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To perform the Fourier Transform of the given function \( h(t) \), let's break it down.

### Given:
\[ h(t) = 
\begin{cases} 
A e^{-\alpha t} \sin(2\pi f_0 t), & t \geq 0 \\
0, & t < 0
\end{cases}
\]

### Fourier Transform Definition:
The Fourier Transform of a time-domain function \( h(t) \) is given by:

\[
H(f) = \int_{-\infty}^{\infty} h(t) e^{-j2\pi ft} dt
\]

Since \( h(t) = 0 \) for \( t < 0 \), the integral becomes:

\[
H(f) = \int_0^{\infty} A e^{-\alpha t} \sin(2\pi f_0 t) e^{-j2\pi ft} dt
\]

### Step-by-Step Solution:

1. Substitute \( h(t) = A e^{-\alpha t} \sin(2\pi f_0 t) \) into the equation:

\[
H(f) = A \int_0^{\infty} e^{-\alpha t} \sin(2\pi f_0 t) e^{-j2\pi ft} dt
\]

2. Use the Euler identity to express the sine function:
\[
\sin(2\pi f_0 t) = \frac{e^{j2\pi f_0 t} - e^{-j2\pi f_0 t}}{2j}
\]

Thus, the integral becomes:
\[
H(f) = \frac{A}{2j} \int_0^{\infty} e^{-\alpha t} \left( e^{j2\pi f_0 t} - e^{-j2\pi f_0 t} \right) e^{-j2\pi ft} dt
\]

3. Split the integral into two parts:
\[
H(f) = \frac{A}{2j} \left( \int_0^{\infty} e^{-\alpha t} e^{j2\pi (f_0 - f)t} dt - \int_0^{\infty} e^{-\alpha t} e^{-j2\pi (f_0 + f)t} dt \right)
\]

4. These integrals have the form:
\[
\int_0^{\infty} e^{-(\alpha - j2\pi \beta)t} dt = \frac{1}{\alpha - j2\pi \beta}
\]
where \( \beta \) represents a frequency term. Applying this to both integrals:

\[
H(f) = \frac{A}{2j} \left( \frac{1}{\alpha - j2\pi (f - f_0)} - \frac{1}{\alpha + j2\pi (f + f_0)} \right)
\]

5. Simplify the expression:
\[
H(f) = \frac{A}{2j} \left( \frac{\alpha + j2\pi(f + f_0) - (\alpha - j2\pi(f - f_0))}{(\alpha^2 + (2\pi (f - f_0))^2)(\alpha^2 + (2\pi (f + f_0))^2)} \right)
\]

### Final Expression for \( H(f) \):
After simplifying, the result will be a rational function of frequency \( f \), depending on \( A \), \( \alpha \), and \( f_0 \).

Would you like more details on the final simplification, or do you have any questions about the steps?

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Here are 5 related questions that expand on this:

1. What are the properties of Fourier transforms for time-shifted functions?
2. How does the Fourier transform change when analyzing an exponentially decaying signal?
3. What is the impact of different \( \alpha \) values on the bandwidth of the resulting Fourier transform?
4. How can this Fourier transform be applied in signal processing or communication theory?
5. What is the inverse Fourier transform of the function \( H(f) \)?

**Tip**: Understanding the Fourier Transform of basic functions, such as sine and exponential, helps in solving more complex cases like this one.

Lakukan Transformasi Fourier dari fungsi h(t) = {Ae^(-αt) sin(2πf_0t) untuk t >= 0, 0 untuk t < 0}

Learn how to perform the Fourier Transform of a damped sine function Ae^(-αt) sin(2πf_0t). This step-by-step solution explores key concepts like exponential decay, signal processing, and the Fourier Transform integral. Suitable for advanced engineering or physics students.

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