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- Indicate which of the statement(s) below is(are) true: (a) y(t) = 3 x(t) is a linear expression %3D (b) y(t) = r(t+2) is a causal system %3D (c) y(t) = K- (t – 2) is memoryless and causal %3D O a. All of them are TRUE O b. (a) is the only TRUE statement O c. (a) and (b) are TRUE O d. (b) and (c) are TRUEThe system shown below contains two linear shift-invariant subsystems with unit sample responses h₁ [n] and h₂ [n], in cascade. x (n) hy (n) 0 w(n) h₁ (n) = 6(n) - 8 (n − 3) 1 h₂ (n) = +! 0 h₂ (n) -1 (.8) u (n) y (n) (a) Let x[n] = u[n]. Find and sketch ya [n] by first convolving x[n] with h₁ [n] and then convolving that result with h₂ [n], i.e. Ya[n] = [x[n] * h₁[n]] * h₂[n] (b) Again let x[n] = u[n]. Find and sketch y₁ [n] by convolving x[n] with the result of convolving h₁ [n] and h₂ [n], i.e. y₁ [n] = x[n] * [h₁ [n] *h₂[n]] (c) What convolution property does this demonstrate?A scientist has measured quantities y, x1,and x,. She believes that y is related to x, and x2 through the equation y = ae+Px2 8, where & is a random error that is always positive. Find a transformation of the data that will enable her to use a linear model to estimate B, and B,.
- 1. Consider the data points (0, 1), (1, 1), (2, 5). (a) Find the piecewise function P(x) = that interpolates the given data points, where Spo(x), x € [0,1]. (p₁(x), x= [1,2], Po(x) = a + be, P₁(x) = c+dx², for some constants a, b, c, d to be determined. (b) Find the natural cubic spline Si(2) S(x) = Jso(x), x= [0,1]. $1(x), x= [1,2], that interpolates the given data points, where so, 81 are cubic functions within their respective intervals. Express the resulting polynomials in monomial form, that is, so (x) = ao + box + cox² +dox³, $₁(x) = a₁ + b₁x + ₁x³ + ₁x³, for some ao, bo, co, do, a1, b1, C₁, d₁. Hint: Recall the formula we derived in class for the cubic splines, 1 hi = oh [(+– z)*M + (z – zi)®M+] - * [+- z)M + (z − )Min] Xi + * [(2 - 2)f(z) + (z – zi)f(+)] for x = [xi,i+1], and i = 0,..., n-1. Solve for the values Mo, M₁, ... by setting up the appropriate system of equations, and use the formula for si to obtain the desired cubic spline.The matrix that projects onto the line y = -x is X 0.6 0.8 0.8 -0.6OC. P(3)= P(4) = P(5) = P(6) = P(7) = P(8) = P(9) = P(10) =, P(11) = %3D %3D %3D %3D O D. P(1)= , P(2) = P(3) = P(4) = P(5) = P(6) = %3D %3D %3D Calculate the expected value of x. E(x) = (Type an integer or a decimal. Do not round.) %3D b. Calculate the variance of x. o =(Simplify your answer.) %3D c. Calculate the standard deviation of x. o, = (Round to three decimal places as needed.) %3D Click to select your answer(s).
- Let h(n) = 3(1/2)² u(n) – 2(1/3)"-1 u(n) be the unit sample response of a linear shift-invariant system. If the input to this system is a unit step, n 20 x(n) = else find lim,o y(n) where y(n) = h(n) * x(n). -3 -2 O 1 01. Find y by Cramer's rule for the following linear system. x+Oy+2z 6 -3x+4y+6z= 30 --2y+3z%=8 rallclenined with sides F(1,-3. -1).Let T(z) = -6z* +z² –1 and U(x) = 2z – 4x + 3. Find T(z) +U(z) %3D OA. -6z4 + 2z3 +z² - 4z + 2 OB. -7z10 +2 OC. -4z4 - 3z² + 2 OD. -427 - 3z + 2
- A random process x(t) is applied to a network with impulse response h(t) u(t) t exp (- bt) where b>0 is a constant. The cross-correlation of x(t) with the output y(t) is known to have the same form. Rxx(t) = u(t) T exp (– bt) (a) Find the auto correlation of y(t) ? (b) What is the average power in y(t) ?Show that Δ2y0 = y2-2y1 + y0.(b) x = 5 + 7t,y = - 3+ 10t, z = 6 – 4t O (x, y, z) = (5, 3, 6) + t(7, 10, 4) ai + bj + ck = (5i + 3j+ 6k) + t(7i + 10j + 4k) %3D O (x, y, z) = (7, 10, – 4) + t(5, – 3, 6) ai + bj + ck = (7i + 10j – 4k) + t(5i – 3j + 6k) O (x, y, z) = (5, – 3, 6) + t(7, 10, – 4) ai + bj + ck = (5i – 3j + 6k) + 1(7i + 10j – 4k) - O (x, y, z) = ( – 5, 3, – 6) + t( – 7, – 10,4) ai + bj + ck = ( – 5i + 3j – 6k) + t( – 7i – 10j + 4k) O (x, y, z) = { – 7, – 10, 4) + t( – 5, 3, – 6) ai + bj + ck = ( – 7i – 10j + 4k) + t( – 5i + 3j – 6k)