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
By Image
Drop file here or Click Here to upload
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)
Related Recommendation
Fourier Transform of Signal g(t) Involving Rectangular and Exponential Functions
Fourier Transform of a Rectangular Pulse
Fourier Transform of Piecewise Function f(t) = { 1−t, 0≤t<1; t−1, 1≤t≤2 }
Fourier Transform of Rectangular Pulse
Fourier Series for Piecewise Function f(x) = {x for 0 < x < 1, 2 - x for 1 < x < 2}