njit-etd2003-081 - New Jersey Institute of Technology
njit-etd2003-081 - New Jersey Institute of Technology njit-etd2003-081 - New Jersey Institute of Technology
A time-frequency distribution is scale covariant if it satisfies: 3.6 Comparison of Time-Frequency Distributions 3.6.1 Short Time Fourier Transform The short time Fourier transform was the first tool devised for analyzing a signal in the time-frequency domain [20]. This is done by extracting a small piece of the signal and taking its Fourier transform, and by continuing this process, one can show the existing frequency components at each instant of time. This idea can mathematically be presented by first designing a window function, h(r — t) which will emphasize the times around the fixed time of interest t . The signal is then multiplied with the window function and Fourier transform is taken: As this process is continued for each particular time, one obtains a different spectrum. The totality of these spectra makes a time-frequency distribution. The energy density of the signal at the fixed time t is where p5p (t, f) is called the spectrogram. The spectrogram can be also written in terms of the Fourier transforms of the signal and window function.
where H(f), S(f) are Fourier transforms of the signal and window function respectively. Note that equation (3.23) can be used to study the behavior of the signal around the fixed frequency of interest f . The spectrogram should not be thought of as a different distribution because it is a member of a general class of distributions [22]. How large should the window be or how should one weigh each piece of the signal To answer these questions one need to understand the time-bandwidth relation, or the uncertainty principle. Now define the duration of a signal s(t) by At : where I. is mean time and is defined as: Now also define the bandwidth of the signal S(f) in the frequency domain by where f is mean frequency and is defined as: The time bandwidth relation is: in /1 \
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where H(f), S(f) are Fourier transforms <strong>of</strong> the signal and window function<br />
respectively. Note that equation (3.23) can be used to study the behavior <strong>of</strong> the signal<br />
around the fixed frequency <strong>of</strong> interest f . The spectrogram should not be thought <strong>of</strong> as<br />
a different distribution because it is a member <strong>of</strong> a general class <strong>of</strong> distributions [22].<br />
How large should the window be or how should one weigh each piece <strong>of</strong> the<br />
signal To answer these questions one need to understand the time-bandwidth relation,<br />
or the uncertainty principle.<br />
Now define the duration <strong>of</strong> a signal s(t) by At :<br />
where I. is mean time and is defined as:<br />
Now also define the bandwidth <strong>of</strong> the signal S(f) in the frequency domain by<br />
where f is mean frequency and is defined as:<br />
The time bandwidth relation is:<br />
in /1 \
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