3. MFR based on ANN trained by wavelet transform

#### 3.1. Theoretical background

Different mathematical guidelines are used in the defined MFR block of this technique. Before moving to the ANN classifier, it allows the features extraction of all received signals. In our study, we chose the wavelet transform for the multiresolution analysis (MRA). In addition, to accomplish the classification mission, the singular value decomposition (SVD), as a factorization tool of matrix, is used. The different blocs of our MFR module are shown in Figure 8.

Figure 8. Different blocs of the optical MFR module.

3. MFR based on ANN trained by wavelet transform

Recognized modulation format

40 Gbps NRZ-DQPSK

10 Gbps NRZ-OOK 100% –– – – 40 Gbps NRZ-DQPSK – 100% – –– 100 Gbps NRZ-DP-QPSK – – 99.99% – 0.01% 160 Gbps DP-16QAM –– – 100% –

10 Gbps NRZ-OOK

18 Advanced Applications for Artificial Neural Networks

Different mathematical guidelines are used in the defined MFR block of this technique. Before moving to the ANN classifier, it allows the features extraction of all received signals. In our study, we chose the wavelet transform for the multiresolution analysis (MRA). In addition, to accomplish the classification mission, the singular value decomposition (SVD), as a factorization tool of matrix, is used. The different blocs of our MFR module are shown in Figure 8.

Table 1. Recognition accuracies of the five used modulation formats using the proposed automatic MFR technique.

Figure 7. The ANN outputs y<sup>i</sup> for 5 used modulation formats in response to the 6200 test cases representing input vectors xi.

100 Gbps NRZ-DP-QPSK

– – 0.02% – 99.98%

160 Gbps DP-16QAM 1 Tbps WDM-Nyquist NRZ-DP-QPSK

3.1. Theoretical background

Current modulation

1 Tbps WDM-Nyquist NRZ-DP-QPSK

format

#### 3.1.1. Wavelet transform and features extraction

Wavelet transform analysis is one of the most popular non-stationary signals processing tool. In this method, continuous wavelet transform (CWT) is used for its ability to construct in time and frequency domain a good representation of treated signals. It is also used to extract the necessary features of each received modulation format [8].

In the following integral, the CWT of the signal f(t) ∈Z is expressed at a scale a > 0 and translational value b∈R:

$$\text{CWT}\left(\mathbf{a}, \mathbf{b}\right) = \int\_{-\infty}^{+\infty} f(t) \Psi\_{\mathbf{a}, \mathbf{b}}^{\*}\left(\mathbf{t}\right) \text{dt} \tag{1}$$

where CWT(a, b) define the wavelet transform coefficients, \* denotes complex conjugate and Ѱa, b(t) is the baby wavelet comes from time-scaling and translation of mother wavelet Ѱ(t) as described in Eq. (3).

$$
\Psi\_{\mathbf{a},\mathbf{b}}(\mathbf{t}) = \frac{1}{\sqrt{\mathbf{a}}} \,\Psi\left(\frac{\mathbf{t} - \mathbf{b}}{\mathbf{a}}\right) \tag{2}
$$

Recently, the selection of the mother wavelet function as well as the decomposition level of signal is the most indispensable challenge in wavelet analysis. It includes Haar, Meyer, Morlet, Symlet, Daubechies and coiflet wavelets [8]. In our case, the Haar wavelet is chosen due to its simple form and also its computation is still easy. It is given in the following equation as a continuous function in both; the time and frequency domain:

$$\Psi(\mathbf{t}) = \begin{cases} 1 & \text{if } 0 \le t \le \frac{T}{2}, \\ -1 & \text{if } \frac{T}{2} \le t \le T \\\ 0 & \text{otherwise}. \end{cases} \tag{3}$$

Give s(t), with 0 < t < Ts, the received optical waveform in a complex form described as:

$$\mathbf{s(t) = f(t) + n(t) = \tilde{f}(t)e^{j\left\{2\pi f\_c t + \theta\_c\right\}} + n(t),\tag{4}$$

where Ts is the symbol duration, θ<sup>c</sup> is the carrier initial phase, fc is the carrier frequency, n(t) is the complex ASE noise and <sup>~</sup>f tð Þ is the complex envelope of the signal f(t) defined indifferently for each modulation format:

• For PSK signals:

$$\tilde{f}\_{\text{PSK}}(t) = \sqrt{S} \sum\_{i=1}^{N} e^{j\phi\_i} h\_{T\_s}(t - iT\_s),\tag{5}$$

with N the number of observed symbols, hTs (t) is the pulse shaping function of duration Ts, S is the average signal power and φi∈{(2π/N)(m � 1) , m =1,2, … , N},

• For QAM signals:

$$\tilde{f}\_{\text{QAM}}(t) = \sum\_{i=1}^{N} (A\_i + jB\_i) h r\_\*(t - iT\_s), \tag{6}$$

where (Ai, Bi) are the assigned QAM symbols.

• For NRZ-OOK signal:

$$\hat{f}\_{\text{NRZ-COK}}(t) = \sum\_{i=1}^{N} d\_i h r\_s(t - iT\_s),\tag{7}$$

with di = {0, 1} symbols.

An example of CWT for the four used modulation formats (40 Gbps NRZ-OOK, 160 Gbps OFDM DP- 16QAM, 400 Gbps DC PDM-QPSK and 1 Tbps WDM-Nyquist NRZ-DP-QPSK) is shown in Figure 9, where we choose a scale a = 100. From the figure, it is clear that each modulation format has its own features in terms of CWT amplitude and number of peaks.

#### 3.1.2. Singular value decomposition

SVD is the most important applicable matrix factorization used for signal processing and statistics. This tool is used to solve the least squares problems, and provides the best way to approximate a matrix with one of lower rank.

Given a real or complex matrix A having m rows and n columns, the matrix product UΣV<sup>∗</sup> is the singular value decomposition for the given matrix A if:


$$\mathbf{\bullet} \quad \mathbf{A} = \mathbf{U}\Sigma\mathbf{V}^{\mathbf{\bullet}}.$$

In addition, if σ is a non-negative scalar, and u and v are nonzero m- and n-vectors, respectively,

$$\mathbf{A}\boldsymbol{\sigma} = \boldsymbol{\sigma}\boldsymbol{\mu} \text{ and } \mathbf{A}\mathbf{u} = \boldsymbol{\sigma}\boldsymbol{\sigma} \tag{8}$$

where σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively.

Modulation Format Recognition Using Artificial Neural Networks for the Next Generation Optical Networks http://dx.doi.org/10.5772/intechopen.70954 21

Figure 9. The continuous wavelet transform of four used modulation formats for only 3000 input symbols at OSNR = 20 dB, residual dispersion = 170 ps/nm and DGD = 10 ps where the scale a = 100.

In fact, the diagonal elements {σi} of <sup>Σ</sup> are the singular values of <sup>A</sup>. The columns f g ui <sup>p</sup> <sup>i</sup>¼<sup>1</sup> of U and vf g<sup>i</sup> <sup>q</sup> <sup>i</sup>¼<sup>1</sup> of V are left and right singular vectors of <sup>A</sup>, respectively.

#### 3.1.3. ANN classifier

• For PSK signals:

• For QAM signals:

• For NRZ-OOK signal:

with di = {0, 1} symbols.

• A = UΣV\*.

respectively.

tively,

3.1.2. Singular value decomposition

approximate a matrix with one of lower rank.

the singular value decomposition for the given matrix A if:

• U and V, respectively, have orthonormal columns;

with N the number of observed symbols, hTs

20 Advanced Applications for Artificial Neural Networks

where (Ai, Bi) are the assigned QAM symbols.

<sup>~</sup><sup>f</sup> PSKðÞ¼ <sup>t</sup> ffiffiffi

the average signal power and φi∈{(2π/N)(m � 1) , m =1,2, … , N},

<sup>~</sup><sup>f</sup> QAMðÞ¼ <sup>t</sup> <sup>X</sup>

N

i¼1

<sup>~</sup><sup>f</sup> NRZ�OOKðÞ¼ <sup>t</sup> <sup>X</sup>

N

i¼1

An example of CWT for the four used modulation formats (40 Gbps NRZ-OOK, 160 Gbps OFDM DP- 16QAM, 400 Gbps DC PDM-QPSK and 1 Tbps WDM-Nyquist NRZ-DP-QPSK) is shown in Figure 9, where we choose a scale a = 100. From the figure, it is clear that each modulation format has its own features in terms of CWT amplitude and number of peaks.

SVD is the most important applicable matrix factorization used for signal processing and statistics. This tool is used to solve the least squares problems, and provides the best way to

Given a real or complex matrix A having m rows and n columns, the matrix product UΣV<sup>∗</sup> is

In addition, if σ is a non-negative scalar, and u and v are nonzero m- and n-vectors, respec-

where σ is a singular value of A and u and v are corresponding left and right singular vectors,

Av ¼ σu and Au ¼ σv (8)

• Σ has non-negative elements on its principal diagonal and zeros elsewhere and

S p X N

i¼1 e

<sup>j</sup><sup>φ</sup>ihTs ð Þ <sup>t</sup> � iTs , (5)

(t) is the pulse shaping function of duration Ts, S is

ð Þ Ai þ jBi hTs ð Þ t � iTs , (6)

dihTs ð Þ t � iTs , (7)

Features extraction of received signals is accomplished using CWT and SVD. In the last step, the pattern recognition method based on ANN is used for our statistical learning model. Its architecture is described in Figure 10. It is structured as an interconnected group of artificial neurons, which use a computational model or mathematical model for information processing. As given in the figure, the ANN is an adaptive system that changes its structure based on external or internal information that flows through the network. Precisely, the architecture of the ANN varies, but generally, it consists of several layers of neurons.

Figure 10. Three-layered feed-forward artificial neural network configuration.

In this technique, we also used the MLP3 for the modulation format identification by assigning output nodes to represent each format. Four output neurons in the output layer represent the 40 Gbps NRZ-OOK, 160 Gbps OFDM DP-16QAM, 400 Gbps DC PDM-QPSK and 1 Tbps WDM-Nyquist NRZ-DP-QPSK modulation formats. As designed previously, before training the network, the multilayer perceptron output can be considered as the highest probability, which represent one from the four modulation types. Eigenvalues after the SVD for each modulation format are represented at the input of the ANN. The multilayer perceptron used in this architecture requires four output neurons representing the number of format types.

#### 3.2. Design of the proposed method

The setup of the proposed MFR technique is shown in Figure 11.

The four used modulation formats are generated with carrier frequency equal to 193.1 THz. Using a variable optical attenuator (VOA), the injected power is tuned and passed through an optical amplifier (OA) that undergo the ASE noise effect at higher gain amplifier. As a result, the OSNR of the signals is adjusted in the range of 10–35 dB (steps of 5 dB). Then, using a CD/ PMD emulator, chromatic dispersion is varied from 85 to 510 ps/nm (steps of 85 ps/nm) to reach 30 km on SMF, and the DGD between 0 and 20 ps (steps of 5 ps). The appropriate carrier to be classified is selected by an OBPF. As described in Figure 8, in MFR block, using the CWT, each received signal is processed by scaling factor up to 128 without amplitude normalization. Moreover, with length equal to 3, the median filter was applied to extract the features set and remove the peaks. In addition, we make the SVD to the time-scale parameters of the wavelet coefficients matrix and obtain the eigenvectors as the final characteristic vectors for each received signal. Finally, we reach the ANN classifier as described in the previous section, where for each modulation format we generate 150 realizations with 65,536 symbols corresponding to different combinations of CD, DGD and OSNR.

Figure 11. Proposed modulation recognition system.

The number of neurons in hidden layer is optimized to be 25 neurons. In addition, 128 input neurons representing the eigenvalues after SVD for each input modulated signal are used, in addition to 4 output neurons to design the modulation formats to be recognized.

In the training process, the input vectors are randomly divided, with 65% used for training, 20% for validation and 15% for testing. On the other side, using LM training algorithm and reducing the MSE (3.71 <sup>10</sup><sup>5</sup> for 10 epochs) optimize the identification rates and minimize the computation time.
