In [7], a one-tap zero forcing equalizer for the FBMC system is considered. In this work, it is assumed that self-interference is ignored. Thus, the output of the FBMC system can be formulated as:
(1.10)
where h is the vector obtain by vectorizing the channel matrix H. Thus, if ĥ represents the channel estimate, the output of the one-Tap equalizer will be obtained as:
(1.11)
Where hardlim represents the hard limiter.
1.4.2 MMSE Block Equalizer
The conventional way to design an MMSE equalizer is to use complex-valued symbols. Unfortunately, this does not work with FBMC-OQAM as this equalizer does not eliminate the imaginary interference. A modified MMSE equalization is formulated to solve this issue by putting real and imaginary parts together using the approach outlined in [11, 13]. This method is termed as full block MMSE equalization whose solution is given by
(1.12)
Where
1.5 Proposed Machine Learning-Based FBMC Equalizer
In our proposed equalization scheme, we employ support vector machine (SVM) [16] to learn the required estimate using available input and output data. The SVM is a kind of kernel machine learning technique that utilizes a nonlinear mapping of the original training data [16]. It is mainly designed for binary classification, which was later extended to the multi-class problem. In supervised learning, when labeled training data is inputted, SVM outputs an optimal hyperplane, which categorizes new examples [17, 18]. The variants of SVM used in this study are Linear SVM, Quadratic SVM, and Cubic SVM [19, 20].
The main idea of the proposed equalizer is to learn the weight matrix for the FBMC equalizer via SVM using available training data {xtrain, ytrain}. Once the equalizer weight matrix is learned, the estimate for the unknown transmitted data xunknown is obtained by processing the received signal ytest through the designed SVM.
1.6 Results and Discussion
This section provides the results of the proposed machine learning-based equalizers for the FBMC system using LSVM, QSVM, and CSVM.
Figure 1.3 BER performance comparison of different equalizers in FBMC system.
Table 1.1 Performance comparison of SVM based FBMC equalizers.
| Performance measure | LSVM | QSVM | CSVM |
| RMSE | 0.005951 | 0.005096 | 0.0048182 |
| Training Time (s) | 86.167 | 84.591 | 68.513 |
In this context, the FBMC system is simulated for 24 subcarriers and 30-time symbols. The prototype filter used is “Hermite” [13]. The results of BER are compared in Figure 1.3, which shows that the proposed SVM based equalizers have better performance than the full block MMSE equalizer of [13]. Moreover, it can be depicted from the results that the CSVM has the best performance among all the proposed methods.
In Table 1.1, the performances of the proposed SVM based equalizers are compared in terms of their testing RMSE and training time in seconds. Again, CSVM is found to be the best among other variants of SVM.
1.7 Summary
In this chapter, we study various methods of FBMC equalizers. We propose machine learning-based equalization techniques for the FBMC system. For this task, we develop the equalizer using three variants of SVM, namely Linear SVM, Quadratic SVM, and Cubic SVM. The BER performances of these methods are compared with that of the well-known One-Tap and Full Block MMSE equalizers. It was found from the simulation results that the proposed SVM based equalizer has better performance than that of the conventional methods. Moreover, the performances of the SVM based equalizers are compared among themselves in terms of RMSE and training time which shows that the CSVM outperformed the LSVM and QSVM in terms of both the RMSE and training time.
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