Understand Fourier Transform

What does Fourier transform do?

  • Find the separate component (cycle ingredient ,i.e., the cycle strength,delay and speed) from a signal.
  • The Fourier transform finds the set of cycle speeds, strength and phases(starting angles) to match any time signal.
  • Fourier transform changes our perspective from “consumer” to “producer”, turning “what did I see” into “how was it made”

The amazing euler’s formula:

\(e^{ix} = cos(x)+isin(x)\)

  • $e^{i\pi}$ means starting at 1 and rotating to $\pi$ get to -1
  • $e^x$ means starting from e and grow continuously at 100% for x seconds
  • real growth: pushing a number in the same, real direction it was going
  • imaginary growth: grows at different direction. Instead of going forward, imaginary growth rotates the number. Taking any number and multiplying by $i$ will not change its magnitude, just the direction it points.

How do we grow $e$ to $a + bi$?

  • it is a mix of real and imaginary growth. The real part $a$ means growing at 100% for $a$ seconds, and the imaginary part $b$ means rotate for $b$ seconds.