Practice Scaling (10.1.3.2) - Fourier Cosine and Sine Transforms

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6 sample questions

Question 1.
What is the definition of Fourier Cosine Transform?
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Review the definition in your notes.
Question 2.
State the linearity property of FCT.
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Think about how we can combine two functions.
3Question 3.
What is the formula for the Fourier Cosine Transform?
- F(s) = \\int f(x) \\cos(sx) dx
- F(s) = \\frac{2}{\\pi} \\int_{0}^{\\infty} f(x) \\cos(sx) dx
- F(s) = \\int_{0}^{\\infty} f(x) \\sin(sx) dx
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Look for the integral limits and constants.
4Question 4.
True or False: The scaling property of the Fourier Cosine Transform states that F{f(ax)} = aF{f(x)}.
- True
- False
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Recall the specific coefficients involved in scaling.
Question 5.
For the function f(x) = x^2 defined on [0, ∞), calculate the Fourier Cosine Transform.
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Remember to integrate by parts!
Question 6.
Discuss how the Parseval's Identity can be applied in a physical context, such as sound waves.
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What does energy conservation mean in this context?

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