To find the Fourier transform $G(f)$ of the signal $g(t)$ using direct integration, we first break down the given signal into its components:

$g(t) = \Pi(t - 1) - 5 \exp(-2|t + 3|)$

Here:

• $\Pi(t - 1)$ is a shifted unit rectangular function centered at $t = 1$, with a width of 1, spanning from $t = 0.5$ to $t = 1.5$.
• $5 \exp(-2|t + 3|)$ is an exponentially decaying signal, symmetrically centered at $t = -3$.

The Fourier transform $G(f)$ is defined by the following integral:

$G(f) = \int_{-\infty}^{\infty} g(t) e^{-j 2\pi f t} \, dt$

Step 1: Fourier Transform of $\Pi(t - 1)$

The unit rectangular function $\Pi(t - 1)$ is a shifted version of the standard rectangular function $\Pi(t)$, and its Fourier transform is well-known.

$\Pi(t - 1) \quad \text{is non-zero for} \quad 0.5 \leq t \leq 1.5$

The Fourier transform of $\Pi(t)$ is:

$\mathcal{F}\left[\Pi(t)\right] = \frac{\sin(\pi f)}{\pi f}$

Since the rectangular function is shifted by 1 unit, the Fourier transform will include a phase shift:

$\mathcal{F}\left[\Pi(t - 1)\right] = \frac{\sin(\pi f)}{\pi f} e^{-j 2\pi f}$

Step 2: Fourier Transform of $5 \exp(-2|t + 3|)$

This is an exponentially decaying function of the form $\exp(-a |t - t_0|)$. The Fourier transform of $\exp(-a|t|)$ is:

$\mathcal{F}\left[\exp(-a |t|)\right] = \frac{2a}{a^2 + (2\pi f)^2}$

For $5 \exp(-2|t + 3|)$, we apply a time shift of $t_0 = -3$. The Fourier transform of a shifted exponential decay is:

$\mathcal{F}\left[5 \exp(-2|t + 3|)\right] = \frac{10}{4 + (2\pi f)^2} e^{j 6 \pi f}$

Step 3: Combine the Results

The Fourier transform of $g(t) = \Pi(t - 1) - 5 \exp(-2|t + 3|)$ is the sum of the Fourier transforms of the individual components:

$G(f) = \frac{\sin(\pi f)}{\pi f} e^{-j 2\pi f} - \frac{10}{4 + (2\pi f)^2} e^{j 6 \pi f}$

This is the Fourier transform of the signal $g(t)$.

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Let me know if you would like a deeper explanation or further steps in the process.

Here are 5 related questions for practice:

1. What is the Fourier transform of $\Pi(t)$ without any shifts?
2. How would you find the Fourier transform of $\cos(2\pi t)$?
3. What is the inverse Fourier transform of $\frac{\sin(\pi f)}{\pi f}$?
4. How does time shifting affect the Fourier transform of a signal?
5. What are the properties of Fourier transforms for even and odd functions?

Tip:

When performing Fourier transforms on shifted signals, remember to account for phase shifts, as they appear as complex exponentials in the frequency domain.