[1]This document is the results of the research project funded by AIMS
CAMEROON with the help of MASTERCARD Foundation.

[orcid=0000-0002-0118-3668] [1]

Conceptualization of this study, Methodology, Software, Data curation, Writing
- original draft

[style=chinese] Software, Result compilation, Writing -review

Software, Result compilation, Writing -review

Supervision, Validation, Writing - review & editing

Supervision, Software, Validation, Writing - review & editing

[cor1]Corresponding author

In this work, we show that Discrete Wavelet Transform is the best method for
feature extraction in vocal signals.

# Boosting the Predictive Accurary of Singer Identification Using Discrete
Wavelet Transform For Feature Extraction

Victoire Djimna Noyum<EMAIL_ADDRESS>School of Mathematical
Sciences, African Institute for Mathematical Sciences, Crystal Garden, Limbe
Younous Perieukeu Mofenjou<EMAIL_ADDRESS>Cyrille Feudjio
<EMAIL_ADDRESS>Alkan Göktug<EMAIL_ADDRESS>School
of Mathematical Sciences, ETH Zurich, Rämistrasse 101, 8092 Zurich,
Switzerland  Ernest Fokoué<EMAIL_ADDRESS>School of Mathematical Sciences,
Rochester Institute of Technology, Rochester, NY 14623

###### Abstract

Facing the diversity and growth of the musical field nowadays, the search for
precise songs becomes more and more complex. The identity of the singer
facilitates this search. In this project, we focus on the problem of
identifying the singer by using different methods for feature extraction.
Particularly, we introduce the Discrete Wavelet Transform (DWT) for this
purpose. To the best of our knowledge, DWT has never been used this way before
in the context of singer identification. This process consists of three
crucial parts. First, the vocal signal is separated from the background music
by using the Robust Principal Component Analysis (RPCA). Second, features from
the obtained vocal signal are extracted. Here, the goal is to study the
performance of the Discrete Wavelet Transform (DWT) in comparison to the Mel
Frequency Cepstral Coefficient (MFCC) which is the most used technique in
audio signals. Finally, we proceed with the identification of the singer where
two methods have experimented: the Support Vector Machine (SVM), and the
Gaussian Mixture Model (GMM). We conclude that, for a dataset of 4 singers and
200 songs, the best identification system consists of the DWT (db4) feature
extraction introduced in this work combined with a linear support vector
machine for identification resulting in a mean accuracy of 83.96%.

###### keywords:

DWT Singer Identification RPCA SVM GMM

##  1 Introduction

Music is a universal art form and cultural activity which can have several
effects on the listener depending on the intention of the artist as well on
the state of mind of the listener. Hence, with music, it is possible to
express critics against politics or society, mobilize people for a course or
to point out feelings arising from love, happiness, sadness, or loneliness.
With the increasing possibilities to access and to share art, the world of
music is becoming more and more vast and diverse. Query fast through this
world and collecting precise information is a challenge data scientists face
today. In this sense, by listening to a song, one could develop interest in
the biography of the artist and may want to access other songs from this
artist. This issue on which this project is based on is known as the
identification of the singer. The identification of the singer is done in
three phases: the separation of the singer’s voice from the background music,
the feature extraction and the identification process using the features
extracted from the vocal signal obtained from the separation procedure.

##  2 Background

A great deal of research has been done in the field of singer identification.
In 2002, (Liu and Huang, 2002) proposed a singer identification technique for
the classification of MP3 musical objects according to their content. They
used phoneme segmentation for signal separation. Unfortunately, the signal of
the singer’s voice at the output of this method still contains a lot of
background music (noise) which make the singer identification difficult.
In 2004, a spectrum-based method of identifying the singer, proposed by
(Bartsch and Wakefield, 2004), worked well only for ideal cases that contained
audio samples with the singer’s voice only. The test set accuracy was 70-80%.
In the ”Identification of the singer based on vocal and instrumental models”
proposed by (Maddage et al., 2004). in the same year, the singer was
identified using both low-level characteristics and knowledge of musical
structure. Using the dataset with 100 popular songs of solo singers, they
obtained an accuracy of over 87%. However, this method was not suitable for
music that was more instrumental than singing.
A systematic approach to identify and separate the unvoiced singing voice from
the musical accompaniment is proposed by (Hsu and Jang, 2009). For the
separation of the singer’s voice, they used the spectral subtraction method.
This method follows the framework of Computer Auditory Scene Analysis (CASA)
which includes the segmentation and clustering steps. This method considerably
improved the clarity of the singing voice signal but was not always sufficient
because, during clustering, a lot of information is lost.
To solve the problem of identifying the singer based on the acoustic variables
of the singer’s voice, (Yang, 2016) used the Gaussian Mixture Method (GMM) and
Support Vector Machine (SVM) in 2016. He obtained accuracies of 96.42% and
81.23% with a dataset of hundred (100) songs of ten (10) singers. For signal
separation, he used Robust Principal Component Analysis (RPCA) which is an
improved version of Principal Component Analysis (PCA) and gives a better
result than NMF. For feature extraction, he used Mel-Frequency Cepstral
Coefficient (MFCC).
In 2017, (Xing, 2017) proposed an effective system of singer identification
with human voice separated from original music. He used first, Robust
Principal Component Analysis (RPCA) to music separation with its high
performance. After the clear enough human voices are extracted, the Linear
Predictive Coding (LPC) method was chosen as the experimental method for
feature extraction. Finally, the singer would be identified by Gaussian
Mixture Model (GMM) with 63.6% of accuracy in a dataset of 100 singers.
In 2019, The work of (NAMEIRAKPAM et al., 2019) had implemented discret
wavelet transform (DWT) as a pre-processing step (denoising) prior to feature
extraction to investigate the performance of singer’s identification with and
without DWT. It is found that after applying wavelet transform the accuracy
result decreases. However, the decrease in percentage accuracy is minimal
(5.79%, 0.72% and 0.72% for 8, 16 and 32 Gaussians respectively). While the
computational time is drastically reduced.

##  3 Problem Statement and Contribution of this Study

The recent and improved research presented above shows that singer’s
identification using DWT for pre-processing and MFCC for feature extraction is
done in a much reduced time, but the accuracy decreases compared to the
results obtained without DWT. Unfortunately, MFCC uses the Fast Fourier
Transform (FFT) for the change from the time domain to the frequency domain.
FFT does not retain the time domain information and results in loss of data
during the change. In this study, we will use DWT for all the feature
extraction process to see if it improves feature extraction more than MFCC
because this method retains time-domain information by its ability to operate
in both in the time and frequency domain simultaneously. The Robust Principal
Component Analysis (RPCA) will be used to separate the singer’s voice from the
background music. To identify the singer, we will apply both the Support
Vector Machine (SVM) and the Gaussian Mixture Model (GMM).

##  4 Study Organization

The objective of this study is to build a model allowing the identification of
the speaker using DWT for feature extraction. To achieve this goal, we present
Robust Principal Component Analysis (RPCA) as the best technique for
separating the singer’s voice and its methodology, followed by the description
and process of feature extraction using Discrete Wavelet Transform (DWT).
Then, we explain the learning techniques such as the Support Vector Machine
(SVM) and the Gaussian Mixture Model (GMM) for singer identification. Finally,
we present the experiments and results. We conclude our research and propose
recommendations for future work.

##  5 Singing Voice Separation Technique: RPCA

Propsed by (Candès et al., 2011), Robust Principal Component Analysis (RPCA)
is a modification of the Principal Component Analysis (PCA) method. RPCA has
been proven to perform well for noise-corrupted data compared to PCA. The idea
of RPCA is to decompose a $V(V\in\mathbb{R}^{n\times p}$) data matrix into two
other matrices, $L$ and $S$, as follows:

$V=L+S$ (1)

In the case of sound, $L(L\in\mathbb{R}^{n\times p}$) is the low-rank matrix
corresponding to the background music and $S(S\in\mathbb{R}^{n\times p}$) is
the sparse matrix characterizing the singing voice. Indeed, in music, the part
of noise incorporated (background music) often varies more slowly with time
compared to the singer’s voice. In other words, the singer’s own voice is
likely to be more non-stationary than the noise. This phenomenon can be easily
observed by analyzing the spectrographs in Figure 1. The spectral structure of
pure noise is usually fixed or slowly varying, while the vocal part changes
rapidly over time. This hypothesis implies that the noise part appears to be
of low-rank, while the pure voice part is sparse (Hung et al., 2018).
Therefore, extracting the sparse component from the music signal matrix tends
to separate the background music from the voice of the speaker. This
separation is made by convex optimization which can be written as (Candès et
al., 2011):

$\displaystyle\mbox{minimize}\ \lVert L\rVert_{*}+\lambda\lVert S\rVert_{1}$
(2) $\displaystyle\mbox{subject to}\ L+S=V,$

where $\lambda>0$ is a trade-off parameter between the rank of $L$ and the
sparsity of $S$, $\lVert.\rVert_{*}$ is the nuclear norm representing the sum
of singular values of matrix entries and $\lVert.\rVert_{1}$ is the
$L_{1}$-norm representing the sum of absolute values of matrix entries (Candès
et al., 2011).

To solve the problem given to the equation 2, we use the Augmented Lagrange
Multiplier (ALM) method. The corresponding formula is given by (Candès et al.,
2011):

$L_{a}(L,S,Y,\mu)=\lVert L\rVert_{*}+\lambda\lVert S\rVert_{1}+\langle
Y,V-L-S\rangle+\frac{\mu}{2}\lVert V-L-S\rVert_{F}^{2}$ (3)

In equation 3, $\mu$ is a penalty parameter (always positive), $Y$ is slack
variable matrix, $\lVert.\rVert_{F}$ is the Frobenius norm. $\langle
Y,V-L-S\rangle$ implies the standard trace inproduct. At the end, we obtain
two matrices: the low-rank matrix $L$ and the sparse matrix $S$ respectively
(Candès et al., 2011).

(a) Original matrix V

(b) Sparse matrix S

(c) Low-rank L

Figure 1: Example RPCA results for Garou2.mp3: (a) the original matrix, (b)
the low-rank matrix, and (c) the sparse matrix.

To obtain the signals of the background music and the singing voice
represented in Figure 2, the Inverse Short-Time Fourier Transform (ISTFT) is
performed to return to the temporal domain.

(a) Original matrix V

(b) Sparse matrix S

(c) Low-rank L

Figure 2: Example of signal after ISTFT for Garou2.mp3: (a) the original
matrix, (b) the low-rank matrix, and (c) the sparse matrix.

The signal separation process is summarized in Figure 3.

Figure 3: Signal separation process (Li and Akagi, 2019)

##  6 Discret Wavelet Transform (DWT) as a feature extraction technique

Extraction of acoustic characteristics plays an essential role in the
construction of a singer identification system. The objective is to select
variables that have a high inter-label range and low discrimination power
within the label. The discriminating power of characteristics or sets of
characteristics indicates the extent to which they can discriminate between
labels. The selection of characteristics is usually done by examining the
discriminative power of the variables. The performance of a set of features
depends on demand. Thus, designing them for a specific application is the main
challenge in building singer identification systems. In this section, we will
present the theoretical background of the Discrete Wavelet Transform (DWT)
method. DWT is based on dividing the signal into several sub-bands before
performing feature extraction.

###  6.1 Introduction of Wavelet Transform (WT)

####  6.1.1 Definition

Wavelet Transform (WT) is a very powerful tool for the analysis and
classification of time series signals. It is unfortunately not known or
popular in the field of data science. This is partly because you need to have
some prior knowledge about signal processing, Fourier Transform and a solid
mathematics background before you can understand the theory underlying the
wavelet transform. However, we believe that it is also due to the fact that
most books, articles and papers are far too theoretical and do not provide
enough practical information illustrating how they could be used.

WT has many applications in the analysis of stationary and non-stationary
signals. These applications include removing noise from signals, detecting
abrupt discontinuities, and compressing large amounts of data (Wang et al.,
2013).

####  6.1.2 Principle of WT

WT decomposes a signal into a group of constituent signals, called wavelets,
each having a well-defined dominant frequency, similar to the Fourier
Transform (FT) in which the representation of a signal is made by sine and
cosine functions of unlimited duration. In WT, wavelets are transient
functions of short duration, i.e. of limited duration centered around a
specific time. The drawback of FT is that, as the time domain transitions to
the frequency domain, information about what is happening in the time domain
is lost. From the observation of the frequency spectrum obtained using FT, it
is easy to distinguish the frequency content of the analyzed signal, but it is
not possible to deduce in what time the signal components of the frequency
spectrum will appear or disappear. Unlike FT, WT allows both time-domain and
frequency-domain analysis, providing information on the evolution of the
frequency content of a signal over time (Montejo and Suárez, 2007). There are
many families of WT but the two principal are:

  * •

Continuous Wavelet Transform (CWT): The values of the scaling and translation
factors are continuous, which means that there can be an infinite amount of
wavelets. It performs a multi-resolution analysis by contraction and
dilatation of the wavelet functions (Aggarwal et al., 2011). Its different
sub-families are:

    * –

Mexican hat wavelet

    * –

Morlet wavelet

    * –

Complex Gaussian wavelets

    * –

Gaussian

  * •

Discrete Wavelet Transformations (DWT): It uses filter banks for the
construction of the multi-resolution time-frequency plane and special wavelet
filters for the analysis and reconstruction of signals (Merry and Steinbuch,
2005). Its different sub-families are:

    * –

Daubechies

    * –

Symlets

    * –

Coiflets

    * –

Biorthogonal

###  6.2 Discrete Wavelet Transformations (DWT)

DWT is defined by the following equation 4:

$\displaystyle W(j,k)=\sum_{n}x(n)2^{-\frac{j}{2}}\psi(2^{-j}n-k),$ (4)

where $\psi(t)$ is a time function with finite energy and fast decay called
the mother wavelet. $W(j,k)$ represents the wavelet coefficients, where $k$
denotes location, and $j$ denotes level.

DWT has four families: (1) Daubechies; (2) Symlets; (3) Coiflets; (4)
Biorthogonal. Each type has a different shape, smoothness, and compactness and
is useful for a different application. Since a wavelet has to satisfy only two
mathematical conditions which are the so-called normalization and
orthogonalization constraints, it is easy to generate a new type of wavelet.

DWT contains three major steps:

  1. 1.

Wavelet Threshold De-Noising

In general, after separation of the voice signal, this signal still contains
some small noises. The elimination of this noise is very important for the
accuracy of the characteristics that will be extracted from the signal.
Indeed, a singer has many different sounds and therefore, if the vocal signal
extracted from these sounds contains noise, the identification will not be
optimal. Donoho has introduced the use of wavelets to denoise the signals. He
developed linear denoising for noises composed of high-frequency components
and non-linear denoising (wavelet shrinkage) for noises also existing in the
low frequencies (Donoho, 1995). Schremmer et al. have developed software for
real-time wavelet noise canceling of audio signals. Noise suppression is
achieved by using soft or hard thresholding of the DWT of the coefficients
(Schremmer et al., 2001). The success criterion for noise suppression is the
difference between the original signal and the denoised signal. A new speech
enhancement system based on a wavelet denoising framework has been introduced
by Fu Qiang and Wan Eric. In this system, noisy speech is first pre-processed
using a generalized spectral subtraction method to initially reduce the noise
level with negligible speech distortion. Then, the decomposition of the
resulting speech signal into critical bands is done using the perceptual
wavelet (Fu and Wan, 2003). Denoising using DWT is developed in (Saric et al.,
2005) where the threshold is given by the equation:

$\displaystyle\lambda=\sigma_{n}\sqrt{(2\log N)},$ (5)

where $\lambda$ is the wavelet threshold, $\sigma_{n}$ is the standard
deviation of the noise, and $N$ is the length of the sample signal.

  2. 2.

Wavelet Decomposition

As illustrated in Figure 4, DWT breaks down a signal into several scales
representing different frequency bands. Short-duration wavelets are used to
extract information from the high-frequency components. Long-duration wavelets
can be used to extract information from low frequencies (Chang et al., 2000).
The process goes on under multiple levels as a subsequent coefficient from the
first level within the approximation. At each process, the frequency
resolution is doubled using the filters while decomposing and reducing the
time complexity to half. In the end, we consider all the high-frequency bands
($H_{1},H_{2},H_{3},H_{4}$) and the last low-frequency band ($L_{4}$).

Figure 4: DWT process level four (dwt, (accessed April 22, 2020))

  3. 3.

Feature Extraction

Multi-resolution analysis (MRA) is used to extract feature vectors from the
signal data. Very common in vocal signal, time-frequency domain DWT based
statistical features for classification include mean average value, standard
deviation, and spectral entropy.

     * •

Mean average value: it defines the mean of each vector of the sub-bands
obtained in the previous step. It is given as,

$\displaystyle\mu=\frac{\sum_{i=1}^{N}x_{i}}{N}$ (6)

     * •

Standard deviation: it defines the variance of the signal. It is given by

$\displaystyle\sigma=\dfrac{\sum_{i=1}^{N}(x_{i}-\mu)^{2}}{N}$ (7)

     * •

Power spectral density: it is calculated in two steps: first, by finding the
Fast Fourier Transform (FFT) $F(\xi_{i})$ of the time series and then, taking
the squared modulus of the FFT coefficients.

$\displaystyle P(\xi_{i})=\frac{|F(\xi_{i})|^{2}}{N}$ (8)

     * •

Spectral entropy: it is the measure of randomness and the information content
of a signal. To calculate the entropy of a given vocal signal, we use the
Shannon entropy formula

$\displaystyle E=-\sum_{i=1}^{N}x_{i}^{2}\log(x_{i}^{2})$ (9)

##  7 Learning Techniques: Classification Models

In the world of machine learning, two main areas can be distinguished:
supervised learning and unsupervised learning. The main difference between the
two lies in the nature of the data and the approaches used to process them. In
this section, we present two learning techniques that are widely used for
audio: Support Vector Machine (SVM) and Gaussian Mixture Model (GMM).

###  7.1 Support Vector Machine (SVM)

Support Vector Machine (SVM) was developed by Cortes and Vapnik in 1995 and
improved by Boser, Guyon, and Vapnik in 1998 (Boser et al., 1992); (Vapnik,
1998) which is useful for solving problems of monitoring classification in
high dimensions. The SVM approach searches directly for a plane or surface of
separation by an optimization procedure that finds the points that form the
boundaries of the classes. These points are called support vectors. Besides,
the SVM approach uses the kernel method to map the data with a nonlinear
transformation to a high-dimensional space and tries to find a separation
surface between the two classes in this new space. When we have two labels
(classes), we use the binary SVM and in cases with more than two labels, we
apply the multi-SVM.

####  7.1.1 Binary SVM

Binary-SVM is used when the data has exactly two classes. For classification,
SVM finds the best hyperplane that separates all data points of one class from
those of the other class as illustrated in Figure 5 by the red line. The best
hyperplane is the one with the largest margin between the classes.

Figure 5: SVM graph (svm, (accessed April 19, 2020)

)

The hyperplane equation is given by

$\displaystyle w^{T}x-b=0,$ (10)

where $w$ is the weight and $b$ the bias. $x$ is an input variable.

There are three cases of Binary-SVM:

  1. 1.

Hard margin

Here, the two classes can be separated linearly (Figure 5). The goal is to
maximize $\frac{2}{\lVert w\rVert}$ which is equivalent to minimizing $\lVert
w\rVert$.

Hence, the problem can be reformulated to $w^{*}=argmin\ \lVert w\rVert^{2}$.
We have two constraints:

    1. (a)

$w^{T}x^{(i)}-b\geq 1,\ if\ y^{(i)}=1$

    2. (b)

$w^{T}x^{(i)}-b\leq-1,\ if\ y^{(i)}=-1$

We combine these two constraints and get

$\displaystyle y^{(i)}(w^{T}x^{(i)}-b)\geq 1,\qquad for\ i=1,...,m,$

where $m$ is the number of samples and $y$ is the label.

So, the optimization problem becomes:

$\displaystyle\begin{cases}w^{*}=\mbox{argmin}\ \lVert w\rVert^{2}\\\
\mbox{subject to}\ y^{(i)}(w^{T}x^{(i)}-b)\geq 1\end{cases}$

The SVM classifier is given by:

$\displaystyle f(\vec{x})=\mbox{sign}(\vec{w}.\vec{x}-b)$ (11)

  2. 2.

Soft margin

This case occurs when the data are not separable in a linear way because there
are dots within the margin. Consequently, the loss of function becomes the
hinge loss:

$\displaystyle\ell(w;x,y)=\mbox{max}[0,1-y^{(i)}(w^{T}x^{(i)}-b)],\
i=1,...,m.\ $ (12)

Having the hinge loss by equation 12, the expected loss is given by

$\displaystyle
L(w)=\frac{1}{m}\left[\sum_{i=1}^{m}\ell(w;x,y)\right]+\lambda\lVert
w\rVert_{2}^{2},$

where $\lambda$ is the trade-off increasing the size of the margin and
ensuring that the data point is on the correct side of the margin.

Hence, the optimization problem becomes:

$\displaystyle\begin{cases}w^{*}=\mbox{argmin}\ L(w)\\\ \mbox{subject to}\
y^{(i)}(w^{T}x^{(i)}-b)\geq 1-\ell\end{cases}$

  3. 3.

Nonlinear classification (kernel SVM)

In Figure 6 a case where the data cannot be separated by a hyperplane is
depicted. We find a map $\phi:x^{(i)}\longrightarrow\phi(x)^{(i)}$ from the
data space to the feature space such that the data are linearly separable in
the feature space by applying the so-called “kernel trick":

$\displaystyle k(x,x_{i})=\phi(x)\phi(x_{i})$ (13)

Kernel function may be any of the symmetric functions that satisfy the
Mercer’s conditions (Brunner et al., 2012).

In the feature space, one can write:

$\displaystyle w^{T}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\alpha_{i}y^{(i)}\phi(x^{(i)})$
$\displaystyle\Longrightarrow w^{T}\phi(x^{(i)})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\alpha_{i}y^{(i)}\phi(x^{(i)})\phi(x^{(i)})$ (14)

Using 13 in 3, we obtain:

$\displaystyle
w^{T}\phi(x^{(i)})=\sum_{i=1}^{n}\alpha_{i}y^{(i)}k(x^{(i)},x^{(i)})$

So,

$\displaystyle y^{(i)}(w^{T}x^{(i)}-b)\Longrightarrow
y^{(i)}\left[\sum_{i=1}^{n}\alpha_{i}y^{(i)}k(x^{(i)},x^{(i)})-b\right]$

There are several functions of the SVM kernel.

    1. (a)

Polynomial kernel: it is a non-stationary kernel. The polynomial kernel is
well suited for problems where all training data are normalized. It is given
by equation 15

$\displaystyle K(x,x_{i})=(\alpha x^{T}x_{i}+c)^{d},$ (15)

where the slope $\alpha$ is the adjustable parameter, $d$ is the polynomial
degree and $c$ is the constant.

The dimension of the feature space vector $\phi(x)$ for the polynomial kernel
of degree $p$ and for the input pattern of dimension $d$ is:

$\displaystyle\frac{(p+d)!}{p!d!}$

    2. (b)

Gaussian kernel: it is an example of radial basis function kernel (RBF). It is
characterized by the equation:

$\displaystyle k(x,x^{\prime})=exp[-\gamma\lVert x-x^{\prime}\rVert^{2}]$ (16)

Usually, $\gamma=\frac{1}{2\sigma^{2}}$. So, the equation 16 becomes:

$\displaystyle k(x,x^{\prime})=exp\left[-\frac{\lVert
x-x^{\prime}\rVert^{2}}{2\sigma^{2}}\right]$ (17)

The adjustable parameter $\sigma$ plays a major role in the performance of the
kernel and should be carefully tuned to the problem at hand. If overestimated,
the exponential will behave almost linearly and the high-dimensional
projection will begin to lose its non-linear power. On the contrary, if
underestimated, the function will lack regularization and the decision
boundary will be highly sensitive to noise in training data (Ramalingam and
Dhanalakshmi, 2014).

SVM classifier is given by:

$\displaystyle f(x)=\mbox{sign}(w^{T}\phi(x)+b)$ (18)

Figure 6: Kernel SVM graph (svm, (accessed April 19, 2020)

####  7.1.2 Multi-Class SVM

SVM was made for binary classification. But in the real world, we deal with
classification problems with more than two classes. Multi-category
classification problems are usually divided into a series of binary problems
so that binary SVM can be directly applied (Mathur and Foody, 2008).

One representative method is the “One-Against-All" approach. Consider an
$M$-class problem, where we have $N$ training samples:
$\\{x^{(1)},y^{(1)}\\},...,\\{x^{(N)},y^{(N)}\\}$. Here,
$x^{(i)}\in\mathbb{R}^{m}$ is a $m$-dimensional feature vector and
$y^{(i)}\in\\{1,2,...,M\\}$ is the corresponding class label.

The One-Against-All approach constructs $M$ binary SVM classifiers, each of
which separates one class from all the rest. The $i^{th}$ SVM is trained with
all the training examples of the $i^{th}$ class with positive labels and all
the others with negative labels.

Mathematically, the $i^{th}$ SVM solves the following problem that yields the
$i^{th}$ decision function $f_{i}(x)=sign(w_{i}^{T}\phi(x)+b_{i})$

minimize $\displaystyle:$ $\displaystyle\qquad
L(w,\xi_{j}^{i})=\frac{1}{2}\lVert w_{i}\rVert^{2}+C\sum_{l=1}^{N}\xi_{j}^{i}$
subject to $\displaystyle:$
$\displaystyle\qquad\tilde{y}_{j}(w_{i}^{T}\phi(x_{j})+b_{i})\geq
1-\xi_{j}^{i},\quad\xi_{j}^{i}\geq 0,$ (19)

where $\tilde{y}_{j}=1$ if $y_{j}=i$ and $\tilde{y}_{j}=-1$ otherwise.

At the classification phase, a sample $x$ is predicted to be in class $i^{*}$
whose $f_{i^{*}}$ produces the largest value

$\displaystyle
i^{*}=\mbox{argmax}f_{i}(x)=\mbox{argmax}(w_{i}^{T}\phi(x)+b_{i}),\quad
i=1,...,M$ (20)

###  7.2 Gaussian Mixture Model (GMM)

Gaussian Mixture Model (GMM) is a parametric probability density function
expressed as a weighted sum of Gaussian component densities. In a biometric
system, GMMs are widely used as a parametric model of the probability
distribution of continuous measurements or features, such as spectral features
related to the vocal tract in a speaker recognition system. GMM parameters are
estimated from training data using the iterative expectation maximization (EM)
algorithm (Reynolds, 2009).

####  7.2.1 K-means clustering

Here, the initialization of the GMM parameters is carried out using the number
of clusters and allowing to form the different centers. In fact, GMM is a
function that is comprised of several Gaussians, each identified by
$k\in\\{1,...,K\\}$ ($K$: number of clusters of the dataset).

Each $k$ has the following parameters (Figure 7):

  1. 1.

A mean vector $\vec{\mu}$ that defines its center.

  2. 2.

A covariance matrix $\Sigma$ that defines its width. This would be equivalent
to the dimensions of an ellipsoid in a multivariate scenario.

  3. 3.

A mixture of weights $w$ that defines how big or small the Gaussian function
will be.

Figure 7: GMM parameters (Reynolds et al., (Reynolds, 2009)).

A mixture of weights must satisfy the constraint that:

$\displaystyle\sum_{k=1}^{K}w_{k}=1$ (21)

GMM is giving by

$\displaystyle p(x|\lambda)=\sum_{k=1}^{K}w_{k}\
g(x|\mu_{k},\Sigma_{k}),\qquad k=1,...,K,$ (22)

where $x$ is the D-dimensional continuous-valued data vector (i.e. measurement
or features), $w_{k}$ are the mixture weights, and $g(x|\mu_{k},\Sigma_{k})$
are the Gaussian densities. Each component density is a D-variate Gaussian
function of the form:

$\displaystyle
g(x|\mu_{k},\Sigma_{k})=\frac{1}{(2\pi)^{D/2}|\Sigma_{i}|^{1/2}}exp\left[-\frac{1}{2}(x-\mu_{k})^{T}\Sigma^{-1}(x-\mu_{k})\right]$
(23)

A collective representation of the parameters is defined as:

$\displaystyle\lambda=\\{w_{k},\mu_{k},\Sigma_{k}\\},\qquad k=1,...,K$ (24)

There are several variants of the GMM presented in equation 24. The
$\Sigma_{k}$ can be full rank or forced to be diagonal. Also, parameters can
be shared or linked between Gaussian components. As an example, by having a
common covariance matrix for all components, the choice of model configuration
(the number of components, full or diagonal covariance and parameter coupling)
is usually determined by the volume of data available for the estimation of
GMM parameters, and by the way in which the GMM is used in a particular
biometric application. It is important to note this because even if the
characteristics are not statistically independent, the Gaussian components act
together to model the overall density of the characteristics. The modeling of
correlations between the vector components of the features can be performed by
the linear combination of the Gaussian diagonal covariance basis. The effect
of employing a set of $K$ full covariance matrices Gaussian can also be
obtained by employing a larger set of Gaussian diagonal covariance matrices.

The use of a GMM to represent feature distributions can also be driven by the
intuitive idea that the densities of individual components can model an
underlying set of hidden classes. For example, in the case of the speaker, it
is reasonable to assume that the acoustic space of the spectral features
corresponding to a speaker’s major phonetic events such as vowels or
fricatives. These acoustic classes reflect certain general configurations of
speaker-dependent vocal pathways that are useful in characterizing speaker
identity. The spectral shape of the $k^{th}$ acoustic class can, in turn, be
represented by the mean $\mu_{k}$ of the $k^{th}$ component density, and
variations in the mean spectral shape can be represented by the covariance
matrix $\Sigma_{k}$. Since not all the characteristics used to form the GMM
are labeled, the acoustic classes are hidden, in the sense that the class of
observation is unknown. The observation density of the feature vectors derived
from these hidden acoustic classes form a Gaussian mixture (assuming that the
feature vectors are independent) (Reynolds, 2009).

####  7.2.2 Expectation-Maximization (EM)

Taking into account the training vectors and a configuration of the GMM, we
wish to estimate the parameters of the GMM, $\lambda$, which in some sense
corresponds best to the distribution of the training vectors. There are
several techniques for estimating the parameters of a GMM (McLachlan and
Basford, 1988). By far, the most popular and best-established method is the
expectation-maximization (EM) algorithm. The objective of expectation
maximization (EM) is to find the model parameters that maximize the
probability of the GMM given the training data. For a sequence of $T$ training
vectors $X={x_{1},....x_{T}}$, the likelihood of the GMM, assuming
independence between the vectors, can be written as

$\displaystyle
p(\vec{x}|\Theta)=\sum_{k=1}^{K}w_{k}p_{k}(\vec{x}|z_{k},\theta_{k}),$ (25)

where $p(\vec{x})$ is the finite mixture model, and
$p_{k}(\vec{x}|z_{k},\theta_{k})$ is the gaussian density for the $k^{th}$
mixture component. $z=(z_{1},...,z_{K})$ is the vector of $K$ binary indicator
variables which are mutually exclusive and exhaustive. $\Theta$ is the
complete set of parameters
($\Theta=\\{w_{1},...,w_{K},\theta_{1},...,\theta_{K}\\}$)

Step 1: E-step

The goal here is to compute the membership weights $w_{ik}$ which are the
probabilities that reflect the uncertainty, given $\vec{x_{i}}$ and $\Theta$.
The membership weight of a data point $\vec{x_{i}}$ in cluster $k$ can be
written as:

$\displaystyle
w_{ik}=p(z_{ik}=1|\vec{x}_{i},\Theta)=\frac{w_{k}p_{k}(\vec{x}|z_{k},\theta_{k})}{\sum_{m=1}^{K}w_{m}p_{m}(\vec{x}_{i}|z_{m},\theta_{m})}$
(26)

Step 2: M-step

This step aims to use the membership weights obtained in equation 26 in
E-step, to calculate new parameter values which are given by equation 27, 28,
29.

$\displaystyle w_{k}$ $\displaystyle=$ $\displaystyle\frac{N_{k}}{N},\ 1\leq
k\leq K,$ (27) $\displaystyle\vec{\mu}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{k}}\sum_{i=1}^{N}w_{ik}\vec{x}_{i},\ 1\leq k\leq K$
(28) $\displaystyle\Sigma_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{k}}\sum_{i=1}^{N}w_{ik}(\vec{x}_{i}-\vec{\mu}_{k})(\vec{x}_{i}-\vec{\mu}_{k})^{T},\
1\leq k\leq K.$ (29)

where $N_{k}=\sum_{i=1}^{N}w_{ik}$ is the column sum of the membership weight
matrix.

##  8 Implementation and Analysis

This section presents the implementation of the techniques studied in the
previous sections for the singer’s identification. We will first give an
overview of the procedure from separation to identification. Then, we will
present all the experimental data used in this study. Finally, we will show
the experiments and results obtained at each step of the singer identification
process with and without feature extraction.

###  8.1 The Block Diagram of the Overall Process of Singer’s Identification

As shown in the Figure 8, the inputs are audio files with the .mp3 extension.
After importing these files, we get the musical signal. We apply the STFT to
each signal to obtain its matrix in the frequency domain. Afterward, the RPCA
technique is applied to this matrix separating it into two matrices: a low
rank and a sparse matrix. After performing the ISTFT on the sparse matrix, the
vocal signal is obtained. This vocal signal obtained from each sound of the
dataset allows building a data-frame. The purpose of this study is to show,
first, the importance of feature extraction and then, to compare the two
techniques DWT and MFCC. Hence, we perform three experiments: (1) Training the
data without feature extraction; (2) Using MFCC for feature extraction; (3)
Using DWT for feature extraction before training with SVM and GMM techniques.

Figure 8: Block diagram of the overall process of singer’s identification

###  8.2 Dataset

We created a database of test recordings by selecting four popular singers:
two men and two women, each with 50 excerpts. These songs go through the pre-
processing phase where missing values are removed and twelve-second singing
voice segments are obtained from each musical recording. As a result, after
separation of signal and after using RPCA, each singer has a total of 263232
singing voice segments which are then introduced into the feature extraction
phase.

###  8.3 Feature extraction using DWT

####  8.3.1 De-noising

For de-noising, we investigate the maximum gain for input signals with
different levels of degradation. The amount of white noise added to the
original signal is controlled with the standard deviation of the noise
$\sigma_{n}$. The maximum gain is obtained by replacing the threshold
$\lambda$ of equation 5 with

$\displaystyle\lambda=k.\sigma_{n}\sqrt{(2\log N)},$ (30)

where $0<k<1$. In fact, the universal threshold t given by equation 5 is too
high for audio signals and it cuts a big part of the original signal. So it is
modified with factor $k$ to obtain a higher quality output signal. The value
of $k$ is changed gradually with steps of 0.1. Finally, we find $k$ that gives
the best result depicted in Figure 9.

Figure 9: De-noising of the signal of Celine’s song

####  8.3.2 Decomposition of signal

According to literature, we use, for this implementation, the DWT Daubechies
four (db4) of level 4. Each signal has the frequency $4160\ Hz$ after de-
noising. In Figure 10, we can easily see the five (05) sub-bands of the
previously de-noised song: $L_{4}(0-260Hz)$, $H_{4}(260-520Hz)$,
$H_{3}(520-1040Hz)$, $H_{2}(1040-2080Hz)$, $H_{1}(2080-4160Hz)$

Figure 10: Decomposition of the signal of Celine’s song previously de-noised

####  8.3.3 Feature extraction strictly speaking

After the decomposition of the signal in sub-bands, we extract the following
features to build our final data-frame:

  * •

Time-Frequency domain: Mean and Spectral entropy.

  * •

Time domain: Mean, Median, and Standard deviation.

  * •

Frequency domain: Power spectral density.

###  8.4 Singer identification

In this step, we first do a feature engineering to see which of the 18
features are correlated and use PCA to take the features which represent
99.99% of variation: 15 features. Then, we separate our data-frame: features
for input and names of singers for output labels. We then check the parameter
of models which can influence the result and take the one which gives us the
best result. Finally, knowing the audio signal, we train the machine learning
models using the fold cross-validation value 10. We also shuffle the data
fifteen (15) times to have realistic results. Then, the general accuracy will
be the mean of the fifteen (15) accuracies.

First, we do the training without feature extraction. We remark that the best
model is delivered by the SVM-RBF model with 36.78% of mean accuracy. It
should be noted that the 263232 columns in the data-frame, without feature
extraction, are considered as input for model training. Then, it becomes
impossible to build the covariance matrix for GMM because the number of inputs
is greater than the number of observations. However, a reduction in size is
not possible because these columns represent the vector of the signal, and
reducing it is equivalent to muting the signal, therefore, the implementation
of this method was not possible.
Second, we do the training using MFCC for feature extraction. We found that,
in general, the SVM model performs better than the GMM. The best model is SVM-
Linear with an mean accuracy of 61.49%.

Finally, we train the models using the final data-frame obtained at the
previous step with DWT for feature extraction. We have determined that the
best model is SVM-Linear with 83.96%.

The results have been summary in Figure 11

Figure 11: Boxplot of performance (accuracies) with DWT

##  9 Conclusion and Recommendation

The objective of this work was to apply the DWT for feature extraction and
compare the results with the MFCC to see which of the two improves the
identification of the singer in term of accuracy. We first gave the physical
and mathematical description of the different techniques ranging from
separation of the vocal signal from the background signal to the singer
identification. Then, we implemented these techniques according to a dataset
of 200 songs (50 songs per singer). RPCA was used for the separation of
signals; DWT and MFCC were used to feature extraction; and SVM and GMM were
used for singer’s identification.

For a set of 200 observations of audio signals, this study shows that:

  * •

The vocal signals are very unstable because the same artist sings in different
styles and uses different nuances that influence the results of the study
using a small number of recordings. As a consequence, low accuracies are
obtained.

  * •

The feature extraction is essential for the singer identification process.

  * •

DWT performs better than MFCC for feature extraction in term of accuracy and
training time.

  * •

SVM performs better than GMM for singer identification.

  * •

The best configuration of techniques for singer identification is DWT + SVM-
Linear with a mean accuracy of 83.96% with a training time of 1 068 s.

However, to generalize these results, it is essential to perform the same
study on a much larger set of recordings.

Since DWT is better at extracting characteristics from audio signals, future
work could be the in-depth study of the different families of DWT to
investigate the effects of their individual properties and to use appropriate
DWTs for different cases of datasets.

Besides, DWT can be used for extended periods to study other non-stationary
signals such as those in the human body (electroencephalogram (EEG),
electrocardiogram (ECG), electro-oculography (EOG)). This will allow
abnormalities to be detected fairly quickly and diseases predicted and treated
before complications arise.

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