11.3.6. Convolution Theorem

Try these first

1. 1.

   What does the convolution of two functions represent?

   Hint

   Think about interactions between two shapes.

2. 2.

   What symbol is used to denote convolution?

   Hint

   It's a simple symbol you would recognize.

3. 3.

   What does the Convolution Theorem state?

   • Convolution in the time domain results in addition in the frequency domain.
   • Convolution in the time domain results in multiplication in the frequency domain.
   • Convolution in the time domain results in transformation in the frequency domain.

   Hint

   Think about how operations translate between domains.

4. 4.

   True or False: The function denoted by f(t) * g(t) refers to their addition.

   • True
   • False

   Hint

   Recall the definitions discussed earlier.

5. 5.

   Given two functions f(t) = cos(2πt) and g(t) = u(t), compute the convolution of these two functions.

   Hint

   Setup the integral for convolution and apply Fourier Transform if necessary.

6. 6.

   Explain how the Convolution Theorem could be utilized in designing filters for audio signals.

   Hint

   Think about signal processing in terms of frequency components.