How Does Increasing 'n' Affect the Cut-off Frequency 'k_cut' in DFT Analysis?

[Silviu]

Homework Statement

I have this function $f(\theta)=cos(n \ sin(\frac{\theta}{2})\pi)$ and I need to take the discrete Fourier transform (DFT) numerically. I did so and I attached the result for $\theta \in [0,2\pi)$ and n =2,4,8,16,32, together with the function for a given n. I need to give a qualitatively explanation of the dependence of $k_{cut}$, the value of k where the $|c_k|$ starts to go down and n.

Homework Equations

The Attempt at a Solution

Based on the plot of the actual function, when you increase n, the period stays the same, $2\pi$ (obviously) but the number of small oscillations within a period increases linearly with n. As for higher and higher n we need higher and higher frequencies, when we do the DFT, the terms with higher and higher frequency will play an important role, compared to the case when n is small, thus, $c_k$ of this high frequency terms (that corresponds to a high k) must have a non-negligible value, thus the descent point goes higher and higher, as n goes higher and higher, in order to keep this high k terms. (again I need a qualitative explanation). Is this enough to explain this behavior, or is there something else happening here that I should consider? Thank you! (ignore all the small oscillations at the end that are due to numerical limitations, I just care about the math behind the first part of the plot)

 

[mighty2000]

Your explanation is partially correct. As n increases, the number of small oscillations within a period also increases, resulting in a higher frequency content in the function. This means that in the DFT, the higher frequency terms will have a larger contribution to the overall function, leading to a higher value of k where the |c_k| starts to decrease.

However, there is another factor at play here. As n increases, the function becomes more and more complex, with more small oscillations and sharper peaks. This means that the function has a larger number of discontinuities and sharp changes, which require higher frequency components to accurately represent in the DFT. This further contributes to the increase in the value of k where the |c_k| starts to decrease.

In summary, as n increases, both the frequency content and the complexity of the function increase, requiring higher frequency components to accurately represent it in the DFT. This is why the value of k where the |c_k| starts to decrease also increases with n.