2.11. Let x(t) 11(1-3)-u(t-5) and h(t) = e-3t11(1). (a) Compute y(i) - x(t) * h(t)
Prob. 5 (a) Let x(t) = u(t) and h(t) = e-looor u(t) + e-lotu(t). -00 <t< oo using graphical convolution(s). Determine y(t) = h(t) * x(t) for Prob. 5 (cont.) (b) Let zln] = uln] and h[n]-G)nuln] + (-))' hnnDetermine vinl -h) rin) for -00n< oo using graphical convolution(s) Prob. 5 (a) Let x(t) = u(t) and h(t) = e-looor u(t) + e-lotu(t). -00
1. Let x(t)-u(t-1) _ u(t-3) + δ(t-2) and h(t)-u(t) _ u(t-1) + u(t-3)-u(t-5) a. Find and sketch x(t-t) and h(t). (Hint: Break x(t) into two signals) b. Find and sketch y(t) - x(t)*h(t) using the quasi-graphical method. Label and show every step (drawings and calculations)
Question # 4 Let x(t) u(t) be a signal, let h(t) = e -5tu(t) be a linear time invariant system (a) Sketch x(t) and h(t) (b) Find the mathematival expression of output of the system y(t) by using convolution. (c) Sketch y(t)
Let X tilda U(0,3) and Y = X^3. Compute P(5 < Y < 12).
Let XN(0, 1) and Y eX. (X) (a) Find E[Y] and V(Y). (b) Compute the approximate values of E[Y] and V(Y) using E(X)(u)+"()VX) and V((X))b(u)2V(X). Do you expect good approximations? Justify your an- Swer Let XN(0, 1) and Y eX. (X) (a) Find E[Y] and V(Y). (b) Compute the approximate values of E[Y] and V(Y) using E(X)(u)+"()VX) and V((X))b(u)2V(X). Do you expect good approximations? Justify your an- Swer
Let u(t) =t^3 i + ln(t) j + e^2t k and v(t) = 1/t^3 i + 2 j + t k 2. Let u(t) - ti+In(t)j+ et k and Compute the derivative of the dot product f u(t)v() in two ways and confirm they agree: Compute the dot product u(t) v(t) first and then differentiate the result. . Alternatively, use the following "Dot Product Rule" u(t) v(t)] u'(t) v(t)+ u(t) v'(t) Aside: It's worth noting that there are other forms...
3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B Find functions g and h such that X, has the same covariance as a Brownian bridge. 3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B...
Problem 5 Let U be an n dimensional vector space and T E L(U,U). Let I denote the identity transformation I(u) = u for each u EU and let 0 denote the zero transformation. Show that there is a natural number N, and constants C1, ..., CN+1 such that C1I + c2T + ... + CN+1TN = 0 (Hint: Given dim(U) = n, what is the dimension of L(U,U)? consider ciI + c2T + ... + Cn+11'" = 0, where...
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.
Problem 1: Let y()- r(t+2)-r(t+1)+r(t)-r(t-1)-u(t-1)-r(t-2)+r(t-3), where r(t) is the ramp function. a) plot y(t) b) plot y'() c) Plot y(2t-3) d) calculate the energy of y(t) note: r(t) = t for t 0 and 0 for t < 0 Problem 2: Let x(t)s u(t)-u(t-2) and y(t) = t[u(t)-u(t-1)] a) b) plot x(t) and y(t) evaluate graphically and plot z(t) = x(t) * y(t) Problem 3: An LTI system has the impulse response h(t) = 5e-tu(t)-16e-2tu(t) + 13e-3t u(t) The input...