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profase

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So I'm trying to intuitively understand the conclusion that a contraction in one domain leads to an expansion in the other domain (and vice-versa). Mathematically, I can see how this would happen, e.g., a bandpass filter with -pi/2 < w < pi/2 would result in a narrower time domain signal than a bandpass filter covering -pi/4 < w < pi/4 because the fourier transform will result in a sinc function with a higher frequency, thereby tapering off quicker; but I can't grasp why. One way I've tried thinking of it is:

If I have a small time-domain signal, the frequency domain will be wide because more frequencies can 'fit' or 'match' the time-signal; whereas if the time-domain signal were long, fewer frequencies would be able to 'fit' or 'match' the signal. Does this make sense? Is it a valid way of thinking about it?

Thanks for your help,

John