4.3.2 Find the Fourier transforms (with f= 0 outside the ranges given) of (a) f(x)= 1 for 0 < x <L (b) f(x)= 1 for x < 0 (c) f(x)= So eikx dk (d) the finite wave train f(x)=sin x for 0<x< 107
(8). The one dimensional neutron diffusion equation with a (plane) source at x-0 is d'f(x) n (2) +002 f (x)-00(x) dx where f(x) is the flux of neutrons (f(x)→0 as x→±o), Q δ (x) is the (plane) source at x-0 (5(x) is the Dirac delta function), and o is a constant. This problem involves finding the solution to this equation using Fourier Transforms. You may use the formulas derived in class for the Fourier Transform of derivatives, but otherwise compute...
4. Let f(x) = 6-2x, 0<x 2 (a) Expand f(x) into a periodic function of period 2, ie. construct the function F(x), such that F(x)-f (x), 0xS 2, and Fx) F(x+2) for all real numbers x. (This process is called the "full-range expansion" of f(x) into a Fourier series.) Find the Fourier series of Fr). Then sketch 3 periods of Fx). (b) Expand fx) into a cosine series of period 4. Find the Fourier series and sketch 3 periods (c)...
реттеу 0 x<-1 2 -1<x<0 fx) -X + 2 0<x<1 0 X>1 The coefficient 4 of Fourier integral representation associated with the above given function can be computed as sin w O A3 + 1 - COSW 2 TW TW COSW OB W sin w Ос 2 TW sin w 1 + COSW OD 3 TW TW COSW sin OE TW
Please show all steps with clear hand writing 3. Consider the periodic function defined by sin(x) 0<x< f(r) = and f(x) = f(x + 27). (a) Sketch f(x) on the interval-3r < r 〈 3T. etch fx on the interva (b) Find the complex Fourier series of f(x) and obtain from it the regular Fourier series. 3. Consider the periodic function defined by sin(x) 0
l etisalat 5:45 PM 슐 moodle.ud.ac.ae 11.1 and 11.2 Fourier Series Q1 Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x Note: Plot the partial sum using MATLAB Hint: Make use of your knowledge of the line equation to find f) from the given graph. 2 Find the Fourier integral representation...
Denote the Fourier series of fr-fx, 1<x< 0 f(x) = { 0, 0SX S1 by F(x). Show that E F(x) = - -_ 2500 cos (2mi) + 2m=0 (2m+1) + 500 + 2n=1 + in sin(nx).
Now suppose f' is continuous and f" is piecewise continuous on (0, L). (b) If f(0) = f(L) = 0, then O f(x) = į bn sin ηπα L 0<x<L, n=1 612 Chapter 11 Boundary Value Problems and Fourier Expansions with bn = 2L n272 S“ r"(a)sin пах dr. L (11.3.5) Solve the initial-boundary value problem. Theorem 11.3.5 (b) will simplify the computa- tion of the coefficients in the Fourier sine series. Uit = 64uze, 0<r <3, t > 0,...
Problem 3: Let x(n) be an arbitrary signal, not necessarily real valued, with Fourier transform X (w). Express the Fourier transforms of the following signals in terms of X() (C) y(n) = x(n)-x(n-1) (d) v(n) -00x(k) (e) y(n)=x(2n) (f) y n even n odd , x(n/2), (n) 0 Problem 4: etermine the signal x(n) if its Fourier transform is as given in Fig. P4.12. X(a) 0 10 10 10 X(o) 0 X(a) Figure P4.12 Problem 3: Let x(n) be an...
let f:[-pi,pi] -> R be definded by the function f(x) { -2 if -pi<x<0 2 if 0<x<pi a) find the fourier series of f and describe its convergence to f b) explain why you can integrate the fourier series of f term by term to obtain a series representation of F(x) =|2x| for x in [-pi,pi] and give the series representation DO - - - 1. Let f: [-T, 1] + R be defined by the function S-2 if-A53 <0...