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 \( g(t) \) 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 \( g(t) \).

---

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.

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|)

This page provides a detailed solution to find the Fourier transform G(f) of the signal g(t) = Π(t-1) - 5exp(-2|t+3|) using direct integration. The solution explains the Fourier transform of a shifted unit rectangular function and an exponentially decaying signal with time shifts. Suitable for undergraduate engineering or physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation