Question

Let $f(x)=x(2 \pi-x)$ be sampled on $n=128$ equally spaced points between 0 and $2 \pi$. Use a random number generator with $-1 \leq r_j \leq 1$ to add noise by replacing each sample value $f_j=f\left(x_j\right)$ by $g_j=f_j+\varepsilon r_j$. Investigate, for different values of $\varepsilon$, how many discrete Fourier modes are required to reconstruct a reasonable denoised approximation to the original signal.

   Let $f(x)=x(2 \pi-x)$ be sampled on $n=128$ equally spaced points between 0 and $2 \pi$. Use a random number generator with $-1 \leq r_j \leq 1$ to add noise by replacing each sample value $f_j=f\left(x_j\right)$ by $g_j=f_j+\varepsilon r_j$. Investigate, for different values of $\varepsilon$, how many discrete Fourier modes are required to reconstruct a reasonable denoised approximation to the original signal.

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 5, Problem 13

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Step 1

- Define the function \( f(x) = x(2\pi - x) \). - Set \( n = 128 \) as the number of equally spaced sampling points. - Calculate the sampling points \( x_j = \frac{2\pi j}{n} \) for \( j = 0, 1, 2, \ldots, n-1 \). Show more…

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Let $f(x)=x(2 \pi-x)$ be sampled on $n=128$ equally spaced points between 0 and $2 \pi$. Use a random number generator with $-1 \leq r_j \leq 1$ to add noise by replacing each sample value $f_j=f\left(x_j\right)$ by $g_j=f_j+\varepsilon r_j$. Investigate, for different values of $\varepsilon$, how many discrete Fourier modes are required to reconstruct a reasonable denoised approximation to the original signal.

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