Understanding the FFT
Starting out learning something new, there's always that initial cloudiness we have until things finally click. They may click in three minutes, or it might take ten years. It just depends on the level of difficulty. This is especially troublesome with math. Remember the first time you saw the definition of a Fourier Transform or the discrete version? You probably thought, like me, what in God's name is that? And the sixth or seventh time you saw it, you were probably still thinking, what in God's name is that?
Maybe your professor made a joke about how Fourier (or Laplace) was clearly on drugs when he came up with his transform. Unfortunately, that doesn't help me. I don't use drugs. And even if I did, I doubt I would see the same thing as Fourier. The really awesome professors will really try to make you understand. First you'll get the definition. Then they'll show you how to get there. I've seen the derivation a few times, and at one point I think I actually understood all the steps.
But being able to follow the steps doesn't give you any intuition about what it all means. The good professors will keep going. They'll explain the time-frequency concepts and give relevant examples that help elucidate what all the crazy math actually means. They'll cover every little step in detail. If there's any hand-waving, it will be done so that you don't even notice.
It wasn't until my senior year of college that the definition of the DFT (and the continuous form) started to make sense to me. Something a professor said in my image compression class finally clicked. It was this, along with my understanding of correlation and orthogonal functions that finally lifted the fog (a little).
Take a look at the definition. Forget about the complex part and treat the exp as a sin or cos. (I'm hand-waving over 90% of the details because I don't understand that well, but you should see the basic idea.) For each summation, you compute one frequency bin. This calculation is essentially a correlation of the input signal and the frequency you're calculating. Notice, one index in the exp traverses the input signal. That's the correlation. The other index determines the frequency you're comparing the signal against. There's also a normalizing factor for the discrete version; ignore that too.
Basically, for each frequency bin, you're comparing the signal against that frequency. That's all the summation is doing. It's just the correlation at zero lag. If there is a strong signal component at that frquency, you'll get a large number in that frequency bin. If there's no signal power, the function is uncorrelated with a sinusoid at that frequency. You don't have to worry about the sinusoid correlating with other frequencies in the signal. One of the reasons sinusoids are chosen as the basis set is because they're orthogonal at integer multiples of the lowest frequency. So the frequency of interest can only correlate with the signal if there is signal power at that frequency. That's it. One way to think about it is that you're correlating the signal at each frequency of interest (at orthogonal frequencies).
It makes so much more sense when you can look at the definition from more than one angle. Another way to boost understanding is by increasing the level of abstraction or complexity. It's similar to how professors give difficult homework problems but easy test problems. The assumption is that if you were able to solve the hard problems, the test problems should be easy. If you've learned the quadratic formula, completing the square, and Newton's Method of root finding, solving x^2+2x+1 the way they taught you in 7th grade should be obvious by now.
If you want to understand more about the DFT, learn some of the variants. There are about a hundred of them, and they're pretty much all related. But there are also generalizations. Take the Fractional DFT, for example. It turns out you can actually take a non-integer transform. What the heck is that? I'll explain in a second.
Try this: take the DFT of a real-valued sequence. Then take it again.* You should get the original sequence time reversed. Hm, that's kind of neat. Ok, now take it two more times (that's four DFTs). Wow, you're right back to where you started. I guess that should be obvious. Now start from the beginning and take the DFT once. Take the conjugate and then take the DFT again. Look at that. You're right back where you started but with only two DFTs this time. This is a trick you can use to get the Inverse DFT using just a conjugate and the DFT. They'll tell you it's because of conjugate symmetry. But there's another reason.
You can actually think of taking the DFT as moving through time-frequency space. If you take it once, you go from the time domain to the frequency domain. If you then take the inverse you move back to the time domain. If instead, you take the DFT again, you end up back on the time domain, but you're on the other side; so, the axis is reversed. The little figure below demonstrates. T_ and F_ are time and frequency reversed respectively.
F | <-^ | | (T -> F) T_------|------T | | F_
So now you see why four DFTs gets you back to where you started. What about the conjugate method? Because of conjugate symmetry, taking the conjugate effectively flips the axis. Thus it's like taking the DFT twice. Try this out in matlab. Move back and forth between the different domains with FFT and IFFT.
Think of taking the DFT twice as DFT^2. The ^2 means applying the operator twice and has nothing to do with squaring the signal. Then x[n] = DFT^4 x[n]. What about the space in between? You can actually take DFT^.3. The result is something that is between time and frequency. There are some interesting applications of moving to this more generalized transform. But it's not just a math quirk. When you optically transform an image, the signal passes through all the fractional transforms on its way from the time domain to the frequency domain.
But I've gotten off track. The point is that once you spend some time understanding the generalized DFT, the one that everyone else is using makes more sense. If you were really to study it in depth, you would probably have a much deeper understanding. Unfortunately, the level of abstraction and complexity is a good one to two orders of magnitude higher. I read the first few chapters of The Fractional Fourier Transform. It's not easy stuff. But look at the insight you've gained from this brief discussion!
*The DFT isn't quite a unitary transform; so, you have to divide the IDFT by the length of the input to get back to the original. So treat every other DFT as the IDFT and divide by another N for each additional two. Also, remember that the negative frequency bins come after the positive ones. And for reasons I don't fully understand, the DC component is not a part of the axis flipping. If you try this, ignore the first element and look at the rest for the reversal.