Understanding Fourier Transform and Analysis 1

Source: Deep Learning on Medium

Switching over from biology to engineering, the name Fourier comes up so often that I had to dive in to it.

I stumbled upon Stanford’s lecture series on Fourier Transform and its Applications and had to share it.

Fourier transform always intrigued me as well because it can interpret any function into bits of smaller functions just like neural networks.

Okay, let’s dig in.

Lecture 1

Fourier series is identified with mathematical analysis of periodic phenomena.

Fourier transform, as limiting case of Fourier series, is concerned with analysis of non-periodic phenomena.

Some ideas in carry over in between Fourier series and transform while some ideas do not.

In engineering, we deal with signals which is at the end of the day just a function.

Analyzing a signal involves breaking it up into simpler constituent parts.

Synthesizing a signal deals with reassembling it from its constituent parts.

Both analysis and synthesis are accomplished by linear operations like integrals and sums.

Periodic phenomena are simply regularly repeating patterns.

There are two types of periodicities:

  1. periodicity in time, which measures frequency: how often something comes to you. And,
  2. periodicity in space, which measures period, the distribution of physical quantities over a region with symmetry, such as heat distribution over a circular ring: how often you walk into a repeating pattern.

A wave, which is a regularly moving disturbance, has both periodicity in time and in space.

It has a frequency 𝛎 and wavelength 𝛌.

Which can be related into a physics equation: distance = rate x time or 𝛌 = velocity x 1/𝛎.

A good thing to notice here is a reciprocal relationship between the frequency and the wavelength.

A mathematical function for modeling periodic phenomena are, of course, sin and cos functions.

cos t and sin t are periodic over a period of 2𝜋.

Meaning, cos t+2𝜋n = cos t and sin t+2𝜋n = sin t, where n = 0, ±1, ±2, ∙∙∙.

These simply functions and there properties can be used to model very complex things.