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If you break a 4 second signal into 0.5 second segments, add them together and then perform an FFT on the resultant 0.5 second data.

Will the FFT / frequency spectrum be the simlar to an FFT on the original 4 second data?

This would mean that you can store long amounts of data in smaller memory if all you cared about was retaining the frequency components... I'm not sure if I am missing something though?

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  • $\begingroup$ No, as Fat32 explained, but what you can do is sum the magnitudes to get a non-coherent gain for the FFT spectrum that you would get in a 0.5 sec data segment. This is similar to getting a finer “Video Bandwidth” on a spectrum analyzer- it does not reduce the noise density as a longer sequence would but it does smooth (average) the noise floor. $\endgroup$ – Dan Boschen Sep 30 '18 at 14:52
  • $\begingroup$ @DanBoschen Hi Dan! Assuming that OP is indeed into spectral analysis of periodic signals (other then signal compression using some esoteric FFT tetchnique which I initially thought of), then I should delete my answer... And then, I consider that OP is asking a puzzle rather then a question and that is quite annoying isn't it :-))) $\endgroup$ – Fat32 Sep 30 '18 at 18:47
  • $\begingroup$ Nothing wrong with your answer or the question. It is very clear that you answered the question so I wouldn't delete it. I was just offering further insight reading between the lines and helping to further the understanding. All good! Never annoying. $\endgroup$ – Dan Boschen Sep 30 '18 at 18:50
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    $\begingroup$ @DanBoschen ok then, lets wait to see if OP clarifies what he actually meant to ask: 1-) Signal compression (seems weird to me now) or 2-) Spectral analysis of periodic signals (a much reasonable application)... $\endgroup$ – Fat32 Sep 30 '18 at 18:54
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That's not true (unless the signal is periodic, and those pieces are the base periods of it).

If you divide a signal $x_L[n]$ of a length $L$ into smaller pieces $x_i[n]$ of length $N$ each,

$$ x_L[n] = x_1[n] + x_2[n-N] + x_3[n-2N] + x_4[n-3N]$$

and sum them into a shorter signal $x_N[n]$ of length N

$$ x_N[n] = x_1[n] + x_2[n] + x_3[n] + x_4[n]$$

and take $N$-point DFT/FFT of their resultant $x_N[n]$, you won't be able to reconstruct the original long singnal $x_L[n]$ back.

Indeed it's not even about FFT. You would not be able to retrieve the components $x_i[n]$ from their sum $x_N[n]$. So since there is no way to reconstruct back $x_L[n]$ in this manner.

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