Optial Modulation Index of the Mach-Zehnder Modulator in a Coherent Optical OFDM yste Eploying Digital redistortion David Rörich, Xiaojie Wang, Michael Bernhard, Joachi peidel Universität tuttgart, Institut für Nachrichtenübertragung {roerich, xiaojie.wang, bernhard, speidel}@inue.uni-stuttgart.de Abstract We study the ipact of nonlinear distortions caused by the external optical odulator in coherent optical OFDM systes and deterine by siulation the optial odulation index that iniizes the ONR penalty introduced by nonlinear distortion and odulation excess loss. To reduce this penalty a digital predistortion device ipleented by a look-up table is proposed and the achievable gain in receiver sensitivity is quantified. These results are copared for the word lengths 6 bit and 8 bit of the digital-to-analog converter and look-up table, respectively. 1 Introduction One widely discussed candidate for optical transission systes that reach data rates beyond Gbit/s is orthogonal frequency division ultiplex (OFDM) cobined with coherent detection (COOFDM). This is due to its resilience to chroatic and polarization ode dispersion [1], [], its tight spectral shape that facilitates the generation of superchannels [3] and its potential to ipleent flexible bandwidth allocation. Its sensitivity to nonlinearities, however, is one of the ajor drawbacks of COOFDM [4]. An iportant origin of nonlinearities is the external optical odulator realized by two Mach-Zehnder odulators (MZMs) that exhibit a sine-shaped electro-optical characteristic. The MZM nonlinear characteristic in COOFDM systes has first been studied by Tang et al. for binary phase shift keying (BK) [4] and in [] for 4- to 18-QAM. Digital predistortion to overcoe the MZM nonlinearity has been investigated in [6] [8] but the achievable receiver sensitivity gain has not been quantified yet. In addition, quantization ipairents have to be taken into account when ipleenting digital predistortion for real-tie COOFDM systes at the envisaged data rates. In [6], the liited word length of the digital-to-analog converter (DAC) has been odeled by quantizing the OFDM transit signal, but quantization errors in the transitter digital signal processing (D) units, especially in the digital predistortion device, have not been studied yet. The ain contribution of this paper is the definition and deterination of the optial odulation index for a COOFDM syste with 4-, 16- and 64-QAM and the investigation of a look-up table (LUT) based predistortion device that allows detailed perforance evaluation of digital predistortion in the presence of transitter quantization errors. Furtherore, the receiver sensitivity gain through digital predistortion is identified and copared for the aforeentioned odulation orders. This paper is organized as follows. The syste odel establishing the basis of our studies is introduced in ection. In ection 3, we define odulation index (MI) and odulation excess loss (EL) and derive an analytical expression for the latter. The optial MI in ters of iniu cobined receiver sensitivity penalty and EL is deterined in ection 4 through nuerical siulations and LUT based digital predistortion is studied in ection. In ection 6 we suarize our findings. yste odel MA OFDM transitter Figure 1 MF IFFT GI EDFA The syste odel used in this work is based on [9] and has been extended by digital predistortion. Its block diagra is depicted in Fig. 1. The OFDM transit- Requant 14 bit M bit M bit 9 Hybrid v I Q Laser on/off redistort. ADC g(v) DAC Block diagra of COOFDM syste. GI 1 I Q u FFT OFDM receiver Laser E in opt. Mod. E out ter consists of apper (MA), inverse fast Fourier transfor (IFFT) and guard interval (GI) insertion. The IFFT diension is 6 resulting in the sae nuber of subcarriers of which 16 are odulated. The IFFT block is odeled by a custo Very High EQ DEMA
peed Hardware Description Language (VHDL) design [] with an output word length of 14 bit. The DAC, working at a sapling rate of 3 Ga/s, usually has a lower word length M of 6 bit to 8 bit. Consequently, the word length of the OFDM transitter output has to be reduced which is realized by the block Requant. It eploys the shift and clip (AC) ethod we have introduced in [9] and is followed by a digital predistortion unit that will be described in ore detail in ection. After digital-to-analog conversion the inphase (I) and quadrature (Q) signals drive two nested MZMs in quadrature configuration, thus odulating aplitude and phase of the laser-generated continuous wave. The electric transfer function of the MZMs is assued to be frequency-flat, since Barros et al. have shown in [7] that the effect of the electric transfer function is negligible. The odulated signal is transitted over 8 k of standard single ode fiber (MF) and therefore affected by chroatic dispersion (CD) whose copensation is enabled by insertion of an GI 8 saples long (3.1% of the basic OFDM sybol length). Nonlinear fiber effects are neglected under the assuption that the signal power fed into the fiber is below 1 W. The received signal is aplified by an erbiu doped fiber aplifier (EDFA) and fed into a 9 hybrid for coherent detection before it is sapled by an analog-to-digital converter (ADC) that likewise operates at 3 Ga/s. As this work is focused on transitter quantization effects the ADC word length is assued to be unliited. iilarly, quantization effects are neglected in the OFDM receiver that consists of GI reoval (GI 1 ), fast Fourier transfor (FFT), equalizer (EQ) and deapper (DEMA). In [9] we have introduced a technique to effectively reove the ipact of transitter quantization errors on channel estiation. Here, perfect channel knowledge is assued for equalization to ake the results independent of the influence that quantization errors have on channel estiation. Using the paraeters entioned above and assuing polarization ultiplex, data rates of 8, 16 and 4 Gbit/s are achieved for 4-, 16- and 64-QAM, respectively. 3 Modulation index and odulation excess loss An MZM biased at iniu transission point exhibits the well-known electro-optical characteristic E out (t) E in (t) = 1 ( ) u(t) π sin, (1) ai where E out (t) and E in (t) are the input and output electric fields, a I 1 is the insertion loss, u(t) is the electric driving voltage and is the half-wave switching voltage. The tie t will be oitted in the reainder of this paper for convenience. It becoes apparent fro (1) that the MZM is approxiately linear for sall u but shows a strong nonlinear characteristic for values of u approaching ±. For quantifying the degree of odulation the odulation index (MI) is introduced: = u rs π, () where u rs is the root ean square value of u. With u rs a statistical quantity is chosen for the definition of MI, because OFDM signals have a noise-like, approxiately Gaussian distribution. pecifically, it has been shown in [11] that the probability density function p(u) of u can be odeled with good approxiation by 1 p(u) = e 1 ( u urs ). (3) u rs π It is therefore of little value to use, for instance, the axiu of u for the definition of MI, since this value occurs rarely. To illustrate the eaning of we define the driving voltage u 9 for which r[ u 9 <u<u 9 ]=.9, where r[ ] stands for probability, and provide soe values of u 9 in Table 1. Fro (1) and Table 1 it can also be seen that E out.1. 1. 1. 1.8. u 9 /..6..79.94 1. Table 1 9% range u 9 in dependence of is sall with respect to E in for sall. In this case the ean optical output power out of the MZM will be reduced significantly copared to the ean optical input power in. The optical power loss caused by odulation is called odulation excess loss and defined by a E = in/a I. (4) out Furtherore, ] out = KE [ Eout = K E a I out = in a I [ sin ( u π sin ( u π ) ] Ein, () where E[ ] is the expectation operator and K is a proportionality constant. The coherent wave E in is statistically independent fro the ter sin (...). In addition, in = KE [ Ein]. Hence, ) p(u)du. (6) Fro (4) using eqs. (), (3) and (6) then follows ) 1 a E = (1 e. (7) As can be seen fro eq. (7) the EL a E neither depends on the nuber of subcarriers nor the subcarrier odulation schee but on the MI alone. It should further be noted that the EL caused by odulation of the optical carrier with an OFDM signal cannot be less than 3 db, because li a E =. To verify (7) the expression is copared with nuerical results that have
been obtained by siulation of the syste described in ection excluding quantization effects. Both results are shown in Fig. and are in good agreeent. log (ae) /db 3 3 1 3 analytical nuerical.1.3..7.9 1.1 1.3 1. 1.7 1.9.1.3 Figure Coparison of analytical and nuerical coputation of EL a E ; liit of a E (dashed line). 4 Optial odulation index To study the effect of MZM nonlinearity on syste perforance the optical signal-to-noise ratio (ONR) required to reach a bit error rate (BER) of -3, γ r (in db), is deterined for different values of by siulation. Quantization effects are neglected in this section in order to isolate the effect of MZM nonlinearity. Moreover, digital predistortion is not applied here. Let γ be the required ONR in db for BER = -3 of an idealized syste with linear external odulator. Then the ONR penalty is defined by γ p = γ r γ. (8) and is shown in Fig. 3 along with the EL. For sall log (ae) /db 3 1 6.1.3..7.9 1.1 1.3 1. 1.7 Figure 3 ONR penalty γ p due to MZM nonlinearity for different odulation orders (solid lines) and EL a E (dashed line). 1 8 6 4 γp/db the ONR penalty is negligible as the driving signal is in the quasi-linear region of the MZM characteristic but shows a steep rise when is further increased. Copared to 4-QAM higher odulation orders are obviously ore sensitive to nonlinearities: for an allowable ONR penalty of db ust be below.6 (.9) for 64-QAM (16-QAM) whereas 4-QAM allows an MI as high as 1.3. Keeping the MI low reduces the ONR penalty significantly. However, this coes at the cost of a high EL which will effectively reduce the available ONR at the receiver side, because the transit laser power in is liited. Therefore, a trade-off between ONR penalty through MZM nonlinearity and ONR penalty through odulation has to be found. For this purpose we define the cobined ONR penalty γ c = γ p +log (a E ) (9) that should be iniized with respect to. Fro Fig. 4 the optial odulation index can be obtained, naely the value of for which γ c is iniu. The γc/db 3 1 6.1.3..7.9 1.1 1.3 1. 1.7 Figure 4 Cobined ONR penalty γ c as a function of for odulation orders 4-, 16- and 64-QAM. higher the odulation order the lower is the optial MI as the sensitivity to nonlinearity increases with increasing odulation order, whereas the EL is independent thereof. For the sae reasons the iniu of γ c increases with increasing odulation order. It should be noted that depending on the optial fiber launch power and the available laser input power in soe EL ight even be desired to avoid fiber nonlinearities. Optiization of MI in the presence of fiber nonlinear effects is still to be investigated and outside the scope of this work. Digital predistortion As deonstrated in the previous section MZM nonlinearity provokes significant reduction of syste perforance, especially for odulation orders 16- and 64- QAM. To itigate this effect a predistortion device
can be used that exhibits the inverse characteristic of the MZM thus yielding an overall linear characteristic. However, perfect copensation of the MZM characteristic by its inverse function is ipossible since (1) is π-periodic and thus the iverse function exhibits abiguities. Values u > cannot be apped to a predistorted value that will result in the desired MZM output value and are therefore clipped. This leads to the predistortion function A if v>1 g(v) = π A arcsin(v) if v 1, () A if v< 1 where v is the input of the predistortion device and A is its axiu output level. An analog ipleentation of () can only approxiate the required arcsine characteristic [1], whereas in the digital doain an LUT is able to ipleent arbitrary functions and can be ade reconfigurable. Hence, the latter has been chosen in this work to ipleent the required predistortion function. Yet a digital realization adds quantization noise that has to be taken into account. In the following we will derive the predistortion function of the LUT and study its perforance. The proposed LUT has an input and output word length of M bit, respectively. It stores an output vector for each possible input vector and is assued to be reconfigurable, so that the ipleented function g(v) can be adapted to the dynaic range of u. The discrete valued LUT input v as well as its output g(v) can be interpreted as signed integer nubers in [ M 1, M 1 1]. WeassueafastLUTdevice so that static and dynaic predistortion characteristics ar identical. Consequently, we can drop tie t in the following and the DAC siply converts g(v) to a voltage u = U g(v). (11) M 1 The MZM characteristic that is supposed to be linearized by g(v) isgivenin(1): E out E in ai =sin (11) =sin ( ) u π ( ) U π g(v) M 1! = Rv, (1) where R is a positive proportionality constant. Fro (1) we derive the predistortion function: g(v) = M 1 arcsin(rv) for Rv 1. (13) U π In order to reduce quantization noise, the range of g(v) has to be fully exploited which leads to the condition g ( M 1)! = M 1 R (13) = 1 M 1 sin ( ). (14) Equations (13) and (14) iply that U which liits the range of. Further increase of can be achieved by (a) finding a predistortion function for U > or by (b) scaling and clipping of v. Ipleenting option (a) would require to reduce the nuber of output words that represent the predistortable range of u, thus leading to an increase in quantization noise. For this reason option (b) has been chosen and ipleented by scaling v with R > 1/ M 1 if the desired MI cannot be achieved otherwise. In this case U = and M 1 1 if Rv > M 1 1 M 1 g(v) = M 1 if Rv < 1. U M 1 π arcsin(rv) otherwise (1) The LUT function is finally obtained by quantizing g(v) given by (1) for all possible input values v. oe exaples of LUT characteristics for M = 8bit are shown in Fig.. Note that the predistortion device in g(v) 17 R =.4/ M 1 U =.6 R =1.8/ M 1 U = R =1/ M 1 U = 18 18 v 17 Figure redistortion characteristic with LUT for M = 8 bit and different values of R. Fig. 1 consists of two identical LUTs for inphase and quadrature signal, respectively. To study the effectiveness of the proposed predistortion device, the ONR required for a BER of -3 has been siulated using the syste odel introduced in ection. Applying the AC ethod for requantization a nuber of ost significant bits is cut so that quantization and clipping noise are iniized before predistortion [9]. The predistorted syste is copared to a syste that does not apply predistortion and the resulting ONR gain as a function of is shown in Fig. 6 for M = 6 bit and 8 bit. The iproveent by predistortion is negligible for sall since the MZM characteristic is approxiately linear in this region. With increasing the ONR gain increases significantly where the sensitivity to nonlinearity of higher odulation orders is indicated by the rise of the ONR gain already at sall copared to 4-QAM. Accordingly, the axiu ONR gain is observed for
4 6 3 6 ONR gain / db 3 1 γc/db 1.dB 1.1.3..7.9 1.1 1.3 1. 1.7 Figure 6 ONR gain through digital predistortion for M = 6bit (dashed line) and 8 bit (solid line)..1.3..7.9 1.1 1.3 1. 1.7 Figure 7 Cobined ONR penalty using LUT predistortion for M = 6 bit (dashed line) and 8 bit (solid line) copared to nonpredistorted syste at M = 8 bit (gray line). 64-QAM, followed by 16-QAM. Fig. 6 also shows that quantization errors introduced by the LUT predistortion can reduce the achievable ONR gain. This effect is especially observed for 6 bit and 64-QAM where quantization noise doinates for sall and thus causes the predistorted syste to perfor even worse than the non-predistorted syste. With increasing distortion doinates again and the predistorted syste exhibits a significant perforance gain of over db. While soe iportant aount of ONR gain is clearly achieved by digital predistortion, the ONR penalty observed in Fig. 3 cannot copletely be eliinated because clipping occurs for large that leads to additional distortion. Furtherore, the residual ONR penalty through MZM nonlinearity as well as the EL have to be taken into account when evaluating the gain through digital predistortion. Thus, the cobined ONR penalty γ c has been deterined and is copared to the non-predistorted syste in Fig. 7. These results show that for 4-QAM the cobined ONR penalty γ c can be reduced for large although the reduction of the iniu γ c is negligible. At 16-QAM and M =8bit the iniu γ c can be reduced by 1. db and the greatest iproveent in receiver sensitivity is achieved for 64-QAM, where γ c is reduced by. db. For M = 6 bit the perforance of the predistorted syste is reduced, especially for 64-QAM. In contrast, 4-QAM is robust against quantization effects. 6 Conclusion We have shown that digital predistortion of the nonlinear MZM characteristic using an LUT can iprove receiver sensitivity in COOFDM systes by. db (1. db) for 64-QAM (16-QAM) by introducing a cobined ONR penalty that takes into account nonlinear distortion effects, transitter quantization errors and odulation excess loss. When 4-QAM is used predistorted and non-predistorted systes exhibit siilar perforance. Our results show, that an LUT input/output word length of 8 bit (6 bit) is sufficient for 64-QAM (16-QAM). Furtherore, we have deterined the optial odulation index in ters of inial cobined ONR penalty for predistorted and non-predistorted systes and have derived an analytical expression for the odulation excess loss in COOFDM systes. 7 Acknowledgent This work was carried out in the fraework of DFG project Elektronische chlüsselbausteine für optische OFDM-ystee hoher Bitrate. The support of DFG is gratefully acknowledged. References [1] W. hieh and C. Athaudage, Coherent optical orthogonal frequency division ultiplexing, Electronics Letters, vol. 4, no., pp. 87 89, 6. [] W. hieh, MD-upported Coherent Optical OFDM ystes, IEEE hotonics Technology Letters, vol. 19, no. 3, pp. 134 136, Feb. 7. [3]. Chandrasekhar and X. Liu, OFDM Based uperchannel Transission Technology, Journal of Lightwave Technology, vol. 3, no. 4, pp. 3816 383, Dec. 1. [4] Y. Tang, W. hieh, X. Yi, and R. Evans, Optiu Design for RF-to-Optical Up-Converter in Coherent Optical OFDM ystes, IEEE hotonics Technology Letters, vol. 19, no. 7, pp. 483 48, 7. [] W. Rosenkranz, A. Ali, and J. Leibrich, Design considerations and perforance coparison of high-order odulation forats using OFDM, in International Conference On Transparent Optical Networks,, pp.. [6] Y. Tang, K.-. Ho, and W. hieh, Coherent Optical OFDM Transitter Design Eploying redistortion, IEEE hotonics Technology Letters, vol., no. 11, pp. 94 96, Jun. 8. [7] D. Barros and J. Kahn, Optical Modulator Optiization for Orthogonal Frequency-Division Multiplexing, Journal of Lightwave Technology, vol. 7, no. 13, pp. 37 378, Jul. 9.
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