Fourier transform and the frequency domain

[Bipolarity]

I understand that the Fourier transform maps one function onto another. So it is a mapping from one function space onto another.

My question is, why is it often referred to as a mapping from time domain to the frequency domain? I don't understand why the image of the Fourier transform represents a signal in the frequency domain. I don't see how the word frequency is even related to the Fourier transform.

What I do know is that the complex exponential and the trigonometric Fourier series both form an orthonormal basis for a certain class of functions under the function inner product. But I don't understand how these ideas are tied to the Fourier transform.

Insight is appreciated. Thanks!

BiP

 

[AlephZero]

$e^{i\omega t} = \cos \omega t + i \sin \omega t$.

In applications of Fourier transforms in physics and engineering, the $\omega$ often corresponds to the physical frequency (cycles / time) of a mechanical vibration or an electrical signal.

 

[Bipolarity]

$e^{i\omega t} = \cos \omega t + i \sin \omega t$.

In applications of Fourier transforms in physics and engineering, the $\omega$ often corresponds to the physical frequency (cycles / time) of a mechanical vibration or an electrical signal.

I see. What you posted is the Euler identity, which connects the interpretation of the frequency of a sine wave with the frequency of a complex exponential function.

But how are these related to the Fourier transform? How exactly is the result of the transform related to the frequencies of signals?

BiP

 

[homeomorphic]

The Fourier transform can be thought as a modified version of Fourier series that can apply to non-periodic functions. I'll just try to give a bird's eye view of this without getting into the details.

Fourier series are a way of writing any "reasonable" periodic function as a sum of sine and cosine functions of different frequencies (or rather, multiples of them). It's an infinite linear combination of sines and cosines, and the coefficients are called Fourier coefficients. The problem with Fourier series is that they only apply to periodic functions.

The idea, then, is to just let the interval of periodicity get bigger and bigger, approaching the whole real number line. With Fourier series, you have discrete frequencies. When you take this limit, these frequencies will get closer and closer together, and in the limit, they become a continuous variable. The Fourier transform is analogous to the Fourier coefficients. You plug in a number that represents the frequency, and out pops the analogue of the Fourier coefficient of that frequency. So, when you apply the Fourier transform to a function, the function that you get is the one that does this. Rigorously, you probably wouldn't formulate the theory in this way, but this is the idea behind it.

 

[Jyan]

Here is a derivation of the Fourier transform from Fourier series. http://www.jpoffline.com/physics_docs/y2s4/cvit_ft_derivation.pdf

Like homeomorphic said, the Fourier transform is obtained from the Fourier series by taking a limit as the period goes to infinity (and since frequency is the inverse of the period, it's the same as taking a limit as frequency goes to 0, which I find more intuitive). Fourier series are easier to understand than the Fourier transform, so thinking about the Fourier transform as a limit of the Fourier series makes it much easier on the brain.

If you are so inclined you can obtain a (electronic probably) copy of "Modern digital and analog communication systems" By B.P. Lathi which has an excellent explanation of the Fourier integral at the beginning of chapter 3.

 

[FactChecker]

I understand that the Fourier transform maps one function onto another. So it is a mapping from one function space onto another.

My question is, why is it often referred to as a mapping from time domain to the frequency domain? I don't understand why the image of the Fourier transform represents a signal in the frequency domain. I don't see how the word frequency is even related to the Fourier transform.

What I do know is that the complex exponential and the trigonometric Fourier series both form an orthonormal basis for a certain class of functions under the function inner product. But I don't understand how these ideas are tied to the Fourier transform.

Insight is appreciated. Thanks!

BiP

Just some non-rigorous notes for intuition:
1) Any time-domain function can be approximated by a sum of frequencies.
2) Because the frequencies are an orthonormal basis, one can just determine how much of each frequency is in the time-domain function and add them together.
3) The Fourier transformation tells how much of each frequency is in the time-domain function.

 

[Edgardo]

Here are some nice videos that explain the time and frequency domain:

Intro to Fourier Transform, see also the website http://www.thefouriertransform.com/#introduction

Fourier Transform Introduction

Fourier transform explained with an oscilloscope