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I guess my question boils down to what "temporal resolution" means?

I'm taking a signal processing class right now and we're learning about DFT and windowing at the moment. We've learned that when we increase the window size, $N$, we have greater "frequency resolution"

enter image description here

From the image we see that the more we increase, $N$, the more we get the actual $\Omega$ we want and the less smear there is.

Isn't this all there is to resolution? We're getting the $\Omega$ that we actually want? Is there something else important that I'm being really oblivious to?

My professor mentions that a larger window, $w[n]$, provides less "temporal resolution"... What does that mean? And how is it different than having good frequency resolution?

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    $\begingroup$ Think of the frequency resolution as being able to distinguish between two frequencies that are very close. If the resolution is low, they will look like a single frequency. Temporal resolution is how long a sine wave needs to last in order to see it clearly in the DFT. $\endgroup$
    – MBaz
    Feb 15, 2021 at 17:31
  • $\begingroup$ @MBaz I think I'm really confused (SORRY!). If we have a larger $N$ window, then wouldn't we get more of the sine wave which means we'll have better temporal resolution? $\endgroup$
    – BigBear
    Feb 15, 2021 at 22:32
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    $\begingroup$ Yes, but consider that a long signal may contain just a few periods of a sine embedded within it. If you observe the signal a long time, those few periods will come up in the spectrum, but you will have large uncertainty about when those occur. $\endgroup$
    – MBaz
    Feb 15, 2021 at 23:21
  • $\begingroup$ @MBaz Why wouldn't we know when they come up? We've sampled a large amount of the signal many times, so don't we have everything? $\endgroup$
    – BigBear
    Feb 16, 2021 at 2:52
  • $\begingroup$ @MBaz if we only take a small piece of a signal (so small $N$ ) then aren't we even worse off? Because we don't know anything a good chunk of the wave? $\endgroup$
    – BigBear
    Feb 16, 2021 at 3:02

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You have probably heard of the uncertainty principles that arise in physics, with the most famous of all being the Heisenberg's uncertainty principle. But the uncertainty principle applies to signals as well. Just as Heisenberg's uncertainty is described by the following inequality: $$ \Delta_x\, \Delta_p \ge \dfrac{h}{4\pi} $$

The above imposes a limit in the level of resolution in the measurement of a particle's position and momentum. In fact their relationship is inverse meaning that as the particle's position measurements precision increases, then its momentum measurement precision decreases or in other words the uncertainty in its momentum measurement increases.

Now in signal processing the goal is to extract information about the signal's structure (if any) usually in the frequency or time domain (it could also be in the spatial domain). But as you correctly described and analyzed in your graphs the more precise we get in the time domain the more uncertain our estimation of its frequency spectrum becomes. You can see it intuitively in the ripple. The smallest temporal window $N=16$ results in the biggest ripple at higher frequencies, taking also a sizeable proportion in the signal's power spectrum, but as $N$ increases so does the ripple increasing our certainty about the signal's frequency spectrum structure.

$$ \sigma_t\, \sigma_f \ge \dfrac{1}{4\pi} $$ where $\sigma_t$ and $\sigma_f$ are the standard deviations of the time and frequency domain's estimations, respectively.

This minimum threshold is called the Gabor limit, and the principle is called Gabor's uncertainty.

Note:   What is meant by "a larger window provides less temporal resolution" is that some of the properties of the signal in the time domain may be less prominent whereas in the frequency domain we would be more certain about its properties and structure.

From a practical perspective this limit is also important because in addition to other factors it could help us choose an appropriate window for our particular application where it could be desirable to extract features from both domains, sometimes we care only for the time domain properties, sometimes for the frequency domain properties and other times for both (also sometimes implications can be made about properties in one domain from properties measured in the other domain).

In conclusion, the more a signal spreads in the time domain, the less it spreads in the frequency domain and vice versa. In analog signals, a classic example of this phenomenon is the a rectangular pulse in the time domain, which Fourier Transform is the sinc function (exhibits spectral leakage).

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    $\begingroup$ @BigBear Indeed a larger window in time domain, large $N$ increases the frequency resolution. It is my bad I used an ambiguous term ('precision'). It is a matter of how well the true frequency spectrum, the one that we would obtain if we had 'infinitely' high resolution. But in reality we can only use $N$ samples at a time, which although can be large in reality is always finite therefore we don't have the exact frequency spectrum (magnitude at each frequency etc.) but an approximation of it. As the window gets larger our approximation of the frequency spectrum gets better. $\endgroup$
    – kalgoritmi
    Feb 15, 2021 at 22:59
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    $\begingroup$ In the time domain what I meant for 'precision' is actually the ability to locate at which part do some interesting properties arise, its as if we cannot tell exactly at which time interval a sudden spike or a large dip actually occurs. $\endgroup$
    – kalgoritmi
    Feb 15, 2021 at 23:03
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    $\begingroup$ What is meant by uncertainty here is that with a smaller window we have a less sharp approximation of the frequency spectrum, we still have the amplitude at each frequency but this is not the true spectrum. $\endgroup$
    – kalgoritmi
    Feb 15, 2021 at 23:13
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    $\begingroup$ For example in digital image processing, an image can be viewed as a signal that accepts 2 input variables $f(x, y)$ and outputs a single value (a greyscale image).These variables denote each sample's row and column in the image. If we had a $N\times M$ image with sampling periods in each dimension (step because now we are in the spatial domain) $(T_x, T_y)$ then $x=i\cdot T_x, \, i=0,1,\ldots, N$ and $y=j\cdot T_y, \, j=0,1, \ldots, M$ $\endgroup$
    – kalgoritmi
    Feb 15, 2021 at 23:40
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    $\begingroup$ No "So the more (x,y) you have the more accurate your image is to the real thing? ", I just referred to spatial domain, because discrete systems do not arise only in time domain. The number of pixels $N\times M$ defines the images resolution. But I think we have already gone off topic. $\endgroup$
    – kalgoritmi
    Feb 15, 2021 at 23:43

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