Modelling of a DFB Laser at Low Bias Directly Modulated with an OFDM Signal for RoF Applications

P. Assimakopoulos, L. C. Vieira, A. Nkansah, D. Wake and N. J. Gomes

Broadband and Wireless Communications Group Department of Electronics

University of Kent Canterbury, UK

pa80@kent.ac.uk, lcv3@kent.ac.uk

F. van Dijk Alcatel-Thals III-V Lab, Joint lab of Bell Labs

and Thales Research & Technology Palaiseau, France

frederic.vandijk@3-5lab.fr

AbstractA DFB laser biased at a low bias point is modeled using a modified AM/AM Rapp model. Distortion effects are simulated and compared to the analytical predictions made by Bussgangs theorem for soft limiter nonlinearity.

I. INTRODUCTION A laser biased at a low bias point offers certain

advantages, namely low power consumption and longer life time. Low power consumption may be particularly important for Power over Fiber (PoF) applications [1]. The RF response at low bias approximates a soft limiter nonlinearity and thus Bussgangs theorem can be applied without requiring the use of numerical solutions since closed form expressions can be readily obtained. Distortion is a particular problem when transmitting signals with large Peak-to-Average Power Ratio (PAPR) such as OFDM. The baseband discrete OFDM symbol is given by [2]:

(1)

The theoretical maximum PAPR for an OFDM signal with QPSK modulation is 10logN, where N is the number of subcarriers. However, this maximum value is statistically insignificant. Through the Central Limit Theorem (CLT), the OFDM signal can be approximated as the sum of Gaussian In-phase and Quadrature components and will therefore have an envelope following a Rayleigh distribution [3]. By exploiting the statistics of the OFDM signal a more practical value can be obtained for the PAPR, of about 12.3 dB for 2048 subcarriers. This value is still high and some form of compensation might be required, the simplest being the power backoff of the nonlinear device. In the case of a low biased laser this option might not be available but depending on system requirements, a certain amount of distortion might be tolerable.

II. THEORY A. Distortion noise

The high PAPR of the OFDM signal means that peaks of the Rayleigh envelope will be clipped if a nonlinear device is used in the signal chain such as an amplifier, DAC etc. In an optical link, the nonlinearity is mainly due to the laser. The distortion from the nonlinear power transfer function of the laser will result in distortion noise that will degrade the Signal-to-Noise Ratio (SNR) of the system.

Using the complex form of Bussgangs theorem, for a Gaussian input x(t), the output y(t) of a nonlinearity will be given by the sum of a useful attenuated term and uncorrelated distortion noise d(t) [4, 5, 6]:

(2)

where is a function of the nonlinearity, proportional to the cross-correlation of the input signal with the output signal, and is given by:

(3)

where f(r) is the nonlinear function, p(r) is the Rayleigh PDF and 22 is the input power to the nonlinearity. It has to be noted that for an AM/AM nonlinearity, will be a real quantity. The Gaussian input assumption is met by assuming independent and identically distributed (iid) frequency samples (Xk). The first assumption is met by randomizing the input symbols and the second by deriving all symbols from the same constellation. The distortion noise can be assumed as zero mean and under certain assumptions as zero mean AWGN [3]. Based on (2) a model can be used, similar to [4] but adapted for an optical link (Fig.1). In this case, n(t) will

= =

1N

0k

Nj22

keXN1x[n]

d(t)x(t)y(t) +=

==

02 p(r)drrf(r)2

1Rxx(0)Rxy(0)

include the main optical link noise contributions namely, Relative Intensity Noise (RIN) from the laser, shot and thermal noises from the Photodiode (PD), d(t) is the distortion noise and h(t) is the impulse response of the optical link.

Fig.1: Optical link equivalent circuit.

At the OFDM receiver, following the Cyclic Prefix (CP) removal and assuming a zero forcing equalization, the estimate for the transmitted symbol at subcarrier k and OFDM symbol m will be given by:

(4)

where includes the other noise contributions scaled by the sampled frequency response of the optical link. The first part on the right hand side of (4) represents the useful signal while the terms in brackets represent the samples of the distortion and other noise contributions at the location of subcarrier k.

For Nyquist rate sampling, the distortion noise will fall in-band and assuming the same average power per subcarrier, the signal to noise ratio at the output of the optical link will be given by:

(5)

where d2 is the distortion noise variance and n2 is the variance of the other noise contributions. The expressions for the attenuation factor () and the distortion noise variance for the soft limiter case can be found in [7]. It has to be noted, that in the case where oversampling is applied in the OFDM system, the result provided by (5) will underestimate the SNR as some of the distortion noise will fall out-of-band.

The EVM can be calculated using:

(6)

III. AM/AM MODELS

We present here an experimental approach to model a directly modulated DFB laser at low bias: that is, for a hard clipping condition. In this approach, a black-box modeling technique is applied to model the laser nonlinearity. More specifically, the nonlinear effects of the link are observed when the input RF power is varied, that is, the nonlinear

amplitude characteristic, known as AM/AM, is obtained from the input-output measured data.

The experimental set-up is depicted in Figure 2. The experimental set-up consisted of a 1550-nm distributed feedback (DFB) laser from Alcatel-Thales III-V Labs, a FC/APC-FC/PC fibre patch cord and an Appointech InGaAs PIN photodiode, with bandwidth of 2.5 GHz and responsivity of 1 A/W. A Mini-Circuits bias-T was used to bias the PD at -5V. The laser was biased at 35 mA and 37 mA through its internal bias circuit, with the temperature maintained at 25oC. These bias currents are near the laser threshold current of 29 mA. A sinusoidal signal of 2.4 GHz was applied by an Agilent E4438C Vector Signal Generator (VSG), and the output power was measured after the photodiode by an Agilent E4440A Vector Signal Analyser (VSA) connected to a laptop with Agilent VSA software. The input power at the laser was varied from -41 dBm to +9 dBm, in order to find a hard clipping laser operation.

Fig.2: Experimental set-up.

The measured data were modeled, at first, by the following 7th-order polynomial

(7)

where A is the input amplitude.

The coefficients of (7) for 35-mA and 37-mA bias current are given in Table I. The AM/AM characteristics of the DFB laser are shown in Figure 3. These models were fitted using a Trust-Region algorithm, there being obtained a root mean square error (RMSE) of 0.0001127 for the 35-mA bias and an RMSE of 9.00410-5 for the 37-mA bias. Table I: Coefficients for the polynomial AM/AM model.

35-mA bias 37-mA bias p1 0.14749 p1 0.16025 p2 3.27611 p2 -1.59453 p3 -105.859 p3 43.116 p4 1137.32 p4 -580.74 p5 -5924.12 p5 3562.69 p6 15150.18 p6 -10172.28 p7 -15239.42 p7 11043.86

[k])n[k](d[k]X[k]X mmmm ++=

][ knm

1/SNREVM =

2n

2d

22

Rx

2SNR+

=

ApApApApApApApg 12

23

34

45

56

67

7 ++++++=

Fig.3: AM/AM characteristic of the DFB laser using 7th-order polynomial model.

Polynomials are commonly used to model directly-modulated lasers [8, 9] because their coefficients can be extracted using straightforward least square fitting algorithms, and they can be directly related to a physical meaning. However, the use of polynomials is generally limited to modeling mild nonlinearities.

Due to the fact that the measurement data comes from a strong nonlinear behavior, a second model was also extracted based on the Rapp model [10] and named here modified Rapp model, given by

(8)

where A is the input amplitude, SAT the saturation and S the smoothness coefficients.

The coefficients of (8) for 35-mA and 37-mA bias current are given in Table II. The AM/AM characteristics are shown in Figure 4. These models were also fitted using a Trust-Region algorithm, there being obtained an RMSE of 8.79510-5 for the 35-mA bias case and of 9.12610-5 for the 37-mA bias.

Table II: Coefficients for the modified Rapp AM/AM model.

35-mA bias 37-mA bias k 0.1760 k 0.1448

SAT 0.0434 SAT 0.0780 a 0.8749 a 0.9172 s 22.27 s 5.84

Fig.4: AM/AM characteristic of the DFB laser using modified Rapp model.

IV. SIMULATION RESULTS The simulation was carried out in MatlabTM -SimulinkTM,

for an OFDM signal with 2048 subcarriers and QPSK modulation. The analytical EVM obtained from the expressions outlined in section 2 was compared to the EVM obtained by simulating an OFDM link with a nonlinearity given by the modified Rapp or polynomial equations of section 3. The sampling was done at the Nyquist rate i.e. no oversampling was performed. The other optical noise contributions were not taken into account. The EVM is plotted against (Fig.5) which is the square root of the input power back-off and is often referred to as the clipping ratio. In Figure 5, is plotted in linear terms (or from 3.5 dB to 9.5 dB).

Fig.5: Simulation results for biases of 35 mA and 37 mA.

The simulation result shows that there is agreement between the analytical and simulated results but the correlation between the two depends, as expected, on the bias point. The result for 37 mA deviates more from the analytical

+

=

SATA as1

kAg2S

1.6Rapp

curve. This is a result of the change in laser response as the bias is increased: more distortion is coming from points in the laser response away from the threshold point and as a result the response is less similar to a soft limiter. However, the deviation between the analytical and simulated curves depends on the AM/AM modeling and how well this fits the measurements. This is shown if Figure 6 as a comparison between the modified Rapp and the polynomial models.

Fig.6: Comparison between modified Rapp and polynomial models at 35 mA with respect to the analytical result.

The deviation from the analytical curve at low clipping ratios is higher for the polynomial model which has a smoother turn instead of a rapid transition into clipping. The deviation at higher clipping ratios which appears like an overshoot from the analytical curve is due to in-part to the smoother transition into the clipping region which affects a higher range of amplitudes of the signal and due to the deviation between the modeled and measured data in the linear region of the DFB laser response. Furthermore, Bussgangs theorem is an approximation in itself and the DFB laser response below threshold, although close, is not a soft limiter response.

V. CONCLUSIONS The results indicate that the analytical approach following

from the complex form of Bussgangs theorem for the case of a soft limiter nonlinearity is a good approximation for the distortion effects of a laser at low bias and can offer a crude estimate of certain metrics such as SNR and EVM. If power

consumption is of prime importance, depending on the system requirements, a certain amount of distortion can be tolerated. The simple analytical approach shown here can be used to define the operational parameters for which the system requirements are fulfilled. That is, a balance point can be defined between low power consumption and tolerable distortion. Furthermore, it has been shown that a modified Rapp model can better model a laser at low bias compared to a polynomial model.

ACKNOWLEDGMENT This work was carried out within the framework of the European Union Integrated Project FUTON (FP7 ICT-2007-215533). Luis C. Vieira is sponsored by the Brazilian Government through CNPq and UTFPR, whose support is gratefully acknowledged. A. Enard, F. Blache and M. Goix from Alcatel-Thales III-V Lab performed the packaging of the laser module.

REFERENCES [1] D. Wake, A. Nkansah, N. J. Gomes, C. Lethien, C. Sion and J. -P.

Vilcot, Optically powered remote units for Radio-Over-Fiber systems, J. Lightw. Technol., vol. 26, pp. 2484-2491, Aug. 2008.

[2] A. Goldsmith, Wireless communications, Cambridge university press, 2005.

[3] J. Tellado, Multicarrier modulation with low PAR-Applications to DSL and wireless, Kluwer academic publishers, 2002.

[4] P. Banelli and S. Cacopardi, Theoretical analysis and performance of OFDM signals in nonlinear AWGN channels, IEEE Trans. Commun., vol. 48, pp. 430 441, Mar. 2000.

[5] P.Zillmann and G. P. Fettweis, On the capacity of multicarrier transmission over nonlinear channels, Veh. Technol. Conf., vol. 2, pp.1148-1152, June 2005.

[6] H. G. Ryu, C. X. Wang and S. B. Ryu, Nonlinear distortion reduction for the improvement of the BER performance in OFDM communication systems, Proc. International Symposium on Communications and Information Technologies, pp. 67-71 Oct. 2007.

[7] J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007.

[8] X. N. Fernando and A. B. Sesay, "Higher order adaptive filter based predistortion for nonlinear distortion compensation of radio over fiber links," Communications, 2000. ICC 2000. 2000 IEEE International Conference on, vol. 1, pp. 367-371 vol.1, 2000.

[9] H. Moon and R. Sedaghat, "FPGA-Based adaptive digital predistortion for radio-over-fiber links," Microprocessors and Microsystems, vol. 30, pp. 145-154, 5/5. 2006.

[10] C. Ciochina, F. Buda and H. Sari, "An Analysis of OFDM Peak Power Reduction Techniques for WiMAX Systems," Communications, 2006. ICC '06. IEEE International Conference on, vol. 10, pp. 4676-4681, 2006.

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 200 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 2.00333 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00167 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing false /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice