# The Use of Correlation Features in the Problem of Speech Recognition
## 3 Materials and Methods
Analyzing the obtained curves, it can be noted that the averaging did not produce a complete smoothing of the spectra. However, almost all the resulting graphs are similar to each other, with the exception of some artifacts. The main features are located in these artifacts. Figure 4 shows examples for all phrases. Figure 3: Spectra of various messages after averaging. Figure 4 develops a search in the voltage spectrum; however, it is not much different from Figure 3. This is due to the fact that the amplitude was displayed on a logarithmic scale. Therefore, further search for preprocessing with a clear separation of classes is required. According to the curves in Figure 4, it is possible to conclude that the shape is absolutely identical with the spectra in relative voltage values. The results of spectrum calculations (2) can be used to estimate the energy spectrum of the signal based on the following expression:  
\[G(F)=|S(F)|_{2}, \tag{3}\]  
where \(G(F)\) is the energy spectrum of the signal, \(|.|\) is the operation of calculating the modulus, and \(S(F)\) is the spectrum of the signal (amplitude). Such a spectrum actually characterizes the energy or power of the signal, so we will use the unit W to evaluate it. In accordance with expression (3), it is easy to calculate the energy spectrum for all the studied speech messages. Obviously, squaring the value, although it forms a nonlinear transformation, taking into account the logarithmic representation, will not greatly affect the visual representation of the spectra. Figure 5 shows the results of such processing for each average phrase. It can be seen that the visualization of the reference energy spectra for each speech command in Figure 5 also does not lead to their clear separation for classification. This is due to the too large scales of the logarithmic scales. However, the energy spectrum is needed to calculate the covariance function. Finally, the search for an autoregressive mathematical model to describe the data can be reduced to the model parameters estimation based on the autocorrelation function The following assumptions can be made for modeling:  
Figure 4: An example of spectra of absolute voltage values after averaging. 1. Let the energy spectrum be symmetrical with respect to the Y-axis, i.e., \(G(-F)=G(F)\). Therefore, the correlation function will also be symmetric (stationary process property). 2. Let the signal that forms an arbitrary correlation function have such a temporal representation that obeys the homogeneity property. This means that the calculation of the covariance function is invariant to the location of the signal. This simplification can be circumvented by using a windowed representation of the signal. Let us write the homogeneity property in terms of the correlation function such as \(B(t_{2}-t_{1})=B(t_{4}-t_{3})=B(\Delta t_{12})\), if \(t_{2}-t_{1}=t_{4}-t_{3}\). 3. Let the normalization of the covariance function be possible after estimating the variance, i.e., construction of the correlation (normalized covariance) \(R(0)=B(0)/\sigma^{2}=1\), where \(\sigma^{2}\) is the variance of the studied speech message. Taking into account the introduced proposals for modeling, the estimate of the correlation function can easily be obtained from the inverse Fourier transform for the energy spectrum (3) and found by the formula:  
Figure 1:0  
where \(\sigma^{2}\) is the maximum transformation value (covariance of identical message points). Calculations of all correlation functions (4) were performed in the Matlab environment through the built-in function of the fast inverse Fourier transform. It is important to note that now the frequency step has been replaced by the time domain step. Figure 1:1 Examples of the radio exchange messages energy spectrum. Figure 6 shows all correlation functions that were calculated for 20 averaged studied voice messages. It should be noted that the \(X\)-axis represents a discrete time interval (\(k\)), and the \(Y\)-axis shows the values of the normalized auto-covariance of messages (\(R\)).