To determine the Continuous-Time Fourier Transform (CTFT) $X(\omega)$ of the given function $x(t)$, we will first decompose $x(t)$ into simpler components. From the plot, $x(t)$ consists of a series of rectangular pulses of varying heights and durations.

Decomposition of $x(t)$

1. Identify segments:
   • From $-4 \leq t < -2$: $x(t) = 0$
   • From $-2 \leq t < -1$: $x(t) = 2$
   • From $-1 \leq t < 2$: $x(t) = 3$
   • From $2 \leq t < 5$: $x(t) = 2$
   • From $5 \leq t < 7$: $x(t) = 0$

Represent $x(t)$ using rectangular functions:

• Rectangular function $\text{rect}\left(\frac{t-a}{b}\right)$ is 1 for $|t-a| < \frac{b}{2}$ and 0 otherwise.

Using the step function, $u(t)$, we can express $x(t)$ as follows:

[ x(t) = 2 \left[u(t+2) - u(t+1)\right] + 3 \left[u(t+1) - u(t-2)\right] + 2 \left[u(t-2) - u(t-5)\right] ]

CTFT of each component:

• CTFT of $u(t)$: $\mathcal{F}\{u(t)\} = \pi \delta(\omega) + \frac{1}{j\omega}$

• CTFT of shifted $u(t - a)$: $\mathcal{F}\{u(t - a)\} = \pi \delta(\omega) + \frac{1}{j\omega} e^{-j\omega a}$

• CTFT of $u(t-a) - u(t-b)$: $\mathcal{F}\{u(t-a) - u(t-b)\} = \frac{1}{j\omega} \left(e^{-j\omega a} - e^{-j\omega b}\right)$

Applying the CTFT to each term:

1. For $2 \left[u(t+2) - u(t+1)\right]$: $X_1(\omega) = 2 \cdot \frac{1}{j\omega} \left(e^{j\omega 1} - e^{j\omega 2}\right)$

2. For $3 \left[u(t+1) - u(t-2)\right]$: $X_2(\omega) = 3 \cdot \frac{1}{j\omega} \left(e^{j\omega 2} - e^{-j\omega 1}\right)$

3. For $2 \left[u(t-2) - u(t-5)\right]$: $X_3(\omega) = 2 \cdot \frac{1}{j\omega} \left(e^{-j\omega 2} - e^{-j\omega 5}\right)$

Final CTFT $X(\omega)$:

$X(\omega) = X_1(\omega) + X_2(\omega) + X_3(\omega)$

Substitute the results: $X(\omega) = \frac{2}{j\omega} \left(e^{j\omega 1} - e^{j\omega 2}\right) + \frac{3}{j\omega} \left(e^{j\omega 2} - e^{-j\omega 1}\right) + \frac{2}{j\omega} \left(e^{-j\omega 2} - e^{-j\omega 5}\right)$

This is the CTFT of the given signal $x(t)$.

Would you like a detailed step-by-step solution for each part, or do you have any specific questions?

Here are some related questions you might find helpful:

1. How is the Fourier transform of a rectangular pulse derived?
2. What properties of the Fourier transform can be used to simplify expressions?
3. How does the shifting property of the Fourier transform work?
4. What is the significance of the impulse function $\delta(\omega)$ in the Fourier domain?
5. How can we use symmetry properties of $x(t)$ to simplify the CTFT?

Tip: Always check the limits and scaling when dealing with piecewise functions to avoid mistakes in the decomposition process.