Wavelets2 - Wavelets Review Fourier Transform and Short...

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Wavelets Review Fourier Transform and Short Term Fourier Transform ( Read the tutorial posted on the web and/or read from either Sayood or Salomon). The Problems with Fourier Transform The Fourier transform hides information about time. It gives unambiguous information about the frequencies that the signal contains but it does not say at what times these frequencies occur. As a result, two signals, one stationary and the other no-stationary, containing the same frequencies will give the same frequency spectrum. Every frequency is considered a global characteristic of the signal. A discontinuity in the local part of the signal is translated in the frequency spectrum over the entire time domain – a local characteristic becomes a global characteristic. This does not mean that the information regarding time is totally lost, it becomes embedded in the ‘phases’ of the components and this is the reason we can reconstruct the original time signal faithfully. The lack of time information makes Fourier transform error prone. If a signal is received correctly for hours and gets corrupted for only a few second, it totally destroys the signal because the frequencies injected spread over the entire time domain and the errors become global. A qualitative explanation of why Fourier transform fails to capture time information is the fact that the set of basis functions ( sines and cosines) are infinitely long and the transform picks up the frequencies regardless of where it appears in the signal. Uncertainty Principle The time and frequency domains are complimentary. If one is local, the other is global. For an impulse signal, which assumes a constant value for a very brief period of time, the frequency spectrum is infinite whereas if a sinusoidal signal extends over infinite time, its frequency spectrum is a single vertical line at the given frequency. We can always localize a signal or a frequency but we cannot do that simultaneously. If the signal has a short duration, its band of frequency is wide and vice versa. Heisenberg’s uncertainty principle was enunciated in the context of quantum physics which stated that the position and the momentum of a particle cannot be precisely determined simultaneously. This principle is also applicable to signal processing where the precise statement is as follows. Let g ( t ) be a function with the property 1 ) ( 2 = dt t g . Then ) ) ( ) ( 2 2 dt t g t t m ( Π 2 2 2 16 1 ) ) ( ) ( ( dt f G f f m
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where t denote average values of t and f ,and G ( f ) is the Fourier transform of g ( t ). m m f , Short-Term Fourier Transform (STFT) (Read the tutorial posted in the course web page.) The STFT is an attempt to alleviate the problems with FT. It takes a non-stationary signal and breaks it down into “windows” of signals for a specified short period of time and does FT on the window by considering the signal to consist of repeated windows over time. This gives a better picture of the frequency content of the signal over the segment
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This note was uploaded on 06/09/2011 for the course CAP 5015 taught by Professor Mukherjee during the Spring '11 term at University of Central Florida.

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Wavelets2 - Wavelets Review Fourier Transform and Short...

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