1. Consider the function f(x)-e- (a) Find its Fourier transform. (b) Use the result of part (a) t...
Consider the function y = x2 for x E (-7,7) . a) Show that the Fourier series of this function is n cos(nz) . b) (i) Sketch the first three partial sums on (-π, π) (ii) Sketch the function to which the series converges to on R . c) Use your Fourier series to prove that 2and1)"+1T2 12 2 2 Tu . d) Find the complex form of the Fourier series of r2. . e) Use Parseval's theorem to prove...
Find the Fourier transform of f(x) = 1–x?, for -1 < x < 1 and f(x) = 0 otherwise. Hence evaluate the integral 6 * * cos sin cos des.
please complete all parts Problem F.7: These are independent problems (a) (5 points) Solve the following integral. (Hint: Think Fourier series.) (cos(nt) - 2sin(5rt)e-Jr dt XCj) (b) (5 points) Find the Fourier transform io of the following signal: 2(t) = sin(4t)sin(30) (c) (5 points) Solve the integral: sin(2t) 4t dt (d) (5 points) Use Parseval's theorem and your Fourier transform table to compute this integral: Problem F.7: These are independent problems (a) (5 points) Solve the following integral. (Hint: Think...
3. If x(t) has the Fourier transform j2π f + 10 Find the Fourier transform of the following signals Hint: use the properties of Fourier transform) a. v(t)-x(1):cos(10π t) d. v(t)X(t) e. v()-e"x(t-1)
Consider the Fourier transform pair e- 12. (a) Use the appropriate Fourier transform properties to find the Fourier transform of te-tl, (b) Use the result from part (a), along with the duality property, to determine the Fourier transform of 4t (1+t2)2
The Fourier Transform of a certain time function, x(t), is shown below F{x(t) x(f) 2.5 7 1.5 1 0.5 -30 -20 -10 10 20 30 f(Hz) equation for X(f A. Write an B. Write an equation for x(t). C. Write and equation for the Fourier Transform of x(2t) and draw a sketch D. Write and equation for the Fourier Transform of x(t) and draw a sketch equation for the Fourier Transform of x()cos(2 E. Write an 15 t and draw...
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)...
(2) Consider the function f(x)- 1 (a) Find the Fourier sine series of f (b) Find the Fourier cosine series of f. (c) Find the odd extension fodd of f. (d) Find the even extension feven of f. (e) Find the Fourier series of fod and compare it with your result -x on 0<a < 1. in (a) (f) Find the Fourier series of feven and compare it with your result in (b)
a) Use MATLAB to find the Fourier Transform F(w) of the following function f(t). b) Plot F(w). Express the x-axis in [Hz]. Plot for f = -8Hz to 8Hz. f(t) = cos(27 (34))e-**" 0.8 0.6 0.4 0.2 f(t) appears to oscillate at 3 cycles/sec 0 -0.2 -0.4 -0.6 0.8 -1 2 -1.5 -0.5 0 0.5 1 1.5 2
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...