## Notes on Preprocessing Dataset

26 Apr 2022# Fourier Transform: back to basics

### Why do we need negative frequencies and what is the meaning of a negative frequency in the first place?

*notes from answer on Quora*

Negative frequencies do not exist in reality. They are constructs in our heads.

- The idea behind it is actually very simple. Mr. Fourier said that any signal is composed of the superposition of sinusoidals with different frequencies.
- Mr. Euler said that a $2sin(2 \pi f.t) = e^{2i ft} + e^{-2i ft}$; $i= \sqrt(-1)$

In other words, a sine is composed of two
**phasors** (arrows rotating with constant speed f /sec, one is rotating counter clock-wise [1st term] and another one
rotating clock-wise [2nd term]). To distinguish whether we are talking about a clock-wise or counter clock-wise
rotation we have a negative sign next to the frequency (which designates the speed of rotation).

Combining points 1 and 2:

- Since any signal can be constructed in time domain from a bunch of sinusoidals [
**Point 1**], and since those sinusoidals are made up of a pair of phasors with opposing frequency signs [**Point 2**] then we get both positive and negative frequencies content when apply FFT to a signal. -
Getting inverse FT, which is defined as:

$x(t) = \int_{-\infty}^{\infty} F(\omega). exp^{2\pi i \omega} d\omega$

Note that the integration goes from $-\infty$ to $\infty$ for the same reason explained above.