10.1.3.2. Scaling

Try these first

1. 1.

   What is the definition of Fourier Cosine Transform?

   Hint

   Review the definition in your notes.

2. 2.

   State the linearity property of FCT.

   Hint

   Think about how we can combine two functions.

3. 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

   Hint

   Look for the integral limits and constants.

4. 4.

   True or False: The scaling property of the Fourier Cosine Transform states that F{f(ax)} = aF{f(x)}.

   • True
   • False

   Hint

   Recall the specific coefficients involved in scaling.

5. 5.

   For the function f(x) = x^2 defined on [0, ∞), calculate the Fourier Cosine Transform.

   Hint

   Remember to integrate by parts!

6. 6.

   Discuss how the Parseval's Identity can be applied in a physical context, such as sound waves.

   Hint

   What does energy conservation mean in this context?