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Meet the FFT How to read the DFT of a single sinusoid

\( y(t) = A \cos{\left(\omega t + \phi\right)} \), fit to data.
Change the amplitude, frequency, and phase. What happens to the FFT \( Y(\omega) \) of \( y(t) \)?

DFT simulator

Questions

  1. Adjust the frequency and look at the FFT. How can you tell the dominant frequency of the signal from the absolute value of the FFT?
  2. With the phase set to zero, giving \( \cos{\omega t} \), which part of the FFT responds to variations in amplitude, real or imaginary?
  3. With the phase set to \( \pi/2 \), giving \( \sin{\omega t} \), which part of the FFT responds to variations in amplitude, real or imaginary?
  4. Set the phase to \( \pi/4 \) and vary the amplitude. How do the real and imaginary components respond, and how do you understand this?
  5. Describe a general technique to determine \( A, B, \omega \) from the FFT of \( y(t) = A \cos{\omega t} + B \sin{\omega t} \).