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I'm a little bit confused as to how to calculate the bandwidth of a signal. A question has me believing that it is correlated with the probability distribution. I am however not sure of this.

Suppose you have the following function:

How exactly do you calculate the bandwidth?

I was thinking of doing a fourier transform, but that seems a little bit cumbersome. In this particular question, an engineering rule of thumb was mentioned so I was also thinking about the Nyquist theorem. But to apply that you have to determine a sampling frequency.

In this problem they also ask for a probability distribution so maybe it has something to do with that. I do, however, not know how to calculate said distribution.

Edit: Thanks for the replies. I cannot answer your comments so I will supply some extra information here: indeed the x should be f(x), that was a typo, my bad. The distribution of x is uniform. I would like to know how to calculate the distribution of f(x) (so p(f(x))). As for the bandwidth. I think infinite bandwidth is technically correct, but I think (this isn't given in the question though) that they really want to know the zero-to-zero bandwidth.

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  • $\begingroup$ Your function definition makes no sense; is it of if I replace the $x=$ with an $f(x)=$? $\endgroup$ – Marcus Müller Jan 5 at 12:35
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    $\begingroup$ Probability distribution of what exactly? under which conditions? Also, you'd have to tell us how your material defines "bandwidth"; if you use the strict definition (based on the support in frequency domain), this has infinite bandwidth, no calculation needed, as it is finite in support in time. $\endgroup$ – Marcus Müller Jan 5 at 12:37
  • $\begingroup$ (I very much presume your question is easy to answer – even for you – once you write down all assumptions that are relevant.) $\endgroup$ – Marcus Müller Jan 5 at 13:03
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    $\begingroup$ i think this is a nonsensical question. it can't be answered factually. $\endgroup$ – robert bristow-johnson Jan 11 at 2:04
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How exactly do you calculate the bandwidth?

You don't need to. The signal has finite length and hence the bandwidth is unlimited. There are also steps in there. These also requires infinite bandwidth.

I do, however, not know how to calculate said distribution.

the function consists of six individual line segments. You probably should calculate the probability density function of each line segment, weigh them by $1/6$ and combine them into a single pdf. That will be easier if you draw it an stare at it for a bit.

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  • $\begingroup$ not agreeing with the second point: a function doesn't have a probability distribution, it's deterministic. What has a probability distribution is the value of the function if you applied the function to a random variable - but then Augustin would have to define the distribution of $x$ to even get something useful. Your assumption that $x$ is uniform on $[-3;3]$ falls out of the bluest of skies! $\endgroup$ – Marcus Müller Jan 5 at 12:52
  • $\begingroup$ Fair point, the problem is poorly formulated and there doesn't seem to be random variable in there. This being said, calculating the PDF of a deterministic signal (assuming it's uniformly sampled) is well defined and quite useful in practice. See for example fil.ion.ucl.ac.uk/~wpenny/course/appendixDE.pdf section 8 $\endgroup$ – Hilmar Jan 5 at 18:10

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