How to visualise Fourier Transform of a function?

Question

I solved many problems on Fourier series,transforms and inverse fourier transforms as part of my academics. And i am aware that FT converts a time domain signal to frequency domain and IFT is vice versa.

How to visualise that FT really does convert a time domain signal to frequency domain?

My Approach:

Actually when i first thought about this i started with Fourier series. A function is expressed as sum of sine and cosine functions.Then i thought why only sine,cosine? which made me realise its related to right angle triangle (to get x and y co-ordinates of a point) and angle is related to the distance of point from origin. This is where omega*t creeps into theta of sine and cosine. And as x axis is time domain and t creeps in here.

Am i in the right path? Please guide me through this...

Answer 1

These are the links which made my visualisation complete..

For the ones with the same problem as i had u suggest them read the following in the same order for clarity

Start with this pdf for an intutution on why Fourier Transform works

Then this for a better clarity on imagination

Answer 2

A Fourier transform represents the amount of oscillation of a particular frequency $\omega$ in a function. A function having one frequency is represented by a spike at that frequency. A periodic function is represented by spikes at an arithmetic sequence of frequencies. In general, higher frequencies represent faster variations in the original function. By performing a low-pass filter, we are cutting off the higher frequencies and "smoothing" out the function.