# 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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Discrete Time Systems

1. a) Find the z-transform of the following system transfer function assuming the input is a unit staircase (i.e. zero order held). The sampling rate is T=0.2.

P(s) = (s+2)/(s+1)(s+5)

b) Given the z-transform of a sequence, U(z) = Z{u},

U(z) = z/(z^2 - 0.8z + 0.6)

i. Find the first 4 terms of u by long division.

ii. Find the underlying signal u(t) assuming that T=1 and no aliasing has occured.

2. a) The system (P_s)(s) = 4/(s^2 + 2s + 3) is subject to a staricase input with T = 0.2 seconds. Calculate the Z domain transfer function (P_z)(z).

b) The system, (P_s)(s) = 1/(s + 2) is subject to a staircase input with T = 0.1 seconds. Calculate the w-domain transfer function (P_z)(w) and comment on the relationship between s- and w-doman transfer functions.

3. a) Given (P_s)(s) = 4/(s + 3), find the w domain description for T = 0.2 and comment on the similarity and difference between (P_s)(s) and (P_w)(w). Use 3 decimal place in your calculations.

bi) Calculate the z-transform of the signal, y)t_ = e^(0t)sin(5t) sampled at T = 1.0.

bii) By finding the inverse z-transform, show that aliasing has occurred and explain why.

c) Find the first 4 terms in the inverse z-transform of Y(z) = 2z/(z^2 + 0.5z + 0.3) by means of long division. Use the final value of the signal to confirm that your results make sense.

4. a) The system, (P_s)(s) = 3/(s^2 + s + 2.5) is subject to a staircase input with T=0.2. Calculate the Z doman transfer function (P_z)(z).

b) A sequence, y_B, has a Z transform,

Y(z) = (z^2 + z)/(z^3 + 2z^2 + 1.4z + 0.3)

Find the first 5 terms in the sequence (y_0 ... y_4) by long division.

c) By considering the initial and final value theorems in the Z domain, find equivalent initial and final value theorems in the W domain and confirm that they are similar to the S domain theorems.

5. Show that aliasing will occur if the signal y(t) = e^-t(sin)(2t) is sampled with T=0.2 by finding the z transform of the sampled sequence and then finding the inverse z transform.

c) If a system has the transfer function P(s) = 1/(s/(3+1)) and T = 0.2, find (P_z)(w).

d) Show that the unit step signal, u(t) = sigma(t)

6. ai) The system, (P_s)(s) = (4s + 7)/(s^2 + 3s + 2) is subject to a staircase input with T = 0.5. Calculate the z domain transfer function, (P_z)(z).

aii) Use the final value theorem to find the final value of the output if the input is a step. Does your answer make sense with respect to the final value of the step response of the continuos system?

b) Given (P_s)(s) = 2/(s + 3), find the w domain description for T = 0.2 and comment of the similarity and difference between (P_s)(s) and (P_w)(w).

c) Given (P_z)(z) = (2z^2 - 0.5z + 1)/(z^2 + 0.2z + 0.5) and u_o = 1, u_1 = 2, u_n = 0, n =/ 0, 1, initioal y = 0.

i) Calculate y_n for n=0, 1, 2, and 3 by writing the difference equation and substituition.

ii) Does the oscillating behaviour make sense in terms of the pole locations in the z plane?

7. a) Find the z-transform of the following system transfer function assuming the input is a unit staircase (i.e. zero order held). The sampling rate is T = 0.2.

P(s) = 3/(s^2 + s + 1)

b) A signal y(t) = e^(-0.t)sin*2pi(t) is sampled at a rate T = 0.5, Show that the z-transform of the sampled sequence is Y(z) = (ze^-0.25)(sin(pi))/(z + e^-0.5)^2 = 0. Make a sketch to explain what has happened.

c) Given P(s) = 1/(s+1), find (P_z)(w) for T = 0.1.

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