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:

\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.