


Convolution probems X(+)=ut)-ult-s)44) X(+)X+). Convolution then plot x C+) and yt). 2ct) = 3(u4) - ut-lo)...
Consider the model, Yt = BO + p1 Yt-1 + Ut, select the assumption(s) that are needed to prove unbiased parameter estimates. (A. E[Ut Us |X, Yt-1, Yt-2, ... ] = 0 B. |p1|< 1 C. E[ Ut? |X, Yt-1, Yt-2, ... ] = su? D. E[ Ut |X, Yt-1, Yt-2, ... ] = 0
Problem 2 ( 50 points) You are given functions x(t) = u(t)-ut-1/2) and y(t) = e(ult) -uſt-1)). 1. Sketch a(t) and y(t). (10 points) 2. Use time-domain integration to find the convolution z(t) = x(6) *y(t). (20 points) 3. Find Zw) using any method of your choice (20 points)
2 (20 marks) Consider the signals: x(t)=u(1+1)-n(1-1), a) Plot x(20) andrespectively (b) Calculate the following convolution, (a)Plot x(21) and«S),respectively. (c) Plot the following signal
- y(t)=r (+)-r(t-1)-rt 3)- u(4-3) +u4-4) where r() is the ramp function. a plot y(t) b) plot z(t) c) plot y' (t) d calculate the energy of y(t) r(t)=t for t o O for tco
(1 point) Solve the nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0, u(T,t) = 0 u(2,0) sin(52) u(x, t) = Steady State Solution lim oo u(a,t) =
Calculate the convolution sum x{n]=x[n]*x,[n]: 3. a). xn] S[n]+36[n-1]+28[n-2], x,[n]- u[n]- u[n-3) b). [n]- S[n]+ d[n=1]+S[n-2]+0.58[n-3]+ S[n-51,x,[n]- x,[2n] 4. An LTI system is described with the following LCCDE: In]=x[n]+2y[n-1] a). Plot a block diagram to show the input-output relationship. b).With the input x[n]= S[n], and known y[0] = 0 . Find out the output sequence In] using recursive calculation. 5. A system is described with the following figure, find out a suitable LCCDE to express the input-output relationship y[n] [n]...
b) Find x(t)= x1(t) * x2(t) using the convolution integral. Write the result by region Show all regions and plots in your calculations. eros 3 x Answer: x(t)= bnien l vo s 1Swans ls AVSV meldoy C) Repeat part b) using Laplace. Write the result in terms of delayed unit steps and verify that it is an equivalent result pnwlle Answer: x(t)= ) 3et-1)u(t) 6(t-2) = c) 3e--1u(t)o()= d) 3e--1)u(t)-8(1) = Hint: Is not the same multiplication by a delayed...
Formuals:
3. A sinusoid eơt s or can be expressed as a sum of exponentials e" and e" with complex ncies s-o +yoo and s* -ju, Locate in the complex plane the complex frequencies of (10 points) the following signals: (a) e cos2t (e) 2 ut) -2t (b) e 3 (c) cos3t (d) e Complex numbers: - R034 1 n even (reje)" rkejke Trignometric Identities sin 2x=2sinxcosx sin2 x+cos2 x = 1 in 1-cos 2 cos2x=1 + cos2x sin(x±y)-sinxcosy±cos x...
Solve the differential Squation. 1 + 4 dt ut? + 14t2 du = +++ +C X 3 t
2.4. Compute and plot y[n] - x[n] * h[n], where x[n] - 0, otherwise 1. 4 sn s 15 0, otherwise h[n] = 2.6. Compute and plot the convolution y[n] - x[n] * h[n], where 2.1. Let x[n] = δ[n] + 2δ[n-1]-δ[n-3] and h[n] = 2δ[n + 1] + 2δ[n-l]. Compute and plot each of the following convolutions: (a) y [n] x[n] * h[n] (c) y3 [n] x[n] * h[n + 2]