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When choosing a window function, window duration, and/or transmission frequency (assuming sampling rate satisfies Nyquist), one may want to understand what sort of spectral leakage would occur at a frequency of interest.

It is known that a finite-duration window corresponds to a non-finite bandwidth frequency response (e.g. rect <-> sinc), and it is also known that a multiplication in the time domain corresponds to a convolution in the frequency domain.

Consider the simple case of a non-windowed, constant-frequency sinusoid in the time domain, which corresponds to a frequency response of two delta spikes centered around 0 Hz.

Applying a window function would convolve the frequency response of the window function (e.g. a sinc function) over the delta spikes.

1. Does this convolution of the window function extend to the negative frequencies when calculating the spectral leakage in the positive frequency components (and vice versa)?

I would say yes based on the above (time domain multiplication <-> frequency domain convolution), and the following two images (source) which I annotated in red. But it leads me to question 2, which I find a bit concerning.

enter image description here

enter image description here

2. If spectral leakage does extend to opposite-sign frequencies, doesn't that imply window functions without a zero crossing at 2x of a frequency of interest would result in constructive and/or destructive interference ("spectral leakage") in the frequency response of the windowed duration at that frequency? I.e., frequencies could interfere with themselves?

Here is an image (source) showing that even for a given window duration, the occurrence of such "self-interference" would depend on the choice of window function:

enter image description here

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Yes spectral leakage includes both negative and positive frequencies; i.e, the convolution is from $-\pi$ to $\pi$.

However its effect depends on what exists at the negative frequency region, when you want to compute the resulting leakage on a given positive frequency.

On a typical application where you have a real valued sine wave of frequency $f_0$ as the windowed signal, then there is a dirac impulse at the negative frequency $-f_0$ as well as the positive frequency $f_0$. Then both impulses shift the spectrum of the window Fourier transform to their own locations and create leakage effects on any other frequency (unless they have zero value at that particular frequency). However the leakage is expected to be smaller from a negative frequency as it's side lobes are already decaying more than the positive frequency contributions. (This can be the opposite when both impulses are close to $\pi$ or close to $0$ radians.)

But if you have a complex signal, such as an analytic signal which has zero negative spectral content, then effectively the negative frequencies will not create leakage.

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  • $\begingroup$ Okay, then I will accept the answer to (1) as yes. And it sounds like you are saying (2) is true as well assuming I am using a physical signal (i.e. non-analytic). However, isn't it a bit odd that a negative frequency component of a signal can interfere with it's positive magnitude response? It seems a bit counter-intuitive, and also makes analytically modeling the magnitude response more complicated. $\endgroup$ – abc Nov 24 '19 at 22:36
  • $\begingroup$ Answer to (1) is yes, answer to (2) is corollary yes. It doesn't sound odd to me either I don't really understand the problem or it's so long that I've accepted it as it is... But some poeple tend to interpret negative frequencies from a physical perspective to deduce that they are some non-real, imaginary kind of things. But I just look at them from an algebraic point of view and do not find an oddity within them. $\endgroup$ – Fat32 Nov 24 '19 at 22:52
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    $\begingroup$ I meant the oddity that a real signal could "self-leak" in the frequency domain, but I think when I look into the sorts of window functions/durations that cause this, it's starting to make more sense. In any case, I appreciate the answer. $\endgroup$ – abc Nov 24 '19 at 23:37
  • $\begingroup$ Hmm I just saw the oddity now ! Yes of course a real sine wave does interfere with itself too! Let's call it self-interference :-)) But that's quite understandable when you decompose a sine wave into two complex exponentials and consider the mutual leakge in between. $\endgroup$ – Fat32 Nov 25 '19 at 0:19
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    $\begingroup$ @CedronDawg Even for the case of an integer number of cycles of a physical sinusoid per window (e.g. 1) with a non-rectangular window (e.g. Hanning), there will be self-interference. $\endgroup$ – abc Nov 25 '19 at 14:27

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