1. Let M (t) be the function being 1 between and 1 1 and 0 2 2 everywhere else, that is, the function in the graph when t = 1. Calculate its Fourier transform, and the convolution N (t) n (t) (the convolution with itself). n(t/T) 1 T 2 2 1. Let M (t) be the function being 1 between and 1 1 and 0 2 2 everywhere else, that is, the function in the graph when t = 1....
5. Consider the 'top-hat' function which is zero everywhere except between -1 and 1 where it takes value 1. Show, using only elementary integration, that the convolution of this function with itself gives a 'triangular function', 0 0 x〉2 which you should sketch Find the Fourier transform of the 'triangular function' f(x) using the result for the Fourier transform of a convolution. 5. Consider the 'top-hat' function which is zero everywhere except between -1 and 1 where it takes value...
Piecewise function f(t) = 1 when 0 < t < 1, and f(t) =-1 when-1 < t < 0. Also f(t) = 0 for any other t (t < 1 or t 2 1). Answer the following questions: 1. Sketch the graph of f (t) 2. Calculate Fourier Transform F(j) 3. If g(t) = f(t) + 1, what is G(jw), ie. Fourier transform of g(t)? 4, extra 3-point credit: h(t) = f(t) + sin(kt), find the Fourier Transform of h(t).
0and / is an odd function of t, find the Fourier sine sin wt d for 0<t< 1 10, (a) If f(t) = for t a 0 transform of f. Deduce thato s if0<t < a. What is the value of the integral for t2 a? for 0 < t < b (b) If g(t)-{ b-t and g is an even function of t, find the Fourier 0 cosine transform of g. Deduce that foo 1-w2bw cosa t dw =...
1 if t>0 Consider the unit step function u(t) if t0 0 if t< The Fourier transform of the unit step function is: U(ω)-Flu (t)]- πδ(w) + 1 , and the graph of the unit step function is shown below: u(t) 1/2 Relate intuitively each term of the Fourier transform U() given above to the corresponding parts f you find it helpful). Explain briefly below. 1 if t>0 Consider the unit step function u(t) if t0 0 if t
0<t<T when Tt< 2 t 2T sin t when 2. Calculate the Laplace transform of the periodic function f(t) 0 f(t-2) when -7s 3. Calculate the inverse Laplace transform of G(s) 3-4e-5 + $2+2s+17 4. Use the Laplace transform to solve each initial value problem: 4y"+ y u2m(t)sin(t/2) y(0)=0 &(0 =0 (a) 0 and /(0) 2 "+4y+13y = 4to(t-T) if y(0) (b) 5. Use the convolution to write a solution of each initial value problem. y"+6y'+10y g(t) 1 y(0) 0...
2TT sinn (1) a) Let x1 [n] = πη Find the Discrete Time Fourier transform of this signal and plot it with all its critical values. (you can use only transform tables from the book). b) Now, define xzlv) = (**) GHS) Using transform properties, find the Discrete Time Fourier transform of x2[n] and plot it with all its critical values. In your calculations be sure to show your steps ! 2TT sinn sinn sinwon c) Let y[n] [( )...
2. Calculate the inverse Fourier transform of X(cfw) = {2 2j 0 <W <T -2j -n<w < 3. Given that x[n] has Fourier transform X(@j®), express the Fourier transforms of the following signals in terms of X(el“) using the discrete-time Fourier transform properties. (a) x1[n] = x[1 – n] + x[-1 - n] (b) x2 [n] = x*[-n] + x[n]
6. (15 %)Let f(t)-31-t 0<t 1, 0 otherwise Find the Fourier transform of FIfl(x) and the deduce that osin"(a) /2) π doo- by using the inverse Fourier transform. 6. (15 %)Let f(t)-31-t 0
Green's function 2 The Green function (10 P) The Fourier transform plays a tantamount role in the theory of inhomogeneous, linear differential equations. If as was shown in the lecture - G is a so called fundamental solution of the differential equation CG(z,z') = δ(z-z') one may calculate a particular solution for an inhomogeneity g by convolution G is called Green function. Since the Fourier transform maps derivatives to multiplications, it simplifies the calcu- lation of the Green function to...