Essay: Wavelet transformation CHAPTER 1

INTRODUCTION
1.1 Introduction
Filters are the manipulation of the amplitude and/or phase response of a signal according to their frequency. These are the basic components of all signal processing and -telecommunication systems. There are two kinds of filters- fixed and tunable. Fixed filters are those in which passband frequencies and stopband frequencies are fixed whereas in case of tunable filters, passband and stopband frequencies are variable. These frequencies can be changed according to the requirement of the applications. Tunable digital filters are widely employed in telecommunications, medical electronics, digital audio equipment and control systems. These filters are also known as variable digital filters [17]. Tunable digital filters are used in telecommunication system in the front end of a receiver to select a particular band of frequencies. In medical electronics, tunable notch filters are used to suppress the power line interference [11]. The bases for the design of the tunable digital filters are the spectral transformation [8] [18]. It is basically used to modify the characteristics of a filter to meet new specifications without repeating the filter design procedure. This modification is done by changing a Low pass(LP) digital filters to Low pass(LP) filters with different cutoff frequencies or to a High pass(HP), Band pass(BP) or Band stop(BS) filters. The variable Band pass (BP) and Band stop (BS) filters are used to eliminate and retrieve some narrow band signals. In [13] variable band pass and band stop filters are shown with high accuracy and independent tuning characteristics.
1.2 Audio Noise Reduction
Audio noise reduction system is the system that is used to remove the noise from the audio signals. Audio noise reduction systems can be divided into two basic approaches. The first approach is the complementary type which involves compressing the audio signal in some well-defined manner before it is recorded (primarily on tape). The second approach is the single-ended or non-complementary type which utilizes techniques to reduce the noise level already present in the source material’in essence a playback only noise reduction system [18]. This approach is used by the LM1894 integrated circuit, designed specifically for the reduction of audible noise in virtually any audio source. Noise reduction is the process of removing noise from a signal. All recording devices, both analogue or digital, have traits which make them susceptible to noise. Noise can be random or white noise with no coherence, or coherent noise introduced by the device’s mechanism or processing algorithms. Their is a Active noise control (ANC), also known as noise cancellation, or active noise reduction (ANR), is a method for reducing unwanted and unprocessed sound by the addition of a second sound specifically designed to cancel the first[26]. Sound is a pressure wave or we can say sound is the analog signals that are processed according to their frequency, which consists of a compression phase and a rarefaction phase. A noise-cancellation speaker emits a sound wave with the same amplitude but with inverted phase (also known as anti phase) to the original sound. The waves combine to form a new wave, in a process called interference, and effectively cancel each other out – an effect which is called phase cancellation.
Modern active noise control is generally achieved through the use of analog circuits or digital signal processing. An Adaptive algorithms are designed to analyze the waveform of the background no neural noise, then based on the specific algorithm generate a signal that will either phase shift or invert the polarity of the original signal. This anti phase is then amplified and a transducer creates a sound wave directly proportional to the amplitude of the original waveform, creating destructive interference [8]. This effectively reduces the volume of the perceivable noise. The transducer emitting the noise cancellation signal may be located at the location where sound attenuation is wanted (e.g. the user’s ear/any music/headphone sound). This requires a much lower power level for cancellation but is effective only for a single user.
1.3 Types of Noises
There are many types and sources of noise or distortions and they include:
1. Electronic noise such as thermal noise and shot noise,
2. Acoustic noise emanating from moving, vibrating or colliding sources such as revolving
Machines, moving vehicles, keyboard clicks, wind and rain,
3. Electromagnetic noise that can interfere with the transmission and reception of voice, image and data over the radio-frequency spectrum,
4. Electrostatic noise generated by the presence of a voltage,
5. Communication channel distortion and fading and
6. Quantization noise and lost data packets due to network congestion.
Signal distortion is the term often used to describe a systematic undesirable change in a signal and refers to changes in a signal from the non-ideal characteristics of the communication channel, signal fading reverberations, echo, and multipath reflections and missing samples [10]. Depending on its frequency, spectrum or time characteristics, a noise process is further classified into several categories:
1. White noise: purely random noise has an impulse autocorrelation function and a flat power spectrum. White noise theoretically contains all frequencies in equal power.
2. Band-limited white noise: Similar to white noise, this is a noise with a flat power spectrum and a limited bandwidth that usually covers the limited spectrum of the device or the signal of interest. The autocorrelation of this noise is sinc-shaped.
3. Narrowband noise: It is a noise process with a narrow bandwidth such as 50/60 Hz from the electricity supply.
4. Coloured noise: It is non-white noise or any wideband noise whose spectrum has a non flat shape. Examples are pink noise, brown noise and autoregressive noise.
5. Impulsive noise: Consists of short-duration pulses of random amplitude, time of occurrence and duration.
6. Transient noise pulses: Consist of relatively long duration noise pulses such as clicks, burst noise etc.
1.4 Introduction to Wavelet Transform
Wavelet transform consists of a set of basis functions that can be used to analyze signals in both time and frequency domains simultaneously. This analysis is accomplished by the use of a scalable window to cover the time-frequency plane, providing a convenient means for the analyzing of non-stationary signal that is often found in most application [8].
Wavelet analysis adopts a wavelet prototype function known as the mother wavelet given as:
?? ( , s) = (1.4)
This mother wavelet in turns generates a set of basis functions known as child wavelets through recursive scaling and translation.
Where, s reflects the scale or width of a basis function,
is the translation that specifies its translated position on the time axis,
is the mother wavelet,
is the normalized factor used to ensure energy across different scale remains the same[10].
1.4.1 Applications of wavelet transform
The standard applications of wavelet transform are:
1. Signal Processing
2. Data Compression
3. Computer Graphics
4. Denoising
1.4.2 Why we use Wavelet Transform
The advantages of wavelet transform are as follows:
1) Space and Time Efficiency.
‘ Low Complexity of DWT.
2) Multiresolution Properties.
‘ Hierarchical Representation and Manipulation.
3) Generality and Adaptability.
‘ Different Basis and Wavelet Functions.
1.4.3 Continuous Wavelet Transform
Continuous wavelet transform of f (t), with respect to the wavelet is defined as
(1.5)
Where t is the translation coefficient and s is the scaling coefficient.
CWT analyzes the signal through the continuous shifts of a scalable function over a time plane. This technique results in redundancy and it is numerically impossible to analysis at infinite number of wavelet sets [10].
1.4.4 Discrete Wavelet Transform
Discrete Wavelet Transform (DWT) is introduced to overcome the redundancy problem of CWT. The approach is to scale and translate the wavelets in discrete steps as given in equation (1.6).
(1.6)
Where is the scaling factor, is the translating factor, k and j are just integers.
Subsequently, we can represent the mother wavelet in term of scaling and translation of a dyadic transform as
(1.7)
Replacing equation, the coefficients of DWT can be represented as [10]:
(1.8)
The Discrete Wavelet Transform is identical to a hierarchical sub band system where the sub bands are logarithmically spaced in frequency and represent octave-band decomposition[8]. By applying DWT, the image is actually divided i.e., decomposed into four sub-bands and critically sub sampled as shown in Figure 1.11:
LL1 LH1 LL2 LH2
LH1
HL2 HH2
HL1 HH1
HL1
HH1
(a) One-Level (b) Two-Level
Figure 1.11: Image Decomposition
These four sub bands arise from separable applications of vertical and horizontal analysis filters for wavelet decomposition as shown in Figure 1.12.
Figure 1.12: One level filter bank for computation of 2-D DWT and Inverse DWT
The filters h and g shown in Figure. 1.11 is one-dimensional Low Pass Filter (LPF) and High Pass Filter (HPF) respectively. Thus, decomposition provides sub bands corresponding to different resolution levels and orientation. These sub bands labeled LH1, HL1 and HH1 represent the finest scale wavelet coefficients i.e., detail images while the sub band LL1 corresponds to coarse level coefficients i.e., approximation image[10]. To obtain the next coarse level of wavelet coefficients, the sub band LL1 alone is further decomposed and critically sampled using similar filter bank shown in Figure 1.12 (a). This results in two-level wavelet decomposition as shown in Figure 1.12(b). Similarly, to obtain further decomposition, LL2 will be used. This process continues until some final scale is reached. The decomposed image can be reconstructed using a reconstruction (i.e., Inverse DWT) or synthesis filter as shown in Figure 1.12(b). Here, the filters g and h represents low pass and high pass reconstruction filters respectively [26].
1.4.5 Rows transformation
Wavelet decomposition in two dimensions begins with the one dimensional wavelet transform on each row of the image f(x,y). The decomposition process begins with convoluting the rows of f(x,y) with low pass filter coefficients to obtain L(x) and down sampled the wavelet coefficients to retain only the even indexed rows of f L(x,y). Next, repeat the process for the rows with high pass filter coefficients to obtain H(x) and similarly retain only the even indexed rows off H(x,y). The need for down sampling by 2 helps to reduce the size of the wavelet coefficients to the original size of f(x,y) as shown in Figure 1.13[10].
Figure 1.13: Down Sampling Process
1.4.6 Columns transformation
After completing the rows transformation, we perform the decomposition on each column of the image f(x,y). The columns of f(x,y) will convolute with low pass filter coefficients to obtain L(x) and down sampled the wavelet coefficients to retain the even indexed columns of fL(x,y). Repeat the same process for the high pass filter coefficients to H(x) and retains only the even indexed column of fH(x,y)[10].
1.4.7 Reconstruction
Wavelet reconstruction is the process of assembling back the wavelet coefficients to the original image without any loss of information. In wavelet domain, this is also known as inverse wavelet transform. Unlike decomposition process described above, the reconstruction process begins with the columns transformation followed by the rows.
To reverse the decomposition process, the reconstruction requires up sampling of the wavelet coefficients. Up sampling is a process that lengthens a signal component by inserting zeros between the samples as shown in Figure [10].
Figure 1.14: Up sampling Process
1.5 Wavelet Based Denoising System
In recent years, there has been a fair amount of research on wavelet-based image denoising. The paper published by Donoho and Johnstone (1994), developed a theoretical framework for denoising signals using Discrete Wavelet Transform (DWT).The method consists of applying the DWT to the original data, thresholding the detailed wavelet coefficients and inverse transforming the set of threshold coefficients to obtain the denoised signal.
Figure 1.15: Block diagram of DWT based denoising framework
Given a noisy signal y = x + n where x is the desired signal and n is independent and identically distributed (i.i.d) Gaussian noise N (0, ??2), y is first decomposed into a set of wavelet coefficients w = W[y] consisting of the desired coefficient q and noise coefficient n. By applying a suitable threshold value T to the wavelet coefficients, the desired coefficient q = T[w] can be obtained; lastly an inverse transform on the desired coefficient q will generate the denoise signal x=w-1[??] [10].
1.6 Wavelet Thresholding
According to wavelet analysis, one of the most effective ways to remove noise without smearing out the sharp edges features of an ideal image is to threshold only high frequency components while preserving most of the sharp features in the image [21]. The approach is to shrink the detailed coefficients (high frequency components) whose amplitudes are smaller than a certain statistical threshold value to zero while retaining the smoother detailed coefficients to reconstruct the ideal image without much loss in its detail. This process is sometimes called wavelet shrinkage since the detailed coefficients are shrunk towards zero [10]. The schemes to shrink the wavelet coefficients, namely the ‘keep-or-kill’ hard thresholding, and ‘shrink-or-kill’ soft thresholding.
The criterions of each scheme are described as follows: given ?? denotes the threshold limit, X denotes the input wavelet coefficients and Y denotes the output wavelet coefficients after thresholding [10].
1.6.1 Hard Thresholding
Y = Thard (X, Y) = {X where |X| ‘ ??
{0 |X| < ?? (1.9)
In the hard thresholding scheme, the input is kept if it is greater than the threshold ??, otherwise it is zero. The hard thresholding procedure removes the noise by thresholding only the wavelet coefficients of the detailed sub-band, while keeping the low-resolution coefficients unaltered [10].
Figure 1.16: Hard Thresholding
1.6.2 Soft Thresholding
Y = Tsoft (X, Y) = {sign{X} (X ‘ ??)+ where |X| ‘ ??
{0 |X| < ?? (1.10)
In soft thresholding scheme, if the absolute value of the input X is less than or equal to ?? then the output is forced to zero. If the absolute value of X is greater than ?? the output is |y| = |x ‘ ??|.
In practice, soft thresholding is more popular than hard thresholding because it reduces the abrupt changes that occurs in hard thresholding and provides more visually pleasant recovered images [10].
Figure 1.17: Soft Thresholding
1.7 Wav
Waveform Audio File Format (WAVE, or more commonly known as WAV due to its filename extension),(also, but rarely, named, Audio for Windows) is a Microsoft and IBM audio file format standard for storing an audio bit stream on PCs. It is an application of the Resource Interchange File Format (RIFF) bit stream format method for storing data in “chunks”, and thus is also close to the 8SVX and the AIFF format used on Amiga and Macintosh computers, respectively[17]. It is the main format used on Windows systems for raw and typically uncompressed audio. The usual bit stream encoding is the linear pulse-code modulation (LPCM) format [2].
1.8 Spectral density
In statistical signal processing and physics, the spectral density, power spectral density (PSD), or energy spectral density (ESD), is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz (Hz), or energy per hertz[16]. It is often called simply the spectrum of the signal. Intuitively, the spectral density measures the frequency content of a stochastic process and helps identify periodicities [7].
1.9 Power spectral density
The above definitions of energy spectral density require that the Fourier transforms of the signals exist, that is, that the signals are integrable/summable or square-integrable/square-summable. (Note: The integral definition of the Fourier transform is only well-defined when the function is integrable. It is not sufficient for a function to be simply square-integrable [16]. In this case one would need to use the Plancherel theorem.) An often more useful alternative is the power spectral density (PSD), which describes how the power of a signal or time series is distributed with frequency. Here power can be the actual physical power, or more often, for convenience with abstract signals, can be defined as the squared value of the signal, that is, as the actual power dissipated in a purely resistive load if the signal were a voltage applied across it[16].
1.10 Wavelet Families
The Wavelet Toolbox’ software includes a large number of wavelets that you can use for both continuous and discrete analysis. For discrete analysis, examples include orthogonal wavelets (Daubechies’ extremal phase and least asymmetric wavelets) and B-spline biorthogonal wavelets. For continuous analysis, the Wavelet Toolbox software includes Morlet, Meyer, derivative of Gaussian, and Paul wavelets.
The choice of wavelet is dictated by the signal or image characteristics and the nature of the application. If you understand the properties of the analysis and synthesis wavelet, you can choose a wavelet that is optimized for your application.
Wavelet families vary in terms of several important properties. Examples include:
‘ Support of the wavelet in time and frequency and rate of decay.
‘ Symmetry or antisymmetry of the wavelet. The accompanying perfect reconstruction filters have linear phase.
‘ Number of vanishing moments. Wavelets with increasing numbers of vanishing moments result in sparse representations for a large class of signals and images.
‘ Regularity of the wavelet. Smoother wavelets provide sharper frequency resolution. Additionally, iterative algorithms for wavelet construction converge faster.
‘ Existence of a scaling function, ??.
For continuous analysis, the Wavelet Toolbox software provides a Fourier-transform based analysis for select analysis and synthesis wavelets. See cwtft and icwtft for details.
For wavelets whose Fourier transforms satisfy certain constraints, you can define a single integral inverse. This allows you to reconstruct a time and scale-localized approximation to your input signal. See Inverse Continuous Wavelet Transform for a basic theoretical motivation. Signal Reconstruction from Continuous Wavelet Transform Coefficients illustrates the use of the inverse continuous wavelet transform (CWT) for simulated and real-world signals. Also, see the function reference pages for icwtft andicwtlin.
Entering waveinfo at the command line displays a survey of the main properties of available wavelet families. For a specific wavelet family, use waveinfo with the wavelet family short name. You can find the wavelet family short names listed in the following table and on the reference page for waveinfo.
Wavelet Family Short Name Wavelet Family Name
‘haar’ Haar wavelet
‘db’ Daubechies wavelets
‘sym’ Symlets
‘coif’ Coiflets
‘bior’ Biorthogonal wavelets
‘rbio’ Reverse biorthogonal wavelets
‘meyr’ Meyer wavelet
‘dmey’ Discrete approximation of Meyer wavelet
‘gaus’ Gaussian wavelets
‘mexh’ Mexican hat wavelet
‘morl’ Morlet wavelet
‘cgau’ Complex Gaussian wavelets
‘shan’ Shannon wavelets
‘fbsp’ Frequency B-Spline wavelets
‘cmor’ Complex Morlet wavelets
1.11 Types of Wavelets
‘ Daubechies wavelets
The Daubechies wavelets, based on the work of Ingrid Daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support. With each wavelet type of this class, there is a scaling function (called the father wavelet) which generates an orthogonal multiresolution analysis. The Daubechies wavelet transforms are defined in the same way as the Haar wavelet transform’by computing running averages and differences via scalar products with scaling signals and wavelets’ the only difference between them consists in how these scaling signals and wavelets are defined. For the Daubechies wavelet transforms, the scaling signals and wavelets have slightly longer supports, i.e., they produce averages and differences using just a few more values from the signal. This slight change, however, provides a tremendous improvement in the capabilities of these new transforms. They provide us with a set of powerful tools for performing basic signal processing tasks. These tasks include compression and noise removal for audio signals and for images, and include image enhancement and signal recognition. Ingrid Daubechies, one of the brightest stars in the world of wavelet research, invented what are called compactly supported orthonormal wavelets ‘ thus making discrete wavelet analysis practicable.
The names of the Daubechies family wavelets are written dbN, where N is the order, and db the “surname” of the wavelet. The db1wavelet, as mentioned above, is the same as Haar wavelet. Here are the wavelet functions psi of the next nine members of the family:
Figure 1.18: db Types
‘ Coiflets
Built by I. Daubechies at the request of R. Coifman. The wavelet function has 2N moments equal to 0 and the scaling function has 2N-1 moments equal to 0. The two functions have a support of length 6N-1. You can obtain a survey of the main properties of this family by typing waveinfo(‘coif’) from the MATLAB command line.
Figure 1.19: coif Types
‘ Symlets
The symlets are nearly symmetrical wavelets proposed by Daubechies as modifications to the db family. The properties of the two wavelet families are similar. Here are the wavelet functions psi.
Figure 1.20: symlet Types
1.12 Filter
Filters are networks that process signals in a frequency-dependent manner. The basic concept of a filter can be explained by examining the frequency dependent nature of the impedance of capacitors and inductors [17]. Filters have many practical applications. A simple, single-pole, low-pass filter (the integrator) is often used to stabilize amplifiers by rolling off the gain at higher frequencies where excessive phase shift may cause oscillations. A simple, single-pole, high-pass filter can be used to block dc offset in high gain amplifiers or single supply circuits[8]. Filters can be used to separate signals, passing those of interest, and attenuating the unwanted frequencies. There are a large number of texts dedicated to filter theory. An ideal filter will have an amplitude response that is unity (or at a fixed gain) for the frequencies of interest (called the pass band) and zero everywhere else (called the stop band). The frequency at which the response changes from passband to stopband is referred to as the cutoff frequency[19].
1.13 Basic Linear Design
The functional complement to the low-pass filter is the high-pass filter. Here, the low frequencies are in the stop-band, and the high frequencies are in the pass band. Figure shows the idealized high-pass filter[11].
Figure 1.21: Types of Filter [16]
If a high-pass filter and a low-pass filter are cascaded, a band pass filter is created. The band pass filter passes a band of frequencies between a lower cutoff frequency, f l, and an upper cutoff frequency, f h. Frequencies below f l and above f h are in the stop band. An idealized band pass filter is shown in Figure 1.1 [16]. A complement to the band pass filter is the band-reject, or notch filter. The idealized filters defined above, unfortunately, cannot be easily built. The transition from pass band to stop band will not be instantaneous, but instead there will be a transition region. Stop band attenuation will not be infinite [8].
Figure 1.22: Low Pass Filter [16]
1.13.1 High Pass Filter
H(s) = (1.1)
Figure 1.23: Frequency Response of Butterworth [16]
1.13.2 Band-Pass Filter
Changing the numerator of the low pass prototype to H0’0 will convert the filter to a band-pass function. The transfer function of a band-pass filter is then:
H(s) = (1.2)
Figure 1.24: Band-Pass Filter
1.13.3 Band-Reject (Notch) Filter
By changing the numerator to H0(S2+’z2), we convert the filter to a band-reject or notch filter. As in the bandpass case, if the corner frequencies of the band-reject filter are separated by more than an octave (the wideband case), it can be built out of separate low- pass and high-pass sections. We will adopt the following convention: A narrow-band band-reject filter will be referred to as a notch filter and the wideband band-reject filter will be referred to as band-reject filter.
H(s) =
Figure 1.25: Band-Reject (Notch) Filter
1.14 Problem Definition
The problem undertaken for the dissertation is ‘Audio Noise reduction using Discrete Wavelet Transformation’. The Current applications include noise propagation problem in industrial air handling systems, noise in aircrafts and tonal noise from electric power, as well as isolation of vibration from noise is one kind of sound that is unexpected or undesired . The noise related problem that I have studied can be divided into non-additive noise and additive noise. The non-additive noise includes multiplier noise and convolution noise, which can be transformed into additive noise through homomorphism transform. The additive noise includes periodical noise, pulse noise, and broadband noise related problems. The noise generated by the engine is one kind of periodical noise while the one generated from explosion, bump, or discharge is pulse noise problem that I have studied in literature survey. There are many kinds of broadband noise, which may include heat noise, wind noise, quantization noise, and all kinds of random noise such as white noise and pink noise.
Statistical relationship between the noise and speech; i.e. uncorrelated or even independent noise, and correlated noise (such as echo and reverberation). In acoustics applications, noise from the surrounding environment severely reduces the quality of speech and audio signals. Therefore, basic linear filters and DWT with thresholding and types of wavelet are used to denoised the audio signals and enhance speech and audio signal quality.
1.15 Chebyshev Type1 Filter
Chebyshev1 filters have a narrower transition region between the passband and the stopband. The sharp transition between the passband and the stopband of a chebyshev filter produces smaller absolute errors and faster execution speeds than a Butterworth filter. The poles of chebyshev1 filter lies on an ellipse [16]. Ripple increase (band), the roll-off becomes sharper (good). The chebyshev filter is completely defined by three parameters-cut-off frequencies, number of poles and passband ripples. The chebyshev response is a mathematical strategy for achieving a faster roll off by allowing ripple in the frequency response [19]. The chebyshev response is an optimal trade-off between these two parameters. The magnitude squared frequency response is given by
|H (??)|2=1/1+ C2CN2(??/?? p)
Here |H (??)|=Magnitude of analog low pass filter.
C=Parameter related to ripples in pass band.
CN(x) = Chebyshev polynomial of order N
The chebyshev1 polynomials are determined by using the equations
CN+1(x) = 2x CN(x) – CN-1(x)
With
C0(x) =1 and C1(x) = x
The following figure shows the frequency response of a lowpass Chebyshev1 filter.
Figure 1.26: Effect of N on Chebyshev1 filter characteristics
1.16 Objectives of Work
This dissertation entitled ‘Audio Noise reduction using Discrete Wavelet Transformation’ aims for the following objectives:
‘ The objective of a noise reduction system is heavily dependent on the specific context and application. In some scenarios, for example, we want to increase the intelligibility or improve the overall speech perception quality.
‘ Study of noise cancellation discrete wavelet transformation with their types and thresholding and filters.
‘ Review of Literature Related to Audio Noise reduction using audio noise filters and MATLAB wavelet toolbox Techniques.
‘ To Implement the Exiting Various Techniques studied as in literature review.
‘ Study and Analyze The Results Being Obtained.
‘ Noise reduction technology is aimed at reducing unwanted ambient sound, and is implemented through two different methods.
There are different types of parameters are calculated that is SNR, PSNR, MSE and the Time to reduce the noise for noisy signals.
1.17 Thesis Outline:
This thesis report has been divided in to the following chapters ‘
‘ Chapter-1 ‘Introduction’, it includes the information about audio , wavelet , and wave thresholding(hard and soft ) and overview of dissertation report. It also includes the problem definition and Objective of the dissertation work.
‘ Chapter-2 ‘Review of Literature’, it provides overview of the work done in this area.
‘ Chapter-3 ‘Methodology’, explain the explicit algorithm and flowchart. It includes the tool used to implement the research work.
‘ Chapter-4 ‘Results and Discussion’, gives the detail about result obtained and discussion about visual results.
‘ Chapter-5 ‘Conclusion and Future Work’, gives the detail of future work that can be done on this de-noising.
‘ References & Appendix
‘ Publications
CHAPTER 2
LITERATURE SURVEY
Previous Research work on ‘Audio Noise reduction using Discrete Wavelet Transformation’ in the literature survey starting from 2010 to November 2013 was studied which helped me to complete my work and enhance my knowledge. I studied so many papers and some of them are given below:
C Mohan Rao, Dr. B Stephen Charles, Dr. M N Giri Prasad[ 2013] have presents a new adaptive filter whose coefficients are dynamically changing with an evolutionary computation algorithm and hence reducing the noise. This algorithm gives a relationship between the update rate and the minimum error which automatically adjusts the update rate. When the environment is varying, the rate is increased while it would be decreased when the environment is stable and the computation complexity of adaptive filter can be significantly reduced. In the simulation, additive white Gaussian noise is added to the randomly generated information signal and efficiently reduced this noise with minimum or no error by using evolutionary computation with Least Mean Square (LMS) algorithms. Adaptive Noise Cancellation is an alternative way of cancelling noise present in a corrupted signal [18].
K.P. Obulesu1 P. Uday Kumar [2013] have studied the audio signals are synthetic signals, in which music or speech, are often corrupted by noise during recording and transmission. Speech enhancement is a long standing problem with numerous applications ranging from hearing aids, to coding and automatic recognition of speech signals etc. and assume that the noise is additive and statistically independent of the signal. Audio denoising procedures are designed to attenuate the noise and retain the signal of interest. Reduction of noise from audio signals has two methods, Diagonal & Non Diagonal audio denoising algorithms. In this paper, Non diagonal method is used in which Block parameters are automatically adjusted to the nature of the audio signal by minimizing a Stein estimator which is calculated analytically from noisy signal values. This Block thresholding method eliminates ‘musical noise’ by grouping Time-frequency coefficients in blocks before being attenuated[15].
Matheel E. Abdulmunim, Rabab F. Abass [2013] have presented the digital videos are often corrupted by a noise during the acquisition process, storage and transmission. It made the video in ugly appearance and also affect on another digital video processes like compression, feature extraction and pattern recognition so video denoising is highly desirable process in order to improve the video quality. There are many transformation for denoising process, one of them are Fast Discrete Wavelet Transform(FDWT) and framelet transform (Double-Density Wavelet Transform) which is a perfect in denoising process by avoiding the problems in the other transformations. In this paper we propose a method named Translation Invariant with Wiener filter (TIW) this method is proposed to solve the shift variance problem and use this method to denoised a noisy video with Gaussian white noise type. It is applied with Two Dimensional Fast Discrete Wavelet Transform (2-D FDWT), Three Dimensional Fast Discrete Wavelet Transform (3-D FDWT), Two Dimensional Double Density Wavelet Transform (2-D DDWT) and Three Dimensional Double Density Wavelet Transform (3-D DDWT)[1].
Raghavendra Sharma, Vuppuluri Prem Pyara [2013] In this author studied a robust DWPT based adaptive bock algorithm with modified threshold for denoising the sounds of musical instruments shehnai, dafli and flute is proposed. The signal is first segmented into multiple blocks depending upon the minimum mean square criteria in each block, and then thresholding methods are used for each block. All the blocks obtained after denoising the individual block are concatenated to get the final denoised signal. The discrete wavelet packet transform provides more coefficients than the conventional discrete wavelet transform (DWT), representing additional subtle detail of the signal but decision of optimal decomposition level is very important. When the sound signal corrupted with additive white Gaussian noise is passed through this algorithm, the obtained peak signal to noise ratio (PSNR) depends upon the level of decomposition along with shape of the wavelet. Hence, the optimal wavelet and level of decomposition may be different for each signal. The obtained denoised signal with this algorithm is close to the original signal [21].
S. N. Sampat, Dr. C. H. Vithalani [2013] have presented the denoising of one dimensional signal using threshold is one of the major applications of wavelet transform. Quadrature Mirror Filter bank method of wavelet transform has many advantages like support for all major orthogonal wavelets, dyadic resolution and adequate retention of energy. Determination of threshold type and threshold value is one of the important tasks in threshold based denoising techniques. Denoising of audio signal is a subjective matter and remains a valid challenge. In this paper, a noisy speech wav file having additive white Gaussian noise is used for denoising to demonstrate features of two stage hard threshold, soft threshold and customized threshold denoising using Quadrature Mirror Filter bank method of wavelet transform . The second stage of denoising uses neighborhood concept where in a set of three wavelet coefficients, threshold is applied to any wavelet coefficient on the bases of the value of the other two neighborhood wavelet coefficients. Eight different denoised files are generated. Various parameters are measured and compared [19].
B. JaiShankar and K. Duraiswamy [2012] have introduced the noises present in communication channels are disturbing and the recovery of the original signals from the path without any noise is very difficult task. This is achieved by denoising techniques that remove noises from a digital signal. Many denoising technique have been proposed for the removal of noises from the digital audio Signals. But the effectiveness of those techniques is less. In this paper, an audio denoising technique based on wavelet transformation is proposed [8].
B. Jai Shankar, K.Duraiswamy [2012] have presented the noises present in signals are difficult to recover using the traditional methods. Now wavelet transform is used for denoising techniques. The thresholding both hard and soft are used in wavelet transform. The technique exposes each and every finest details contributed by the grouped set of blocks and also it protects the vital and unique features of every individual block. The blocks are filtered and replaced in their original positions from where they are detached. Their implementation results reveal that the proposed technique achieves a state-of-the-art denoising performance in terms of signal-to-noise ratio[9].
Eric Martin [2012] have introduces an adaptive audio block thresholding algorithm. The denoising parameters are computed according to the time-frequency regularity of the audio signal using the SURE (Stein Unbiased Risk Estimate) theorem. The author studied unlike the diagonal estimators, the adaptive audio block thresholding algorithm based on a non-diagonal estimator is very much elective with white noise. However there are some defects. The sounds which are like a white Gaussian noise will be deleted. For instance, it’s impossible to hear cymbals from a drum kit after a denoising [12].
J. Jebastine, Dr. B. Sheela Rani [2012] In this paper the author describes the development of an adaptive noise cancellation algorithm for effective recognition of speech signal and also to improve SNR for an adaptive step size input. An adaptive filter with Fast Block Least Mean square Algorithm is designed for noise free audio (speech/music) signals. The signal input used is a audio speech signal which could be in the form of a recorded voice. The filter used is adaptive filter and the algorithm used is Fast Block LMS algorithm. A Gaussian noise is added to this input signal and given as a input to the Fast Block LMS [10].
Kai Siedenburg, Monika D??orfler [2012] in this paper the author considers the denoising problem from the viewpoint of sparse atomic representation. A general framework of time-frequency soft-thresholding is proposed which encompasses and connects well-known shrinkage operators as special cases. In particular, the ground breaking idea of exploiting signal sparsity in the framework of redundant representations is extended to incorporate knowledge about structural properties of the observed signals. Convergence of the corresponding algorithms is numerically evaluated and their performance in denoising real-life audio signals is compared to the results of similar existing approaches. The novel approach is competitive with respect to signal to noise ratio and improves the state of the art in terms of perceptual criteria[23].
Rajeev Aggarwal, Jai Karan Singh Vijay, Kumar Gupta [2011] In this paper the author describe the Discrete-wavelet transform (DWT) based algorithm are used for speech signal denoising. Here both hard and soft thresholding are used for denoising. Analysis is done on noisy speech signal corrupted by babble noise at 0dB, 5dB, 10dB and15dB SNR levels. Output SNR (Signal to Noise Ratio) and MSE(Mean Square Error) is calculated & compared using both types of thresholding methods. Soft thresholding method performs better than hard thresholding at all input SNR levels. Hard thresholding shows a maximum of 21.79 dB improvement whereas soft thresholding shows a maximum of 35.16 dB improvement in output SNR[2].
Romain Serizel, Marc Moonen [2010] ‘ has presented the combined active noise control and noise reduction schemes for hearing aids to tackle secondary path effects and effects of noise leakage through an open fitting. Such leakage contributions affect the noise signals. The result of these signals appears to have a non-negligible impact on the final signal-to-noise ratio. The author studied a noise-reduction algorithm and an active noise control system in cascade may be efficient as long as the causality margin of the system is large enough. A Filtered-x Multichannel Wiener Filter is presented and applied to integrate noise reduction and active noise control. The cascaded scheme and the integrated scheme are compared experimentally with a Multi channel Wiener Filter in a classic noise reduction framework without active noise control, where the integrated scheme is found to provide the best performance [21].
Guoshen Yu, St??phane Mallat [2008] have studied the removing noise from audio signals requires a non diagonal processing of time-frequency coefficients to avoid producing ‘musical noise.’ Non diagonal time-frequency estimators are more effective than diagonal estimators to remove noise from audio signals because they introduce less musical noise. These non diagonal estimators are derived from a time-frequency SNR estimation performed with parameterized filters applied to time-frequency coefficients. This paper introduces an adaptive audio block-thresholding algorithm that adapts all parameters to the time-frequency regularity of the audio signal. The adaptation is performed by minimizing a Stein unbiased risk estimator calculated from the data [26].
CHAPTER 3
METHODOLOGY & IMPLEMENTATION
3.1. Methodology
The dissertation is removes noise from the audio signal. It is based upon GUI (graphical user interface) in MATLAB. It is an effort to further grasp the fundamentals of MATLAB and validate it as a powerful application tool. There are basically different files. Each of them consists of m-file and figure file. These are the programmable files containing the information about the filter and figure files are the way to analyze the given audio and enter the various filter related data. Now open these files in the Matlab individually. Now run the first file and then filter file and then filtered sound file to remove the noise.
In this work we will firstly upload the sound in the format .wav in the given window. Listen the sound which will appear to be noisy .In the GUI we will take the add noise button and when click on that button noise is added. After this select the wavelet type to denoise the audio signal with hard and soft threshold. If u want to save the signal then click on save button . After this play the denoised signal and find the Noisy SNR and Denoised SNR and elapsed time of the signal.
3.2 Graphical User Interface (GUI)
MatLab provides Graphical User Interface Development Environment (GUIDE).A MatLab tool used to create GUI’s. Decide between using GUIDE or writing the code from scratch GUI’s give the user a simplified experience running a program. Associates a ‘function(s)’ with components of the GUI.GUI should be consistent and easily understood. Provide the user with the ability to use a program without having to worry about commands to run the actual program.
3.3 Components of GUI
1. Push button. 2. Edit text. 3. Static text.
4. Slider. 5. Checkbox. 6. Pop-up-menu.
7. Radio button. 8. Panel. 9. List box.
10. Button group. 11. ActiveX control. 12. Toggle button.
13. List box.
This chapter contains the stepwise, detailed of two proposed algorithms that are followed while denoising audio signals using filters and wavelet transforms. For better and easy understanding, a complete flowchart of the proposed algorithms has been shown at the end of this chapter.
3.4 Discrete Wavelet transforms Algorithm
Step 1: Load an original wave signal.
Step 2: Noise is added to the original wave signal read in above step using the Gaussian noise and produces the noisy wave signal.
Step 3: The Gaussian original wave signal on which logarithmic transform is performed firstly.
Log J(x, y) = log I(x, y) + log ??(x, y)
Step 4: A multilevel decomposition is performed on the log transformed signal using wavelet transform.
Step 5: Apply the wavelet types.
Step 6: Apply thresholding to the noisy coefficients using bayes shrinkage method.
Step 7: After the decomposed signal coefficients are thresholded using the thresholding technique, denoised image is reconstructed as IR(x,y) using inverse wavelet transforms- IDWT.
Now apply the filter based on statistics estimated from a local neighborhood around each pixel. Filter reconstructed image IR(x,y) according to following formula:
Where, ?? is the local mean, ??2 the variance in 3×3 neighborhoods around each pixel and v2 is the average of all estimated variances of each pixel in the neighborhood.
Step 8: Take exponent of the signal obtained in above step and obtained the denoised signal.
Step 9: Now we get the denoised signal and different parameters.
The above algorithm is used to denoised the audio signals with DWT. In this algorithmic work DWT wavelet types like coif5 and sym4 and other types are used to denoised the signals .The Gaussian type noise is removed from the signals with hard and soft threshold techniques. In this the denoised signal may be saved in the database. The denoised and noisy signal SNR is calculated with SNR formulas. After that the efficiency and threshold values is also calculated. The elapsed time is calculated that shows the total processing time of the noisy signal and denoised signals.
Flow Chart for the Work
CHAPTER 4
RESULTS & DISSCUSSION
4.1. RESULT & ANALYS??S
This Chapter shows the implementation results of the dissertation work.Their are different figures that shows how the signals are processed and how the system tools works in MATLAB.
DWT Results:
Figure 4.1: DWT GUI starting window
Figure 4.2: Browse the original signal
Figure 4.3: Play the noisy signal
Figure 4.4: Denoised Signal with Coif wavelet
Table 4.1: Using Coif5 wavelet type with Soft Thresholding
Name of Signal Nosiy SNR value
Denoised SNR value
Total Time Elapsed value(in sec.)
1N.wav 1.00144 12.1227 31.354
2N.wav 1.01068 21.2134 8.87863
3N.wav 0.985718 19.4941 8.86212
Table 4.2 : Using Coif5 wavelet type with Hard Thresholding
Name of Signal Nosiy SNR value
Denoised SNR value
Total Time Elapsed value(in sec.)
1N.wav 2.00151 3.5589 79.0825
2N.wav 1.96955 4.9958 31.4523
3N.wav 1.97672 20.943 21.1168
Table 4.3 : Using sym4 wavelet type with soft Thresholding
Name of Signal Nosiy SNR value
Denoised SNR value
Total Time Elapsed value(in sec.)
1N.wav 2.03813 7.88699 18.1931
2N.wav 1.95963 6.55297 9.95348
3N.wav 1.99192 8.75724 18.282
Table 4.4 : Using sym4 wavelet type with Hard Thresholding
Name of Signal Nosiy SNR value
Denoised SNR value
Total Time Elapsed value(in sec.)
1N.wav 2.00151 11.2783 32.7486
2N.wav 1.96955 9.24276 8.0365
3N.wav 1.97672 9.03269 8.02389
CHAPTER 5
CONCLUSION AND FUTURE WORK
This chapter concludes the work in this thesis in terms of the various parameters that have been considered while denoising audio using various methods of filters ad DWT. Here it also provides with a look up in the future scope of our work area.
5.1. CONCLUS??ON
The different filters with different frequencies are used to remove noise. It can be concluded that for different center frequencies, order of the filter always remains same. By changing its center frequencies filters are being tuned to different frequencies. We used wavelet transform for denoising speech signal corrupted with Gaussian noise. Speech denoising is performed in wavelet domain by different types of wavelet with different thresholding. By using this we can get the better results of de-noising, especially for low level noise. During different analysis we found that soft thresholding is better than hard thresholding because soft thresholding gives better results than hard thresholding. Higher threshold removes noise well, but the part of original signal is also removed with the noise. It is generally not possible to filter out all the noise without affecting the original signal. We can analyze the denoised signal by signal to noise ratio (SNR), Threshold values and elapsed time analysis.
In the DWT Coif wavlet with hard threshold and soft threshold and Sym4 hard and soft threshold is implemented and compared with each others. In this Coif wavelet with soft threshold is best as compared to coif hard threshold and Sym4 wavelet with hard and soft threshold.In DWT soft threshold results are has been best as compared to hard threshold.
5.2. FUTURE WORK
Future work might involve a real time implementation of the system so that the maximum noise is reduced form the audio signals and videos. In the future anybody can extent the order of the different filters and works on higher amplitude signals. They can calculate the efficiency of the filters that they have to implement. In the DWT we are using coif and sym4 with hard and soft threshold but in the future different types of wavelet is implemented with different types of thresholding techniques or hybrid techniques is designed with the help of filters and wavelets and thresholding techniques. Other things in future the results may be improved in the filters and DWT techniques.
REFERENCES
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[27] BOOKS REFFERED
a)digital signal processing by S Salivahanan
b)digital signal processing by SK Mitra
Appendix
Glossary of terms
A
Audio 1,2,3,20,21
Active noise control 2 ,26
Active noise reduction 2,26
Adaptive 22,23,25,27,47,50
B
Band Pass 1, 3,4,5 ,28.30,31
Band Stop 1, 3,4,5,7,9, 28.30,31
Butterworth 9,10,11,21,35,36,37,39
C
Compressing 2
Complementary 2
Communication 3,24,47
Chebyshev 12 ,39,40
D
Dedicated 4
Distortion 3
DWT 23,26,28,30
E
Electromagnetic 3
Electrostatic 3
H
Higher 11,46
I
Interference 1,2
Integrated 26
K
Kind 1,28
L
Linear 5,20,22,28,29
Large 4,26
Least Mean Square 22
N
Narrow 1,3,7,11
Noise 3,4,18,22,23
P
Power spectral density 20, 21
Energy 13,20
Energy spectral density 20
R
Remove 18, 19, 24,27,30,47
Reduction 2, 22, 26,28,29,48
S
Stopband 1, 4, 11, 30
T
Transient 4,9,49
Transition 5, 9, 10, 11
Thresholding 26, 28, 29
U
Used 1,3,4,12,13, 16, 20, 23
W
Wavelet 24,25,26,28
8.2. Abbreviations
ANC – Active noise control
ANR- Active noise reduction
BP- Band Pass
BS-Band Stop
DWT ‘ Discrete wavelet Transformation
PSD – Power Spectral Density
ESD – Energy spectral density
LMS – Least Mean Square
SURE -Stein Unbiased Risk Estimate
Publications
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