The function in the figure below is only non-zero for -1.5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Function Analysis and Representation
#### Problem Statement
The function in the figure is non-zero only for the interval \(-1.5 \leq x \leq 1.5\). Outside that range, the function is equal to zero. A question arises: Would you use a Fourier series or Fourier transform to represent this function?
#### Graph Description
The graph illustrates a piecewise function plotted on a Cartesian plane:
- The x-axis ranges from \(-3\) to \(3\).
- The y-axis ranges from \(-1.5\) to \(1.5\).
- There are two horizontal lines depicted:
- The first line is at \(y = 1.0\) for \(-1.5 \leq x \leq -0.5\).
- The second line is at \(y = -1.0\) for \(0.5 \leq x \leq 1.5\).
- The function is zero outside the interval \([-1.5, 1.5]\).
#### Analysis
- **Fourier Series**: Suitable for periodic functions. This approach is ideal if the function could be extended periodically over an interval.
- **Fourier Transform**: Suitable for non-periodic functions. Since this function is not inherently periodic, a Fourier transform would be more appropriate to represent it across the entire real line.
In conclusion, given the nature of the function's defined interval and zero values outside, using a Fourier transform is more appropriate for such a discontinuous, non-periodic representation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65ba51bd-93fe-4637-8f7c-e52d3fdaa23e%2F99576aaa-ff48-45de-9c80-735f86893da6%2Fk1eawco_processed.png&w=3840&q=75)
Transcribed Image Text:### Function Analysis and Representation
#### Problem Statement
The function in the figure is non-zero only for the interval \(-1.5 \leq x \leq 1.5\). Outside that range, the function is equal to zero. A question arises: Would you use a Fourier series or Fourier transform to represent this function?
#### Graph Description
The graph illustrates a piecewise function plotted on a Cartesian plane:
- The x-axis ranges from \(-3\) to \(3\).
- The y-axis ranges from \(-1.5\) to \(1.5\).
- There are two horizontal lines depicted:
- The first line is at \(y = 1.0\) for \(-1.5 \leq x \leq -0.5\).
- The second line is at \(y = -1.0\) for \(0.5 \leq x \leq 1.5\).
- The function is zero outside the interval \([-1.5, 1.5]\).
#### Analysis
- **Fourier Series**: Suitable for periodic functions. This approach is ideal if the function could be extended periodically over an interval.
- **Fourier Transform**: Suitable for non-periodic functions. Since this function is not inherently periodic, a Fourier transform would be more appropriate to represent it across the entire real line.
In conclusion, given the nature of the function's defined interval and zero values outside, using a Fourier transform is more appropriate for such a discontinuous, non-periodic representation.
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