## 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.