https://file.aiquickdraw.com/mathbot/file/17266684309113fs2ep38.jpg

Let's solve the questions one by one.

### 1. Fourier Transform of \( h(t) \)
The given function is:
\[
h(t) = 
\begin{cases} 
A e^{-\alpha t} \sin(2\pi f_0 t) & t \geq 0 \\
0 & t < 0
\end{cases}
\]

The Fourier transform \( H(f) \) of \( h(t) \) is calculated as:
\[
H(f) = \int_{-\infty}^{\infty} h(t) e^{-j2\pi ft} dt
\]

Given \( h(t) = 0 \) for \( t < 0 \), the integral reduces to:
\[
H(f) = \int_0^{\infty} A e^{-\alpha t} \sin(2\pi f_0 t) e^{-j2\pi f t} dt
\]

This expression involves both an exponential term and a sinusoidal term. The solution involves using the Fourier transform of the sine function modulated by an exponential decay. After some simplifications, this Fourier transform results in a complex frequency-domain representation.

---

### 2. Real and Imaginary Parts, Amplitude Spectrum, and Phase
After obtaining \( H(f) \), the real and imaginary components can be derived by separating the terms.

1. **Real Part**: \( \Re(H(f)) \)
2. **Imaginary Part**: \( \Im(H(f)) \)
3. **Amplitude Spectrum** \( |H(f)| \): This is the magnitude of the complex function \( H(f) \).
4. **Phase** \( \Theta(f) \): The phase is the argument of the complex function \( H(f) \).

---

### 3. Inverse Fourier Transform
The given equation for \( H(f) \) is:
\[
H(f) = \frac{\alpha^2}{\alpha^2 + (2\pi f)^2}
\]

The inverse Fourier transform is applied to return to the time-domain signal \( h(t) \). This can be solved using standard inverse Fourier techniques.

Would you like detailed steps for each of these processes?

Here are some related questions:
1. What is the general Fourier transform of an exponential function multiplied by a sinusoidal term?
2. How do you calculate the magnitude and phase from a complex Fourier spectrum?
3. What is the significance of the inverse Fourier transform in signal processing?
4. How does the parameter \( \alpha \) influence the decay in time-domain and frequency-domain?
5. Can you apply the Fourier transform to other piecewise functions?

**Tip**: Understanding the relationship between time-domain decay and frequency-domain spread is crucial when dealing with exponential signals.

1. Lakukan Transformasi Fourier untuk fungsi h(t) yang diberikan.
2. Tentukan bagian Real, Imaginer, amplitudo spektrum |H(f)|, dan fase Θ(f) dari Transformasi Fourier h(t).
3. Tentukan Inversi Transformasi Fourier dari H(f).

Learn how to calculate the Fourier Transform of an exponential sinusoidal function and find its real and imaginary parts, amplitude spectrum, and phase. Also, explore the steps to derive the inverse Fourier transform for a given frequency-domain function. Suitable for undergraduate students in Electrical Engineering or Physics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation