Solution 10 Computational Intelligence Lab 2016

Note: The solution can now be found on the CIL website of FS16: Master solution (VPN required)

Problem 1

Question 1

See Solution 7 from FS15.

Question 2

1. Formula (linear transformation or change of basis) applied to original signal x, given U: ${\displaystyle z=U^{T}x}$
2. Inverse formula to obtain reconstructed signal applied to ${\displaystyle {\hat {z}}}$: ${\displaystyle {\hat {x}}=U{\hat {z}}}$ (due to orthogonality of U)
1. In both cases, we reconstruct the signal using only the frequencies of high magnitude, discarding the other components. So the rectangle spans over the whole frequency range, from a threshold value up to plus infinity. Here, we keep the lower frequencies (the ones on the left of the spectrum), so we get a low-pass filter. This de-noises the signal.
2. False, because the signal on the top row is a continuous, sine-like signal, which is better suited for a Fourier Basis (global support).
3. True, because the signal on the bottom row is localized.
4. The peaks correspond to the most dominant frequencies of the original signal.