[SOLVED] ECE250 Assignment 3

0. Theory Problems:
   1. 1)  (8 points) Consider a periodic signal s(t) with fundamental time-period T. A portion of that signal, i.e., for 0 < t < T/4 is known and depicted in Fig. 1.
      Determine the complete signal s(t), i.e., for the interval 0 < t < T, if the signal follows the following conditions:

      (a) (4 pts) s(t) is an even function, and its Fourier series comprises of only odd-harmonics. (b) (4 pts) s(t) is an odd function, and its Fourier series comprises of only odd-harmonics.

   2. 2)  (14 points) A rectangular pulse p(t) of total width T1 and height unity is given. It is also known that the pulse is symmetric about the origin.
      1. (a)  (1 pt) Using the given information, plot the signal p(t).
      2. (b)  (1 pt) A new signal q(t) is defined as a periodic repetition of p(t) with time-period T0 = 3T1/2. Plot

         the signal q(t).

      3. (c)  (4 pts) Compute P(jω), i.e. the Fourier transform of p(t) and sketch the magnitude |P(jω)| for

         |ω| ≤ 6π/T1.

      4. (d)  (4 pts) Compute the Fourier series coefficients ak of the periodic signal q(t) and sketch the coefficients ak fork=0, ±1, ±2, ±3.

1

Figure 1: Signal s(t) (problem-1).

(e) (4 pts) Using the obtained relations, specify the relationship between X(ω) and ak. Justify how the Fourier series for signal q(t) can be determined considering that the Fourier transform of p(t) is known. Note that p(t) is one time-period of the periodic signal q(t).

3) (6 points) A system S generates the output signal y(t) related to Fourier transform of the provided input x(t) as depicted in the Fig. 2(a). If the same input is passed through the system configuration as shown in Fig. 2(b), compute the output v(t).

Figure 2: Block diagram for system S (problem-3).

4. 4)  (8 points) Consider a continuous-time LTI system described by:

   dy(t) + 2y(t) = x(t) (1) dt

   Using the Fourier transform, find the output y(t) to each of the following input signals; (a) (4 pts) x(t) = e−tu(t)

   (b) (4 pts) x(t) = u(t)

5. 5)  (12 points) Compute the Fourier transform of the following signals:

   (a) (4 pts) [e−αtcos(ω0t)]u(t), α > 0 (b) (4 pts) e−3|t|sin(2t)

(c) (4 pts)

􏰄 􏰅􏰄 􏰅

sinπt sin2πt πt πt

Programming Problem:

1. (7 points) For the discrete time signals defined below, compute the Discrete Time Fourier Transform (DTFT) and plot the following:

• the signal x[n]
• real part of the complex DTFT signal
• imaginary part of the complex DTFT signal • magnitude spectrum of the DTFT signal

(a) (3pts) Signal-1: Unit impulse signal or x1[n] = 1 for the value of n = [0] and zero otherwise. (b) (4pts) Signal-2: x2 [n] = 1 for the values of n = [−4, −3, −2, −1, 0, 1, 2, 3, 4] and zero otherwise.

For plots use n ∈ [−1000, 1000] and ω ∈ [−2π, 2π]. Add all relevant plots in your report and comment about the periodicity of the obtained DTFT signals.

Hint: To simulate continuous signals use appropriate discretization (wherever required).