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- Consider the function f(x) coefficients. (a) do (b) a6 (c) ag (d) b7 (e) bg = 8 + 5 sin²(4x) restricted to the interval [-л,л]. Calculate the following FourierQ1) Find Fourier transform for the following functions: a) x(t) = 3 + (sin2wt)? b) y(t) = 5. (e-6t). 8(t – 8) + 6u(t + 8) %3D(1+x 1" sin x dx 0, If f(7)
- • Let: X1(Jw) = 2T8(w) + n8(w – 4n) + n8(w + 47) • Use the inverse Fourier Transform to find x;(t) and finally express x1(t) as a cosine function.If x(t) <---> X(f) and y(t) <---> Y(f), using the Fourier table and Fourier transform pairs, determine the Fourier transform of the following expressions. Identify and list the properties that you have used in the solution. Hint: You are not supposed to solve manually.using laplace transform, solve the equation: y'' + 4y = 8cos(2t) - 8*exp(-2t) y(0) = -2 y'(0) = 0
- Obtain the Fourier transform of the function where f(x) = 0 for x < 0 and f(x) = e−axsin(bx) for x > 0.f(t) satisfies the integral equation: f(t) = 4e-3t H(t)-7/f(t-u) e-6u H(u) du Find the solution of the integral equation using Fourier transforms. Your answer should be expressed as a function of t using the correct syntax. f(t) =Find the Fourier transform specified in part (a) and then use it to answer part (b). (a) Find the Fourier transform of " sin pt t>0, t 0) and p are constant parameters. (b) The current I(t) flowing through a certain system is related to the applied voltage V(t) by the equation I(t) = K(t – u)V(u) du, where K(t) = a¡f(71, P1,t) + azf(y2, P2, t). The function f(y, p, t) is as given in (a) and all the a;, y¡ (> 0) and p, are fixed parameters. By considering the Fourier transform of I(t), find the relationship that must hold between a, and az if the total net charge Q passed through the system (over a very long time) is to be zero for an arbitrary applied voltage.