use direct integration to find the Fourier transform G(f) of the signal g(t) (Note: the unit rectangular function has a range between -1/2 and 1/2): g(t)= Π(t-1)-5exp(-2|t+3|)

To find the Fourier Transform \( G(f) \) of the signal \( g(t) = \Pi(t - 1) - 5 \exp(-2|t + 3|) \), we will use direct integration. 

### Step 1: Breakdown of the signal \( g(t) \)
The given signal is a combination of two components:
1. **\( \Pi(t - 1) \)**: The unit rectangular function shifted to the right by 1 unit, with a range between -1/2 and 1/2.
2. **\( -5 \exp(-2|t + 3|) \)**: An exponentially decaying function centered at \( t = -3 \).

We need to find the Fourier transform of each component individually and then combine the results, since the Fourier transform is linear.

### Step 2: Fourier Transform of \( \Pi(t - 1) \)
The unit rectangular function \( \Pi(t) \) is defined as:
\[
\Pi(t) =
\begin{cases}
1, & \text{for} \ |t| \leq \frac{1}{2}, \\
0, & \text{otherwise}.
\end{cases}
\]
The Fourier transform of \( \Pi(t) \) is a sinc function:
\[
\mathcal{F}\{\Pi(t)\} = \text{sinc}(f) = \frac{\sin(\pi f)}{\pi f}.
\]
For a shifted rectangular function \( \Pi(t - 1) \), the Fourier transform includes a phase shift factor:
\[
\mathcal{F}\{\Pi(t - 1)\} = \text{sinc}(f) \cdot e^{-j2\pi f}.
\]

### Step 3: Fourier Transform of \( -5 \exp(-2|t + 3|) \)
The term \( \exp(-2|t + 3|) \) is an exponential decay centered at \( t = -3 \). The Fourier transform of \( \exp(-a|t|) \) is:
\[
\mathcal{F}\{\exp(-a|t|)\} = \frac{2a}{a^2 + (2\pi f)^2}.
\]
For \( \exp(-2|t + 3|) \), we apply the time-shifting property:
\[
\mathcal{F}\{\exp(-2|t + 3|)\} = \frac{4}{4 + (2\pi f)^2} \cdot e^{j6\pi f}.
\]
Multiplying by the constant -5, we get:
\[
\mathcal{F}\{-5 \exp(-2|t + 3|)\} = -5 \cdot \frac{4}{4 + (2\pi f)^2} \cdot e^{j6\pi f}.
\]

### Step 4: Combine the results
Now, combining the Fourier transforms of both components, we get the total Fourier transform \( G(f) \) of the signal \( g(t) \):
\[
G(f) = \text{sinc}(f) \cdot e^{-j2\pi f} - 5 \cdot \frac{4}{4 + (2\pi f)^2} \cdot e^{j6\pi f}.
\]

This is the Fourier transform \( G(f) \) of the given signal \( g(t) \).

Would you like further details or explanations?

Here are five related questions:
1. How does the time-shifting property affect the Fourier transform of a signal?
2. What is the significance of the sinc function in signal processing?
3. How can you derive the Fourier transform of a Gaussian function?
4. How do phase shifts appear in the Fourier domain, and what is their physical meaning?
5. How would you compute the inverse Fourier transform of a given function?

**Tip:** When working with the Fourier transform, always remember the duality between time shifts and phase shifts in the frequency domain.

Use direct integration to find the Fourier transform G(f) of the signal g(t) (Note: the unit rectangular function has a range between -1/2 and 1/2): g(t) = Π(t - 1) - 5exp(-2|t + 3|).

Explore the process of finding the Fourier transform G(f) of the signal g(t) = Π(t - 1) - 5exp(-2|t + 3|) using direct integration. This problem covers key concepts like the Fourier transform, time-shifting property, and exponential functions. Ideal for undergraduate students.

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