## Evaluation of the computational effort for chromatic dispersion compensation in coherent optical PM-OFDM and PM-QAM systems

Optics Express, Vol. 17, Issue 3, pp. 1385-1403 (2009)

http://dx.doi.org/10.1364/OE.17.001385

Acrobat PDF (283 KB)

### Abstract

Recently, coherent-detection (CoD) polarization multiplexed (PM) transmission has attracted considerable interest, specifically as a possible solution for next-generation systems transmitting 100 Gb/s per channel and beyond. In this context, enabled by progress in ultra-fast digital signal processing (DSP) electronics, both multilevel phase/amplitude modulated formats (such as QAM) and orthogonal-frequency-division multiplexed (OFDM) formats have been proposed. One specific feature of DSP-supported CoD is the possibility of dealing with fiber chromatic dispersion (CD) electronically, either by post-filtering (PM-QAM) or by appropriately introducing symbol-duration redundancy (PM-OFDM). In both cases, ultra-long-haul fully uncompensated links seem to be possible. In this paper we estimate the computational effort required by CD compensation, when using the PM-QAM or PM-OFDM formats. Such effort, when expressed as number of operations per received bit, was found to be logarithmic with respect to link length, bit rate and fiber dispersion, for both classes of systems. We also found that PM-OFDM may have some advantage over PM-QAM, depending mostly on the over-sampling needed by the two systems. Asymptotically, for large channel memory and small over-sampling, the two systems tend to require the same CD-compensation computational effort. We also showed that the effort required by the mitigation of polarization-related effects can in principle be made small as compared to that of CD over long uncompensated links.

© 2009 Optical Society of America

## 1. Introduction

1. R. Noé, “Phase Noise-Tolerant Synchronous QPSK/BPSK Baseband-Type Intradyne Receiver Concept With Feedforward Carrier Recovery,” J. Lightwave Technol. **23**, 802–808 (2005). [CrossRef]

11. J. Renaudier, G. Charlet, M. Salsi, O. B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver,” J. Lightwave Technol. **26**, 36–42 (2008). [CrossRef]

12. W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express **16**, 841–859 (2008). [CrossRef] [PubMed]

18. A. J. Lowery and J. Armstrong, “Orthogonal Frequency Division Multiplexing for Dispersion Compensation of Long-Haul Optical Systems,” Opt. Express **14**, 2079–2084 (2006). [CrossRef] [PubMed]

*OP*). Another fundamental assumption that we made is that the same DSP technology is used for both classes of systems and, in particular, the same FFT/IFFT technology is available to both PM-QAM and PM-OFDM. This makes it possible to carry out a fair and meaningful comparison. Note that, from an implementation viewpoint, additions and multiplications have different complexities. However, with both formats, by far the bulk of the computational effort consists of FFTs/IFFTs, as we shall see later on. Therefore, for both formats the DSP ratio of additions to multiplications is essentially established by the common FFT/IFFT technology. Consequently, as far as a comparative analysis of the

_{b}*relative*computational effort of PM-QAM to PM-OFDM is concerned, discerning additions from multiplications would not change the result.

*N*of subcarriers that are necessary to keep the PM-OFDM cyclic prefix overhead at an acceptable level. In Sect. 3, we compute the

_{SC}*OP*needed by PM-OFDM, given NSC found in the previous section.

_{b}*N*of the finite-impulse-response (FIR) filters needed to compensate for CD in PM-QAM. In Sect. 5 we compute the

_{F}*OP*needed by PM-QAM, based on

_{b}*N*as found in the previous section.

_{F}*OP*for either PM-OFDM and PM-QAM, we take into account various important implementation aspects such as zero-padding, over-sampling and the efficiency of FIR filter implementation.

_{b}*R*: total channel bit rate [Tb/s];_{b}*L*: link length [km];*λ*: channel wavelength [nm];*D*: fiber dispersion [ps/(nm km)];*c*: speed of light, 299792 [km/s];*R*: symbol rate [TBaud]; in the case of PM-OFDM it is the rate of the so-called ‘OFDM symbols’, for PM-QAM it is the standard symbol rate;_{s}*T*: symbol duration [ps], equal to_{s}*R*^{-1};_{s}

*M*: number of bits per subcarrier and per PM-OFDM symbol, or number of bits per PM-QAM symbol.

## 2. Number of Subcarriers for PM-OFDM

*N*needed to support a given system-design target amount of uncompensated CD.

_{SC}*f*

_{1}and

*f*[16

_{NSC}16. S. L. Jansen, I. Morita, T. C. W. Schenck, N. Takeda, and H. Tanaka ‘Coherent Optical 25.8-Gb/s OFDM Transmission Over 4160-km SSMF,’ J. Lightwave Technol. **26**, 6–15 (2008). [CrossRef]

^{2}· s)/km]). We will henceforth use the following close approximation for Eq. (2): Δ

*τ*=

_{g}*8*|

*D*(

*fN*-

_{SC}*f*

_{1})|

*L*. Assuming that the subcarrier frequency is expressed in [THz], and every other quantity follows the units given in the bulleted list at the end of the previous section, then Δ

*τ*conveniently results in [ps].

_{g}*cyclic prefix*needs to be added in order to preserve a suitable common sampling window at the Rx, of duration

*T*, good for the OFDM symbols on all subcarriers. This avoids symbol discontinuities and prevents loss of subcarrier orthogonality [28], [12

_{s}12. W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express **16**, 841–859 (2008). [CrossRef] [PubMed]

*T*, the actual time taken to transmit one symbol is now

_{s}*T*′

_{s}> T

_{s}and the actual OFDM symbol rate goes down to:

*T*, the subcarriers are at minimum spacing when they are spaced

_{s}*R*and trying to pull them closer would generate loss of orthogonality. Therefore, even though

_{s}*R*is no longer the symbol rate, it still remains the minimum frequency separation between adjacent OFDM subcarriers.

_{s}*R*. To restore the original bit rate, it is necessary to add more carriers, i.e., to increase

_{b}*N*to a greater

_{SC}*N*′

*. Unfortunately, increasing the number of subcarriers in turn increases Δ*

_{SC}*τ*, which would require a longer cyclic prefix and eventually an even greater

_{g}*N*′

*, and so on. The problem must then be solved in a combined way. First, we remark that:*

_{SC}*N*

*, which contains Δ*

_{SC}*τ*. We then use the approximated form of Eq. (2) to eliminate Δ

_{g}*τ*. Remembering that

_{g}*T*=

_{s}*R*

^{-1}

_{s}, we further obtain the following intermediate result:

*N*′

*=*

_{SC}*N*

*· (1 + 8 |*

_{SC}*D*|

*LN*′

_{SC}*R*

^{2}

*. Using Eq. (1) to replace*

_{s}*R*with

_{s}*R*/(

_{b}*MN*), we then get:

_{SC}*N*′

*is the increased number of subcarriers needed to support the cyclic prefix while keeping*

_{SC}*R*constant.

_{b}*N*is essentially an initial guess of the needed number of subcarriers. Once a value for

_{sc}*N*has been somehow decided, then Eq. (7) tells us how many subcarriers

_{sc}*N′*are actually needed to cope with dispersion. However, the fact that

_{sc}*N′*depends on an arbitrary initial guess makes Eq. (7) somewhat unsatisfactory. It would be desirable to eliminate

_{sc}*N*and directly find the actually needed number of subcarriers

_{sc}*N′*.

_{sc}*N*.

_{SC}*overhead*, as

*k*≥ 1. This quantity is crucial because the cyclic prefix has two detrimental effects on the system, both of which are directly expressed in terms of

*k*.

*ρ*for the bandwidth efficiency in the absence of cyclic prefix. When using the cyclic prefix, we have shown that we need more subcarriers to transmit the same bit rate. Since the spacing among subcarriers must remain the same, this means that the use of the cyclic prefix lowers

_{B}*ρ*to a new value

_{B}*ρ′*which is given by:

_{B}*k*is a fundamental system design parameter, in the sense that fixing

*k*we can

*a priori*set a limit to both the loss of bandwidth efficiency and the system OSNR penalty.

*N′*vs. the OSNR penalty ∆OSNR

_{SC}_{dB}defined in (10). The plot is drawn using a set of ‘reference system parameters’ which are summarized in Table 1. The plot shows that, to keep the OSNR penalty below 1 and 0.5 dB, about 1900 and 3200 subcarriers are needed at 3000 km, respectively. Typically, for FFT implementation efficiency, these numbers would have to be rounded up to the next power-of-two and therefore would become 2048 and 4096, respectively. The rounding is almost exact for 0.5 dB penalty at 1000 and 2000 km, where 1024 and 2048 subcarriers would suffice, respectively.

*k*= 2, i.e., for 3 dB OSNR penalty and 50% reduction of the bandwidth efficiency. The plots in Fig. 3 confirm it. On the other hand, the OSNR penalty at such minimum is too large for

*k*= 2 to be a solution of practical interest.

24. W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express **16**, 6378–6386 (2008). [CrossRef] [PubMed]

*K*OFDM sub-bands within the same Tx channel, and then demodulating each sub-band as a separate OFDM signal. This way, each sub-band occupies a smaller bandwidth and the group delay difference between a sub-band extreme subcarriers can be made much smaller than the group delay difference between the extreme subcarriers of the overall Tx channel (ideally

_{sub}*K*

_{sub}^{-1}times smaller). In addition, the speed of DACs and ADCs can be likewise reduced, since each sub-band approximately carries a fraction

*K*

_{sub}^{-1}of the payload. This is a very interesting concept but it has also drawbacks: the TX and Rx must use 4

*K*DACs and ADCs (though slower), and must perform 2

_{sub}*K*IFFTs and FFTs. Moreover, both the TX and RX must make use of perfectly synchronous RF oscillators and RF mixers to perform sub-band upconversions and downconversions. Other techniques for mo/demodulation are also possible but added complexity is always present, in other forms.

_{sub}## 3. Operations per bit for OFDM

*N′*of necessary subcarriers. A higher-order IFFT can be used to increase the number of time-samples per OFDM symbol that the DACs use to create the modulating waveforms. This simplifies the removal of aliases off the spectrum and makes the analog electrical modulating waveforms into the electro-optical modulators more ideal. The increase in the IFFT order can be obtained through zero-padding, by imposing zero-amplitude coefficients to ‘ghost subcarriers’, which may reside on either side of the payload subcarrier spectrum. We assume that the Tx IFFT is of order

_{SC}*n*

_{Tx}

*N′*, where the ‘oversampling factor’

_{SC}*n*

_{Tx}≥ 1 takes zero-padding into account. Note that zero-padding requires faster DACs and the speed of DACs is one of the most critical aspects of OFDM implementation. Therefore a key design goal is to try and keep it as small as possible.

*N′*, like for the Tx IFFT. Oversampling can be operated at the Rx as well. The FFT would then process a larger number of samples than just

_{SC}*N′*. This would allow some spectral margin against aliasing, specifically to protect the subcarriers at the channel band edges. In the calculations that follow, the FFT is assumed to be performed over

_{SC}*n*

_{Rx}

*N′*samples, where

_{SC}*n*

_{Rx}≥ 1 takes oversampling into account. It is interesting to notice that the oversampling factors at the Tx and Rx,

*n*

_{Tx}and

*n*

_{Rx}, are independent of each other. They can be separately optimized according to the specific Tx and Rx individual optimization constraints.

*n*

_{Rx}both in the context of PM-QAM (see Section 4) and of PM-OFDM. In both cases it gives an indication of the oversampling that is carried out at the Rx, though the specific definition is somewhat different. Since the two different contexts of use will always be clearly defined, we keep the same notation.

*n*

_{Tx}and

*n*

_{Rx}, the number of arithmetic operations required to demodulate a single bit of the payload, or

*OP*, is given by:

_{b}*OP*and

_{s,Tx}*OP*are the total number of operations needed to process a full OFDM symbol at the Tx and Rx , respectively. Such processing requires computing two IFFTs and two FFTs (one per polarization) over

_{s,Tx}*n*

_{Tx}

*N′*and

_{SC}*n*

_{Rx}

*N′*complex numbers, respectively.

_{SC}*N*complex numbers requires a number of operations

*OP*:

39. P. Duhamel and H. Hollmann, “Split-radix FFT algorithm,” Electron. Lett. **20**, 14–16 (1984). [CrossRef]

*q*≃ 4. The well-known Cooley-Tukey algorithm has a slightly larger

*OP*count, but essentially behaves similarly. In [25], it was conservatively assumed

*q*= 5. However, though important from a system design viewpoint, the actual value of

*q*becomes largely irrelevant within a relative comparison of PM-OFDM with PM-QAM, if we assume that the same FFT technology is available to both systems. Actual implementation details may also deeply affect the on-chip performance of a certain FFT algorithm, but by the same reasoning they are unimportant to the effect of a comparison between the two formats, as long as they are using the same technology.

*OP*

_{s,Rx}one only needs to change the subscripts ‘Tx’ with ‘Rx’. Note that the factor ‘2’ in front of the right-hand side is due to the fact that two separate IFFTs are needed, one per polarization.

12. W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express **16**, 841–859 (2008). [CrossRef] [PubMed]

29. Xingwen Yi, W. Shieh, and Yan Tang, “Phase Estimation for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. **19**, 919–921 (2007). [CrossRef]

## 4. Number of FIR filter taps for PM-QAM

30. J. H. Winters, “Equalization in Coherent Transmission Systems Using a Fractionally Spaced Equalizer,” J. Lightwave Technol. **8**, 1487–1491 (1990). [CrossRef]

31. M. G. Taylor, “Coherent Detection Method Using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments,” IEEE Photon. Technol. Lett. **16**, 674676 (2004). [CrossRef]

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. **25**, 2033–2043 (2007). [CrossRef]

33. S. J. Savory, “Digital Filters for Coherent Optical Receivers,” Opt. Express **16**, 805–817 (2008). [CrossRef]

*τ*depends on the channel memory, that we will call

_{F}*μ*. We will express both

*τ*and

_{F}_{μ}in number of symbol intervals. They are different quantities by definition but, if we constrain the FIR filters to exactly compensate for the channel memory induced by CD, then from well-known results of signal theory and filtering theory it follows

*τ*≃

_{F}*μ*.

*μ*depends both on the accumulated dispersion

*D*·

*L*and on the symbol rate

*R*. It also depends on the actual Tx pulse spectral/temporal shape. The smoother the pulse, the smaller μ and consequently

_{s}*τ*. Therefore, determining the actual length of the FIR filter is not simple. In fact, it should be done “a posteriori”, based on a penalty constraint. One should gradually reduce the length of the FIR filter till a certain target penalty is incurred. However, different FIR impulse-response synthesis/optimization techniques could be used, which may lead to different results. Also, different Tx pulse shapes intrinsically generate different

_{F}*μ*’s and therefore require different FIR lengths.

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. **25**, 2033–2043 (2007). [CrossRef]

30. J. H. Winters, “Equalization in Coherent Transmission Systems Using a Fractionally Spaced Equalizer,” J. Lightwave Technol. **8**, 1487–1491 (1990). [CrossRef]

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. **25**, 2033–2043 (2007). [CrossRef]

*R*. Assuming operation at

_{s}*λ*= 1550 nm, Eq. (17) becomes:

*τ*≃ 8.5 · |

_{F}*D*| ·

*LR*

^{2}

_{s}(the number ‘8.5’ has dimensions [(nm

^{2}· s)/km]).

*μ*, simply by adding more memory to the Viterbi processor itself till the system-required OSNR closely approaches its asymptotic minimum vs. processor memory. Such processor memory (in number of symbols) gives an accurate estimate of

*μ*.

34. G. Bosco, P. Poggiolini, and M. Visintin, “Performance Analysis of MLSE Receivers Based on the Square-Root Metric”, J. Lightwave Technol. **26**, 2098–2109 (2007). [CrossRef]

36. P. Poggiolini, G. Bosco, Y. Benlachtar, S. J. Savory, P. Bayvel, R. I. Killey, and J. Prat, “Long-Haul 10 Gbit/s Linear and Non-Linear IMDD Transmission over Uncompensated Standard Fiber Using a SQRT-Metric MLSE Receiver,” Opt. Express **16**, 12919–12936 (2008). [CrossRef] [PubMed]

*λ*=1550 nm, then:

*μ*≃ 8 |

*D*| ·

*L*·

*R*

^{2}

_{s}. Since, as mentioned, the FIR filter impulse response duration should ideally match the channel memory, (18) also gives us an estimate of

*τ*for a FIR filter capable of compensating for the set amount of CD.

_{F}**25**, 2033–2043 (2007). [CrossRef]

36. P. Poggiolini, G. Bosco, Y. Benlachtar, S. J. Savory, P. Bayvel, R. I. Killey, and J. Prat, “Long-Haul 10 Gbit/s Linear and Non-Linear IMDD Transmission over Uncompensated Standard Fiber Using a SQRT-Metric MLSE Receiver,” Opt. Express **16**, 12919–12936 (2008). [CrossRef] [PubMed]

*τF*. In the following, we will therefore take Eq. (18) as the estimate of

*μ*, and hence of

*τ*.

_{F}*N*=

_{F}*τ*·

_{F}*n*

_{Rx}, where

*n*is the number of samples per symbol taken by the Rx analog-to-digital converter (ADC).

_{Rx}*n*

_{Rx}is critical for system design and it is currently being debated how low it can practically be made. The value

*n*

_{Rx}= 2 guarantees good performance whereas the value

*n*

_{Rx}= 1 is the theoretical lower limit. It is also possible to use intermediate values, such as

*n*

_{Rx}= 1.5. The lower

*n*

_{Rx}, the lower is the FIR computational effort. However, operating close, or at,

*n*

_{Rx}= 1 may cause large penalties due to aliasing and other problems [32

**25**, 2033–2043 (2007). [CrossRef]

*n*

_{Rx}, it is mandatory that there is a digital clock-recovery circuit, possibly using an interpolation/decimation stage, that eventually provides one single sample per symbol to carry out decision. As stated in the introduction, we consider this topic outside of the scope of the paper. On the other hand, we point out that CD compensation can and should occur before clock recovery, also because the clock recovery circuit would not lock on a highly dispersed signal. Current system prototypes, such as Nortel’s PM-QPSK at 43 Gb/s, follow this scheme. As a result, the computational effort of the FIR filters for CD compensation is correctly estimated using

*n*

_{Rx}as the number of samples per symbol.

*n*

_{Rx}.

*n*

_{Rx}= 2, then Eq. (19) yields

*N*≃ 620.

_{F}*μ*= 8|

*D*|

*L*·

*R*

^{2}

_{b}/

*M*

^{2}appears in the latter, too. This could be expected, since it is clearly ‘channel memory’ that drives the need for the cyclic prefix in PM-OFDM as well. Even though

*μ*in (18) was defined and estimated in the context of PM-QAM, we will use it both in PM-QAM and PM-OFDM formulae to simplify the equations and ease comparisons. For instance, Eq. (11) can be rewritten in compact form as:

*OP*for PM-OFDM can be likewise simplified and becomes:

_{b}## 5. Operations per bit for PM-QAM

*N*[27]. A straightforward count of operations per bit of the two FIR filters shown in Fig. 2, assuming TD, leads to:

_{F}*μ*whereas Eq. (21) scales as log

_{2}(

*μ*). The difference is striking and it shows that the TD approach would result in PM-OFDM having a far superior performance than PM-QAM.

*P*of samples of the incoming signal and then block-process them together. The minimum value for

*P*is

*N*, but choosing

_{F}*P*>

*N*improves the algorithm efficiency. We will come back to the choice of

_{F}*P*later. The filter output at each iteration will consist of a block of

*P*samples of the dispersion-compensated signal. Note that the duration of an iteration is

*P*·

*T*/

_{s}*n*

_{Rx}.

*P*to

*P*+

*N*. As a result, at every iteration, the overlap-and-add algorithm needs to perform:

_{F}- one FFT over
*P*+*N*samples_{F} *P*+*N*complex multiplications of the FFT output times the channel transfer function_{F}- one IFFT over
*P*+*N*samples_{F} - NF complex sums.

*OP*is then:

_{i}*OP*operations are needed to process a whole block of

_{i}*P*input samples, over both polarizations. Then, we observe that each of these dual-polarization blocks carries

*P*/

*n*

_{Rx}symbols, corresponding to (

*M*·

*P*)/

*n*

_{Rx}bits. As a result, the number of operations per bit is:

*P*=

*pN*. Using this equation and then (19) to relate

_{F}*N*to the system parameters and CD, we get:

_{F}*μ*to simplify the notation:

## 6. Comparison of PM-OFDM and PM-QAM computational effort for CD compensation

*M*= 4 and so, as PM-QAM format, we actually assume PM-QPSK.

- The PM-OFDM cyclic-prefix overhead parameter
*k*is 1.122, corresponding to an OSNR penalty of 0.5 dB and a 12.2% bandwidth efficiency loss. - The PM-OFDM Tx and Rx oversampling factors are
*n*_{Tx}=*n*_{Rx}= 1.25. - The PM-QPSK Rx oversampling factor is
*n*_{Rx}= 1.5, which appears to be reachable without incurring substantial penalties [32**25**, 2033–2043 (2007). [CrossRef] - The PM-QPSK overlap-and-add block-length parameter
*p*is set to 7.5, i.e., the total length of the sample block that is processed by the Rx FFT/IFFT is*P*= (1 +*p*)*N*= 8.5·NF. This choice of_{F}*p*makes the FFT block-length identical between PM-OFDM and PM-QPSK. This makes the comparison more fair because the FFTs and IFFTs used by the two formats then have the same complexity. Note that*p*could be set as low as 1, allowing for a much smaller block-length for PM-QPSK. However, the total number of operations per bit would actually increase, since decreasing*p*makes the overlap-and-add algorithm less efficient.For both systems we chooseFig. 4.Operations per transmitted bit for PM-OFDM and PM-QPSK, vs. link length at 111 Gb/s (top) and vs. bit rate at 1000 km (bottom). D=16.7 ps/(nm·km). Other parameters: for PM-OFDM, k = 1.122 and n Tx = n Rx = 1.25; for PM-QPSK, p = 7.5 and n Rx = 1.5. The FFT parameter q was set to 5.*q*= 5, as it was suggested in [25]. As mentioned before,*q*has little impact on a comparison between PM-QPSK and PM-OFDM, but it is necessary to set it in order to achieve an approximate estimate of the actual total number of operations per bit.

*OP*

_{b,PM-OFDM}and

*OP*

_{b,PM-QPSK}, obtained using the above parameters. The top plot assumes

*R*= 111 Gb/s whereas

_{b}*L*ranges between 10 and 3000 km. Both systems use an identical FFT/IFFT block length of 4000 samples at 3000 km, linearly decreasing to zero as

*L*goes to zero. For the sake of clarity we refrained from rounding up the FFT/IFFT block lengths to powers of 2.

*p*= 1, it could be as small as 1000 samples; however, the number of operations per bit would go up to 160, vs. 106 when using

*p*= 7.5. Throughout the plot, PM-OFDM requires about 28% fewer operations per bit than PM-QPSK. This result is essentially independent of the link length

*L*. Due to the logarithmic law in both (21) and (26), the increase in operations per bit between 1000 km and 3000 km, despite the tripling of distance, is small (only about 15%).

*OP*vs. the bit rate, assuming a fixed link length

_{b}*L*= 1000 km (Fig. 4, bottom plot). Again, PM-OFDM requires about 28% fewer operations per bit, essentially independently of

*R*. Going from 42.7 to 111 Gb/s requires only about 33% more operations per bit, thanks to the logarithmic impact of

_{b}*R*.

_{b}*p*can be made large enough that

*p*= 7.5. If we now also assume that the channel memory

*μ*is ‘large’, either because the link length or the symbol rate, or both, are large, then the only term in the square brackets that is significant is log

_{2}(

*μ*). Note that the case-study example did have a quite large

*μ*= 310, at 3000 km, so this circumstance is not unrealistic. Under all these assumptions it turns out that, in the asymptotic sense defined above, we have:

## 7. Impact of polarization-related effects

3. Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express **13**, 7527–7534 (2005). [CrossRef] [PubMed]

10. C. R. S. Fludger, et al., “Coherent Equalization and POLMUX-RZ-DQPSK for Robust 100-GE Transmission,” J. Lightwave Technol. **26**, 64–72 (2008). [CrossRef]

**25**, 2033–2043 (2007). [CrossRef]

33. S. J. Savory, “Digital Filters for Coherent Optical Receivers,” Opt. Express **16**, 805–817 (2008). [CrossRef]

*μ*is given by (18) and

_{CD}*μ*=

_{DGD}*τ*·

_{DGD}*M*/

*R*. In turn,

_{b}*τ*is the channel memory due to the maximum differential group delay (DGD) that the system is designed to handle, measured in [ps]. We remind the reader that

_{DGD}*μ*is measured in number of PM-QAM symbols but is also used as a channel memory parameter for PM-OFDM (see end of Sect. 4). In long uncompensated systems, however,

*μ*can be expected to be small as compared to

_{DGD}*μ*, so that in many practical cases

_{CD}*μ*≃

*μ*.

_{CD}10. C. R. S. Fludger, et al., “Coherent Equalization and POLMUX-RZ-DQPSK for Robust 100-GE Transmission,” J. Lightwave Technol. **26**, 64–72 (2008). [CrossRef]

3. Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express **13**, 7527–7534 (2005). [CrossRef] [PubMed]

*μ*, the computational effort for the FFTs and IFFTs needed for frequency-domain FIR filtering remains the same as for CD alone. We now have four transfer functions rather than two, plus two more complex sums. Considering the overall butterfly filtering, we easily get:

*M*to 7 + 8/M.

*n*

_{Rx}= 1.5) as compared to the 310 symbols of channel memory which are dealt with by the CD-compensating first stage. The very different performance is due to the extreme efficiency of FIR filtering in frequency-domain, with the overlap-and-add algorithm, as opposed to the great inefficiency of time-domain FIR filtering.

## 8. Conclusion

37. Xingwen Yi, W. Shieh, and Yiran Ma, “Phase Noise Effects on High Spectral Efficiency Coherent Optical OFDM Transmission,” J. Lightwave Technol. **26**, 1309–1316 (2008). [CrossRef]

38. H. C. Bao and W. Shieh, “Transmission of Wavelength-Division-Multiplexed Channels With Coherent Optical OFDM,” IEEE Photon. Technol. Lett. **19**, 922–924 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | R. Noé, “Phase Noise-Tolerant Synchronous QPSK/BPSK Baseband-Type Intradyne Receiver Concept With Feedforward Carrier Recovery,” J. Lightwave Technol. |

2. | S. Tsukamoto, D. S. Ly-Gagnon, K. Katoh, and K. Kikuchi, “Coherent Demodulation of 40-Gbit/s Polarization-Multiplexed QPSK Signals with 16-GHz Spacing after 200-km Transmission,” in Proc. OFC 2005, PD paper 29, Anaheim (USA), March. 6–11, (2005). |

3. | Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express |

4. | D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent Detection of Optical Quadrature Phase-Shift Keying Signals With Carrier Phase Estimation,” J. Lightwave Technol. |

5. | S. J. Savory et al., “Digital Equalisation of 40 Gbit/s per Wavelength Transmission over 2480km of Standard Fibre without Optical Dispersion Compensation,” in Proc. ECOC 2006, paper Th2.5.5, Cannes (FR), Sept. 24-28, (2006). |

6. | C. R. S. Fludger, T. Duthel, T. Wuth, and C. Schulien, “Uncompensated Transmission of 86 Gbit/s Polarization Multiplexed RZ-QPSK over 100km of NDSF Employing Coherent Equalisation,” in Proc. ECOC 2006, PD paper Th4.3.3, Cannes (FR), Sept. 24-28, (2006). |

7. | K. Roberts, “Electronic Dispersion Compensation Beyond 10 Gb/s,” in Proc. of IEEE LEOS Summer Topical Meetings, Portland (USA), paper MA2.3, Jul. 23-25, (2007). |

8. | G. Charlet et al., “12.8 Tbit/s transmission of 160 PDM-QPSK (160X2X40 Gbit/s) channels with coherent detection over 2550 km,” Proc. ECOC 2007, paper PD 1.6, Berlin (D), Sept. 16-20, (2007). |

9. | C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, Han Sun, and M. OSullivan, “WDM Performance and PMD Tolerance of a Coherent 40-Gbit/s Dual-Polarization QPSK Transceiver,” J. Lightwave Technol. |

10. | C. R. S. Fludger, et al., “Coherent Equalization and POLMUX-RZ-DQPSK for Robust 100-GE Transmission,” J. Lightwave Technol. |

11. | J. Renaudier, G. Charlet, M. Salsi, O. B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver,” J. Lightwave Technol. |

12. | W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express |

13. | W. Shieh and C. Athaudage, “Coherent Optical Orthogonal Frequency Division Multiplexing,” Electron. Lett. |

14. | W. Shieh, X. Yi, and Y. Tang, “Transmission Experiment of Multi-Gigabit Coherent Optical OFDM Systems over 1000 km SSMF Fibre,” Electron. Lett. |

15. | S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4160-km SSMF Enabled by RF-pilot Tone Phase Noise Compensation,” Proc. OFC 2007, Anaheim (CA), paper PDP 15, March 25-29, (2007). |

16. | S. L. Jansen, I. Morita, T. C. W. Schenck, N. Takeda, and H. Tanaka ‘Coherent Optical 25.8-Gb/s OFDM Transmission Over 4160-km SSMF,’ J. Lightwave Technol. |

17. | B. Goebel, B. Fesl, L. D. Coelho, and N. Hanik, “On the Effect of FWM in Coherent Optical OFDM Systems,” in Proc. OFC |

18. | A. J. Lowery and J. Armstrong, “Orthogonal Frequency Division Multiplexing for Dispersion Compensation of Long-Haul Optical Systems,” Opt. Express |

19. | A. J. Lowery, “Improving Sensitivity and Spectral Efficiency in Direct-Detection Optical OFDM Systems,” in Proc. OFC 2008, paper OMM4, San Diego (CA), Feb. 24-28, (2008). |

20. | S. L. Jansen, I. Morita, and H. Tanaka, “16x52.5-Gb/s, 50-GHz Spaced, POLMUX-CO-OFDM Transmission over 4,160 km of SSMF Enabled by MIMO Processing,” in Proc. ECOC 2007, paper PD 1.3, Berlin (D), Sept. 16-20, (2007). |

21. | S. L. Jansen, I. Morita, and H. Tanaka, “10x121.9-Gb/s PDM-ODFM Transmission with 2-b/s/Hz Spectral Efficiency over 1,000 km of SSMF,” in Proc. OFC 2008, paper PDP2, San Diego (CA), Feb. 24-28, (2008). |

22. | Yiran Ma, W. Shieh, and Qi Yang, “Bandwidth-Efficient 21.4 Gb/s Coherent Optical 2x2 MIMO OFDM Transmission,” in Proc. OFC 2008, paper JWA59, San Diego (CA), Feb. 24-28, (2008). |

23. | E. Yamada, et al., “Novel No-Guard-Interval PDM CO-OFDM Transmission in 4.1 Tb/s (50x88.8 Gb/s) DWDM Link over 800 km SMF Including 50-Ghz Spaced ROADM Nodes”, in Proc. OFC 2008, paper PDP8, San Diego (CA), Feb. 24-28, (2008). |

24. | W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express |

25. | H. Bulow, B. Franz, A. Klekkamp, and F. Buchali, “40 Gb/s Distortion Mitigation and DSP-Based Equalisation,” in Proc. ECOC 2007, Berlin, Germany, Sept. (2007). |

26. | A. V. Oppenheim and R. V. Schafer, |

27. | S. W. Smith, |

28. | L. Hanzo, M. Munster, B.J. Choi, and T. Keller, |

29. | Xingwen Yi, W. Shieh, and Yan Tang, “Phase Estimation for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. |

30. | J. H. Winters, “Equalization in Coherent Transmission Systems Using a Fractionally Spaced Equalizer,” J. Lightwave Technol. |

31. | M. G. Taylor, “Coherent Detection Method Using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments,” IEEE Photon. Technol. Lett. |

32. | E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. |

33. | S. J. Savory, “Digital Filters for Coherent Optical Receivers,” Opt. Express |

34. | G. Bosco, P. Poggiolini, and M. Visintin, “Performance Analysis of MLSE Receivers Based on the Square-Root Metric”, J. Lightwave Technol. |

35. | P. Poggiolini, G. Bosco, and M. Visintin, “MLSE Receivers and Their Applications in Optical Transmission Systems”, in Proc. of The 20th Annual Meeting of the IEEE LEOS, Lake Buena Vista, Florida (U.S.A.), 21-25 Oct., pp. 216–217, (2007). |

36. | P. Poggiolini, G. Bosco, Y. Benlachtar, S. J. Savory, P. Bayvel, R. I. Killey, and J. Prat, “Long-Haul 10 Gbit/s Linear and Non-Linear IMDD Transmission over Uncompensated Standard Fiber Using a SQRT-Metric MLSE Receiver,” Opt. Express |

37. | Xingwen Yi, W. Shieh, and Yiran Ma, “Phase Noise Effects on High Spectral Efficiency Coherent Optical OFDM Transmission,” J. Lightwave Technol. |

38. | H. C. Bao and W. Shieh, “Transmission of Wavelength-Division-Multiplexed Channels With Coherent Optical OFDM,” IEEE Photon. Technol. Lett. |

39. | P. Duhamel and H. Hollmann, “Split-radix FFT algorithm,” Electron. Lett. |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems

(060.4080) Fiber optics and optical communications : Modulation

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: September 4, 2008

Revised Manuscript: January 2, 2009

Manuscript Accepted: January 14, 2009

Published: January 22, 2009

**Citation**

P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, "Evaluation of the computational effort for chromatic dispersion compensation in coherent optical PM-OFDM and PM-QAM systems," Opt. Express **17**, 1385-1403 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1385

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### References

- R. Noé, "Phase Noise-Tolerant Synchronous QPSK/BPSK Baseband-Type Intradyne Receiver Concept with Feedforward Carrier Recovery," J. Lightwave Technol. 23, 802-808 (2005). [CrossRef]
- S. Tsukamoto, D. S. Ly-Gagnon, K. Katoh, K. Kikuchi, "Coherent Demodulation of 40-Gbit/s Polarization-Multiplexed QPSK Signals with 16-GHz Spacing after 200-km Transmission," in Proc. OFC 2005, PD paper 29, Anaheim (USA), March. 6-11, (2005).
- Y. Han and G. Li, "Coherent optical communication using polarization multiple-input-multiple-output," Opt. Express 13, 7527-7534 (2005). [CrossRef] [PubMed]
- D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, "Coherent Detection of Optical Quadrature Phase-Shift Keying Signals With Carrier Phase Estimation," J. Lightwave Technol. 24, 12-21 (2006). [CrossRef]
- S. J. Savory et al., "Digital Equalisation of 40 Gbit/s per Wavelength Transmission over 2480km of Standard Fibre without Optical Dispersion Compensation," in Proc. ECOC 2006, paper Th2.5.5, Cannes (FR), Sept. 24-28, (2006).
- C. R. S. Fludger, T. Duthel, T. Wuth, and C. Schulien, "Uncompensated Transmission of 86 Gbit/s Polarization Multiplexed RZ-QPSK over 100km of NDSF Employing Coherent Equalisation," in Proc. ECOC 2006, PD paper Th4.3.3, Cannes (FR), Sept. 24-28, (2006).
- K. Roberts, "Electronic Dispersion Compensation Beyond 10 Gb/s," in Proc. of IEEE LEOS Summer Topical Meetings, Portland (USA), paper MA2.3, Jul. 23-25, (2007).
- G. Charlet et al., "12.8 Tbit/s transmission of 160 PDM-QPSK (160X2X40 Gbit/s) channels with coherent detection over 2550 km," Proc. ECOC 2007, paper PD 1.6, Berlin (D), Sept. 16-20, (2007).
- C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, Han Sun, M. OSullivan, "WDM Performance and PMD Tolerance of a Coherent 40-Gbit/s Dual-Polarization QPSK Transceiver," J. Lightwave Technol. 26, 168-175 (2008). [CrossRef]
- C. R. S. Fludger, et al., "Coherent Equalization and POLMUX-RZ-DQPSK for Robust 100-GE Transmission," J. Lightwave Technol. 26, 64-72 (2008). [CrossRef]
- J. Renaudier, G. Charlet, M. Salsi, O. B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, "Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver," J. Lightwave Technol. 26, 36-42 (2008). [CrossRef]
- W. Shieh, H. Bao, and Y. Yang, "Coherent Optical OFDM: Theory and Design," Opt. Express 16, 841-859 (2008). [CrossRef] [PubMed]
- W. Shieh and C. Athaudage, "Coherent Optical Orthogonal Frequency Division Multiplexing," Electron. Lett. 42, 587-589 (2006). [CrossRef]
- W. Shieh, X. Yi, and Y. Tang, "Transmission Experiment of Multi-Gigabit Coherent Optical OFDM Systems over 1000 km SSMF Fibre," Electron. Lett. 43, 183184 (2007). [CrossRef]
- S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, "20-Gb/s OFDM Transmission over 4160-km SSMF Enabled by RF-pilot Tone Phase Noise Compensation," Proc. OFC 2007, Anaheim (CA), paper PDP 15, March 25-29, (2007).
- S. L. Jansen, I. Morita, T. C. W. Schenck, N. Takeda, and H. Tanaka "Coherent Optical 25.8-Gb/s OFDM Transmission Over 4160-km SSMF," J. Lightwave Technol. 26, 6-15 (2008). [CrossRef]
- B. Goebel, B. Fesl, L. D. Coelho and N. Hanik, "On the Effect of FWM in Coherent Optical OFDM Systems," in Proc. OFC 2008, Anaheim (CA), paper JWA58, San Diego (CA), Feb. 24-28, (2008).
- A. J. Lowery and J. Armstrong, "Orthogonal Frequency Division Multiplexing for Dispersion Compensation of Long-Haul Optical Systems," Opt. Express 14, 2079-2084 (2006). [CrossRef] [PubMed]
- A. J. Lowery, "Improving Sensitivity and Spectral Efficiency in Direct-Detection Optical OFDM Systems," in Proc. OFC 2008, paper OMM4, San Diego (CA), Feb. 24-28, (2008).
- S. L. Jansen, I. Morita and H. Tanaka, "16x52.5-Gb/s, 50-GHz Spaced, POLMUX-CO-OFDM Transmission over 4,160 km of SSMF Enabled by MIMO Processing," in Proc. ECOC 2007, paper PD 1.3, Berlin (D), Sept. 16-20, (2007).
- S. L. Jansen, I. Morita and H. Tanaka, "10x121.9-Gb/s PDM-ODFM Transmission with 2-b/s/Hz Spectral Efficiency over 1,000 km of SSMF," in Proc. OFC 2008, paper PDP2, San Diego (CA), Feb. 24-28, (2008).
- Y. Ma, W. Shieh, and Qi Yang, "Bandwidth-Efficient 21.4 Gb/s Coherent Optical 2x2 MIMO OFDM Transmission," in Proc. OFC 2008, paper JWA59, San Diego (CA), Feb. 24-28, (2008).
- E. Yamada, et al., "Novel No-Guard-Interval PDM CO-OFDM Transmission in 4.1 Tb/s (50x88.8 Gb/s) DWDM Link over 800 km SMF Including 50-Ghz Spaced ROADM Nodes," in Proc. OFC 2008, paper PDP8, San Diego (CA), Feb. 24-28, (2008).
- W. Shieh, Q. Yang, and Y. Ma, "107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing," Opt. Express 16, 6378-6386 (2008). [CrossRef] [PubMed]
- H. Bulow, B. Franz, A. Klekkamp, and F. Buchali, "40 Gb/s Distortion Mitigation and DSP-Based Equalisation," in Proc. ECOC 2007, Berlin, Germany, Sept. (2007).
- A. V. Oppenheim and R. V. Schafer, Digital Signal Processing, (Prentice-Hall Inc., Englewood Cliffs, NJ, 1975), pp. 110-113.
- S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, California Technical Publishing, San Diego, CA, 1997) Chap. 18.
- L. Hanzo, M. Munster, B. J. Choi, and T. Keller, OFDM and MC-CDMA, (John Wiley and Sons, Hoboken, NJ, 2003).
- X. Yi, W. Shieh, and Y . Tang, "Phase Estimation for Coherent Optical OFDM," IEEE Photon. Technol. Lett. 19, 919-921 (2007). [CrossRef]
- J. H. Winters, "Equalization in Coherent Transmission Systems using a Fractionally Spaced Equalizer," J. Lightwave Technol. 8, 1487-1491 (1990). [CrossRef]
- M. G. Taylor, "Coherent Detection Method using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments," IEEE Photon. Technol. Lett. 16, 674676 (2004). [CrossRef]
- E. Ip and J. M. Kahn, "Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion," J. Lightwave Technol. 25, 2033-2043 (2007). [CrossRef]
- S. J. Savory, "Digital Filters for Coherent Optical Receivers," Opt. Express 16, 805-817 (2008). [CrossRef]
- G. Bosco, P. Poggiolini, and M. Visintin, "Performance Analysis of MLSE Receivers Based on the Square-Root Metric", J. Lightwave Technol. 26, 2098-2109 (2007). [CrossRef]
- P. Poggiolini, G. Bosco, and M. Visintin, "MLSE Receivers and Their Applications in Optical Transmission Systems", in Proc. of The 20th Annual Meeting of the IEEE LEOS, Lake Buena Vista, Florida (U.S.A.), 21-25 Oct., pp. 216-217, (2007).
- P. Poggiolini, G. Bosco, Y. Benlachtar, S. J. Savory, P. Bayvel, R. I. Killey, and J. Prat, "Long-Haul 10 Gbit/s Linear and Non-Linear IMDD Transmission over Uncompensated Standard Fiber Using a SQRT-Metric MLSE Receiver," Opt. Express 16, 12919-12936 (2008). [CrossRef] [PubMed]
- Xingwen Yi,W. Shieh, and Yiran Ma, "Phase Noise Effects on High Spectral Efficiency Coherent Optical OFDM Transmission," J. Lightwave Technol. 26, 1309-1316 (2008). [CrossRef]
- H. C. Bao and W. Shieh, "Transmission of Wavelength-Division-Multiplexed Channels With Coherent Optical OFDM," IEEE Photon. Technol. Lett. 19, 922-924 (2007). [CrossRef]
- P. Duhamel and H. Hollmann, "Split-radix FFT algorithm," Electron. Lett. 20, 14-16 (1984). [CrossRef]

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