Denoising Audio Signal from Various Realistic Noise using Wavelet Transform

Table of Contents

List of Figures

List of Tables

Abbreviations

Chapter 1: Introduction and Objectives of the Project
1.1 Introduction
1.2 Project Summary.

Chapter 2: Theoretical Examination of Wavelet Transform
2.1 Principle of Wavelet Transform
2.2 Development of Wavelet Transform
2.3 Wavelet transform
2.3.1 Continuous wavelet transform
2.3.2 Discrete wavelet transform
2.4 Multiresolution Analysis (MRA)
2.4.1 Philosophy of Multiresolution Analysis:
2.4.2 Features of MRA
2.4.3 Properties of Scale and Time- Frequency Resolution

Chapter 3: Literature review
3.1 Overview
3.2 Short time Fourier transform
3.3 From Fourier Transform to Wavelet Transform
3.4 Comparison of wavelet transforms with Fourier transform
3.5 Wavelet Functions (WF)
3.6 Applications of Wavelet Transform
3.7 Audio Signal Denoising Using Wavelet Transform
3.8 Examples of Wavelet Based Noise Analysis

Chapter 4: Audio Signal Denoising Using Wavelet Transform
4.1 Digital Audio Signal
4.2 Audio Signal Denoising
4.2.1 Decomposition
4.2.2 Threshold selection
4.2.3 Reconstruction

Chapter 5: Experimental Results
5.1 Noise Analysis Using MATLAB
5.2 Critical examination of results

Chapter 6: Conclusion
6.1 Conclusion and Observation
6.2 Further Development and Future Work

References

Edinburgh Napier University School of Engineering and the Built Environment

This is to certify that the project report entitled “Performance and Comparative Analysis of Wavelet Transform in Denoising Audio Signal from Various Realistic Noise” prepared by Bharath Kumar Munegowda, is a record of an original work conducted under the guidance of Dr. Murray MacCallum, and submitted in partial completion of the requirements for the award of the degree of M.Sc. in Electronic and Electrical Engineering. This project report also meets all the requirements and standards detailed by the University

Certified by :

________________________

Dr. Murray MacCallum (Supervisor)

_________________________

Dr. David Binnie (II Marker)

_________________________

Dr. Lourdes Alwis (Module Leader)

Date:

ABSTRACT

Audio signals are more frequently polluted with various types of realistic noises. So, periods ago in order to reduce the noise level, some filtering approach will be used. But, presently there are many transform based techniques to estimate the noisy audio signal. One of the transform technique known as wavelet transform will be used for denoising an audio signal from realistic noise. Predominantly, the objective of this proposed research is to characterise discrete wavelet transform (DWT) towards denoising a one dimensional audio signal from common realistic noise. Moreover, the idea is to implement the audio signal denoising techniques such as decomposition, thresholding (soft) and reconstruction in the MATLAB simulation software, and elaborate a comparative analysis based on choice of wavelet transform over Fourier transform. Likewise, for the different level of decomposition, signal to noise (SNR) will be estimated .To sum up, in this research, different circumstances has been measured to elect best wavelet function and its level, based on its response of signal to noise ratio (SNR) in denoising audio signal

Acknowledgments

First of all, I would like to express my sincere gratitude to Dr. Murray MacCallum from bottom of my heart for his guidance, continuous support, motivation, and immense knowledge over the period of my M.Sc. Dissertation course

Besides my supervisor, I would take this opportunity to propose a special gratefulness to my M.Sc. project module leader Lourdes Alwis for her support, and providing precise information about stats throughout the research period

Also, I would like to thank school of Engineering and the built environment, Edinburgh University, Edinburgh, Scotland for providing their support throughout my project

I would love to thank my father Mr. Munegowda and my mother Mrs. Bhagyalakshmi Munegowda for their support and inspiration that made me to study M.sc in United Kingdom (Edinburgh)

Last but not the least, I would also like to thank all my friends for supporting me spiritually throughout my project

List of Figures

Figure 1: Method of Decomposition

Figure 2: Method of Reconstruction

Figure 3: Bandwidth Separation at each level

Figure 4: An illustration of Time-Frequency Resolution (R.Polikar 1994)

Figure 5: Result of the signal in Fourier transform (Gao & Yan 2011)

Figure 6: (a): Time series. (b) Fourier transform (Vetierli & RIOUL 1991)

Figure 7: Short-time Fourier transform (STFT), time-frequency resolution of window sizes τ and τ/2(Gao & Yan 2011)

Figure 8: Time and Frequency resolution of the wavelet transform (Gao & Yan 2011).

Figure 9: Harr wavelet φ (t) (Kaiser 2011)

Figure 10:.Biorthogonal wavelet (Kaiser 2011)

Figure 11: Symelt wavelet (Kaiser 2011)

Figure 12: Meyer wavelet (Kaiser 2011)

Figure 13: Shannon wavelet. (Kaiser 2011)

Figure 14: Different Coiflet wavelets (Sai et al. 2014)

Figure 15: Different types of Daubechies wavelet

Figure 16: Process flow diagram of denoising

Figure 17: Level 1 Decomposition

Figure 18: Level 3 Decomposition

Figure 19: Level Two Decomposition Filter Bank Tree

Figure 20: Level 2 Decomposition Flow Diagram (Misiti et al. 2015)

Figure 21: Level 2 simulation results of Decomposition

Figure 22: Graphical representation of Original thresholding(Aggarwal et al. 2011)

Figure 23: Graphical representation of hard thresholding(Aggarwal et al. 2011)

Figure 24: Graphical representation of soft thresholding(Aggarwal et al. 2011)

Figure 25: Level two Approximation filter bank tree

Figure 26: Level 2 simulation results of Reconstruction (output after Thresholding)

Figure 27: Representation of different categories of signals processed (MATLAB).

Figure 28: Illustration of Approximation and Detail coefficients (MATLAB)

Figure 29: Illustration of Denoising Audio Signal by Thresholding

Figure 30: SNR simulation results (MATLAB)

Figure 31: SNR of contaminated signal Vs. SNR of denoised signal at level 1

Figure 32: SNR denoised Vs. SNR achieved at level-2

Figure 33: SNR of contaminated signal Vs. SNR of denoised signal at level 2

Figure 34: SNR denoised Vs. SNR achieved at level-2

Figure 35: SNR of contaminated signal Vs. SNR of denoised signal at level -3

Figure 36: SNR Denoised vs. SNR achieved at level-3

Figure 37: Average SNR results at each level

List of Tables

Table 1: SNR results for selected wavelets at level-1 Decomposition

Table 2: Estimated SNR for selected wavelets at level- 2 Decomposition

Table 3: Results of SNR for level-3 Decomposition

Table 4: Resultant best outcome of level and wavelet

Abbreviations

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Chapter 1: Introduction and Objectives of the Project

1.1 Introduction

The wavelet transform is one of the operative transforms for processing one-dimensional audio signals, where it provides outstanding results on audio signals even if wavelets transform is operating on multiresolution frequencies. Furthermore, one-dimensional wavelet transform can allow the signal to deposit more efficiently during the signal processing. In contrast to this, Fourier transform offers a poor form of spectrum information over the time. Moreover, due to the under capability of Fourier transform to perform for non-periodic signals made short time Fourier transform (STFT) to come into force. However, the problem with the advanced short time Fourier transform was regardung the use of fixed window size. In contrast to it, wavelet came in force to provide variable window sizes for analysing different frequency components inside an audio signal .in addition to it there are many other abilities of wavelet transform those makes to choose wavelet transform over Fourier transform and short-time Fourier transform (STFT). The wavelet transform also has the special capability to denoise an audio signal by using some wavelet predefined techniques. The objective of audio denoising is attenuating noise. Since, all non-stationary signal processing applications are continuously troubled by various types of realistic noises, which creates major damage to the non-stationary signals. This noise may be in any form; it may be in white noise, pink noise, realistic noise or another different kind of noises. In such cases, wavelet transform helps in the analysing irregular signal by decomposing the signal from the fundamental process called as scaling , where the information will get partitioned intensely by shrinking the magnitude of the audio signal. Also, decomposition process generates various approximation and detail coefficients .but the detail coefficient still contains noise residuals . Hence to reduce noise level, the hard or soft thresholding techniques can be used .After that, finally the original audio signal is reconstructed using reconstruction process. Moreover, all these various techniques of denoising audio signal using wavelet transform are examined using a signal to noise ratio (SNR) .As a result, to increase the quality of signal during denoising process , the signal to noise ratio of the denoised signal have to be higher than noisy input signal. Based on signal to noise ratio finest choice wavelet for particular audio signal de-noising can be made.

1.2 Project Summary.

This proposed research will cover some of the chapters which provide unique knowledge for the readers. This research will comprise of six various chapters created for the study on comparative analysis of selected wavelets in denoising an audio signal from realistic noise .so the each chapter in this proposed research will deliver,

Chapter 1

Chapter 1 will introduce wavelet transform to the readers and delivers some important features objective of this investigation.

Chapter 2

This chapter will provide the information on the development of wavelet transform, different types of wavelet transform which are related to this research and the impact of multiresolution analysis on wavelets.

Chapter 3

This chapter will all about the literature review, which takes readers towards the wavelets transform starting from, its comparison with Fourier transform, various wavelet functions and some wavelet application areas .moreover, this chapter will also provide some knowledge about denoising audio signal and illustration on realistic noises.

Chapter 4

This chapter will describe the different phases of denoising the one-dimensional audio signal. Mainly, the way how these phases influence in attaining original audio signal will be discussed.

Chapter 5

This chapter will be a complete examination of results obtained by programming in the MATLAB.it is consider to be a major chapter for this particular research, where all results are analysed comprehensively with their different signal to noise ratio(SNR).

Chapter 6

After analysing the selected number of wavelets at various levels of decomposition through MATLAB, it is very crucial to use translate those (SNR) results achieved into appropriate conclusion.so this chapter will deliver conclusions for this research with finest choice of wavelet with its corresponding level. Further, this chapter will also provide basic information about further developments and future work.

Chapter 2: Theoretical Examination of Wavelet Transform

2.1 Principle of Wavelet Transform

One the ultimate solution for representing a signal in both time and frequency information concurrently is Wavelet transform which is desirably capable, to make it simple, it is possible to pass the time domain signal through several low and high pass filters , indeed which gives the filtered portions of the signal. Lastly, it is necessary to repeat the process again and again , where some of the time portions of the signal corresponding to that particular frequency has been removed from the signal.

The complete theory and theorems regarding wavelet transform has been completely described by writer Robi Polikar in the second edition of “Fundamentals concepts and an overview of the wavelet theory”.

2.2 Development of Wavelet Transform

The idea of wavelets has been not the idea which came into existence recently. The idea started since the early nineteenth century when Joseph Fourier discovered the superposition of sine and cosines to represent the functions. Before 1930, the leading branch of mathematics regarding wavelets began with the help of Fourier synthesis, which is the sum of any - periodic function is as given below equation.

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Where, and are the coefficients can be calculated by 2.1.1

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This Fourier synthesis played an essential role in the evolution of the ideas for Fourier series convergences and orthogonal systems. Furthermore, the mathematicians founded a new analysis rather than the frequency analysis which can analyse the periodic function by creating mathematical structures that vary scale(Graps 1995).

The first mention of wavelets appeared in addition to the thesis of Harr (1909) through his concept called with compact support. However, Harr wavelet was somewhat not continuously differentiable and Harr wavelet had limited applications. Further, several researchers discovered scale varying functions for representation of periodic functions known as Harr basic function. With the help of Harr basic function in 1930 physicists, Paul Levy invented Brownian motion, which was superior to Fourier basic function(Graps 1995).

Later in between 1960 to 1980 the two mathematicians known as Guido Weiss and Ronald R Coifman came up with the simplest way of expressing atoms using elements of space functions , it allowed reconstruction of all elements with the help of common function and assembly rules(Graps 1995).In 1982, a French geophysicist called as Jean Morlet familiarised wavelets with his study of seismic signal analysis. Within few years, another geophysicist known as Alex Grossmann came up with a formula for inverse wavelet transform, thus, with the residuals of old concepts, the two geophysicists collaborated to give the study on various applications of wavelets transforms for decomposing a signal. The wavelet analysis is originally introduced to detect and analyse unexpected change in signal. Because, in Fourier transform, the time-frequency analysis of signal has a drawback of not retaining the local information. So, the windowed Fourier transform was introduced by Dennis Gabor(Sifuzzaman et al. 2009).

It can be said from the views of Barford, Fazzio and Smith (1992) the disadvantage of using Fourier transform is a limitation to the stationary signal analysis , hence, the short time Fourier transform was introduced to exploit nonstationary signal analysis although it was limited to analysing time restricted elements. Their views also explain that wavelet transform has more similarities with the short time Fourier transform.it has been expressed by Barford et al. (1992) that the multiresolution analysis has been the motivation to find the limitation in analysing non-stationary signal.

Another limitation of Fourier transform according to the Mallat (2009) is gathering information in sharp spikes is impossible .whereas with the help of wavelet transform it is a practical method.

Although the short time Fourier transform has similarity to Hossain, Amin (2011) states that the wavelet has the additional feature of varying window width for spectrum analysis with the help of mother wavelet and scaling function.

In 1985, Stephane Mallat through the knowledge of digital signal processing gave additional features to wavelets such as the relationship between quadrature filters, orthogonal wavelets, and pyramid algorithms. From these results wavelets y. Mayer developed non-trivial wavelets, which is continuously differentiable but non for compact support(Graps 1995).

After a couple of years, Debuchehies used Mallats work to develop wavelet orthogonal basis function, which is one of the most efficient and has more applications in the field of digital signal processing even at present(Graps 1995a).

2.3 Wavelet transform

It has been already debated in the previous sections that the wavelet transform was driven by the concept of Fourier transform signal analysis. Likely, to decompose the signal and reconstruct it Joseph Fourier used vector basis function. The basis vector function concept delivers that any function could be represented by the direct product of any basis function and its equivalent coefficients. Further, this basis function of complex sinusoidal windowed by the function g (t) and centred around was used in the case of short-time Fourier transform.

So the function f (t) of short-time Fourier transform (STFT) would be represented by means of basis vector as

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The vector basis function in the equation (3.6) is =. Here, these windowed basis vector functions are famed by their position and its frequency . In the very similar way the wavelet transform can also be illustrated in terms of basis function by substituting frequency variable with scale variable and position variable with shift variable b.so, the transformed equation for basis function is written as

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2.3.1 Continuous wavelet transform

To overcome from the resolution limit of the short time Fourier transform, it is believed that the time and frequency planes have to be varied with respect and during filter analysis. In contrast to this, after many years research on wavelets Gabor, Morlet and Grossman came up with an ultimate coloration of theoretical physics and signal processing, which in result, assisted to formalise continuous wavelet transform. Further, it was derived by this analysis that any wavelet wordlist is constructed from the mother wavelet.moreover, the mother wavelet constantly equal to average value of zero.

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By using equation (3.6.1), the theoretical expression for continuous wavelet transform can be driven as,

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Where, is as similar to the equation 3.6.1 with a substitute of a and b in the place of and .that is,

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This equation (2.5.1) is also called as mother wavelet function for generating wavelet function.

In contrast, the main reason behind the choice of discrete wavelet transform over continuous wavelet transform is in the stage of information redundancy is more in continuous wavelet transform. Also, the overlapping time-frequency windows of takes place. As a result, insignificant failure in signal feature extraction. Additional information regarding disadvantage of using continuous wavelet transform could be read from the book written by Olkkonen (2011) on “discrete wavelet transform for non-stationary signal processing”.

2.3.2 Discrete wavelet transform

Discrete wavelet transform has many valuable properties that are beneficial in the time series data removal field. Therefore, it is essential completely to realise the underpinning information about discrete wavelet transform thoroughly.as a result, in this section, the whole benefits and functionalities of discrete wavelet transforms would be discussed. However, the relevant information regarding this subject is described in the books of authors Jensen and Cour-Harbo (2001), Kaiser (1995).

Discrete wavelet transform (DWT) uses a set of basis function came from the wavelets, to convert a time series signal .the vital reason behind the renovation is to reduce the noise. To make it simple. First, the time series signal is operated on a set of a mathematical function of wavelets to decay signal into different constituents and then discrete wavelet transform separates these constituents into a different frequency at various scales. Moreover, in time-frequency plane, the less information redundancy can be achieved by transforming the original signal discretely (Olkkonen 2011).Likewise, it is necessary to guarantee the detailed reconstructed of original signal x (t) based on, in what way to sample the coefficients a and however, there are different level of decomposition based on numerous methods of wavelets. Further,in coming to few pages, the examination of reducing in redundancy and reconstruction method of discrete wavelet transform would be detailed.

To perform frequency scaling, First of all, let scaling factor , assume that and.

Likely, for scaling function to be in dyadic arrangement, is also assumed to be equal to zero (0).

As a result, the function will be moreover, is a dyadic wavelet that is,

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In conclusion, the corresponding wavelet transform will be dyadic wavelet transform.

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Further, it is necessary to develop duplicate wavelet, which has similar scale and time shift as exclusive wavelet Moreover, need to combine with unique wavelet. Because it is essential to recover original signal x (t). Accordingly, the duplicate wavelet transform is specified as

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Thus the relationship between and is given by

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Here the relationship factor in equation (2.7) is Fourier transform of duplicate wavelet of equation (2.6.1).

Consequently, the original signal can be easily recovered as.

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In order to recover original signal x (t), it is essential to provide condition called as stability condition to the relationship factor by assuming it to be within the constants and .given as,

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Furthermore, if the time domain signal required to be sampled, formerly assume, note that the coefficient would be chosen in such way that the original signal must be able to recover. In addition, the interval must be able to increase up to which can be completed by decreasing bandwidth and central frequency by times.

Therefore, the wavelet function for this case is given by

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Therefore, its wavelet transform will be

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Equation (2.11) is called as discrete wavelet transform for decomposition. In a similar way, to recover the original signal just have to do a summation of both continuous time and frequency plane.

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Where, is called as detailed wavelet coefficients

=dual wavelet

In contrast to this, there will be many demands might occur based whether is efficient to designate the as arbitrary signal and whether there is any data redundancy while decomposing the signal .but for all this, Daubechies emanated with the solutions with her phenomenal studies on wavelet frame theory.

Let us consider continuous function frame and containing two constants and both greater than zero (0).note that conditions for arbitrary signal is given by

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By any chance if = then the frame is said to very tight, if = then the frame is called as orthogonal basis function and the redundancy can also be measured in terms of frame bounds if frame vectors are normalised that is,.

Likely, frame operator is considered to be which is equal to

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So, it can be easily said by looking at equations (2.13.2) and (2.13.3) that.also, if = then,.In contrast, if there is a case, where the reconstruction is not detailed, the dual wavelet is given by

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In addition to it, the reconstructed signal can be formulated as,

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Here in the equation (3.15), is an error which is given by

Prospectively, from this formulations, it can be known that the greater information redundancy could be achieved by setting smaller values on constants and.

Now, it is clear to know that the equation (3.11) is the equation for discrete wavelet transform and the equation (3.12) used as signal reconstruction equation is additionally called as inverse discrete wavelet transform. Usually, will be considered as zero and to construct orthogonal wavelet basis normally the function space has to be adopted.

This is given by, and

Where, orthogonal basis vector is contains details of the function and the orthogonal filters gives decomposed original signal. Moreover, orthogonal wavelet basis can be constructed by orthogonal basis vector.this can be achieved by considering standard orthogonal theorem family as moreover, is given by the standard equation,

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Here, is scale function and is Fourier transform of .in case if is considered to the orthogonal basis of the vector then, the above equation it can be assumed that

By equation of, (2.16)

Similarly, equation in frequency is also given by,

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Where,

For the formulation of orthogonal basis function of order ,it is very crucial to consider another condition for evaluation of .because, to satisfy the basic condition.

Then, equations (3.17) and (3.18) can be represented using these condition as,

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Similarly,

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If all these equations are pooled as for one equation as of and .with respect to these conditions one can, be able to achieve following conclusions for orthogonal basis functions, those are given as.

and

and

After analysing completely this two results would provide that, H is low pass filter and G is band pass filter. In addition to it, by using the equation (2.15.1) and also the condition that has been specified in early is the orthogonal basis, so now the resultant arbitrary function holds same for both for H and G, which are described below.

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Likely, now it can also come to know that, there is orthogonality between and so,

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Remember, equation (2.22) certainly has to fulfil the properties specified in orthogonal basis function .therefore, in a direction to fulfil equation (2.22) only one solution has been achieved and this solution also represents the constructive method of an orthogonal wavelet basis function. Solution is given by,

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In addition to all this, to evaluate discrete wavelet transform of any given signal, for instance; suppose if signal has coefficients of and are estimated on and . As solution for these, Mallet came up with new innovated method of expressing the given signal by using fast algorithmic conducts. Note that: signal and is same as in equation (2.13).

Therefore, decomposition of given signal and reconstruction of the given original signal can be represented based on Mallets fast algorithm as,

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Indeed, equation (2.23) is simplified as.

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The equations (2.23) to (2.24.1) are used to evaluate approximated coefficients and detailed coefficients (Decomposition). In contrast, equation (2.25) is used to reconstruct the decomposed signal.

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Figure1: Method of Decomposition

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Figure 2: Method of Reconstruction

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Figure 3: Bandwidth Separation at each level

Referring to figure 1, it explains the method of decomposition by level two. The frequency bandwidth of original signal has to be separated into two equal amounts; simultaneously one portion has to pass through low pass filter h (k) and another portion has to pass through high pass filter g (k). Flowingly two times of downsampling has to conduct for both portioned signals, down sampling is because to take alternative samples from the signals. Further, the outputs from down samplings are considered to be detailed and approximated coefficients .in coming sections the method of decomposition will be clarified completely.

Similarly, referring to the figure 2, the detailed and approximated coefficients have to go through two time’s upper sampling process to combine each alternative samplings. Flowingly has to push through two filters to obtain original reconstructed signal back. Additional details about the reconstruction method will be explained in later sections.

Figure 3 clarifies that how the signal would divide into two equal parts from the high pass and low pass filters at each level of decomposition.in addition the same process is carried out in opposite way for reconstruction process which will describe in coming sections.

After discussing the analysis and progression of discrete wavelet transform, one can obviously note that it has many advantages over continuous wavelet transform such are easy to implement, improvement in redundancy, frequency analysis of signal based on different resolutions is great.

2.4 Multiresolution Analysis (MRA)

2.4.1 Philosophy of Multiresolution Analysis:

The faultless frequency resolution obtained by many window outcomes of Fourier transform will not be able to provide complete information redundancy on time for non-stationary signals. Fourier transform can fulfil this apptitude through two different properties of windows for only stationary signals.

- Narrow window – Basically, delivers better time resolution but the outcome of frequency resolution is impoverished.
- Wide window- provide rich frequency resolution and reduced time resolution.

So, with the help of both these windows, it is possible to achieve time-frequency resolution efficiently.However, it has also been noticed that there are chances where a wide window might disrupt to the stationary signal. Therefore, care has to be taken while choosing the method of processing signal or image. Due to this damage of stationary signal from the wide window, there is necessary to construct efficient window which is best for processing time-frequency resolution. As a result, the multiresolution analysis came into presence.

Mallat and Meyer first introduced the concept of multiresolution analysis. Moreover, it is the framework for constructing orthonormal wavelet bases which are disused under the section of discrete wavelet transforms. Hence, it is clearly noticeable that the Heisenberg uncertainty standard is satisfied, the time and frequency resolution will have a problem with every transform used till. However, there is only one substitute for this kind of difficulties that is by using multiresolution analysis (MRA).As the concept of multiresolution itself specifies it is used to inspect the numerous frequencies containing disparate resolutions in the original signal. Importantly the time localization of spectral components can be extracted with the help of wavelet multiresolution analysis.

2.4.2 Features of MRA

It is known that due to better performance and desirable properties of time-frequency resolution and information redundancy of discrete wavelet transform has forced to solve a different kinds of problems in analysis .Moreover, the concept multiresolution utilizes two sets of function to analyse signal of different resolution with various frequencies.one is the scaling function and other is wavelet function which is connected to low pass filter and high pass filters correspondingly. Indeed, the scaling function and wavelet are obtained by the weighted sum of shifted and scaled components. Furthermore, the different scaling translation of the wavelet function will make possible to get several time and frequency localizations of the original signal.

Additionally, multiresolution analysis is intended to sustain its features of maintaining decent time resolution and reduced frequency resolution at higher frequencies and good frequency resolution and poor time resolution at low frequencies. Moreover, this kind of approach makes sense for the real time signals, where there is an extended period of low-frequency constituents and fewer period of high-frequency constituents. Therefore, in the process of analysing the original signal different frequency with various resolutions, it is a benefit to practising the concept of multiresolution analysis over Fourier windows particularly for practical signals(Polikar et al. 2007).

2.4.3 Properties of Scale and Time- Frequency Resolution

2.4.3.1 The Scale:

The scale properties deliver that if the signal is scaled by the greater amount, then the outcome would be less detailed coefficients of the signal. In contrast, lower the magnitude of signal would effect in better-detailed coefficients and this process is called as signal pyramids(Vetierli & Rioul 1991). Correspondingly, less the scale at high frequency will compress the information for a shorter period, and more scale at low frequency will deliver comprehensive data.

Provided the original sequence of the signal, it is evident to drive the lower resolution signal through low pass filter. Because, according to Nyquist’s rule, the scaling in the signal has to double to get rid of detailed coefficients by losing high-frequency details. Moreover, this will yield in the resolution variation of detailed information, because, as if the signal is reserved under mathematical examination, it illuminates that the change of scale is due to down sampling of signal where the signal is shifted and passed through low pass filter. Similarly, the scaled signal is sent through high pass filter to yield shifted version of approximated coefficients(Polikar et al. 2007).

Probably, it is evidently seen by the definition of the discrete wavelet transform, the scaling factor (S) is in the denominator segment of parity. As a result, the broadened value of S is seen for the condition satisfying S>1.likewise, for S<1 condition , the scale value will shorten the signal for any kind wavelet applications.(Debnath 2003) (Polikar et al. 2007).

In summary, it is examined that the choice of scale parameter depends on the property of signal .however.it is efficient to use lower scales parameter provides improved scale resolution and reduced frequency resolution. Correspondingly, for more ambitious higher scales parameter gives lower frequency for better frequency resolution.

2.4.3.2 Time and frequency resolution (TFR)

In multiresolution analysis time and frequency resolution of the wavelet transform is identical parameter .in order to view the capability of time frequency resolution an illustration has been showed in Figure 4 is normally used to illuminate the idea regarding the interpretation of time frequency resolution on wavelet transform.

The following figure expresses that the complete area is constant, and each block in the area characterises time-frequency plane of equal measurement. In case of lower frequencies, For the purpose of larger scaling that is to obtain better frequency resolution the height of each are tinier, and their width are larger.In contrast to it, for lesser scaling purpose at a higher frequency, each block has extra heights and reduced widths.

Consequently, in any case, if the figure is compared with earlier section of short time Fourier transform, where it uses window function for analysis, both the time and frequency is constant through the entire window. However, in the case of wavelet transform can use different mother wavelet function for extracting the time-frequency resolution from various areas. However, there is a limitation in adjusting or changing the size of the blocks due to Hisenburgh uncertainty principle. However, within this restriction, it is possible to change by varying the dimension of mother wavelet without harming the entire area(R.Polikar 1994).

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Figure 4: An illustration of Time-Frequency Resolution (R.Polikar 1994)

Chapter 3: Literature review

3.1 Overview

Wavelets are functions used for the representation of certain signals and mathematical requirements. The knowledge of analysing audio signals and scale that is used for the wavelets analysis makes it effective in processing these signals. For many periods of the same use of sine and cosine functions for the representations of signals, scientists needed more appropriate functions other than sine and cosine. After doing some researches, the Fourier transform came into replacements to analyse the lost signal.However, due to the drawback of Fourier transform offering poor performance on sharp signals the wavelet transform came into existence.it has been thought that the wavelet transform can approximate the digital signal and images even at the sharp spikes, later it has been observed wavelet analysis become unsuccessful in approximating the signals in electronic data. However,,the successful use of wavelet transform for signal analysis has been most used on time scaled applications even at present criteria. Furthermore, by following in this chapter the advancement of wavelet transform will be described, and, about de-noising audio signals with the use of various features of wavelet transform will be brought into concern and finally, some information about the realistic noises (their existence) will be discussed in the following part.

3.2 Short time Fourier transform

Short time Fourier transform can be defined as analysing the signal with fixed resolution .the instantaneous frequency has to be often considered as a way to familiarise depending on time.in order to obtain continuous frequency components, the two-dimensional time and frequency s (t, f) representation of signal x(t) composed by time dependence spectral characteristics is necessary. Therefore, mathematician Gobar adapted Fourier transform successfully for the first time to represents (t, f).

If the signal x (t) is been considered as a stationary and it has finite extent g (t) through time location.then the Fourier transform of windowed signals will predominantly give short time Fourier transform which can be stated as follows(Vetierli & Rioul 1991).

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Where, (t, f) is two-dimensional function of time an frequency plane denotes the signal is window function

Furthermore, expect the addition of window function all the other functions are same in short time Fourier transform (STFT).

A possible drawback of short time Fourier transform is related to the time and frequency resolution that is by considering the capability of the short time Fourier transform whose Fourier transform of a windowed function is G (f) and it is under two pure sinusoidal functions. Then the bandwidth of the filter is given as.

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Where denominator term is the energy of g (t).

Similarly, the spread in time is given as

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In this case, the condition of discrimination for two sinusoidal will be satisfied only if they are apart between the two periods for the given function. Now, it is to be noted that that the bandwidth product of time and frequency must be lower bounded, which is.

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As a result, this is called as Heisenberg uncertainty principle, which states that one can only measure only time resolution for frequency resolution for finite interval or the other way, which means it is hard to know what spectral component resists in the particular period (Hill 2013).

3.3 From Fourier Transform to Wavelet Transform

Fourier transform is one of most commonly used signal processing tool. It exposes the frequency composition of time series signal x (t) by converting into the frequency domain. In 1807 French mathematician Joseph Fourier establishes that it is possible to represent the weighted summation of sine and cosine function using any periodic functions.in the book titled as , The analytical theory of heat (Fourier 1822), prolonged his research by representing the signal by series of integrals of sine and cosine functions known as a periodic signal x(f) (Gao & Yan 2011).

By using inner product arrangement, the Fourier transform of a signal x (t) can be represented as

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Similarly, the inverse Fourier transform of the signal x (t) can be stated as

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Figure 5: Result of the signal in Fourier transform (Gao & Yan 2011)

However, by viewing Figure 5, it can be said that the frequency signal components are not continuously present during the observation to time components. Thus, the Fourier transform is not appropriate for examining nonstationary signals. Moreover, it is essential to switch to the new signal processing technique, which is been described in the following section, for dealing with nonstationary signals.

3.4 Comparison of wavelet transforms with Fourier transform

In this specific segment, the operative capabilities of wavelets will be compared with Fourier transform grounded on their resemblances and differences. The Fourier transform has the feature of generating a data structure equal to log2 n segments similar to discrete wavelet transform.in addition to it, both basis functions of discrete wavelet transform and Fourier transform are restricted in frequency, which indeed makes some mathematical tools to take part in spectrum analysis (Graps 1995). Similarly, both wavelet transform and Fourier transform are represented using integral function, but Fourier customs a correlation process with exponential function and wavelet transform customs a correlation process with the transformation of any analysing wavelet. Besides, both Fourier transform and wavelet transform can consume real and complex valued function. However, the output of Fourier transform limits to complex only. whereas, wavelet transform provides both real and complex values as output (Barford et al. 1992).Fourier spectrum analysis is the leading systematic instrument for frequency domain analysis. But, Fourier transform provides very poor form of information with spectrum variation with respect to time. In contrast, wavelet transform provides the best time-frequency information with respect to time. Moreover, neither Fourier transform nor short time Fourier transform has ability to deal with the non-stationary signals. Alternatively, wavelet transform has sufficient capability to perform with the non-periodic audio signals with different transient (Mihov et al. 2009).Fourier transform is localised to only single function and it only scale only one function .on the other side, using wavelet transform it is possible to shift and scale the function simultaneously.

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Figure 6: (a): Time series. (b) Fourier transform (Vetierli & RIOUL 1991)

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Figure 7: Short-time Fourier transform (STFT), time-frequency resolution of window sizes τ and τ/2(Gao & Yan 2011)

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Figure 8: Time and Frequency resolution of the wavelet transform (Gao & Yan 2011).

Figure 6(a) shows the time series representation of any signal x (t) whereas Figure 6 (b) Represents Fourier transform of any signal x (t).Additionally, it can be seen from Figure 6 (b) there is a lack of frequency information redundancy with respect to time. That is, by any chance one need to know about frequency in particular time it will be impossible. Due to this limitation of Fourier transform, short time Fourier transform came into life where it uses a window function to decompose a time domain signal to time-frequency representation.

Figure 7 illustrates the effect of short time Fourier transform and Gaussian window size on time and frequency resolution .in the analysis of signal x (t), irrespective of the actual window size the total product of the time and frequency resolutions are same. That is, during the signal analysis, the time and frequency resolutions will remain fixed over chosen window function. In contrast, the width of the window in fast time Fourier transform cannot be altered on modification in time simultaneously. Moreover, frequency information () with respect time () is not much appreciable in short time Fourier transform. Because, much redundancy of frequency information is limited to lower frequencies. Where as in continues wavelet transform the frequency redundancy is greater even at higher frequencies and lower frequencies. But, as far as the figure 8 of continuous wavelet transforms is concerned, the time and frequency resolutions, where and are constants, are directly or inversely proportional to scaling parameters. Additionally, figure 8 illustrates that at higher frequencies, wavelet transform with its time shifting of base wavelet function, is capable of mining the coefficients through entire spectrum of time series by the service of small scales and alternatively uses large scales at lower frequencies.

3.5 Wavelet Functions (WF)

This section will provide information about various kinds of discrete wavelet transform functions depending on their characteristics. There numerous types of wavelets used to analyse a signal coefficients. Furthermore, it is time to know about operations carried after setting mother wavelet functions. Every wavelet functions are set initially at zero time, the first scale wavelet function is combined with the input signal and constant for normalisation purpose, before estimating final equation .later, is shifted based on values of t. Likewise, they are described below.

- Harr wavelet: wavelet basis of a Harr wavelet is made using the rescaled sequence of square shaped functions(Sai et al. 2014). Harr wavelet mother wavelet and scaling function are described as,
- Biorthogonal wavelet: it is a kind of wavelet is not certainly orthogonal but the associated wavelet transform is invertible. Moreover, a graphical representation of biorthogonal wavelet is shown in Figure10 (Sai et al. 2014).
- Symlet wavelet (SYM): it is vital to know that symlet wavelet is defined for any positive integer value (n). Wavelet function and scaling function will always have solid support measurement of 2n.symelt wavelet may be also called as “smallest distorted” wavelet. Likewise, its graphical representation has been shown in Figure 11.
- Meyer wavelet: Mayer wavelet is discrete wavelet function uses Fourier transform to derive discrete time wavelet coefficients. The graphical way of expressing Meyer wavelet is shown in Figure 12.
- Shannon wavelet: Shannon wavelets are defined by ideal band pass filter during signal analysis and wavelet may be a real value or complex value in the analysis(Sai et al. 2014).It is also great to know that both Shannon and Harr wavelets are Respectively Fourier duals. Shannon wavelet graph is shown in Figure 13.
- Coiflet wavelet: it usually has N number of fading movements for both scaling functions and wavelet. There are various levels in Coiflet using which the signal analysis is conducted. For Example, coif1, coif 2, coif3, coif 4, coif5 so on , this graphical representation can be seen in the below Figure14.
- Daubechies wavelet (dB): this mother wavelet was introduced by mathematician Daubechies fits into orthogonal wavelets, and it always defines a discrete wavelet transform. Surprisingly, Daubechies wavelet has the most amount of fading movements, and each of Debucheies class has own scaling functions for producing an orthogonal multiresolution analysis(Sai et al. 2014). Similar to Coiflets even Daubechies has various classes which are shown in Figure 15.

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Figure 15: Different types of Daubechies wavelet

3.6 Applications of Wavelet Transform

This particular section is all about the various fields in where wavelets are implied for accomplishing the purpose of specific work. Moreover, some of the fine explorers have used wavelets in various areas of applications, where the wavelet transform plays a crucial role in specific applications.

Application of Wavelets Analysis to Turbulence:

Several scholars like Vincent, Meneguzzi and Ashurst have researched for many years on turbulence for developing high resolution direct numerical simulations .that indeed, helps fundamentally to analyse flows in turbulence in all three dimensions.in contrast, it became difficult to analyse turbulence model in the basic way of flow. Due to this problem further, academics including Frohlich ,Meneveau developed wavelet analysis to inspect intrinsic flow in turbulence(Debnath 1998). Most importantly, the wavelet transform of two-dimensional to characterise plain arrangements are implemented.

Wavelet Transform Application on Non-Stationary Electrocardiogram (ECG) Signal Processing:

Electrocardiogram (ECG) signal characterises variation in electrical potential in the course of cardiac cycle recording among the exterior conductors and the human body. Likewise, the outcome signal is the result of propagations between the heart and cardiac muscle. In this case, wavelet analysis plays a crucial role characterise oscillations of heartbeat rate below adjustable physiological circumstances. Additionally, the wavelet analysis helps the signal to split the original signal to various scales and according to analyst called Hammer states that, analysis on blood pressure signal is accomplished with the support of time-frequency properties of the wavelet transform. Even for de-noising electrocardiogram (ECG) signal wavelet functions such as Db4, coif5, sym7, and thresholding techniques plays very imperative roles. That is with the help of major thresholding effect it is possible to remove the noises in electrocardiogram (ECG) signals .for instance: noise removal methods such as signal to interference, noise power and percentage root mean square difference uses wavelet thresholding rule for efficient de-noising electrocardiogram (ECG) signals(Debnath 1998).

In addition to this, there are many other fields where the statistical tool called as wavelet transform can be applied in existing creations, specifically

- Information compression
- Fingerprint authentication
- Image de-noising and flattening
- DNA analysis
- Speech recognition
- Computer visuals and multifractal analysis
- Edge and Angle estimation
- Several area in the field where physics is applicable such as molecular dynamics, quantum mechanics and turbulence mechanics.

3.7 Audio Signal Denoising Using Wavelet Transform

The signal processing applications will always be distracted by different types of noises as the unwanted signal gets an overlay on the actual signal .for an instance, during the transitions of the audio signal over the communication lines, results in signal impurity. The signal that has been polluted need to be removed completely, however, it is challenging to remove impurities in the original signal without harming the original signal(Jaishankar and Duraiswamy 2012). There are numerous applications in denoising audio signal such as music reestablishment and speech restoration. Furthermore, diagonal and non-diagonal estimations are two methods used for de-noising an audio signal .by making use of magnitudes of time-frequency depiction of signal the spectral audio de-noising method could be used. However, the realistic noise is located in all frequencies so it is hard to remove it from the original audio signal. In this case, the signal has to be represented in high amplitude with the use of discrete wavelet transform(Badkul and Chourasiya 2015).

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