4. Compute and plot the results of each of the following convolutions: (a) ut) u(t- 2)...
2(a). Compute and plot the convolution of ytryh)x where h(t) t)-u(t-4), x(t)u(t)-u(t-1) and zero else b). Compute and plot the convolution y(n) h(n)*x (n) where h(n)-1, for 0Sns4, x(n) 1, n 0, 1 and zero else.
5.5 Starting with the Fourier transform pair 2 sin(S2) X(t) = u(t + 1) – ut - 1) = X(92) = S2 and using no integration, indicate the properties of the Fourier transform that will allow you to compute the Fourier transform of the following signals (do not find the Fourier transforms): (a) xz(t) = -u(t + 2) + 2u(t) – u(t – 2) (b) xz(t) = 2 sin(t)/t (C) X3 (t) = 2[u(t + 0.5) - ut - 0.5)]...
Problem 1 Compute graphically and plot x[n] *h[n] and x[n] *h[n] (convolutions) for (a). Find a way to derive x[n] *h[n] and x[n] * ñ[n] for (b) without any computation, by using your result of (a) and the properties of convolution. State which property you use. 0 1 2 3 4 5 6 | * 3-2-10 1 2 3 Notation: In the following problems, x[n]={a.b.c) means that_x[-1)=a, x[0]=b, x[1]=c and x[n]=0 otherwise.
Signals and system Class
2.15 Compute the following convolutions without computing any integrals (b) 11(1) * [211 (t) _ 211 (t-3)] (c) u(t) [(t - 1) u(t - 1)]
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t) 2, u(1, t) = 4 и (х, 0) — 2(1 — х). where s(t) describes the source term (a) Find a series solution for u(x, t) with s(t) = e"1. (b) What is the convergence criteria for the transient extension function if s(t) = 0.
Problem 4: Consider the following problem for the heat equation (1)...
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t) 2, u(1, t) = 4 и (х, 0) — 2(1 — х). where s(t) describes the source term (a) Find a series solution for u(x, t) with s(t) = e"1. (b) What is the convergence criteria for the transient extension function if s(t) = 0.
Problem 4: Consider the following problem for the heat equation (1)...
Find the general solution of jutt + 2 ut + 2 u 3 u(0,t)ut)-0for all t s o ater for all x E (0, π), t > 0 Be sure to clearly indicate the following steps in your solution: 1. 2. 3. How to use separation of variables How to solve the resulting elgenfuiction/eigenvalue problem How the superposition principle is used.
6. Plot the following functions and then find the Fourier transforms: (a) f (t) - Kt[u(t +a/2) - u(t - a/2)]. What is the value of F(0)? (b) f(t)- A cos (t)[u(t 2) ut -2)] (c) f(t) -Ae-2M-Du(t -1). 2(t-1)
Use the important property L{f + g}=L{f(t)}.L{g(t)} of convolutions to compute the Laplace transform of sø (t - 1)? cos(2t) dt. a. 1 s'(s2 + 4) 1 b. s (s2 + 4) 2 s3 (32 + 4) 2 d. 5° (s2 + 4)
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...