Convolution and the Gaussian

(1 point) The following function is called a Gaussian Function. Gaussian functions are used in statistics, image processing, signal processing, and in mathematics to solve heat and diffusion equations. (x-6)2 f(x) = de 22 Find the first and second derivative of the given Gaussian function when a = 3, b = 1, and c = 2 f' ( x ) = f" (x) = NOTE: It will be much easier to type your answer if you simplify the derivative functions (but is not necessary).*17.3 Convolution and the Gaussian. The function that has the same form as its Fourier transform is the Gaussian. Specifically if f(x)=e x/, its FT is given by h(k)= e-(c'/4). Using this fact, show that the convolution (f * f, )(x) of two 12 Gaussian functions f(x)=e-x/ and f,(x)=ex/ is proportional to the Gaussian function excite?). [Hint: you need not calculate any integrals to do this problem.]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).+ 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