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Wavelet transform is young and rapidly gaining popularity transform, discovered by the scientists in the era of the signal analysis. The significance is because it enables time-frequency representation of the signal and made easy for it analysis. Signal can be represented with time at one axis and frequency at another axis. Even though there is so many traditional methods of analyzing the signals they got their limitations as well, one of them is the Fourier analysis. The Fourier analysis gives only frequency spectrum of the signal but not the time information, another drawback is, it cannot be applicable to the non stationary signals which are non periodic and varying frequencies with time. Another famous transform known as the short term fourier transform suffers with the problem of frequency resolution. Hence the ultimate solution which over come the problems in representing the signal in time-frequency domain is wavelet transform. This project endeavors to critically analyze the wavelet transform.
Wavelet is well recognized by its nature now days. It is the most frequently used transform in most of the applications for the analysis especially in the medical fields. It grabbed the attention in recent era of signal processing. Due to its demand lots of research is still going on its usage and loads of methods evolved in deducing the wavelet transform. Much architecture is emerged for its implementation. Lots of ready made tools are available which use the wavelet transforms for the analysis.
This project proposes the applications of the discrete wavelet transform. Different architectures of implementing the Discrete wavelet transform is explained, by designing the filter coefficients of the debauches wavelet filter theoretically and using the filter coefficients discrete wavelet transform is implemented in Matlab. An application is also is taken for carefully explaining the significance of the wavelet transform. All the operation procedures in simulink are clearly explained to construct discrete wavelet transform application. The major task in the project is to design the orthogonal filters such that a perfect reconstruction is obtained.
DWTDiscrete wavelet transform
DFTDiscrete Fourier transform
FFTFast Fourier transform
TFRTime frequency representation
MRAMulti resolution analysis
STFTShort term Fourier transform
HUPHeisenberg Uncertainty Principle
The transform of a signal is viewing the signal in different perspective and need of the transform is to obtain the information which can't be viewed with the normal representation. The usual representation of the signal is the time-domain representation. That is what ever the signal measuring is measured as a function of time, but it swallows most of the detailed information lying inherently. With the time-domain signal we cannot obtain the frequency information of the signal directly, so there is a need of transforming the signal where the application requires the frequency information. You can easily obtain the frequency spectrum of the required signal by simply applying the Fourier transform which converts time domain signal into the frequency domain.
With the implementation of the Fourier transform one can easily obtain the frequency information of the signal. Fourier transform is the one of the example of the frequency transform, but even this transform also fails to give some of the details like the time of occurrence of frequency. In order to get the solution for the problem people had invented the window concept named as Short term Fourier transform, but it suffers with the resolution problem. Finally wavelet analysis will be the better solution to represent the signal in Time-frequency representation.
Wavelet transform provides the Time-frequency representation of the signal, the disadvantages in the short term Fourier transform will overcome by the wavelet transform. Wavelet transform can be implemented on the non-stationary signals and another most important property of the wavelet which is lacking the in the short term Fourier transform is the resolution. The short term Fourier transform uses the constant resolution for the analysis of the signal where as the wavelet uses the multi resolution technique where different resolutions are used to analyze different frequencies.
A wave is time or space function which is oscillating and is periodic and wavelets are localized waves, they have their energy concentrated in time or space and are suited to analysis of the transient signals. The Fourier transform and short term Fourier transform uses
waves to analyze signals where as wavelets in wavelet transform which is of finite energy. There is a similarity between the short term Fourier transform and wavelet transform, it is like
the multiplying the wave with wavelet in case of the wavelet transform similar to the window in short term Fourier transform before computing the transform of the each obtained segment.
However in wavelet transform the width of the wavelet function is varied according to the spectral component. Wavelet transform gives good frequency resolution and poor time resolutions at low frequencies and at high frequencies it gives good time resolution and poor frequency at high frequency.
Heisenberg Uncertainty Principle
The Heisenberg Uncertainty principle state was originally stated in physics, and claims that it is impossible to simultaneously know both the position and velocity of a particle. However it has an analog in signal processing. In terms of signals, it is impossible to know both the frequency and the time at which they occur.
In most of the signal processing applications the frequency details are much important even though the fourier transform serves with frequency spectrum, applications such as electrocardiograph and electromyography the signals needed to be analyzed with time resolution also. As we know that Fourier transform is unable to give good results to the non stationary signals the wavelet transform is introduced.
The method of converting the time domain signal to the frequency domain is called the fourier transform. The time-domain waveform is displayed with voltage on one axis and time on the other axis. The frequency domain of the given signal can be viewed with magnitude or power at one axis and frequency at the other axis which is called the frequency spectrum. Fourier transforms are categorized into 3 ways according to the nature of the application. The following are the general Fourier transforms
- Continuous Fourier transform (CFT)
- Discrete Fourier transform (DFT)
- Fast Fourier transform (FFT)
Fourier transform is the summation of amplitudes of all the frequency components present in the signals. The frequency axis ranges from 0 to ¥. The following equations explain the Fourier transform in much easier way.
Any periodic function can be expressed as an infinite sum of periodic complex exponential functions. That means the fourier transform decomposes the signals in to complex exponential functions of different frequencies. In the above equations t represents the time and f represents the frequency and x represents the signal which is in the time domain,
X represents the signal in frequency domain. The first equation represents the Fourier transform of x(t), and the second equation represent the Inverse Fourier transform of X(f).If you carefully examine the first equation the signal is multiplied with the exponential
term with frequency f and integration is done to the obtained results. That is the integration is done over all the times. When the value of f in the summation results in the large value that means the spectrum is dominant with that frequency.
If it results in relatively small quantity that means it has very less components of that frequency if it results 0 then there is no such frequency component. With the above explanation it is clearly visible that it is hard to get the time details of the frequency almost impossible because it is the summation over -¥ to +¥. That is we can obtain the frequency information present in the required signal which is called the frequency spectrum and it is not possible to obtain the information that at which time that frequency has occurred. It is useful when the signal is stationary (the signal is vibrating periodically with fixed pattern) but it wont be useful with the non-stationary waveforms. The following is the fourier transform of the signal x(t)
x(t) = cos(2pt) + cos(10pt) + cos(20pt) Fourier Transform, X(f), of x(t)
It seems to be perfect solution to finding a relation between frequency, time, and amplitude. The Fourier transform is missing one of the three elements that were necessary to analyze a signal. Fourier transform gives no details of occurrence of the frequency. It assumes that each
Frequency exists over all time. The above example shows the case of the assumption. if we took the non-stationary signal then the real challenge for the Fourier transform persists.
cos(2pt), for t = [0,2) cos(10pt), for t = [2,4)Fourier Transform, X(f), of x(t)
cos(20pt), for t = [4,6 )
The above example of the Fourier transform of the signal which is not stationary is also similar to the previous example where the signal is stationary except some noise and amplitude. The small amplitudes represent the frequencies which are rarely occurred during the time interval and the noise is due to the sudden changes in the time signal. Hence we can say though it is useful to obtain the frequency spectrum but it is missing the time details in the transformation which is not neglected in most of the applications where the time details helps a lot in the performance.
Applications of Fourier transform
- As FT transforms a signal to be analyzed in the frequency domain, it is possible to block certain frequencies within a signal, while leaving others. The process of Fourier transform is to convert the signal into frequency domain with convoluting with the complex equation the unwanted frequencies are eliminated. The inverse FT transforms the signal back to the time domain .
- Industry applications include filtering data in geologic seismic analysis, and filtering known frequencies of noise from wireless signals. However, due to a phenomenon known as aliasing, the sample rate of this procedure is limited, limiting its precision .
- The discrete cosine transform compresses the image by detecting the frequencies un detectable to the naked eye and eliminates them. This is how the compression is achieved in the JPEG the well know image format.
- It can't be used to analyze the non-stationary signals
- It can't be used where the time details of the frequency is essential
- The Fourier transform fails to distinguish the spectrums produced by stationary and non stationary signals.
In order to overcome these drawbacks and to introduce the time analysis in the signal scientist introduced the short time Fourier transform where time frequency representation is partially achieved.
Short Term Fourier Transform (STFT)
The short term fourier transform is like the revised edition of the fourier transform. The concept of the STFT is to assume the non-stationary signals into small segments such that the stationary is applicable to the small segment. In detail we can say like assuming some part of a non stationary signal is stationary. A window function is choosen such that the width of the window is equal to the segment of the signal. For that segment the stationary is applicable.
There is no restriction for the width of the segment to be it can be as small as we want and as big. The operation of the STFT (short term Fourier transform) can explained as firstly the window function is located at t=0 and the window width is ‘T', at the first instant the window function is overlap with the first T/2 seconds with the signal. If you clearly explain the phenomenon with in the signal length the window function is multiplied with the signal with varying times of overlapped window and the output is also in the form of a wave submitted to the fourier transform. As we already stated that this small portion of the signal is assumed stationary. Similarly the window is shifted to the next step that is t=1 again the same procedure is repeated to get the fourier transform of the product of window function and the signal. Below is the equation which shows the entire process of the STFT.
Where x(t) is the signal itself, w(t) is the window function, and * is the complex conjugate. As you can see from the equation, the STFT of the signal is nothing but the Fourier transform of the signal multiplied by the window function. As we know the fact that we are starving for the time-frequency representation, we are aware that it can't be achieved completely which is the ideal case.
The best way of understanding this concept is by taking an example. Take a non stationary signal which is non periodic and contains the frequencies of 300Hz, 200Hz, 100Hz and 50Hz respectively. The following is the non stationary wave form and the corresponding short term fourier transform the figures are clearly explained as.
The STFT obtained is 3 dimensional where the axis is namely time, frequency and amplitude which is shown as in the figure. If we carefully examine waveform and the generated STFT we can understood so many hidden components in the short term fourier transform. As we stated that there are four frequency components in the given signal which is non stationary the four peaks in the figure represent those four frequencies.
This can be - termed as a time-frequency representation but unfortunately this got a problem called the resolution problem and is explained as we can only know what frequency band exists at certain time intervals that means it is not giving any exact frequency component of the signal at that particular time interval but it is narrowing the hunt by giving the shorter bvand of frequencies of probability. The frequency resolution is the very disappointing drawback in the short term fourier transform.
The reason for the resolution problem can explained as the window function in the fourier transform which lasts for all the times that means the summation over all the times from minus infinity to plus infinity giving all the frequency information without any frequency resolution problem. But where as for the STFT the window function is finite and it spans only small amount of the signal which raises the frequency resolution problem. Poorer frequency resolution results in the lacking of exact knowledge of the frequency components present in the signal which is a drawback for the short term fourier transform.
It is a reciprocal if you increase the width of the window you will lose the time information of the signal and if you made it narrow then you will face the frequency resolution problem. Clearly to say in the short form if you go for the narrow window then we have to lose the frequency information with the expense of the time details and if you got for the wider window you will loose the time information for the expense of the frequency information in both the cases it is not appreciated.
Hence there is only one ultimate solution to reduce the problems of time-frequency representation which is our anticipated transform that is wavelet transform. Before going to the wavelet transform we have to know few fundamentals which follow.
Multi resolution analysis
So far we are dealing with the transforms which got drawback of frequency resolution due to the fixed length of the window function. The multi resolution analysis is the phenomenon which uses different resolutions for different frequencies. That means each spectral component is not resolved equally.
The multi resolution analysis is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. It gives better results especially when the signals contain long duration of the low frequencies and short duration of the high frequencies. Accidentally the real time world signals are like that only for example the following figure shows the signal of such kind.
The scale is the important parameter used in the wavelet transform. Scale is similar to the time axis in the time domain representation, low frequencies that means high scales correspond to global information of a signal. We know that most of the real time signal spans with low frequencies. Where as high frequencies that means low scales correspond to detailed information of a hidden pattern in the signal usually spans very little duration of the real world signals. If you take scale as a mathematical operation, it either dilates or compresses a signal. If you increase the scale that means you are increasing the dilation of the signal, further the scale further stretch out of the signal and small scale corresponds to compressed signals. If f(t) is a given function f(st) is the function depends on the scale S that means if you want to dilate signal then you need to put s<1 if you want to compress the signal then s>1. But in the wavelet transform the scaling factor lies in the denominator so the definition which stated above is inverted that is for s>1 the signal gets dilated and for s<1 it gets compressed. This is all because the scale factor lies in the denominator of the wavelet function.
A wave is a periodic oscillating function of time or space. In contrast, wavelets are localized waves. They have their energy concentrated in time or space and are suited to analysis of transient signals. We know both Fourier transform and Short term Fourier transform use waves, while to compare with the signal in wavelet transform, it utilizes wavelets of finite energy to analyze. The theory behind wavelets is to analyze the real world signal according to scale. In wavelet analysis, we use scale to look at data plays a special role. With wavelet analysis, we can use approximating functions that are contained neatly in finite domains. Wavelets are well-suited for approximating data with sharp discontinuities .
Wavelet theory is also a form of mathematical transformation, similar to the FT in that it takes a signal in time domain, and represents it in frequency domain. Wavelet functions are distinguished from other transformations in that they not only dissect signals into their component frequencies, they also vary the scale at which the component frequencies are analyzed. Therefore wavelets, as component pieces used to analyze a signal, are limited in space. In other words, they have definite stopping points along the axis of a graph, they do not repeat to infinity like a sine or cosine wave does. As a result, working with wavelets produces functions and operators that are "sparse" (small), which makes wavelets excellently suited for applications such as data compression and noise reduction in signals. The ability to vary the scale of the function as it addresses different frequencies also makes wavelets better suited to signals with spikes or discontinuities than traditional transformations such as the Fourier transform .
Applications of wavelets
- The Joint Photographic Experts Group (JPEG) has approved the next standard for image compression, known as JPEG2000, based on wavelet compression algorithms. By setting the "mother wave" for image compression and decompression ahead of time as a part of the standard, JPEG2000 will be able to provide resolution at a compression of 200 to 1, equivalent to current JPEG at 5 to 1.