S vs. DTF-S OFDM NL performance for MN = 128 tones (also vs. CE-SC and plain OFDM discrete-points). Plain OFDM NL performance is exceeded with both DFT-S and WAV-S systems, but only DFT-S exceeds the single-carrier (CE-SC) performance. (b) 16-PSK channel over $L=2\times 100Km$with MN = 32 tones simulated analytically and also with MC trilinear Volterra series. The analytic WAV-S curve accurately tracks the MC-Trilinear simulation and indicates improved performance relative to plain OFDM, but worse performance relative to single-carrier. The DFT spread performance at its optimized peak (occurring for M = 2 sub-bands) exceeds the performance of all other systems (but requires higher complexity)

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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

We derived an analytic model rigorously accounting for the improved nonlinear tolerance of DFT-S OFDM, precisely predicting the optimal DFT-S sub-band dimensioning required in order to maximize the link non-linear tolerance, exceeding the NL performances of plain OFDM and (cyclically extended) single-carrier transmission. The NLT advantage of DFT-S OFDM was traced to the particular statistical dependency introduced among the OFDM sub-carriers by means of the DFT spreading operation. We further extended DFT-S to the unitary-spread generalized modulation format, such that DFT-S and WAV-S OFDM become special cases. We have found that the performance improvement attained by the hardware efficient WAV-S relative to plain OFDM is more modest than that achieved by DFT-S, which remains at this point the preferred format for maximizing NL performance, however if the emphasis is on low complexity then WAV-S should be preferred over plain OFDM.

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.