Math Problem Statement
Solution
To find the Fourier Transform of the given function , we start by defining the function mathematically:
1 & \text{if } |x| < a \\ 0 & \text{if } |x| > a \end{cases}$$ ### Step-by-Step Solution The Fourier Transform $$F(k)$$ of $$f(x)$$ is given by: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx$$ Since $$f(x)$$ is zero outside $$|x| < a$$, we can limit the integration from $$-a$$ to $$a$$: $$F(k) = \int_{-a}^{a} 1 \cdot e^{-ikx} \, dx$$ 1. **Integrate $$e^{-ikx}$$:** $$F(k) = \int_{-a}^{a} e^{-ikx} \, dx$$ To integrate $$e^{-ikx}$$, we use: $$\int e^{-ikx} \, dx = \frac{e^{-ikx}}{-ik} = -\frac{1}{ik} e^{-ikx}$$ 2. **Evaluate the integral:** $$F(k) = \left[ -\frac{1}{ik} e^{-ikx} \right]_{-a}^{a}$$ Substitute the limits: $$F(k) = -\frac{1}{ik} \left( e^{-ika} - e^{ika} \right)$$ 3. **Simplify using Euler's formula:** $$F(k) = -\frac{1}{ik} \left( \cos(ka) - i \sin(ka) - \cos(ka) - i \sin(ka) \right)$$ Notice that $$e^{-ika} - e^{ika} = -2i \sin(ka)$$: $$F(k) = -\frac{1}{ik} (-2i \sin(ka))$$ Simplify further: $$F(k) = \frac{2 \sin(ka)}{k}$$ So, the Fourier Transform of $$f(x)$$ is: $$F(k) = \frac{2 \sin(ka)}{k}$$ This is a form of the sinc function, which is common in Fourier Transforms of rectangular pulse functions. --- Would you like additional details or have any questions? Here are some follow-up questions that might help expand your understanding: 1. How is the sinc function related to the Fourier Transform of rectangular functions? 2. What are the applications of Fourier Transforms in real-world signal processing? 3. How would the Fourier Transform change if $$f(x)$$ had a different constant value instead of 1? 4. What is the significance of the result in terms of frequency components? 5. How does the parameter $$a$$ influence the Fourier Transform $$F(k)$$? **Tip:** Remember that the Fourier Transform of a rectangular function in the time domain corresponds to a sinc function in the frequency domain. This relationship is fundamental in signal processing and communications.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Fourier Transform
Rectangular Pulse
Sine Function
Formulas
Fourier Transform: F(k) = ∫_{-∞}^{∞} f(x) e^{-ikx} dx
Integral of e^{-ikx}: ∫ e^{-ikx} dx = -1/ik e^{-ikx}
Simplification: e^{-ika} - e^{ika} = -2i sin(ka)
Theorems
Fourier Transform of Rectangular Functions
Euler's Formula
Suitable Grade Level
Undergraduate or Advanced High School (Calculus Level)
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