# Homework 02 Solution

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1. (12 pts) Analyze whether the following systems have these properties: memory, stability, causality, linearity, invertibility, time-invariance. Provide your answer in detail.

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P

(a) (6 pts) y[n] = x[n k]

k=1

1. (6 pts) y(t) = tx(2t + 3)

1. (13 pts) Consider an LTI system given by the following block diagram:

 x(t) + Z y(t)   5

1. (3 pts) Find the di erential equation which represents this system.

1. (10 pts) Find the output y(t), when the input x(t) = (e t + e 3t)u(t). Assume that the system is initially at rest.

1. (15 pts) Evaluate the following convolutions.

(a) (10 pts) Given x[n] = 2 [n] + [n + 1] and h[n] = [n 1] + 2 [n + 1], compute and draw y[n] = x[n] h[n].

1. (5 pts) Given x(t) = u(t 1) + u(t + 1) and h(t) = e t sin(t)u(t), calculate y(t) = dxdt(t) h(t).

1. (20 pts) Evaluate the following convolutions.

1. (10 pts) Given h(t) = e 2tu(t) and x(t) = e tu(t), nd y(t) = x(t) h(t).

(b) (10 pts) Given h(t) = e3tu(t) and x(t) = u(t) u(t 1), nd y(t) = x(t) h(t).

5. (20 pts) Solve the following homogeneous di erence and di erential equations with the speci ed initial conditions.

(a) (10 pts) 2y[n + 2] 3y[n + 1] + y[n] = 0, y = 1 and y = 0.

(b) (10 pts) y(3)(t) 3y00(t) + 4y0(t) 2y(t) = 0, y00(0) = 2, y0(0) = 1 and y(0) = 3.

6. (20 pts) Consider the following discrete time LTI system which is initially at rest:

 x[n] h0[n] w[n] h0[n] y[n] 1 where w[n] w[n 1] = x[n]. 2   1. (10 pts) Find h0[n].

1. (5 pts) Find the overall impulse response, h[n], of this system.

1. (5 pts) Find the di erence equation which represents the relationship between the input x[n] and the output y[n].

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