# Mathematical Background and Algorithms of a Collection of Android Apps for a Google Play Store Page
## 3 Problem Solving
### Vocal SMS Writing and Reading App for Visually Impaired People
When the message length \(L\) is more than 160 characters, it must be divided into segments. Each segment can contain a maximum of 153 characters because some characters are used for segmentation details. Equation (11) provides the value of the number of parts \(N_{p}\). Figure 1:6  
Error-correcting codes like Reed-Solomon codes are useful for maintaining message integrity. In a Reed-Solomon code \(RS(n,k)\), with \(n\) as the block length and \(k\) as the message length, the quantity of errors that can be corrected \(t\) is given by Equation (12). Figure 1:7  
Communicating with the SMSC (Short Message Service Center) requires the use of protocols such as SMPP (Short Message Peer-to-Peer) to transmit messages. The overall length of a message \(L_{msg}\), which encompasses both the header and the body, is represented in Equation (13). Figure 1:8  
where \(L_{header}\) represents the length of the SMPP header, and \(L_{body}\) indicates the length of the encoded message. Modulation and demodulation processes are describable utilizing signal-processing techniques. The modulated signal, denoted as \(s(t)\), is shown in Equation (14). Figure 1:9  
In which:  
* \(A\) represents the amplitude;
* \(m(t)\) denotes the message signal;
* \(f_{c}\) stands for the carrier frequency;
* \(\phi\) symbolizes the phase. The function represents the TTS system as follows: \[\text{SendTextSMS}(\text{recipient},\text{text})\]0 (15) Where:
* \(x\) serves as the input text. * \(F(x)\) is the feature extraction function responsible for transforming text into linguistic features. * \(S\) is the synthesis function responsible for creating the audio signal \(y(t)\) using the features. The synthesis function \(S\) is usually represented as \[\text{SendTextSMS}(\text{recipient},\text{text})\]1 (16) Where:
* \(a_{i}\) represents the amplitude for the \(i\)-th sine wave. * \(f_{i}\) denotes the frequency. * \(\phi_{i}\) denotes the phase offset. * \(N\) represents the count of sine waves used in the synthesis process. Figure 2 shows the sine wave components in TTS. Figure 3 shows the spectrogram of audio signal. The STT system can be characterized as follows:  
\[\text{SendTextSMS}(\text{recipient},\text{text})\]2  
Where:  
* \(y(t)\) represents the audio signal input. * \(G\) is the function responsible for translating the audio signal into text. The recognition function, denoted by \(G\), operates through multiple layers:  
\[\text{SendTextSMS}(\text{recipient},\text{text})\]3  
Where:  
\[\text{SendTextSMS}(\text{recipient},\text{text})\]4 Spectrogram of audio signal. \[\text{SendTextSMS}(\text{recipient},\text{text})\]5 Sine wave components in TTS. Enc is the function responsible for extracting features from the audio signal. * Dec is the function responsible for converting features into text. A vocal SMS application enables users to both send and receive text messages through voice commands. The functionality of this system depends on different mathematical models that address signal processing, energy usage, and data transfer. To represent these critical components of the app, we will employ techniques from differential equations, linear algebra, analytic geometry, and mathematical analysis. An auditory signal can be described as a time-dependent function \(x(t)\), with \(t\) representing time and \(x(t)\) indicating the signal's amplitude at that moment. This audio signal undergoes sampling at specific time points \(t_{n}=n\Delta t\), where \(\Delta t\) denotes the interval between samples. The sampled values of the signal are represented by \(x_{n}=x(t_{n})\). The Fourier transform is employed to change a signal from the time domain to the frequency domain, a process crucial for compression and transmission.