Fourier Transform is well developed transformation theory with lots of approximation made. In Fourier Transform, signals in time domain is transformed or mapped to frequency domain which can give frequency information about signal that is global information of analysed signals without time (local) information that is Fourier Transform can not give time or time range of occurrence of particular frequency signal or band of signals.
Fourier Transform is given as:
$$ S(\omega)= \int f(t) e^{-j\omega t}dt $$
Fourier Transform is given as:
$$ S(\omega)= \int f(t) e^{-j\omega t}dt $$
STFT is given as:
$$ S(\omega)= \int f(t) g(t-\tau) e^{-j\omega t}dt $$
$$ \text{ Thus STFT is simply the FT of } f(t)g(t-\tau)$$ Because of windowing function being approximated on window width, some of important information in the signal might be lost if the informaiton content frequency signal is out of integration range of windowing function.
if the windowing function has small width it will have good time resolution but poor frequency resolution and will have poor time resolution and good frequncy resolution at large windowing function width as stated in Heisenberg Principle stated as multiple of change in time and change in frequency is always constant.
To overcome these resolution problem of with windowing function in STFT, Wavelet Transform is introduced which maps the time signal into frequency-time domain. Because of changing window size of analysis functions, Wavelet Transform gives good frequency resolution at lower frequency range and good time resolution at high frequency range.
In the fig. above has best illustration why wavelet transform is preferable. In STFT change in time and change in frequency is always same to keep product of change in time and change in frequency is constant according to Heisenberg Principle. Wavelet Transform also keep Heisenberg Principle true by corresponding increase in change in time correspond to change in frequency. This change in time and frequency results good time resolution at high frequency range and good frequency resolution at low frequency range. Thus resolving the problem in STFT, Wavelet Transform is better than STFT.
In the fig. above has best illustration why wavelet transform is preferable. In STFT change in time and change in frequency is always same to keep product of change in time and change in frequency is constant according to Heisenberg Principle. Wavelet Transform also keep Heisenberg Principle true by corresponding increase in change in time correspond to change in frequency. This change in time and frequency results good time resolution at high frequency range and good frequency resolution at low frequency range. Thus resolving the problem in STFT, Wavelet Transform is better than STFT.
Time Domain Signal
In time domain, signal can analysed if the signal amplitude is high enough at the time of consideration. But in time domain we can not know which which signal is information carrying signal and which signal is noise signal. So to identify and filter out unwanted noise signal, time domain signal is transformed into frequency domain so that we can filter out noise signals. The Fourier Transform is only capable of giving information about frequency information but not time instant at which that particular signal is occurred.
Transformed Signal
In transformed signal, transformation coefficient is important. Transformation coefficient is treated as to be carrying information. In the analysis and coding of information signal, transformation coefficient is analysed and coded for further processing.
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