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Signals and Systems

Please see the attachment for all the questions with proper symbols/notations.

1. Compute the Laplace transform of e^(-10t)cos(3)u(t) .

2. Compute the z-transform of the discrete time signal defined by:
x[n] = δ[n] + 5δ[n - 1]

3. Compute the inverse Laplace transform of X(s) = (s+2) / (s^2+7s+12) .

4. Determine if the signal given is linear, time-invariant, causal, and/or memoryless.
y(t) = [sin(6t)]x(t)

5. A continuous time signal x(t) has the Fourier transform X(ω) = 1 / (jω+b), where b is a constant. Determine the Fourier transform for v(t) = x(5t - 4).

6. Compute the unit-pulse response h[n] for the discrete-time system
y[n + 2] -2y[n + 1] + y[n] = x[n] (for n= 0, 1, 2, 3)

7. Determine the inverse DTFT of X(Ω) = sin(Ω)cos(Ω).

8. Determine if the signal given is linear, time-invariant, causal, and/or memoryless.
y(t) = d^(x(t))

9. Determine if x(n) = cos ( PI n / 4) cos ( PI n / 4) is periodic; if periodic, calculate the period.

10. Compute the DTFT of the discrete-time signal x[n] = (0.8)^n u[n].

11. Compute the impulse response h(t) for (dy(t) / dt) - 3y(t) = x(t).

12. Compute the inverse Fourier transform for X(ω) = sin^2(3ω).

13. Determine if x(t) = cos(3t + π / 4) is periodic; if periodic, calculate the period.

14. Determine if the linear time-invariant continuous-time system defined is stable, marginally stable, or unstable.
(s - 1) / (s^2 + 4s + 5)

15. For a discrete-time signal x[n] with the DTFT X(Ω) = 1 / (e^jΩ + b), where b is an arbitrary constant, compute the DTFT V(Ω) of v[n] = x[n - 5].

16. For the RC circuit shown in the figure (in the attachment), find the input/output differential equation.

17. Solve the differential equation (dy(t) / dt) + 2y(t) = x(t); where x(t) = u(t), and y(0) = 4.

19. Compute the Fourier Transform for the convolution of sin(2t)*cos(2t) .

20. Use the Laplace Transform to compute the solution to the differential equation defined by
(dy / dt) + 2y = u(t) where y(0) = 0.

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