Fourier Transform is given as:
$$ F(\omega) = \int f(t)e^{-j\omega t}dt $$
Though Fourier Transformation is highly developed transform, as given in transformation equation transformed output is function of frequency only, it is unable to give the information about time of occurrence of particular frequency signal or range of signals, transformation of signal with known frequency band is better analysed without need of time of occurrence. That is it can give global information about frequency of signal occurred but can not give the instant of occurrence of particular signals that is local information. In real world, most of signals are non stationary and with need of time of occurrence of particular signal or range of signals.
Heisenberg Principle states that time and frequency information can not be extract at the same time, in contrast time range of occurrence can be extract for certain band frequency not for particular frequency. Short Time Fourier Transform is the solution which can give time range of occurrence for frequency bands which is given as:
$$ S(\omega)= \int f(t) g(t-\tau) e^{-j\omega t}dt $$
Though Short Time Fourier Transform(STFT) is solution for the problem to analyse time of occurrence of signal or signals, is not good enough for signal analysis as windowing function has fixed width and fixed hight with constant product of change in time and change in frequency. Being fixed change in time and fixed change in frequency in STFT analysis functions, it is good for the signal where more information is content in the range neither high frequency range signal nor low frequency range signal because STFT has not good time analysis at high frequency and has not good frequency analysis at low frequency range but optimum selection of window width, time frequency analysis can be good to some extend.
Now as wavelet function is of varying change in time and change in frequency with constant product of change in time and change in frequency (Heisenberg Principle), wavelet function will have high change in time at low frequency range and high change in frequency at high frequency with scaling functions and wavelet functions. Thus wavelet function is better transformation function for non-stationary signals and defined as;
$$ S(a, b)= \int f(t) \psi{(\frac{t-a}b}) dt $$
where a is translation that is shift of wavelet in time domain and b is inverse of frequency which gives the frequency information.
Admissibility of Wavelets
To be classified a function as wavelet function some mathematical criteria must be satisfied and the criteria is admissibility condition. Two major admissibility conditions are:
1. Wavelet function should be of finite energy.
$$ E= \int_{-\infty}^\infty |\psi(t)|^2 dt < \infty $$
2. Wavelet function should be of non zero frequency components that is zero mean function.
Fourier transform of wavelet is given as:
$$ \psi(f) = \int_{-\infty}^\infty \psi(t) e^{-2 \pi f t}dt $$
then admissibility constant Cg is given as:
$$ Cg = \int_0 ^\infty \frac{|\psi(f)|^2}f df < \infty $$
This implies that wavelet has no zero frequency component ie. zero frequency component should not be exist.
Applications
Being existence of infinite number of wavelet functions, wavelet functions can be chosen depending upon the signal to be analysed. All the real signals are non-stationary signal which can be analysed with less approximation for transformation using wavelet function as analysis function. Hardly we can choose the frequency range of signal to be analysed assuming signals as noise out of the that range. For example we use filter of lower band to be 20 Hz to 20 KHz as upper band limit for voice signal and so always we are meant to analyse the signal in that range. In addition to that analysis does not mean to look magnitude of particular frequency signal but also time of occurrence as well. So wavelet function is better analysis function which can give the information about the frequency range of signals and its occurrence time ranges.
Recently wavelet transform is used in analysis of climate data signals, biomedical signals and wavelet transformation as data compress technology as lossless compression method. Wavelets are using in astronomy, acoustics, nuclear engineering, neurophysiology, MRI, fractals, turbulence, radar, earth-quake predictions etc. In fact wavelet transform has huge amount of applications.
note: wavelet has large number of application for data analysis but the challenge is appropriate wavelet function selection which can extract almost all information required.
$$ F(\omega) = \int f(t)e^{-j\omega t}dt $$
Though Fourier Transformation is highly developed transform, as given in transformation equation transformed output is function of frequency only, it is unable to give the information about time of occurrence of particular frequency signal or range of signals, transformation of signal with known frequency band is better analysed without need of time of occurrence. That is it can give global information about frequency of signal occurred but can not give the instant of occurrence of particular signals that is local information. In real world, most of signals are non stationary and with need of time of occurrence of particular signal or range of signals.
Heisenberg Principle states that time and frequency information can not be extract at the same time, in contrast time range of occurrence can be extract for certain band frequency not for particular frequency. Short Time Fourier Transform is the solution which can give time range of occurrence for frequency bands which is given as:
$$ S(\omega)= \int f(t) g(t-\tau) e^{-j\omega t}dt $$
Though Short Time Fourier Transform(STFT) is solution for the problem to analyse time of occurrence of signal or signals, is not good enough for signal analysis as windowing function has fixed width and fixed hight with constant product of change in time and change in frequency. Being fixed change in time and fixed change in frequency in STFT analysis functions, it is good for the signal where more information is content in the range neither high frequency range signal nor low frequency range signal because STFT has not good time analysis at high frequency and has not good frequency analysis at low frequency range but optimum selection of window width, time frequency analysis can be good to some extend.
Now as wavelet function is of varying change in time and change in frequency with constant product of change in time and change in frequency (Heisenberg Principle), wavelet function will have high change in time at low frequency range and high change in frequency at high frequency with scaling functions and wavelet functions. Thus wavelet function is better transformation function for non-stationary signals and defined as;
$$ S(a, b)= \int f(t) \psi{(\frac{t-a}b}) dt $$
where a is translation that is shift of wavelet in time domain and b is inverse of frequency which gives the frequency information.
Admissibility of Wavelets
To be classified a function as wavelet function some mathematical criteria must be satisfied and the criteria is admissibility condition. Two major admissibility conditions are:
1. Wavelet function should be of finite energy.
$$ E= \int_{-\infty}^\infty |\psi(t)|^2 dt < \infty $$
2. Wavelet function should be of non zero frequency components that is zero mean function.
Fourier transform of wavelet is given as:
$$ \psi(f) = \int_{-\infty}^\infty \psi(t) e^{-2 \pi f t}dt $$
then admissibility constant Cg is given as:
$$ Cg = \int_0 ^\infty \frac{|\psi(f)|^2}f df < \infty $$
This implies that wavelet has no zero frequency component ie. zero frequency component should not be exist.
Applications
Being existence of infinite number of wavelet functions, wavelet functions can be chosen depending upon the signal to be analysed. All the real signals are non-stationary signal which can be analysed with less approximation for transformation using wavelet function as analysis function. Hardly we can choose the frequency range of signal to be analysed assuming signals as noise out of the that range. For example we use filter of lower band to be 20 Hz to 20 KHz as upper band limit for voice signal and so always we are meant to analyse the signal in that range. In addition to that analysis does not mean to look magnitude of particular frequency signal but also time of occurrence as well. So wavelet function is better analysis function which can give the information about the frequency range of signals and its occurrence time ranges.
Recently wavelet transform is used in analysis of climate data signals, biomedical signals and wavelet transformation as data compress technology as lossless compression method. Wavelets are using in astronomy, acoustics, nuclear engineering, neurophysiology, MRI, fractals, turbulence, radar, earth-quake predictions etc. In fact wavelet transform has huge amount of applications.
note: wavelet has large number of application for data analysis but the challenge is appropriate wavelet function selection which can extract almost all information required.
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