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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3354–3362
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Wideband dynamic behavioral modeling of reflective semiconductor optical amplifiers using a tapped-delay multilayer perceptron

Zhansheng Liu, Manuel Alberto Violas, and Nuno Borges Carvalho  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3354-3362 (2013)
http://dx.doi.org/10.1364/OE.21.003354


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Abstract

In this paper, we propose a wideband dynamic behavioral model for a bulk reflective semiconductor optical amplifier (RSOA) used as a modulator in colorless radio over fiber (RoF) systems using a tapped-delay multilayer perceptron (TDMLP). 64 quadrature amplitude modulation (QAM) signals with 20 Msymbol/s were used to train, validate and test the model. Nonlinear distortion and dynamic effects induced by the RSOA modulator are demonstrated. The parameters of the model such as the number of nodes in the hidden layer and memory depth were optimized to ensure the generality and accuracy. The normalized mean square error (NMSE) is used as a figure of merit. The NMSE was up to −44.33 dB when the number of nodes in the hidden layer and memory depth were set to 20 and 3, respectively. The TDMLP model can accurately approximate to the dynamic characteristics of the RSOA modulator. The dynamic AM-AM and dynamic AM-PM distortions of the RSOA modulator are drawn. The results show that the single hidden layer TDMLP can provide accurate approximation for behaviors of the RSOA modulator.

© 2013 OSA

1. Introduction

The accurate mathematical model will contribute to optimally designing of devices and predicting the characteristics of devices. In [6

6. M. J. Connelly, “Wide-band steady-state numerical model and parameter extraction of a tensile-strained bulk semiconductor optical amplifier,” IEEE J. Quantum Electron. 43(1), 47–56 (2007). [CrossRef]

], a steady-state physical model of SOA was developed. The Levenberg-Marquardt algorithm was used to extract the parameters of the physical model. In [7

7. M. J. Connelly, “Reflective semiconductor optical amplifier pulse propagation model,” IEEE Photon. Technol. Lett. 24(2), 95–97 (2012). [CrossRef]

], a simple physical model for pulse propagation in RSOA was proposed to predict its gain dynamics, spatial dependence of the pulse shape, and dynamic chirp. Previously we have also developed a multi-section physical model of the bulk RSOA used as external modulator in radio over fiber systems [9

9. Z. Liu, M. Sadeghi, G. de Valicourt, R. Brenot, and M. Violas, “Experimental validation of a reflective semiconductor optical amplifier model used as a modulator in radio over fiber systems,” IEEE Photon. Technol. Lett. 23(9), 576–578 (2011). [CrossRef]

]. The model was implemented by using symbolically defined devices (SDD) component in advanced design systems (ADS). The model parameters were extracted by fitting the measured data based on the ADS optimization tools. All of these models require the knowledge of physical parameters, which are difficult to be measured.

Unlike physical models, behavioral models, also called the black box, focus on the relation between the input and output signals of devices or systems. Artificial neural network (ANN) based behavioral models have been widely used to model the radio frequency (RF) power amplifiers [10

10. Y. Cao and Q. J. Zhang, “A new training approach for robust recurrent neural-network modeling of nonlinear circuits,” IEEE Trans. Microw. Theory Tech. 57(6), 1539–1553 (2009). [CrossRef]

], [11

11. F. Mkadem and S. Boumaiza, “Physically inspired neural network model for RF power amplifier behavioral modeling and digital predistortion,” IEEE Trans. Microw. Theory Tech. 59(4), 913–923 (2011). [CrossRef]

], erbium-doped fiber amplifier (EDFA) [12

12. N. Vijayakumar and S. N. George, “Design optimization of erbium-doped fiber amplifiers using artificial neural networks,” Opt. Eng. 47(8), 085008 (2008). [CrossRef]

] and quantum-dot SOA (QD-SOA) [13

13. J. I. Ababneh and O. Qasaimeh, “Simple model for quantum-dot semiconductor optical amplifiers using artificial neural networks,” IEEE Trans. Electron. Dev. 53(7), 1543–1550 (2006). [CrossRef]

] because of its generality and accuracy. In [13

13. J. I. Ababneh and O. Qasaimeh, “Simple model for quantum-dot semiconductor optical amplifiers using artificial neural networks,” IEEE Trans. Electron. Dev. 53(7), 1543–1550 (2006). [CrossRef]

], the pulse amplification and four-wave mixing (FWM) characteristics of the QD-SOA has been investigated by using a multilayer perceptron (MLP) model.

In this paper we attempt to use a simple tapped-delay MLP (TDMLP) to model the RSOA modulator which can be used in the RoF uplink. The model has been trained and validated by the measured sampled input and output data of wideband modulated signals. The dynamic nonlinear distortion and memory effects of the RSOA modulator have been investigated based on the TDMLP model.

2. TDMLP based behavioral model for RSOA

2.1 TDMLP model

In order to investigate the memory effect of the RSOA modulator, a single hidden layer TDMLP based model is proposed as shown in Fig. 1
Fig. 1 TDMLP based model with 2 × (m + 1) input nodes, m is the number of memory depth, z−1 is for unit delay operation, M hidden nodes and two output nodes.
. The transformation from the input layer to the hidden layer is nonlinear, while the transformation from the hidden layer to the output layer is pure linear. bh,j (j=1,2,...,M) are the biases in the hidden layer. f()is the activation function in the hidden layer, which will be introduced (in Subsection 2.3). bo,k (k=1,2 in this case) are the biases in the output layer.

The output uj(n) of the jth node in the hidden layer is depicted by
uj(n)=f(i=0mωi,jx(ni)+bh,j)
(1)
where the ωi,j is the weighting coefficient from the ith node in the input layer to the jth node in the hidden layer. The x can be xI or xQ.

The output of the kth node in the output layer is given by
yk(n)=j=1Mωj,kuj(n)+bo,k
(2)
where the ωj,k is the weighting coefficient from the jth node in the hidden layer to the kth node in the output layer. ykis yI or yQ in our case.

All of the weighting coefficients ωi,j, ωj,kand bias parameters bh,j, bo,k need to be determined during the training phase. The number of the hidden layer can be increased if needed. The mathematical results, however, have been shown that the single hidden layer MLP is capable of approximating uniformly any continuous multivariate function to any desired degree of accuracy [14

14. K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989). [CrossRef]

]. This implies that any failure of a function mapping must arise from inadequate choice of parameters or an insufficient number of hidden nodes. More than one hidden layer is often used to reduce the number of total nodes in hidden layers.

2.2 Overfitting

2.3 Activation function

The effects toward obtaining the best performance of the ANN based model also need to focus on the neuronal activation scheme. Different choices of the activation functions result in different network models. In general, the activation function in the hidden layer is a nonlinear function. The anti-symmetric functions often yield faster convergence. The most common choice of the activation function in the hidden layer is the anti-symmetric hyperbolic tangent, which is mathematically described as:

f(x)=tanh(x)=exexex+ex
(3)

Among the reasons for this popularity are still its boundedness in the unit interval, the function’s and its derivative’s fast computability. It is differentiable everywhere.

2.4 Back propagation learning algorithm (BPLA)

In order to obtain the optimal weight and bias parameters, one will minimize the average error function, which is represented as [16

16. S. Haykin, Neural Networks: A Comprehensive Foundation (Prentice-Hall, 1999).

], [17

17. M. T. Hagan and M. B. Menhaj, “Training feedforward networks with the Marquardt Algorithm,” IEEE Trans. Neural Netw. 5(6), 989–993 (1994). [CrossRef] [PubMed]

]
V(ω)=12Pp=1Pi=1N[tip(ω)yip(ω)]2=12Pp=1Pi=1N[eip(ω)]2
(4)
where P is the number of training patterns. N is the number of the output nodes (N is equal to 2 in our model). t is the target output and y is the model output. e is the network errors.ω is the parameters such as weights and biases to be extracted. In our case, t and y choose the corresponding in-phase parts and quadrature parts when i is equal to 1 and 2, respectively.

For the LM algorithm, the weight and bias parameters are updated by [17

17. M. T. Hagan and M. B. Menhaj, “Training feedforward networks with the Marquardt Algorithm,” IEEE Trans. Neural Netw. 5(6), 989–993 (1994). [CrossRef] [PubMed]

]
ωn+1=ωn[JTJ+μI]1JTe
(5)
where ()Tdenotes the transpose operation. n is the number of iteration. ωdenotes the vector of all training parameters (weights and biases).Jis the Jacobian matrix that contains first derivatives of the net errors with respect to the weights and biases. eis the vector of the network errors. Iis the identity matrix and μis the learning parameter.

3. Experimental and training

3.1 Experimental setup

The experimental test bench was set up as shown in Fig. 2
Fig. 2 The experimental test bench.
. The test baseband signals were designed in MATLAB and fed into a vector signal generator (VSG) to be up-converted to RF domain with a carrier frequency of 1 GHz. A commercial broadband amplifier (AMP), ZHL-42W, was used to drive the device under test (DUT). A vector signal analyzer (VSA) was used to capture the baseband input and output signals of the DUT. This was realized by a switch. For instance, the direct connection between the output of AMP and VSA enables us to obtain the measurements of the input signals for the DUT. The VSG and VSA were connected to a computer controller using general purpose interface bus (GPIB) cables. A 10 MHz reference and trigger signals between the VSG and VSA were used in order to synchronize and trigger the measurement events. Any error in the generation acquisition synchronization could lead to poor repeatability of measurements.

A commercial distributed feedback (DFB) laser, which was biased at 30 mA and had a wavelength of 1550 nm, was used as the seed light of the RSOA. By adjusting a tunable optical attenuator, the input optical power of the RSOA was set to −7 dBm, which made the RSOA working under saturation condition [9

9. Z. Liu, M. Sadeghi, G. de Valicourt, R. Brenot, and M. Violas, “Experimental validation of a reflective semiconductor optical amplifier model used as a modulator in radio over fiber systems,” IEEE Photon. Technol. Lett. 23(9), 576–578 (2011). [CrossRef]

]. The RSOA was biased at 90 mA. The forward and reverse signals of the RSOA were separated by an optical circulator. A photodiode (PD) with a responsivity of 0.8 A/W was used as a detector to convert optical signals to electrical ones.

3.2 TDMLP training

The training of the TDMLP involves choosing a set of weights and biases to optimize the accuracy of the mapping. The TDMLP based model can be easily trained by the BPLA in MATLAB. In order to train the TDMLP, a random 64 quadrature amplitude modulation (QAM) signal with 20 Msymbol/s, which were filtered with a square root raised cosine (RRC) filter with the roll-off factor of α=0.22, were created at baseband in MATLAB. The 64 QAM signal with a sampling rate at 80 MHz was fed into the VSG memory to be up-converted to the RF band at 1 GHz and finally to be passed through the DUT. In our case the average RF input power of the DUT was set up to 13 dBm, which is near the input 1 dB compression point of the RSOA. The complex baseband input and output signals of the DUT were captured from the VSA. After alignment and pre-processing, around 40 000 samples were captured to train, validate and test the TDMLP model.

Using the cross-validation technique, the training, validation and testing of the TDMLP were carried out using different segments of the measured input and output signals. The first 10 000 samples were used for training the nets. The second 20 000 samples, which were different with ones for training, were used to validate the model. The remaining 10 000 different samples were used for testing of the capacity of the model to predict the output of the RSOA modulator.

To evaluate the performance of the TDMLP based model in time domain, the normalized mean square error (NMSE) (in dB), which is the total power of the error vector between the measurement and model outputs, normalized to the measured signal power, is explicitly given as follows:
NMSE=10log10{n=1N([tI(n)yI(n)]2+[tQ(n)yQ(n)]2)n=1N([tI(n)]2+[tQ(n)]2)}
(6)
where the tI(n)and tQ(n)are the in-phase and quadrature parts of the real output of the RSOA, respectively. yI(n) and yQ(n)are the in-phase and quadrature parts of the output of the model, respectively.

4. Results and discussion

By varying the number of hidden nodes and the memory depth to determine the optimal values, several particular instances were trained and validated. The performance of the TDMLP model versus memory depth and the number of hidden nodes is shown in Table 1

Table 1. TDMLP model performance versus memory depth and the number of hidden nodes

table-icon
View This Table
.

The performance increases with memory depth increasing. It, however, degrades when increasing memory depth further. The performance of the TDMLP model also increases with the number of hidden nodes increasing. By further increasing the hidden nodes, however, the performance starts to degrade due to overfitting the training data set which reduces the generality of the TDMLP. This issue can be solved by increasing the training data set. In our work we chose the hidden nodes of 20 and memory depth of 3 for following analysis even though there are several slightly better points and the more weighting coefficients are required.

The training, validation and testing performance under the hidden nodes of 20 and memory depth of 3 is shown in Fig. 3
Fig. 3 The training performance of the TDMLP based model for the RSOA modulator with memory depth of 3 and hidden nodes of 20.
. It is shown that the performance of the TDMLP improves with increasing number of training epochs. The error on the validation and testing data sets starts off decreasing as the underfitting is reduced, but then it eventually begins to increase again as overfitting occurs. A solution to ensure the best generality of the ANN is to use the procedure of early stopping. One trains the network on the training data set until the performance on the validation data set starts to deteriorate and then stops the training phase (in our work we set maximum validation failures to 10).

The time domain waveforms of in-phase and quadrature parts of the RSOA outputs are shown in Fig. 4
Fig. 4 Measured and modeled outputs in-phase and quadrature parts in time domain.
, where only first 250 samples are shown for clarity. They indicate that the measured data points were fitted by the modeled ones well. This can also be proved in frequency domain. The measured and modeled power spectral densities are shown in Fig. 5
Fig. 5 Normalized power spectral density of measured and modeled outputs.
. After demodulation process, the constellation for symbols of the transmitted, measured output and TDMLP model output are depicted in Fig. 6
Fig. 6 Normalized constellation (red ‘ + ’ for the transmitted symbol, blue ‘·’ for the measured output, and green ‘x’ for the TDMLP model output).
. The error vector magnitudes (EVMs) for measured and modeled output symbols are 5.70% and 5.64%, respectively, calculating by the following formula.
EVM=n=1N|s(n)r(n)|2n=1N|r(n)|2
(7)
where r(n) is the reference transmitted symbol, s(n) is the measured or modeled output symbol, N is the number of symbols.

The nonlinear distortion and memory effect can be described by plotting the dynamic AM-AM and dynamic AM-PM characteristics as shown in Fig. 7
Fig. 7 Dynamic AM-AM and AM-PM characteristics of the RSOA modulator.
. We can clearly see that the distortion including static nonlinearities and memory effects was induced by the RSOA modulator. The results of the model match with the measurements quite well. The phase changes were up to 7°. The TDMLP based model with memory can accurately predict the dynamic behaviors of the RSOA modulator.

5. Conclusion

In this paper, a TDMLP based model has been proposed to predict accurately the dynamic nonlinear distortion and memory effects of the RSOA modulator in RoF links. The cross-validation early stopping guarantees the generalization of the TDMLP based model while avoiding overfitting during the training phase. The NMSE between measurement and simulation is up to −44.33 dB when the number of hidden nodes and memory depth were set to 20 and 3, respectively. The measurement and simulation results show that a single hidden layer in TDMLP is enough to model the RSOA modulator. The dynamic AM-AM and dynamic AM-PM conversion characteristics of the RSOA modulator have also been demonstrated. The phase shift at high input power was up to 7°.

Acknowledgments

References and links

1.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007). [CrossRef]

2.

A. J. Seeds, “Microwave Photonics,” IEEE Trans. Microw. Theory Tech. 50(3), 877–887 (2002). [CrossRef]

3.

D. Wake, A. Nkansah, and N. J. Gomes, “Radio over fiber link design for next generation wireless systems,” J. Lightwave Technol. 28(16), 2456–2464 (2010). [CrossRef]

4.

D. Wake, A. Nkansah, N. J. Gomes, G. de Valicourt, R. Brenot, M. Violas, Z. Liu, F. Ferreira, and S. Pato, “A comparison of radio over fiber link types for the support of wideband radio channels,” J. Lightwave Technol. 28(16), 2416–2422 (2010). [CrossRef]

5.

T. Durhuus, B. Mikkelsen, C. Joergensen, S. Lykke Danielsen, and K. E. Stubkjaer, “All-optical wavelength conversion by semiconductor optical amplifiers,” J. Lightwave Technol. 14(6), 942–954 (1996). [CrossRef]

6.

M. J. Connelly, “Wide-band steady-state numerical model and parameter extraction of a tensile-strained bulk semiconductor optical amplifier,” IEEE J. Quantum Electron. 43(1), 47–56 (2007). [CrossRef]

7.

M. J. Connelly, “Reflective semiconductor optical amplifier pulse propagation model,” IEEE Photon. Technol. Lett. 24(2), 95–97 (2012). [CrossRef]

8.

N. Nadarajah, K. L. Lee, and A. Nirmalathas, “Upstream access and local area networking in passive optical networks using self-seeded reflective semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 19(19), 1559–1561 (2007). [CrossRef]

9.

Z. Liu, M. Sadeghi, G. de Valicourt, R. Brenot, and M. Violas, “Experimental validation of a reflective semiconductor optical amplifier model used as a modulator in radio over fiber systems,” IEEE Photon. Technol. Lett. 23(9), 576–578 (2011). [CrossRef]

10.

Y. Cao and Q. J. Zhang, “A new training approach for robust recurrent neural-network modeling of nonlinear circuits,” IEEE Trans. Microw. Theory Tech. 57(6), 1539–1553 (2009). [CrossRef]

11.

F. Mkadem and S. Boumaiza, “Physically inspired neural network model for RF power amplifier behavioral modeling and digital predistortion,” IEEE Trans. Microw. Theory Tech. 59(4), 913–923 (2011). [CrossRef]

12.

N. Vijayakumar and S. N. George, “Design optimization of erbium-doped fiber amplifiers using artificial neural networks,” Opt. Eng. 47(8), 085008 (2008). [CrossRef]

13.

J. I. Ababneh and O. Qasaimeh, “Simple model for quantum-dot semiconductor optical amplifiers using artificial neural networks,” IEEE Trans. Electron. Dev. 53(7), 1543–1550 (2006). [CrossRef]

14.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989). [CrossRef]

15.

E. B. Baum and D. Haussler, “What size net gives valid generalization?” Neural Comput. 1(1), 151–160 (1989). [CrossRef]

16.

S. Haykin, Neural Networks: A Comprehensive Foundation (Prentice-Hall, 1999).

17.

M. T. Hagan and M. B. Menhaj, “Training feedforward networks with the Marquardt Algorithm,” IEEE Trans. Neural Netw. 5(6), 989–993 (1994). [CrossRef] [PubMed]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(230.4110) Optical devices : Modulators
(250.5980) Optoelectronics : Semiconductor optical amplifiers

ToC Category:
Optoelectronics

History
Original Manuscript: October 18, 2012
Revised Manuscript: December 18, 2012
Manuscript Accepted: January 2, 2013
Published: February 4, 2013

Citation
Zhansheng Liu, Manuel Alberto Violas, and Nuno Borges Carvalho, "Wideband dynamic behavioral modeling of reflective semiconductor optical amplifiers using a tapped-delay multilayer perceptron," Opt. Express 21, 3354-3362 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3354


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References

  1. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics1(6), 319–330 (2007). [CrossRef]
  2. A. J. Seeds, “Microwave Photonics,” IEEE Trans. Microw. Theory Tech.50(3), 877–887 (2002). [CrossRef]
  3. D. Wake, A. Nkansah, and N. J. Gomes, “Radio over fiber link design for next generation wireless systems,” J. Lightwave Technol.28(16), 2456–2464 (2010). [CrossRef]
  4. D. Wake, A. Nkansah, N. J. Gomes, G. de Valicourt, R. Brenot, M. Violas, Z. Liu, F. Ferreira, and S. Pato, “A comparison of radio over fiber link types for the support of wideband radio channels,” J. Lightwave Technol.28(16), 2416–2422 (2010). [CrossRef]
  5. T. Durhuus, B. Mikkelsen, C. Joergensen, S. Lykke Danielsen, and K. E. Stubkjaer, “All-optical wavelength conversion by semiconductor optical amplifiers,” J. Lightwave Technol.14(6), 942–954 (1996). [CrossRef]
  6. M. J. Connelly, “Wide-band steady-state numerical model and parameter extraction of a tensile-strained bulk semiconductor optical amplifier,” IEEE J. Quantum Electron.43(1), 47–56 (2007). [CrossRef]
  7. M. J. Connelly, “Reflective semiconductor optical amplifier pulse propagation model,” IEEE Photon. Technol. Lett.24(2), 95–97 (2012). [CrossRef]
  8. N. Nadarajah, K. L. Lee, and A. Nirmalathas, “Upstream access and local area networking in passive optical networks using self-seeded reflective semiconductor optical amplifier,” IEEE Photon. Technol. Lett.19(19), 1559–1561 (2007). [CrossRef]
  9. Z. Liu, M. Sadeghi, G. de Valicourt, R. Brenot, and M. Violas, “Experimental validation of a reflective semiconductor optical amplifier model used as a modulator in radio over fiber systems,” IEEE Photon. Technol. Lett.23(9), 576–578 (2011). [CrossRef]
  10. Y. Cao and Q. J. Zhang, “A new training approach for robust recurrent neural-network modeling of nonlinear circuits,” IEEE Trans. Microw. Theory Tech.57(6), 1539–1553 (2009). [CrossRef]
  11. F. Mkadem and S. Boumaiza, “Physically inspired neural network model for RF power amplifier behavioral modeling and digital predistortion,” IEEE Trans. Microw. Theory Tech.59(4), 913–923 (2011). [CrossRef]
  12. N. Vijayakumar and S. N. George, “Design optimization of erbium-doped fiber amplifiers using artificial neural networks,” Opt. Eng.47(8), 085008 (2008). [CrossRef]
  13. J. I. Ababneh and O. Qasaimeh, “Simple model for quantum-dot semiconductor optical amplifiers using artificial neural networks,” IEEE Trans. Electron. Dev.53(7), 1543–1550 (2006). [CrossRef]
  14. K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw.2(5), 359–366 (1989). [CrossRef]
  15. E. B. Baum and D. Haussler, “What size net gives valid generalization?” Neural Comput.1(1), 151–160 (1989). [CrossRef]
  16. S. Haykin, Neural Networks: A Comprehensive Foundation (Prentice-Hall, 1999).
  17. M. T. Hagan and M. B. Menhaj, “Training feedforward networks with the Marquardt Algorithm,” IEEE Trans. Neural Netw.5(6), 989–993 (1994). [CrossRef] [PubMed]

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