The 2D FFT is admittedly more confusing, but to answer your question about nyquist and sample rate for a 2D fft, in particular with regard to images, you have to think in terms of “spatial frequencies”. For example, how often does some feature repeat and what is the smallest feature that you need to resolve. In an image, sample rate corresponds to the number of pixels in a given spatial unit. So, for example, if you have a mosquito in a picture, and the mosquito fills a space smaller than 2 pixels, it will fall below the Nyquist sampling limit. Indeed, you may see a speck in the image, but there will be no information about the size of that speck - only that it is there. Further, you would need at least 2 pixels covering the smallest feature of the mosquito - say a small stripe on its back or it’s antennae - to resolve these either. So, to summarize, for an image, sampling frequency is pixels per unit length and nyquist defines the resolution limit.
Thank you very much!! fallen in love with your brain…
Great article - really helpful and empowering
very nice work!
but if you could provide intuitive explanations about forward FFT that will be great!
The demonstrations here seem to be inverse FFT to me.
Nice introduction!
EXCELLENT site! My mind has opened.
I recall my days at secondary school in Wales being taught, parrot-fashion: “The square on the hypotenuse is the sum of the squares on the opposite two sides. Now, get on with it!” - Cue blank looks from us all.
Where were you dude?!!!
I think I will use these lessons for my kids, that is how valuable I think your site is. Well done Sir and thank you for taking the time to make this available.
BTW, I would love to see your insights into Z-Transforms and how to APPLY the damn things! :0)
Andy
Why did you end your tutorial just when you started explaining the math part of the math?
All you’ve done is teach the why. What about the how? That’s the part that’s difficult.
You rock!
Bamm!! Your explanation rocks! 
Excellent explanation, really really useful. Cheers!
It helps me very much! Up to now, I hardly recognize anyone even my professors as a real teacher. Today, you are the real one. Thank you!
Thanks An!
Awesome!
This is probably the most useful thing I’ve seen on the internet so far. I don’t know I would have been able to understand this sh*t without you. Thanks a lot.
This is really useful. I studied Fourier Transformations thirty years ago as an undergraduate and longed to understand what I was doing!
We used them for Crystallography: a FT of a crystal structure shows you what its x-ray picture will look like… but I still don’t understand why? Can you explain?
Thank you very much for such a beautiful explanation and for all the effort to make it so intuitive! Very inspiring!
Epic
Just got introduced to Fourrier Transform and i was lucky enough to read your article. Very insightful on many level.
I just have to work a bit on the math now, but the intuition is there.
Thank you for this.
Hello,
I do not understand teh differances between continuous fourier transform and discrete fourier transform?
Examples from real life are need!
What are the differances between these two?
This is a phenomenal article. Should be used in universities worldwide. I feel much better about becoming an electrical engineer. Thanks.
Very good work!