lem 2 15 pts) (t) = u(t + 2)-a(t-2) and your calculations. ut2) and h(t) u(t...
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)
1. Problem 1: (20 pts) Let 3(t) = u(1 – t) and h(t) = tſu(t) - t - 2)). (a) Sketch h(t), 3(1), 2(t - T) and carefully label the values on the axes. (6 pts) (b) Determine y(t) = 3(t)h(t) by performing graphical convolution. No need to sketch y(t). (14 pts)
Determine the system response y(t) for h(t)=u(t)+u(t-2) and x(t)=u(t). [Hint: use Laplace Transform multiplication: [[x(+) * h(t)] = X(s)H(s).] y(t) = tu(t)-(t - 2)u(t-2) y(t) = tu(t)+(t-2)u(t-2) y(t) = tu(t)-(t +2)u(t+2) y(t) = tu(t) + (t-2)u(t+2)
6. Signal x()- exp(-t) u() and signal ho) is as shown. (a) Express h(t) in terms of ramp functions only 2 O2 3 4 (b) Find y(t) x(t)*h(t) 0)
6. Signal x()- exp(-t) u() and signal ho) is as shown. (a) Express h(t) in terms of ramp functions only 2 O2 3 4 (b) Find y(t) x(t)*h(t) 0)
Determine the system response y(t) for h(t)=u(t)+u(t-2) and x(t)-u(t). [Hint: use Laplace Transform multiplication: C[x(t) * h(t)] = x(s)H(s). y(t) = tu(t)-(t - 2)u(t - 2) y(t) = tu(t)-(t + 2)u(t+2) y(t) = tu(t) + (t - 2)u(t - 2) y(t) = tu(t) + (t - 2)u(t + 2)
Problem 3, (25 pts) Consider the integral y(t)x(t) dr where x(t)-ult +1)-u(t -1) Find the Fourier transform Y(au) by using the differentiation and the integrati domain properties. Reduce your answer t o the simplest form possible as a function of sinc(u). sin(θ)sene-o siren Formulas: sine(θ)
5. Fourier Transform and System Response (12 pts) A signal æ(t) = (e-t-e-3t)u(t) is input to an LTI system T with impulse response h(t) and the output has frequency content Y(jw) = 3;w – 4w2 - jw3 (a) (10 pts) Find the Fourier transform H(jw) = F{h(t)}, i.e., the frequency response of the system. (b) (2 pts) What operation does the system T perform on the input signal x(t)?
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Determine the system response y(t) for h(t)=u(t)tu(t-2) and x(t)=u(t). [Hint: use Laplace Transform multiplication: C[x(t) *h(t)) = X(s)H(s). y(t) = tu(t)-(t +2)u(t + 2) y(t) = tu(t) + (t - 2)u(t + 2) y(t) = tu(t) + (t - 2)u(t - 2) y(t) = tu(t)-(t - 2)u(t - 2) Question 8 (10 points) What is the Fourier Transform of f(t) = 55(t - 1)? ew 5e-sw 5e-510 را که م
1. Prove that h(t) * (t) = (t) *h(t) 2. A system has an impulse function h(t) = sinº (3t)u(t). Find the unit step (NOTE: an integral table is posted on D2L.) 3. Consider a system with input (t) and output y(t). Let r(t) y(t) = 1 + x(t-1) Is this system linear? Is it causal? Is it BIBO stable? Justify your answer
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.