## 3 FMMs for Discrete Channel Modeling
### Estimation of FMM Parameters  
After the model and its parameters have been identified and empirical data have been gathered, the essential next step is to evaluate the models' fit to the experimental data. This evaluation will determine whether or not the model accurately represents the data. The model parameter value that best fits or reflects the experimental data can be used to assess fitness utilizing an approach known as parameter estimation [41]. The maximum likelihood estimate (MLE) and the least-square estimate (LSE) are frequently employed methods in determining the parameter value; this study chose the MLE framework for FMM parameter estimation. Parameter estimation using the MLE algorithm was chosen due to the algorithm's favorable properties [41] in these areas: the MLE estimator has sufficient and accurate information about the parameter of interest; the actual parameter value generated by the data can be consistently recovered asymptotically for data with sufficiently large samples; and estimates of the parameter can be obtained asymptotically. When determining model parameter values, the model employs a statistical technique known as the maximum likelihood estimate (MLE). This approach selects values to maximize the probability that the model-represented process produces the observed data. Advantages such as increased efficiency, improved numerical accuracy, consistency, and invariance of the parameter estimation process have been linked to the use of the MLE method to estimate FMM parameters [41]. Direct numerical maximization (DNM) using a Newton-type minimization algorithm and expectation maximization (EM) [42] using the popular iterative Baum-Welch algorithm (BWA) are two ways to achieve the MLE of FMM parameters. A reliable method for the assessment of the FMM parameters is the BWA [33; 43; 44]. One uses an iterative algorithm and dynamic programming techniques to estimate the complete parameter set \(\Gamma=(A,B,\pi)\) that represents the measured or simulated error sequence [33]. The BWA is an unsupervised machine learning method used to evaluate the FMM's parameters, corresponding to measured bit error sequences. The algorithm's training and input data are the experimentally measured error sequences and the initialized FMM parameters. The algorithm modifies the entire set of initialized FMM parameters \(\Gamma=(A,B,\pi)\) to obtain the most likely parameters representing the measured error sequences. To estimate the model parameters, it uses the maximum likelihood estimation method [42]. Developing the BWA requires the estimation of two different probabilities. The first is the forward-path probability, which is defined as the probability of generating a partial observation sequence in the forward path (starting from the beginning of the data) and arriving at a particular state at a particular frame; it is referred to as the probability of arriving at a particular state. On the other hand, the backward-path probability is the probability of producing a partial observation sequence in the opposite direction from the final data frame, given that the state sequence starts at a particular time. The BWA can be classified as a specific expectation-maximization (EM) algorithm instance. This algorithm's various phases include the initial phase, the forward phase, the backward phase, and the update phase. The E-step encompasses both the forward and backward phases of the EM algorithm, whereas the M-step corresponds to the update phase. The forward and backward formulas and computations are employed during the E-step to predict the anticipated hidden states. These predictions are based on the empirical data and the parameter matrices utilized before adjustments. Then, the M-step update formulas are used to improve the parameter matrices so that the observed data and the predicted latent states match up as well as possible. For a comprehensive and detailed mathematical exposition of the BWA and the modeling technique utilized in this research, please refer to [31; 33].