6. Discrete wavelet transform (dwt)

Wavelet Transform (WT) theory is centered around signal analysis using varying scales in the time and frequency domains [45]. With the support of theoretical physicist Alex Grossmann, Jean Morlet introduced wavelet transform which permits high-frequency events identification with an enhanced temporal resolution [45–47]. A wavelet is a waveform of effectively limited duration that has an average value of zero. Many wavelets also display orthogonality, an ideal feature of compact signal representation [46]. WT is a signal processing technique that can be used to represent real-life non-stationary signals with high efficiency [33, 46]. It has the ability to mine information from the transient signals concurrently in both time and frequency domains [33, 45, 48].

Continuous wavelet transform (CWT) is used to split a continuous-time function into wavelets. However, there is redundancy of information and huge computational efforts is required to calculate all likely scales and translations of CWT, thereby restricting its use [45]. Discrete wavelet transform (DWT) is an extension of the WT that enhances the flexibility to the decomposition process [48]. It was introduced as a highly flexible and efficient method for sub band breakdown of signals [46, 49]. In earlier applications, linear discretization was used for discretizing CWT. Daubechies and others have developed an orthogonal DWT specially designed for analyzing a finite set of observations over the set of scales (dyadic discretization) [47].

#### 6.1. Algorithm description, strength and weaknesses

Wavelet transform decomposes a signal into a group of basic functions called wavelets. Wavelets are obtained from a single prototype wavelet called mother wavelet by dilations and shifting. The main characteristic of the WT is that it uses a variable window to scan the frequency spectrum, increasing the temporal resolution of the analysis [45, 46, 50].

WT decomposes signals over translated and dilated mother wavelets. Mother wavelet is a time function with finite energy and fast decay. The different versions of the single wavelet are orthogonal to each other. The continuous wavelet transform (CWT) is given by [33, 45, 50]:

$$\mathcal{W}\_x(a,b) = \frac{1}{\sqrt{a}} \int\_{-\infty}^{\infty} x(t) \psi^\* \left(\frac{t-b}{a}\right) dt\tag{11}$$

Thus, ψm,p is defined as:

parameter <sup>b</sup> <sup>¼</sup> nb0am

of positive integers).

Figure 5. Block diagram of DWT.

<sup>ψ</sup>m,p <sup>¼</sup> <sup>1</sup>

dilated and translated discretely by selecting the scaling parameter <sup>a</sup> <sup>¼</sup> am

The DWT of a discrete signal is derived from CWT and defined as:

decomposition is called dyadic decomposition [33].

boundaries because of the discontinuities at the boundaries [50].

ð Þ DWT ð Þ¼ <sup>m</sup>; <sup>k</sup> <sup>1</sup>

ffiffiffiffiffi am 0

ffiffiffiffiffi am 0 <sup>p</sup> <sup>X</sup> n

where g(\*) is the mother wavelet and x[n] is the discretized signal. The mother wavelet may be

The scaling and wavelet functions can be implemented effectively using a pair of filters, h[n] and g[n], called quadrature mirror filters that confirm with the property g n½ �¼ �ð Þ<sup>1</sup> <sup>1</sup>�<sup>n</sup>

The input signal is filtered by a low-pass filter and high-pass filter to obtain the approximate components and the detail components respectively. This is summarized in Figure 5. The approximate signal at each stage is further decomposed using the same low-pass and highpass filters to get the approximate and detail components for the next stage. This type of

The DWT parameters contain the information of different frequency scales. This enhances the speech information obtained in the corresponding frequency band [33]. The ability of the DWT to partition the variance of the elements of the input on a scale by scale basis is an added advantage. This partitioning leads to the opinion of the scale-dependent wavelet variance, which in many ways is equivalent to the more familiar frequency-dependent Fourier power spectrum [47]. Classic discrete decomposition schemes, which are dyadic do not fulfill all the requirements for direct use in parameterization. DWT does provide adequate number of frequency bands for effective speech analysis [51]. Since the input signals are of finite length, the wavelet coefficients will have unwantedly large variations at the

<sup>p</sup> <sup>ψ</sup> <sup>t</sup> � pb0am

x n½ �∙g

<sup>0</sup> (with constants taken as a<sup>0</sup> > 1, b<sup>0</sup> > 1, while m and n are assigned a set

0 am 0

> <sup>n</sup> � nb0am 0 am 0

� � (15)

Some Commonly Used Speech Feature Extraction Algorithms

http://dx.doi.org/10.5772/intechopen.80419

� � (16)

<sup>0</sup> and translation

h n½ �.

13

where ψð Þt is the mother wavelet, a and b are continuous parameters.

The WT coefficient is an expansion and a particular shift represents how well the original signal corresponds to the translated and dilated mother wavelet. Thus, the coefficient group of CWT (a, b) associated with a particular signal is the wavelet representation of the original signal in relation to the mother wavelet [45]. Since CWT contains high redundancy, analyzing the signal using a small number of scales with varying number of translations at each scale, i.e. discretizing scale and translation parameters as <sup>a</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> and b <sup>¼</sup> <sup>2</sup><sup>j</sup> k gives DWT. DWT theory requires two sets of related functions called scaling function and wavelet function given by [33]:

$$\phi(t) = \sum\_{n=0}^{N-1} h[n]\sqrt{2}\phi(2t - n) \tag{12}$$

$$\psi(t) = \sum\_{n=0}^{N-1} \mathbf{g}[n] \sqrt{2} \phi(2t - n) \tag{13}$$

where ϕð Þt is the scaling function, ψð Þt is the wavelet function, h[n] is the an impulse response of a low-pass filter, and g[n] is an impulse response of a high-pass filter.

There are several ways to discretize a CWT. The DWT of the continuous signal can also be given by [45]:

$$(DWT)(m, p) = \int\_{-\infty}^{+\infty} x(t) \cdot \psi\_{m, p} dt \tag{14}$$

where ψm, <sup>p</sup> is the wavelet function bases, m is the dilation parameter and p is the translation parameter.

Thus, ψm,p is defined as:

discretization was used for discretizing CWT. Daubechies and others have developed an orthogonal DWT specially designed for analyzing a finite set of observations over the set of

Wavelet transform decomposes a signal into a group of basic functions called wavelets. Wavelets are obtained from a single prototype wavelet called mother wavelet by dilations and shifting. The main characteristic of the WT is that it uses a variable window to scan the

WT decomposes signals over translated and dilated mother wavelets. Mother wavelet is a time function with finite energy and fast decay. The different versions of the single wavelet are orthogonal to each other. The continuous wavelet transform (CWT) is given by [33, 45, 50]:

> ffiffi a p ð ∞

�∞

The WT coefficient is an expansion and a particular shift represents how well the original signal corresponds to the translated and dilated mother wavelet. Thus, the coefficient group of CWT (a, b) associated with a particular signal is the wavelet representation of the original signal in relation to the mother wavelet [45]. Since CWT contains high redundancy, analyzing the signal using a small number of scales with varying number of translations at each scale, i.e. discretizing

> h n½ � ffiffiffi 2

> g n½ � ffiffiffi 2

where ϕð Þt is the scaling function, ψð Þt is the wavelet function, h[n] is the an impulse response

There are several ways to discretize a CWT. The DWT of the continuous signal can also be

þ ð∞

�∞

where ψm, <sup>p</sup> is the wavelet function bases, m is the dilation parameter and p is the translation

x tð Þψ<sup>∗</sup> <sup>t</sup> � <sup>b</sup> a � �

dt (11)

k gives DWT. DWT theory requires two sets

<sup>p</sup> <sup>ϕ</sup>ð Þ <sup>2</sup><sup>t</sup> � <sup>n</sup> (12)

<sup>p</sup> <sup>ϕ</sup>ð Þ <sup>2</sup><sup>t</sup> � <sup>n</sup> (13)

x tð Þ∙ψm,pdt (14)

frequency spectrum, increasing the temporal resolution of the analysis [45, 46, 50].

Wxð Þ¼ <sup>a</sup>; <sup>b</sup> <sup>1</sup>

of related functions called scaling function and wavelet function given by [33]:

N X�1 n¼0

N X�1 n¼0

ϕðÞ¼ t

ψðÞ¼ t

of a low-pass filter, and g[n] is an impulse response of a high-pass filter.

ð Þ DWT ð Þ¼ m; p

where ψð Þt is the mother wavelet, a and b are continuous parameters.

scale and translation parameters as <sup>a</sup> <sup>¼</sup> <sup>2</sup><sup>j</sup> and b <sup>¼</sup> <sup>2</sup><sup>j</sup>

given by [45]:

parameter.

scales (dyadic discretization) [47].

6.1. Algorithm description, strength and weaknesses

12 From Natural to Artificial Intelligence - Algorithms and Applications

$$
\psi\_{m,p} = \frac{1}{\sqrt{a\_0^m}} \psi \left(\frac{t - pb\_0 a\_0^m}{a\_0^m}\right) \tag{15}
$$

The DWT of a discrete signal is derived from CWT and defined as:

$$(DWT)(m,k) = \frac{1}{\sqrt{a\_0^m}} \sum\_{n} \mathbf{x}[n] \cdot \mathbf{g}\left(\frac{n - nb\_0 a\_0^m}{a\_0^m}\right) \tag{16}$$

where g(\*) is the mother wavelet and x[n] is the discretized signal. The mother wavelet may be dilated and translated discretely by selecting the scaling parameter <sup>a</sup> <sup>¼</sup> am <sup>0</sup> and translation parameter <sup>b</sup> <sup>¼</sup> nb0am <sup>0</sup> (with constants taken as a<sup>0</sup> > 1, b<sup>0</sup> > 1, while m and n are assigned a set of positive integers).

The scaling and wavelet functions can be implemented effectively using a pair of filters, h[n] and g[n], called quadrature mirror filters that confirm with the property g n½ �¼ �ð Þ<sup>1</sup> <sup>1</sup>�<sup>n</sup> h n½ �. The input signal is filtered by a low-pass filter and high-pass filter to obtain the approximate components and the detail components respectively. This is summarized in Figure 5. The approximate signal at each stage is further decomposed using the same low-pass and highpass filters to get the approximate and detail components for the next stage. This type of decomposition is called dyadic decomposition [33].

The DWT parameters contain the information of different frequency scales. This enhances the speech information obtained in the corresponding frequency band [33]. The ability of the DWT to partition the variance of the elements of the input on a scale by scale basis is an added advantage. This partitioning leads to the opinion of the scale-dependent wavelet variance, which in many ways is equivalent to the more familiar frequency-dependent Fourier power spectrum [47]. Classic discrete decomposition schemes, which are dyadic do not fulfill all the requirements for direct use in parameterization. DWT does provide adequate number of frequency bands for effective speech analysis [51]. Since the input signals are of finite length, the wavelet coefficients will have unwantedly large variations at the boundaries because of the discontinuities at the boundaries [50].

Figure 5. Block diagram of DWT.
