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Can anyone experienced in signal processing and STFT explains to me why the window function in the below posted image is from (t-t'), given that t is the total time and t' is the width of the window?

I can not figure it out because, initially, the window is located at t=0, and if the window length for an example is 3, then the window will spans from t=0 -> t=3, and if the total time T = 10 for an example then the window function will be like w(T-3), which is 7?! I really can not understand it and I believe there is any hidden thing I can not comprehend.

Kindly please clarify it and guide. Thanks Image: enter image description here

Andrestand
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rmaik
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2 Answers2

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note that, the width of the winow function is constant throughout the entire STFT process. and the time (t) in the function g(t-t') indicate sthat, t: is the current time on the time axis and it is variable each time the window is moved/shifted to the righ to overlap the main signal.

in other words, and i hope this clarifies better, the "t" at the end of the time axis is NOT the "t" in the function g(t-t'). as i stated earlir in the function g(t-t'), t: is the current time on the time axis and it is variable for each shift of the window function and t': is the width of the window and it is constant throughout the entire STFT process.

Amrmsmb
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  • t is your time variable, not the total time.
  • t' is not the width of the window, it is the integration variable in the integral, and the integral is missing a dt' at the right end.
  • g(x) is the window function, and the width of it is not defined above, but represented as the width of the light blue bell in the figure.

The image may have a different interpretation, but it may be wrong; if you apply these adjustments:

  • Swap the labels t and t' on the horizontal axis.
  • Change x(t) with x(t') on the vertical axis.

you are now looking at x(t') (black line) and at g(t-t') (upper contour of the light-blue zone) for a FIXED time t. The bell-shaped window function is centered around t, and the product of the bell and of the signal is the function of which you are calculating the Fourier transform in the equation, and it is non-zero only in proximity of the fixed value of t. Consistently, the quantity is the 'local', i.e. short-time, Fourier transform of the signal, in the vicinity of the fixed time t.

You can do the same for all values of t (with a different figure for each value, with a bell shifted to the left/right), and obtain the STFT.

gg349
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  • thanks for the answer, but now can i consider t is the width of the bell-shaped window function? if not what is the width of the window function? – rmaik Dec 30 '14 at 14:13
  • no you can't, t is just a variable. You are the one to choose the window function and its length. For example, you could choose a [Hann](http://en.wikipedia.org/wiki/Window_function#Hann_.28Hanning.29_window) window, with `N=100` samples. – gg349 Dec 30 '14 at 14:50