Math Problem Statement
Find the Fourier transform for the rectangular pulse.
Solution
This problem involves finding the Fourier transform of a rectangular pulse function , which is defined as follows:
\begin{cases} 1 & \text{if } |t| < T_p/2, \\ 0 & \text{if } |t| \geq T_p/2 \end{cases}$$ Or equivalently, $$x(t) = \text{rect}\left(\frac{t}{T_p}\right)$$. The solution typically proceeds as follows: 1. **Fourier Transform Definition**: The Fourier transform of $$x(t)$$ is defined as: $$X(f) = \int_{-\infty}^\infty x(t)e^{-j2\pi ft} \, dt.$$ 2. **Substitute $$x(t)$$**: For the rectangular pulse: $$X(f) = \int_{-T_p/2}^{T_p/2} 1 \cdot e^{-j2\pi ft} \, dt.$$ 3. **Simplify the Integral**: Evaluate: $$X(f) = \int_{-T_p/2}^{T_p/2} e^{-j2\pi ft} \, dt.$$ The result of this integral is: $$X(f) = \frac{\sin(\pi f T_p)}{\pi f}.$$ 4. **Introduce Scaling**: Incorporating scaling from $$T_p$$: $$X(f) = T_p \cdot \text{sinc}(fT_p),$$ where $$\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$$. This result demonstrates that the Fourier transform of a rectangular pulse is a sinc function. The width of the rectangular pulse in time determines the main lobe width of the sinc function in frequency. --- Would you like me to provide step-by-step details for the evaluation of the integral or explain any part of this further? Here are some follow-up questions related to this concept: 1. How does the width of the pulse $$T_p$$ affect the Fourier transform? 2. What is the physical significance of the sinc function in signal processing? 3. How does the Fourier transform relate to bandwidth in this case? 4. Can you explain the duality principle between time-domain and frequency-domain functions? 5. How can this result be used in applications like communication systems? **Tip**: When working with Fourier transforms of piecewise functions, always pay attention to the limits of integration and any symmetries in the function to simplify calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Fourier Transform
Rectangular Pulse
Signal Processing
Formulas
X(f) = ∫_{-∞}^∞ x(t)e^{-j2πft} dt
x(t) = rect(t/T_p)
X(f) = T_p · sinc(fT_p)
sinc(x) = sin(πx)/(πx)
Theorems
Fourier Transform Definition
Time-Frequency Duality
Suitable Grade Level
Undergraduate