1. Use Fourier Transforms to x(t + to) ⓧh(t-to)-y(t) show that if )Sh)-y0), thern
1) (Fourier Transforms each of the following signals (a - c), sketch the signal x(t), and find its Fourier Transform X(f) using the defining integral (rather than "known" transforms and properties) (a)x(t) rectt 0.5) from Definition)- For (c) r(t) = te-2, 11(1) (b) x(t)-2t rect(t)
1) (Fourier Transforms each of the following signals (a - c), sketch the signal x(t), and find its Fourier Transform X(f) using the defining integral (rather than "known" transforms and properties) (a)x(t) rectt 0.5) from...
4.30p Find the Fourier transform of δ(t-t) and use it to find the Fourier transforms of: a) 8(t-2) - 8t 2) b) cos(5t)
Answers are:
10. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 40 w2 - 13iw (b) Use Fourier transforms to solve dy -5t + 8y — 9е эH (). dt 15t H(t) 1 8t (а) (1) Н, 9 Then solve for 5iw (b) Apply the Formula of transform of derivatives to get: (iw+8)Y(w) Y (w) and take the inverse transform to have -8t у(0) — Зе 5 н(t) —...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
3. Use Fourier Transforms to solve u(0, )sin(ar) -o0 o0, t > 0,
3. Use Fourier Transforms to solve u(0, )sin(ar) -o0 o0, t > 0,
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms Fu(x, t) _ u(x, t) + g(x) =-a ди (x, t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u(x, 0) 0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following...
8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 1 12 8iw w2 _ (b) Hence, determine the inverse Fourier transform of -iw 12 8iw -w2' (c) Use Fourier transforms to solve d2y ,dy + dt2 12y (t 1 8
8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 1 12 8iw w2 _ (b) Hence, determine the...
3) (Fourier Transforms Using Properties) - Given that the Fourier Transform of a signal x(t) is X(f) - rect(f/ 2), find the Fourier Transform of the following signals using properties of the Fourier Transform: (a) d(t) -x(t - 2) (d) h(t) = t x( t ) (e) p(t) = x( 2 t ) (f) g(t)-x( t ) cos(2π) (g) s(t) = x2(t ) (h)p()-x(1)* x(t) (convolution)
3) (Fourier Transforms Using Properties) - Given that the Fourier Transform of a signal...
8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the Fourier transform of sgn(t 1)e4t-1| (b) Hence, find the Fourier transform of 5i sgn(t 1) eit-4e-1| (Simplify your answer.
8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the Fourier transform of sgn(t 1)e4t-1| (b) Hence, find the Fourier transform of 5i sgn(t 1) eit-4e-1| (Simplify your answer.
2) (Fourier Transforms Using Properties) - Given that the Fourier Transform of x(t) e Find the Fourier Transform of the following signals (using properties of the Fourier Transform). Sketch each signal, and sketch its Fourier Transform magnitude and phase spectra, in addition to finding and expression for X(f): (a) x(t) = e-21,-I ! (b) x(t)-t e 21 1 (c) x(t)-sinc(rt ) * sinc(2π1) (convolution) [NOTE: X(f) is noLI i (1 + ㎡fy for part (c)]
2) (Fourier Transforms Using Properties)...