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fourier analysis: signals and filters 245Decompose this into its components; then check that there are three ofthem in the ratio 5:2:1 (or 25:4:1 if a power spectrum is plotted) and thatthey resum to give the input signal.5. Sample the signaly(t) = 5 + 10 sin(t +2).Compare and explain the results obtained by sampling (a) without the5, (b) as given but without the 2, and (c) without the 5 and withoutthe 2.6. In our discussion of aliasing, we examined Figure 10.2 showing thefunctions sin(πt/2) and sin(2πt). Sample the function( π)y(t) = sin2 t + sin(2πt)and explore how aliasing occurs. Explicitly, we know that the true transformcontains peaks at ω = π/2 and ω =2π. Sample the signal at a ratethat leads to aliasing, as well as at a higher sampling rate at which thereis no aliasing. Compare the resulting DFTs in each case and check if yourconclusions agree with the Nyquist criterion.Highly nonlinear oscillator: Recall the numerical solution for oscillations of aspring with power p =12[see (10.1)]. Decompose the solution into a Fourierseries and determine the number of higher harmonics that contribute at least10%; for example, determine the n for which |b n /b 1 | < 0.1. Check that resumingthe components reproduces the signal.Nonlinearly perturbed oscillator: Remember the harmonic oscillator with anonlinear perturbation (9.2):V (x)= 1 (2 kx2 1 − 2 )3 αx , F(x)=−kx(1 − αx). (10.43)For very small amplitudes of oscillation (x ≪ 1/α), the solution x(t) willessentially be only the first term of a Fourier series.1. We want the signal to contain “approximately 10% nonlinearity.” Thisbeing the case, fix your value of α so that αx max ≃ 10%, where x max isthe maximum amplitude of oscillation. For the rest of the problem, keepthe value of α fixed.2. Decompose your numerical solution into a discrete Fourier spectrum.3. Plot a graph of the percentage of importance of the first two, non-DCFourier components as a function of the initial displacement for 0