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1. The Gaussian function f(x) = e" is widely used in mathematics, statistics, and engineering. We will derive the Fourier transform of the Gaussian function f(x) = e- by the following steps. (a) Evaluate the improper integral I = edr. Hint: First square the integral and then use the polar coordinate to evaluate the double integral, i.e, P=orderdy = eydady. (b) Solve the ODE y' = =y with the initial condition y(0) = yo. (c) Find the Fourier transform of f(x) = e . Hint: Let f(w) = F[f](w) be the Fourier transform of f(x). Then take the derivative of f(w). You may need to use the property that F[f](w) = awF[f](w).6. The (normalized) Gaussian function is defined as 1 -12 g(r) = OV2x where (_g(x)dr = 1. Note that this implies that the "DC component" of its Fourier transform G(0) = 1. Prove that the Fourier transform is also a Gaussian function of the form -w202 G(w) = e Hint: Use the differentiation with respect to t, and multiplication properties of the Fourier transform, and then integrate both sides.+ 2 Fit to page [ Page view A') Read aloud Question 4 (a) Solve the difference equation In+2 + 3En+1 + 21 = 0, 10 = 1, $1 = 2 using the Z-transform. (b) Calculate the Fourier transform of the Gaussian function f (t) = e- +212 You may find useful to know that the integral of a Gaussian function is given as Change my default Don't ask again