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Periodic rectangular function

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Settler

Structural
May 22, 2010
88
Question...

Can anybody tell me what happens to this equation when:

1. tau->0
2. T->infinity


I'm talking about the Fourier Series... The answer is not zero I guess. But I don't know what it is...

Please help!
 
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Thank you very much! BUT!!!

I wasn't saying tau~0 and T-> infinity at the same time :(

Plus I'm not talking about the FFT but the Fourier Series!
 
What prompts you to ask? Is there an application or are you just curious? You have to admit, it looks a bit like homework...

To understand what happens in (1) and (2) (which are equivalent after timescaling), consider the parts of the FS equation:

1) the DC component, tau/T, goes to zero because the average content of the signal gets small as the pulse width gets small.

2) the cosine amplitudes, (2/n*pi)*sin(pi*n*tau/T), flatten out and get smaller. Plot the amplitude, replacing n with x and varying tau to see this happening - the graph will be continuous but the FS amplitudes are at periodic points on this line. As Greg indicates, the spectral components are "smearing" out to a flat line.

3) the cosines, cos(2*pi*n*t/T), are simply shifted to lower frequencies the bigger T is.

 
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