To solve the problem, we will use the given Fourier Transform (FT) of the triangular pulse $f(t)$ in Fig. (a) and apply time-shifting properties for the other figures (b), (c), and (d).

The FT of the triangular pulse $f(t)$ in Fig. (a) is given as: $F(\omega) = \frac{1}{\omega^2} \left( e^{j\omega} - j\omega e^{j\omega} - 1 \right)$

Step 1: Time-Shifting Property of Fourier Transform

If $f(t)$ has the Fourier Transform $F(\omega)$, then the time-shifted signal $f(t - t_0)$ has the Fourier Transform: $F(\omega) e^{-j\omega t_0}$

Here:

• $t_0$ is the amount of shift in time.

Step 2: Solving for Figures

(b): Fig. (b) - $f_b(t)$

• In Fig. (b), the triangular pulse is shifted to the right by $t_0 = 1$.
• Using the time-shifting property: $F_b(\omega) = F(\omega) \cdot e^{-j\omega}$

(c): Fig. (c) - $f_c(t)$

• In Fig. (c), the triangular pulse consists of two shifted pulses:
  • One shifted to the left by $t_0 = -1$
  • One shifted to the right by $t_0 = 1$
• The signal can be expressed as: $f_c(t) = f(t + 1) + f(t - 1)$
• Using the time-shifting property, the FT of $f_c(t)$ is: $F_c(\omega) = F(\omega) e^{j\omega} + F(\omega) e^{-j\omega}$
• Simplify: $F_c(\omega) = F(\omega) \left( e^{j\omega} + e^{-j\omega} \right)$
• Using Euler's formula $e^{j\omega} + e^{-j\omega} = 2\cos(\omega)$, we get: $F_c(\omega) = 2F(\omega) \cos(\omega)$

(d): Fig. (d) - $f_d(t)$

• In Fig. (d), the triangular pulse is shifted to the left by $t_0 = -1$.
• Using the time-shifting property: $F_d(\omega) = F(\omega) e^{j\omega}$

Final Results

The Fourier Transforms for the given signals are:

1. Fig. (b): $F_b(\omega) = F(\omega) e^{-j\omega}$
2. Fig. (c): $F_c(\omega) = 2F(\omega) \cos(\omega)$
3. Fig. (d): $F_d(\omega) = F(\omega) e^{j\omega}$

Let me know if you would like a more detailed explanation or any clarification!

Here are 5 related questions to expand your understanding:

1. What is the Fourier Transform of a shifted rectangular pulse?
2. How does the time-scaling property of FT affect a signal?
3. What happens to the FT if the triangular pulse is scaled in amplitude?
4. How can we use convolution properties to solve FTs of composite signals?
5. How does the phase shift appear in the FT when time-shifting is applied?

Tip: The time-shifting property of Fourier Transform introduces a phase factor $e^{-j\omega t_0}$, which affects only the phase, not the magnitude.