Introduction

Sound is a wave that arises from changes in atmospheric pressure caused by vibration [1]. Combined with artificial intelligence systems, it can have broad applications in various fields. In medicine, image recognition using deep learning is utilized. In [2], models are used to help specialists diagnose diseases more quickly. It is worth noting that sound contains a lot of information. Based on sound, certain abnormalities can be detected. In [3], the use of heart sounds for early disease detection is excellently demonstrated, allowing for earlier treatment. In article [12], there is another medical application, namely recognizing people with Parkinson's disease from recorded voice samples. The average accuracy of this method is around 90%.

Therefore sound should be converted into a spectrogram, and then a model for image recognition should be used. A spectrogram is a visual representation of the intensity of a signal over time, with respect to different frequencies present in a given waveform. The evaluation of spectrograms involves transforming the signal from the time domain to the frequency domain using the Fourier transform [4]. In [4], it is shown that sound can also be used in the food industry to identify various food products. In the following articles [8] [9][10] [11], various techniques utilizing sound recognition are described, such as Environmental Sound Recognition (ESR) and Automatic Sound Recognition (ASR), which can be used in a smart home. A smart home, along with artificial intelligence methods, can provide support for people, reduce exploration costs, and improve energy efficiency [13] [14]. Therefore, this field also utilizes sound recognition. This mechanism can be used as one of the biometric security measures for homes [15]. However, the sound processing scheme is the same.

All these articles demonstrate that data analysis is very important for the application of neural networks. In particular, sound must be properly processed. Sound, especially human speech or music, has certain features that can be used for its characterization, such as a unique human voice, communication method-specific noise, or the use of similar instruments in musical pieces of the same genre [1]. Therefore, to extract the most important features of a sound signal in the form of a coefficient matrix, the MFCC algorithm, which will be described in detail in this article. Later in this article, there will be a comparison of two classifiers: KNN and Naive Bayes.

Methodology

The MFCC Algorithm (Mel-frequency cepstral coefficients)

Before describing the MFCC algorithm itself, certain concepts need to be defined:

1. Mel scale -a scale of pitches that measures the perceived frequency of sound, in contrast to the objective frequency scale measured in hertz.

The function for converting a frequency in hertz to the Mel scale:

The inverse function:

𝑚(𝑓 ) = 1125 log (1 + 𝑚 𝑓 700 ) (1)𝑓 (𝑚) = 700(exp 1125 -1) (2)

2. Window function is a function that takes non-zero values only within a specified interval. That functions are used to filter signals. For the purposes of mathematical description, let's introduce the following notation: Description of the MFCC algorithm:

1. Let:

• 𝑛 be the number of samples in the input signal,

• 𝑋 = {𝑥 0 , 𝑥 1 , ..., 𝑥 𝑛−1 } * be the input signal,

• ℎ be the sampling frequency of the input signal in hertz,

• 𝑡 be the number of triangular filters,

• 𝑢 be the number of numbers transformed by the discrete Fourier transform,

• 𝑙 be the length of the window in samples,

• 𝑠 be the number of samples by which the window is shifted and 0 < 𝑠 ≤ 𝑙,

• 𝑐 be the number of cepstral coefficients.

2. Filters are applied to remove noise. This step is optional, but noise removal improves accuracy, so a high-pass filter is used in the form of:

𝑦 𝑘 = 𝑥 𝑘 − 0.97𝑥 𝑘+1 𝑘 ∈ {0, 1, 2, ..., 𝑛 − 2} *(3)

3. Triangular filters are created to extract desired features from the input signal while omitting unnecessary ones. These filters are distributed on a frequency scale between 0 and ℎ . Initially, the boundaries of this scale are converted from frequency in hertz to the Mel scale.

𝑎 = 𝑚(0) = 0 𝑏 = 𝑚( ℎ ) = 1125 log (1 + ℎ ) (4)

2 1400

An array of 𝑡 + 2 numbers is created, evenly distributed between 𝑎 and 𝑏.

𝑈 = {𝑎 + 𝑘Δ𝑥 : 𝑘 ∈ {0, 1, 2, ..., 𝑡 + 1} * } * 𝑏 − 𝑎 (5) Δ𝑥 = 𝑡 + 1

Another array is created containing the scaled elements of array 𝑈 , converted from the Mel scale to the frequency scale in hertz and rounded down.

𝐵 = { ⌊︂ 𝑢 + 1 𝑓 (𝑥) ⌋︂ : 𝑥 ∈ 𝑈 } * (6)

Array 𝐵 contains non-linearly distributed numbers from 0 to ⌊︂ 𝑢 . ⌋︂

2 ⎨ 𝑠 {︃ 𝑙 𝑠

Another array is created to contain the triangular filters mentioned at the beginning.

𝐹 = { { 𝑔( , 𝑘 𝑗) : 𝑗 ∈ {0, 1, 2, . . . , ⌊︂ 𝑢 ⌋︁ } * } * :(7)

where:

𝑘 ∈ {0, 1, 2, . . . , 𝑡 − 1} * } * 𝑔( , 𝑘 𝑗) = ⎪ ⎪ ⎧ ⎪ ⎪ ⎩ 𝑗−𝐵[𝑘] 𝐵[𝑘+1]−𝐵[𝑘] 𝐵[𝑘+2]−𝑗 𝐵[𝑘+2]−𝐵[𝑘+1] , ⌊𝐵[𝐵]⌋︂ ≤ 𝑗 ∧ 𝐵 < ⌊𝐵[𝑘 + 1]⌋︂ , ⌊︂ 𝐵[𝑘 + 1]⌋︁ ≤ 𝐵 ∧ 𝐵 < ⌊𝐵[𝑘 + 2]⌋︂ 0 , for other 𝐵 4.

The input signal is divided into windows, where a window is defined as:

𝑊 (𝑖) = {𝑝(𝐵) : 𝑠𝑖 ≤ 𝐵 < 𝑠𝑖 + 𝑙} * 𝑖 ∈ {0, 1, 2, ..., ⌊︂ 𝑛 + 𝑠 − 𝑛 mod 𝑠 ⌋︁ − 1} *(8)

where:

𝑖 is the window index.

𝑝(𝑘) = 𝑥 𝑘 , 0 ≤ 𝑘 < 𝑛 0 , 𝑛 ≤ 𝑘

The power spectrum is calculated, i.e., the discrete Fourier transform (DFT) of the first 𝑢 elements from the array 𝑊 (𝑖), then square each number in the resulting array and scale these elements by 1 .

𝑃 (𝑖) = 1 𝑆(DFT(𝑊 (𝑖), 𝑢))(9)

𝑙 where 𝑃 (𝑖) denotes the i-th power spectrum for the i-th window 𝑊 (𝑖) and

𝑆({𝑠 0 , 𝑠 1 , ..., 𝑠 𝑛 } * ) = {|𝑠 0 | 2 , |𝑠 1 | 2 , ..., |𝑠 𝑛 | 2 } *

Absolute values are required in function 𝑆 as 𝑠 𝑘 can be complex numbers.

The previously calculated filters are then utilized to filter the power spectrum via the matrix product of 𝑃 (𝑖) and the transpose of matrix 𝐹 .

𝐶(𝑖) = 𝑃 (𝑖)𝐹 F (10)

The final step is to compute the natural logarithm for each element of 𝐶(𝑖) and transform these logarithms using the discrete cosine transform (DCT) of type II.

𝑅(𝑖) = DCT(𝐿(𝐶(𝑖)))[: 𝑐](11)

where 𝐿({𝑥 0 , 𝑥 1 , ..., 𝑥 𝑛 } * ) = {ln 𝑥 0 , ln 𝑥 1 , ..., ln 𝑥 𝑛 } * .

The result of the algorithm is a matrix:

𝑅 = {𝑅(0), 𝑅(1), 𝑅(2), ... , 𝑅( ⌊︂ 𝑛 + 𝑠 − 𝑛 mod 𝑠 ⌋︁ − 1)} * (12) ∏︁ ∏︁

KNN (k-Nearest-Neighbours)

The K-Nearest Neighbors (KNN) algorithm is a classification and regression method that utilizes the similarity between data points. It operates by finding the nearest neighbors (data points) to a new point and uses their information to predict the class or value for that point [5]. Before describing the KNN algorithm itself, certain concepts need to be defined: Value of k -The number of neighbors to be considered during classification or regression. Mahalanobis distance -It considers the correlations between two vectors x and y with covariance matrix S and scales distances depending on the distribution of data. It is given by the formula:

𝐷 𝐵 ( , 𝑥 𝑦) = √︁ (𝑥 − 𝑦) ⊤ 𝑆 −1 (𝑥 − 𝑦)(13)

Naive Bayes classifier

The Naive Bayes classifier is a machine learning method used for classifying data into decision classes.

𝑛 𝑑 𝑁𝐵 = arg max 𝑃 (𝑑 𝑖 ) • 𝑃 (𝑥 𝑗𝑘 | 𝑑 𝑖 )(16)

𝑑𝑖∈𝑉𝐷

Experiments

𝑘=1

The example of MFCC algorithm will be conducted using the file classical.00000.wav. 2. After applying the high-pass filter, the samples look as follows on Figure The array 𝐵 looks as follows: As can be seen, the array 𝐵 contains numbers ranging from 0 to 𝑢 = 256.

3⇒ 𝐵 = {0,

As can be seen, these numbers are not evenly distributed, meaning the differences between consecutive numbers increase as the elements progress. This is because the boundary frequencies

⇒ 𝑔(0, 𝑗 ) = , 1 ≤ 𝑗 < 3 ⎪ ⎩ ⎪ ⎩

𝑎 and 𝑏 were converted from the frequency scale in Hertz to the Mel scale, which is nonlinear. The reason why a change to a nonlinear scale was required will be explained later in the example.

Triangular filters are created, again, according to the formula from the description:

𝐹 = {{𝑔( , 𝑘 𝑗) : 𝑗 ∈ {0, 1, 2, ..., 256} * } * : 𝑘 ∈ {0, 1, 2, ..., 25} * } *

For 𝑘 = 0 the function 𝑔(0, 𝑗) will look like: After calculating all the values 𝑔(𝐵, 𝑗), all 𝑡 = 26 filters can be represented on a graph on figure 5 Due to the application of the Mel scale to distribute these filters, the highest density is at the beginning, and the lowest at the end. The reason for needing the Mel scale is that it accurately represents how humans perceive sound. It turns out that most useful information is in the lower frequencies, not the higher ones. Therefore, it makes sense to place more filters at the beginning, which was achieved by converting the frequency scale in Hertz to the Mel scale. Without this, all filters would be evenly distributed across the entire scale.

𝑔(0, 𝑗) = ⎧ ⎪ ⎨ ⎪ 𝑗−𝐵[0] 𝐵[1]−𝐵[0] 𝐵 [2] − 𝑗 𝐵[2]−𝐵[1] , ⌊︂ 𝐵[0]⌋︂ ≤ 𝐵 < ⌊︂ 𝐵[1]⌋︂ , ⌊︂ 𝐵[1]⌋︂ ≤ 𝐵 < ⌊︂ 𝐵[2]⌋︂ 0 , for other 𝐵 ⎧ ⎪ ⎨ 𝑗 , 0 ≤ 𝐵 < 1 0 ,

{︃

It was what would happen if evenly distributed filters were used, and it degraded the accuracy by about 5%. The input signal is divided into windows.

𝑊 (𝑖) = {𝑝(𝑘) : 220𝑖 ≤ 𝑘 < 220𝑖 + 551} * , 𝑖 ∈ {0, 1, ..., 3008} * 𝑝(𝑘) = 𝑥 𝑘 , 0 ≤ 𝑘 < 661794 0 , 661794 ≤ 𝑘

For the window 𝑖 = 0: Having the power spectral density, the matrix product 𝑃 (𝑖) and 𝐹 F is calculated, which will be the operation of filtering frequencies according to the triangular filters previously established.

𝑊 (0) = {𝑝(𝑘) : 0 ≤ 𝑘 < 551} * ⇒ 𝑊 (0) = {𝑥 0 , 𝑥 1 , 𝑥 2 , ..., 𝑥 550 } * ⇒ 𝑊 (0) = {−𝐶(𝑖) = 𝑃 (𝑖)𝐹 F (17)

For 𝑖 = 0 we get: Analysis of the results is conducted for 6 classes of abstraction, with the following music genres:

𝐶(0) = 𝑃 (0)𝐹 F ⇒ 𝐶(0) = {10695.• Classical music, • Disco, • Hip-hop, • Metal, • Blues, • Country.

For each genre, there are 100 assigned tracks, each lasting 30 seconds. The split between training and test data is 70:30.

Before conducting a detailed analysis, it is important to determine the most effective value of k for the KNN classifier. According to the Table 2, it can be seen that the most effective value is k = 5, therefore this value should be adopted for the analysis. The next step is to evaluate the obtained matrices with the KNN classifier and Naive Bayes. Performance evaluation metrics such as accuracy, loss, precision, recall, and F1 score will be used to assess the effectiveness of these methods [4]. These metrics are essential for evaluating the performance of machine learning models and are described by the following equations:

• TP (True Positive), which is the number of cases where the model correctly classified positive instances. As we can see from the Table 2, 3, the metric values are very good for KNN with 6 classes of abstraction, whereas Naive Bayes performs significantly worse. In terms of accuracy for the entire test set, KNN achieved 80.56%, while the naive Bayes classifier achieved 36.67%. On the Figure 6, 7 we observe the confusion matrix. An ideal confusion matrix has 100% on the diagonal, and the rest should be 0%. For KNN, the matrix is nearly ideal. The classifier performed worst for the disco music genre. However, for Naive Bayes, the confusion matrix does not resemble the ideal one. Nevertheless, the algorithm performed best for the metal genre.

Conclusion

This MFCC algorithm allows for highly efficient classification of music genres with the KNN classifier. The covariance matrix effectively extracted features from the audio signal and could be used for commercial purposes such as in medicine. In the case of classifiers, KNN uses distance metrics that can be very effective in measuring similarities between musical pieces. Additionally, it does not assume any specific form of the classification function, relying instead on local similarities, which is why it worked perfectly here. On the other hand, the advantages of using the naive Bayes classifier are its simplicity and speed compared to KNN. Naive Bayes assumes that the features are independent, which is rarely true for audio data where different features can be strongly correlated. In this project, only one feature was used, namely the mean of the sum of all elements of the matrix, which may have influenced the low accuracy compared to KNN. To achieve high accuracy, more advanced methods such as neural networks like CNN and RNN should be used. In the future, based on the matrix obtained from the MFCC algorithm, a spectrogram can be created and the given algorithm can be tested on more complex models to achieve better results.