Math Problem Statement
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).
Solution
Let's solve the questions one by one.
1. Fourier Transform of
The given function is:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Fourier Transform
Inverse Fourier Transform
Complex Numbers
Signal Processing
Formulas
Fourier Transform: H(f) = ∫_{-∞}^{∞} h(t) e^{-j2πft} dt
Inverse Fourier Transform: h(t) = ∫_{-∞}^{∞} H(f) e^{j2πft} df
Amplitude Spectrum: |H(f)|
Phase: Θ(f) = arg(H(f))
Theorems
Fourier Transform Properties
Euler's Formula
Suitable Grade Level
Undergraduate - Electrical Engineering or Physics