Modular Arithmetic and Wavelets for Speaker Verification

INTRODUCTION

Automatic Speaker Verification (ASV) refers to the mission of verifying speaker’s identity by means of the speaker-specific information contained in speech signal. Speaker verification methods are absolutely divided into text-dependent and text-independent applications. When the same text is used for both training and testing, the system is called to be text-dependent but for text-independent process, the text used to train and test of the ASV system is totally unconstrained. The text-independent speaker verification necessitates no restriction on the type of input speech. In contrast, the text-independent speaker verification generally gives less performance than text-dependent speaker verification, which requires test input to be the same utterances as training data (Nemati and Basiri, 2010; Xiang and Berger, 2003).

Speaker verification has been the topic of active research for many years, and has many important applications where propriety of information is a concern (Lamel and Gauvain, 2000). Applications of speaker verification can be found in biometric person authentication such as an extra identity check in credit card payments over the Internet while, the potential applications of speaker identification may be utilized in multi-user systems. For example, in speaker tracking the task is to locate the segments of given speaker (s) in an audio stream (Kwon and Narayanan, 2002; Lapidot et al., 2002; Martin and Przybocki, 2001). It has also potential applications in an automatic segmentation of teleconferences and helping in the transcription of courtroom discussion.

There has been a wide spectrum of proposed approaches to speaker verification starting with very simplistic models such as those based on long term statistics. The most sophisticated methods rely on large vocabulary speech recognition with phone-based HMMs.

Feature extraction is a key stage in speaker verification systems. Speech extracted features used in a speaker verification system drop within two classes based on their related space. One class includes features defined in an unconditional or absolute and irrelative space, while the other includes features defined in a relative space. For the first class, depiction of a speaker in the feature space is not related to any reference speaker (Naini et al., 2010). While there is a momentous body of literature on features in the absolute space, very little research has been conducted for studying the properties of features extracted in the relative space. Mel-frequency cepstral coefficient (MFCC), Linear Prediction Cepstrum Coefficient (LPCC), wavelet coefficients (Afshari, 2011; Avci and Akpolat, 2006), etc., are among the most common speech features in absolute space. In recent times, Campbell et al. (2006) used Maximum A Posteriori (MAP) adapted GMM mean super vectors as an absolute feature with Support Vector Machine (SVM) as a discriminative model for speaker verification (Campbell et al., 2006). For features defined in a relative space, each speaker in the feature space is described relative to some reference speakers. As well as, extracted features in the relative space can be applied in conjunction with any other set of techniques from the verification phase menu that are deemed more suitable. Kuhn et al. (2000) introduced the eigenvoices concept and represented each new speaker relative to eigenvoices (Kuhn et al., 2000; Thyes et al., 2000). Afterwards, other researchers used a different approach where they introduced the idea of space of anchor models to represent enrolled speakers in verification systems and to verify a test speaker in a relative feature space (Naini et al., 2010; Mami and Charlet, 2002, 2003, 2006).

Speech features are often extracted by Fourier Transform (FT) and short time Fourier transform (STFT). Unfortunately, they accept signals stationary within a given time frame and may therefore lack the ability to represent localized events properly. Recently, Wavelet Transform (WT) has been proposed for feature extraction. The particular benefit of wavelet analysis possesses is the characterizing signals at different localization levels in both time and frequency domains (Derbel et al., 2008; Wu and Ye, 2009; Zheng et al., 2002). Furthermore, the WT is well suited to the analysis of non-stationary signals (Daqrouq et al., 2010; Daqrouq, 2011). It provides an alternative to classical linear time-frequency representations with better time and frequency localization characteristics. In earlier studies, these properties were applied in speaker recognition, particularly Wavelet Packet Transform (WPT) (Wu and Lin, 2009; Lung, 2004; Avci, 2009).

Artificial neural network performance is depending mainly on the size and quality of training samples (Visser et al., 2003). When the number of training data is small, not representative of the possibility space, standard neural network results are poor. Incorporation of neural fuzzy or wavelet techniques can improve performance in this case, particularly, by input matrix dimensionality decreasing. Artificial Neural Networks (ANN) are known to be excellent classifiers but their performance can be prevented by the size and quality of the training set (Qureshi and Jalil, 2002).

The specific aim of the present study was to address this question by developing and evaluating the number of repeating the remainder (by modular arithmetic) for speech based text-independent verification of the speaker from imposters. For better investigation two other verification methods are proposed; Gaussian Mixture Model method and K-Means clustering method.

FEATURES EXTRACTION BY WAVELET PACKET

Wavelet packet: The wavelet packet method is a generalization of wavelet decomposition that offers a richer signal analysis. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position and scale as in wavelet transform decomposition and frequency. In the following, the wavelet transform is defined as the inner product of a signal x (t) with the mother wavelet ψ (t):

(1)

(2)

where, a and b are the scale and shift parameters, respectively. The mother wavelet may be dilated or translated by modulating a and b.

The wavelet packets transform performs the recursive decomposition of the speech signal obtained by the recursive binary tree (Fig. 1). Basically, the WPT is very similar to Discrete Wavelet Transform (DWT) but WPT decomposes both details and approximations instead of only performing the decomposition process on approximations. The principle of WP is that, given a signal, a pair of low pass and high pass filters is used to yield two sequences to capture different frequency sub-band features of the original signal. The two wavelet orthogonal bases generated form a previous node are defined as:

(3)

(4)

where, h [n] and g [n] denote the low-pass and high-pass filters, respectively.

Fig. 1:	Wavelet packet at depth 3

In Eq. 3 and 4, ψ [n] is the wavelet function. Parameters j and p are the number of decomposition levels and nodes of the previous node, respectively (Wu and Lin, 2009).

Modular arithmetic: Modular arithmetic is referenced in number theory, group theory, ring theory, knot theory, abstract algebra, cryptography, computer science, chemistry and the visual and musical arts. Modular arithmetic provides less computational complexity (Cormen et al., 2001). It performs efficiently on large numbers. Wong and Blow (2006) presented a logical design of an all-optical processor that performs modular arithmetic. Modular arithmetic could be used to process the header of all-optical packets on the fly (Wessing et al., 2002).

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value. Time-keeping on a clock introduces an example of modular arithmetic (If the time is 7:00 now, then 8 h later it will be 3:00). The Swiss mathematician Leonhard Euler pioneered the modern method to congruence about 1750, when he explicitly introduced the idea of congruence modulo a number N. Modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801 (Encyclopedia Britannica, 2010).

In fact, the notion of modular arithmetic is related to that of the remainder in division. The operation of extracting the remainder is sometimes referred to as modulo operation. Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is well-matched with the operations of the ring of integers: addition, subtraction and multiplication. For a positive integer n two integers a and b are assumed to be congruent modulo, written:

(5)

if their difference a-b is an integer multiple of n. The number n is called the modulus of the congruence.

In computer science discipline, it is the remainder operator that is usually indicated by either "%" (e.g. in C, Java, JavaScript, Perl and Python) or "Mod" (e.g. in BASIC, SQL, Haskell and Matlab). These operators are commonly pronounced as "mod", however, it is specifically a remainder that is computed. The remainder operation can be represented using the floor function.

If a ≡ b (mod n), where n > 0, then if the remainder b is calculated:

(6)

where, is the largest integer less than or equal to then a≡b (mod n) and 0≤b<n

If instead a remainder b in the range -n ≤ b < 0 = is required, then:

(7)

In this work, the WP tree consists of three stages and therefore has G_High = 2³ high pass nodes (with original signal node) and G_Low = 2³ low pass nodes with original signal node. More generally, for a q-stage tree, there are G = (G_High + G_low) = 2^q+1-1 nodes.

For a given orthogonal wavelet function, a library of wavelet packet bases is generated. Each of these bases offers a particular way of coding signals, preserving global energy and reconstructing exact features. The wavelet packet is used to extract additional features to guarantee higher recognition rate. In this study, WPT is applied at the stage of feature extraction, but these data are not proper for classifier due to a great amount of data length. Thus, it is essential to seek for a better representation for the speech features. Previous studies showed that the use of entropy of WP as features in recognition tasks is efficient. Avci and Akpolat (2006) proposed a method to calculate the entropy value of the wavelet norm in digital modulation recognition. In the biomedical field, Behroozmand and Almasganj (2007) presented a combination of genetic algorithm and wavelet packet transform used in the pathological evaluation and the energy features are determined from a group of wavelet packet coefficients. Kotnik et al. (2003) proposed a robust speech recognition scheme in a noisy environment by means of wavelet-based energy as a threshold for de-noise estimation. In the study of Wu and Lin (2009) the energy indexes of WP were proposed for speaker identification. Sure entropy is calculated for the waveforms at the terminal node signals obtained from DWT (Avci, 2009) for speaker identification. Avci (2007) proposed features extraction method for speaker recognition based on a combination of three entropies types (sure, logarithmic energy and norm). In this study, the modular arithmetic to decrease the number of features in each WP node is proposed. For a speech signal in a WP node.

{u (t)} = {u (t₁), u (t₂),..., u (t_M)}, where M is the length of u (t), the remainder b is written as mod (n)_{u (t)} = 0,1,2,..., n-1, for n>2. In this study, the number of repeating the remainder mod (n)_u(t) is utilized as a modular arithmetic wavelet speech signal feature vector:

WR (n) = r (0), r (1), r (2), ..., r (n-1). Where r is the number of repeating the remainder mod (n) applied for a speech signal without WP and R denotes the same but without WP (Fig. 2).

Verification: Depending on the application, the universal area of speaker recognition is divided into two particular tasks: identification and verification. In speaker identification, the goal is to decide which one of a group of known voices best matches the input voice sample. This is also referred to as closed-set speaker identification. Applications of pure closed-set identification are limited to cases where only enrolled speakers will be encountered but it is a useful means of studying the separability of speakers’ voices or finding similar sounding speakers, which has applications in speaker-adaptive speech recognition. In verification, the task is to decide from a voice sample if a person is who he or she claims to be. This is sometimes referred to as the open-set problem, because this task requires distinguishing a claimed speaker’s voice known to the system from a potentially large group of voices unknown to the system (i.e., imposter speakers) (Reynolds et al., 2000).

The basis for presented verification systems is the modular arithmetic wavelet method MWM used to represent speakers. More specifically, the feature vectors extracted from a person’s speech is modeled by speaker model. For a D-dimensional feature vector denoted as x, the model for speakers is defined as the average vector calculated for fifteen different utterances of a same speaker. This average vector is used to present the hypothesized speakers to be compared to background speakers, which present the imposter speakers. Background speakers are defined as the average vector calculated for fifteen different utterances of a many speakers.

Hypothesized Speaker Model: The basis for verification systems is to create feature vector pattern used to represent speakers. Therefore, in this study, WR for hypothesized speaker model presenting is utilized. More specifically, the average of WR feature vectors extracted from fifteen different person’s utterances is used to represent each speaker pattern. This feature vector is compared to claimed speaker WR using verification functions.

Background speaker model: The background speakers should be selected to signify the population of expected imposters, which is in general application specific. Two issues that come up with the use of background speakers are the selection of the speakers and the number of speakers to use. Ideally the number of background speakers should be as large as possible to better model the imposter population but practical considerations of computation and storage state a small set of background speakers.

Fig. 2:	First 150 coefficients of the feature vector extracted by MWM for two speakers

In the verification experiments, the number of background speakers is set to three. In proposed scenario, it is assumed that imposters will attempt to gain access only from similar sounding or at least same-sex speakers.

The single-speaker detection task can be defined as a basic hypothesis test between:

The optimum test to choose between these two hypotheses is a following ratio test given by:

(8)

where, V (Y, H_i ), I = 0,1, is the verification function for the hypothesis H_i evaluated for the observed speech segment Y.

Verification functions: In this study three verification functions are proposed:

(9)

where, MW_Y is WR taken for observed speech segment Y, at level three of WP and MWM_Hi is the average of WR vectors calculated for fifteen different utterances of a same speaker (hypothesized speaker H₀) or background model (H₁). The decision threshold for accepting or rejecting is determined by:

(10)

(11)

The decision threshold for accepting or rejecting is determined by:

(12)

(13)

Verification by gaussian mixture model: Gaussian Mixture Model GMM recently has become the dominant approach in text-independent speaker identification and verification. One of the influential attributes of GMMs is their capability to form smooth approximations to arbitrarily formed densities. As a typical model based approach, GMM has been utilized to characterize speaker’s voice in the form of probabilistic model. It has been reported that the GMM approach outperforms other classical methods for text-independent speaker recognition. In the following paragraph briefly mathematical development of the GMM based speaker verification scheme is introduced:

Given Y = {Y₁, Y₂, ... Y_k} where, Y = {y_t1 = T₁, y_t2 = T₂, ..., y_tj = T_j} is a sequence of T_j feature vectors in jth cluster R^j, the complete GMM for speaker model λ is characterized by the mean vectors, covariance matrices and mixture weights from all component densities. The parameters of the speaker model are denoted by:

λ = {P_j,t, _tu_j,t, _tΣ_j,t}, i = 1, 2, ..., M_j and j = 1, 2, ..., Then, the GMM likelihood can be written as:

(14)

Equation 14, is the Gaussian mixture density for jth cluster and defined by a weighted sum of M_j component densities (Lung, 2007). The Expectation Maximization (EM) algorithm to create an object of the Gaussian mixture distribution class restraining maximum likelihood estimates of the parameters in a Gaussian mixture model with k components for data in the n-by-d matrix X, where n is the number of observations and d is the dimension of the data is used. In this study, verification is performed by building Gaussian mixture model by EM with 2 components of MWM vectors of two speakers feature vectors GMMW. Then the GMM likelihood is used as the verification decision whether accept or reject. This is accomplished by determining empirical threshold for decision performing.

Verification by K-mean clustering method: In this section, a brief outline of K-Means clustering algorithm and verification by this method is presented. Clustering in N dimensional Euclidean space R^N is the process of partitioning a given set of n points into a number, say K, of clusters based on some similarity metric which establishes a rule for assigning patterns to the domain of a particular cluster centroid as seen at Fig. 3. Let the set of n points {x₁, x₂, ..., x_n} be represented by the set S and the K clusters is represented by C₁ C₂, ..., C_k (Bandyopadhyay and Maulik, 2002). Then C_i ≠ φ for i = 1, ..., k, C_i∩ C_j = φ for i =1, ..., K, j = 1, ..., K, where:

K-Means (Hemalatha and Vivekanandan, 2008; Wagstaff et al., 2001) is one of the commonly used clustering techniques, which is an iterative hill climbing algorithm. It consists of the following steps:

•	Choosing K initial cluster centroids Z1, Z2, ..., Zk, randomly from the n points {x1, x2, ..., xn}
•	Assigning point xi, i = 1, 2, ..., K to cluster Cj, j ε {1, 2, ..., k} where, ||xi-zj||≤|| xi-zp|| p =1, 2, …, k and j≠ p
•	Calculating new cluster centroids: , where

i = 1, 2, ..., K, where n_i is the number of elements belonging to cluster C_i

•	If then end. Otherwise continue from 2

K-means is a common clustering algorithm that has been used in a variety of application disciplines, such as image clustering and information retrieval, as will as speech and speaker recognition.

Fig. 3:	K-Means data clustering with K=4 (This paragraph with equations and Fig. 3 are essential for understanding of contribution of the paper)

Different types of clustering algorithms that are based on K-Means, are mentioned by Wagstaff et al. (2001), such as the modified version for background knowledge, a genetic algorithm and the syllable contour that is classified into several linear loci that serve as candidates for the tone-nucleus using segmental K-Means segmentation algorithm.

Here is an investigation of a new speaker verification system that based on K-Mean feature extraction method KMM taken from speech signals. More specifically, the presented verification method by K-Means clustering consists of two main stages:

In total, 64 coefficients vector are extracted by this method for each speaker for five signal frames of 1 second duration (totally 320 coefficient is extracted for each speaker). The verification decision is taken based on the method presented in by Eq. 10, 12 and 13.