Skip to main content
Signal Processing

You are not logged in. Your edit will be placed in a queue until it is peer reviewed.

We welcome edits that make the post easier to understand and more valuable for readers. Because community members review edits, please try to make the post substantially better than how you found it, for example, by fixing grammar or adding additional resources and hyperlinks.

Required fields*

8
  • $\begingroup$ I’m assuming the explanation you are looking for is the edge discontinuities that are induced by the DFT assuming your signal is $N$-periodic? $\endgroup$ Commented Mar 31 at 2:55
  • $\begingroup$ There are so many things that you're getting right. I don't understand how that leads you to this question: "why does tapering your signal to zero (via some non-rectangular window function) mean that it is possible to recreate your signal with a tighter group of frequencies?" - - - - - It's sorta nonsensical. So I dunno how to answer it. What do you mean by "a tighter group of frequencies"? $\endgroup$ Commented Mar 31 at 3:51
  • $\begingroup$ @Baddioes Yep, and I'm fully comfortable and on board with the fact that the DFT assumes your signal is $N$-periodic, but I'm looking for exactly what about this periodic assumption leads to this specific behavior. I usually see an intuitive but hand-wavey argument that roughly goes "discontinuity=bad=spectral leakage", but I'm looking for a reasonably rigorous argument (but from the DFS representation perspective, rather than the other one I mention in my post) for why this discontinuity leads to the specific phenomenon of leakage, and why smoothing out the discontinuity reduces the leakage. $\endgroup$ Commented Apr 5 at 21:57
  • $\begingroup$ @robert The DFT calculates the DFS representation of your signal. If your signal is a pure sinusoid w/ freq between bins, the magnitude distribution in the DFS rep. will be more tightly concentrated around the bins closest to the true frequency when you taper with a non-rectangular window function, and more spread out when you don't. The DFS representation has to use all frequencies to recreate your signal, but it uses less (magnitude) of those far away frequencies to recreate your signal when you use a tapering window. It seems like there should be a nice intuitive explanation for this. $\endgroup$ Commented Apr 5 at 22:04
  • $\begingroup$ I guess the answer to your original question is that tapering the signal to zero prevents a discontinuity when the signal is periodically extended. That discontinuity is more high frequency in content which, when modulated up to the sinusoid's frequency (call that a "carrier frequency"), causes components that are further away from that carrier frequency. $\endgroup$ Commented Apr 5 at 23:33