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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...
. Let n (t) be the function being 1 between 1 and and 0 2 2...
. Let n (t) be the function being 1 between 1 and 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). (t/T) 1 T 2 T2
. Let n (t) be the function being 1 between 1 and and 0 2 2 everywhere else, that is, the function in the graph when T= 1. Calculate its Fourier transform,...
5. Consider the 'top-hat' function which is zero everywhere except between -1 and 1 where it take...
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 <...
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).
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-...
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...
1 if t>0 Consider the unit step function u(t) if t0 0 if t< The Fourier transform of the unit ste...
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
1. Solve the ODE/TVP: y" +2y'+y=5(1-2),y(0)-0.7(0) =0. Use the Convolution Theorem everywhere possible, in parts (b)...
1. Solve the ODE/TVP: y" +2y'+y=5(1-2),y(0)-0.7(0) =0. Use the Convolution Theorem everywhere possible, in parts (b) and (c). (a) Find Y(s), the Laplace Transform of y(t), (b) Express y(t) in terms of the convolution product ONLY with explicit functions of t, e.g., f(t)-g(t) or f(t) g(t) * h(t), but do not evaluate any of the convolution product(s); (c) Obtain y(t) by working out completely the convolution product(s) in part (b), show all your intermediate work and results, and simplify your...
6. (15 %)Let f(t)-31-t 0
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 inhomog...
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...
Write the time domain function r(t) of the graph below as the sum of two rectangular...
Write the time domain function r(t) of the graph below as the sum of two rectangular pulse functions, then compute the fourier transform X(w) in terms of sinc function. (Reminder: A rectangular pulse centered 50,427 at the origin with width 27 is defined as II(t) = 11, -T<t<+T and it has the fourier transform II(t) sincwr)) 1 0 Amplitude -1 -2 0 2 3 8 9 10 11 5 6 7 Time-
Suppose, we let g(t) of problem 1 be periodic (i.e., g(t) is 9T (t) according to the notation using). To be prec...
Suppose, we let g(t) of problem 1 be periodic (i.e., g(t) is 9T (t) according to the notation using). To be precise let A 4Volts, let the pulse width T-0.1 seconds and let the 0.2 seconds. Find its continuous Fourier transform. Hint: gr. (t) is now that we are fundamental period To periodic and hence you can first find the Fourier series coefficients (C,) and relate those coefficients to the continuous Fourier transform of a periodic signal. Accurately sketch the...
. Let n (t) be the function being 1 between 1 and 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). (t/T) 1 T 2 T2
. Let n (t) be the function being 1 between 1 and and 0 2 2 everywhere else, that is, the function in the graph when T= 1. Calculate its Fourier transform,...
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).
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...
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
1. Solve the ODE/TVP: y" +2y'+y=5(1-2),y(0)-0.7(0) =0. Use the Convolution Theorem everywhere possible, in parts (b) and (c). (a) Find Y(s), the Laplace Transform of y(t), (b) Express y(t) in terms of the convolution product ONLY with explicit functions of t, e.g., f(t)-g(t) or f(t) g(t) * h(t), but do not evaluate any of the convolution product(s); (c) Obtain y(t) by working out completely the convolution product(s) in part (b), show all your intermediate work and results, and simplify your...
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...
Write the time domain function r(t) of the graph below as the sum of two rectangular pulse functions, then compute the fourier transform X(w) in terms of sinc function. (Reminder: A rectangular pulse centered 50,427 at the origin with width 27 is defined as II(t) = 11, -T<t<+T and it has the fourier transform II(t) sincwr)) 1 0 Amplitude -1 -2 0 2 3 8 9 10 11 5 6 7 Time-
Suppose, we let g(t) of problem 1 be periodic (i.e., g(t) is 9T (t) according to the notation using). To be precise let A 4Volts, let the pulse width T-0.1 seconds and let the 0.2 seconds. Find its continuous Fourier transform. Hint: gr. (t) is now that we are fundamental period To periodic and hence you can first find the Fourier series coefficients (C,) and relate those coefficients to the continuous Fourier transform of a periodic signal. Accurately sketch the...
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