So I got a exercise for signal processing (calculating Gibbs phenomenon overshoot of a Fourier approximation of a simple step function:
(3) [5 points] Here is a way to compute the magnitude of the overshooting.
Using a finite number of terms, we have:
. . . . .\(\displaystyle f(x)\, \approx\, S_N (x)\, =\, \displaystyle{ \sum_{t\, =\, 0}^N \,} b_k \, \sin[kx]\)
Differentiate \(\displaystyle \, S_N (x)\, \) with respect to x and compute the summation analytically.
Find the first zero for \(\displaystyle \,S_N^{'} (x)\, =\, \dfrac{dS_N (x)}{dx}\,\) for x > 0.
Let's assume the first zero is x = x0 such that \(\displaystyle \,S_N^{'} (x_0)\, =\, 0.\)
(4) [5 points] To compute \(\displaystyle \,S_N (x_0),\, \) we integrate:
. . . . .\(\displaystyle S_N(x_0)\, =\, S_N(0)\, +\, \displaystyle{ \int_0^{x_0} \,} S_N^{'}(x)\, dx\)
Again you get an analytical solution. Now you can take the limit for \(\displaystyle \, N\, \rightarrow\, \infty\,\)
and you can compute the amount of overshooting. You might need the fact that:
. . . . .\(\displaystyle \dfrac{\sin(t)}{t}\, \rightarrow\, 1\, \mbox{ as }\, t\, \rightarrow\, 0\)
I got all the Fourier coefficient integrals, but i am not sure how to start this problem really. I pumped it through Mathmatica but the derivative of the starting function just comes back zero so I am kinda confused. Any help understanding how to work through this would be greatly appreciated.
(3) [5 points] Here is a way to compute the magnitude of the overshooting.
Using a finite number of terms, we have:
. . . . .\(\displaystyle f(x)\, \approx\, S_N (x)\, =\, \displaystyle{ \sum_{t\, =\, 0}^N \,} b_k \, \sin[kx]\)
Differentiate \(\displaystyle \, S_N (x)\, \) with respect to x and compute the summation analytically.
Find the first zero for \(\displaystyle \,S_N^{'} (x)\, =\, \dfrac{dS_N (x)}{dx}\,\) for x > 0.
Let's assume the first zero is x = x0 such that \(\displaystyle \,S_N^{'} (x_0)\, =\, 0.\)
(4) [5 points] To compute \(\displaystyle \,S_N (x_0),\, \) we integrate:
. . . . .\(\displaystyle S_N(x_0)\, =\, S_N(0)\, +\, \displaystyle{ \int_0^{x_0} \,} S_N^{'}(x)\, dx\)
Again you get an analytical solution. Now you can take the limit for \(\displaystyle \, N\, \rightarrow\, \infty\,\)
and you can compute the amount of overshooting. You might need the fact that:
. . . . .\(\displaystyle \dfrac{\sin(t)}{t}\, \rightarrow\, 1\, \mbox{ as }\, t\, \rightarrow\, 0\)
I got all the Fourier coefficient integrals, but i am not sure how to start this problem really. I pumped it through Mathmatica but the derivative of the starting function just comes back zero so I am kinda confused. Any help understanding how to work through this would be greatly appreciated.
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