Notes on Preprocessing Dataset

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.

1. The idea behind it is actually very simple. Mr. Fourier said that any signal is composed of the superposition of sinusoidals with different frequencies.
2. 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.