## 3 FMMs for Discrete Channel Modeling  
For reliable data transmission over a channel, a typical digital communication system uses a source encoder, a channel encoder, a modulator, a demodulator (both of which are signal processing subsystems), a channel decoder, and a source decoder. In practice, achieving reliable transmission through the channel is invariably hampered by errors caused by destructive variables such as noise and channel distortion. These variables change the relationship between the input and output data sequences, making it more difficult to achieve reliable transmission. Consequently, mathematical channel models must be used to estimate the transmission reliability based on a set of probabilities that relate the output sequences of the channel to the inputs and accurately describe how channel errors affect transmission. This is done to choose a method that effectively protects digital signals from disruptive factors.  
The main components of a discrete communication system [31; 33] are shown in Figure 3. All system components located between \(P\) and \(Q\) in Figure 3 are referred to as \"discrete channels\". Point \(P\) denotes the transmitter side channel encoder's output, and point \(Q\) denotes the receiver side channel decoder's input. The waveform channel can be found between the two points \(P^{1}\) and \(Q^{1}\). Moreover, a symbolic sequence \(X\) defined as \(X=\{x_{1},x_{2},\ldots,x_{k}\}\) is input to point \(P\), and a symbolic sequence \(Y\) defined as \(Y=\{y_{1},y_{2},\ldots,y_{k}\}\) is output from point \(Q\), and the two sequences are related. The impairments caused by the physical PLC channel, non-ideal modulator and demodulator, and other devices in the PLC network can be seen as the insertion of an error sequence that exists between the input and output, which is defined as \(O=\{o_{1},\ldots,o_{T}\}\).  
Discrete channel modeling (DCM) is a well-established statistical method for the identification and simulation of communication channel impairments and memory activity in digital communication. Similarly, DCM facilitates the development of error correction and coding methods that work well with a real channel. DCM is perfect for modeling digital communication transmission error patterns because digital alphabets are discrete, and a set of probabilities can be used to link the output sequence of the channel to the input sequence to show how channel-induced errors [33; 34] affect the transmitted data. These two factors combine to make DCM an ideal candidate for this modeling task.  
Discrete communication channels often use a discrete-time Markov sequence to model the mechanisms by which errors are generated. In this model, the model's state defines the channel's different states, and a set of transition probabilities represents transitions between states [23; 35]. The model's parameters and structure can be derived from a measured or simulated sequence of errors [33], and multiple symbolic transitions exist between states. DCM first generates a random number before transmitting one symbol to determine the channel state; it then generates another set of numbers for the input-to-output transition.  
In the past, DCM approaches were used to analyze and model a wide range of communication channels, such as high-frequency (HF) radio channels [36], indoor PLC channels [23], wireless channels [37], underwater acoustic channels, and, more recently, hybrid PLC-VLC channels [38]. This research combines Markov model concepts with the long burst error characteristics of the transmission medium to gain a deeper understanding of the PLC channel and conduct a thorough analysis of it. In addition, data transfer over the power line channel is more negatively affected by impulse noise than data transmission over other communication channels; therefore, the DCM technique will be effective in describing and characterizing data transmission over power line channels.