A few overview questions about FFT. Relationships between input, buffer size, and expected output?

Question

I have a FFT set up and I've found that it responds very differently based on the buffer size and input. The output varies wildly, with magnitude of the peak frequency ranging from 500 to <1.

I have two tests in particular I'm running, the first being a sine wave at amplitude 1.0 with an LFO sweeping it's frequency at 0.1Hz. I've found that smaller buffers are more accurate in their ability to track the frequency. The magnitude of the frequency playing reaches 300 pretty easily.

The next test is with a mic, and it responds incredibly differently. If I whistle or hum, it will pick up that frequency. If I play music, rather than getting an increase in magnitudes of all frequencies, everything becomes weaker, including any whistling or humming. Changing the buffer size to be smaller helps, but at most iterations of the FFT on a buffer it results in very tiny magnitudes across the board.

In general, the things I'm finding are that smaller buffers are more likely to get a very strong magnitude of a frequency, but the more frequencies there are with a (magnitude > 1), the less likely that any of them are very strong (> 200). But none of this really means anything if it's a product of a broken implementation, and I can't seem to find any specific numeric information on what my expectations should be.

So, given an audio input and an output of frequencies and magnitudes (calculated by $\sqrt{\frac{\text{real}^2 + \text{imag}^2}{\frac{\text{numSamples}}{2}}}$), my questions are as follows:

1. Given a sole, unmoving sine wave, with a relatively small buffer size what kind of magnitudes are to be expected on what frequencies?

2. Given a sole, unmoving sine wave, with a relatively large buffer size what kind of magnitudes are to be expected on what frequencies?

3. Given a sole, sweeping frequency sine wave, with a relatively small buffer size what kind of magnitudes are to be expected on what frequencies?

4. Given a sole, sweeping frequency sine wave, with a relatively large buffer size what kind of magnitudes are to be expected on what frequencies?

5. Given static, with a relatively small buffer, what kind of magnitudes are to be expected?

6. Given static, with a relatively large buffer, what kind of magnitudes are to be expected?

I know this is a lot of questions, but the results I'm getting from these tests are not what I was expecting, and I want to start by making sure it's not my expectations that were wrong.

Answer 1

So, I'm not an expert at all, but I feel that I've perhaps learned enough to be somewhat helpful to anyone who ends up here with little introduction to FFT. If any of this is wrong, see the "P.S" at the bottom. tl;dr I'd rather share than leave this question unanswered entirely.

A few things are important to note: Parseval's theorem, bin frequencies, and normalization:

Parseval's theorem is something to the ends that the energy of the signal will be the energy of FFT bins (bins being each frequency/magnitude pair). So, what this means in practice (I think) is that a small buffer size will make the bin magnitude of a sine wave frequency vary based off the amount of high amplitude wave is in the buffer. It won't be consistent.

Bin Frequencies are the frequency of each bin from your FFT. They aren't going to be the exact numbers you are looking for without a bit of work, but they should be close enough to give you some info.

Normalization is something that helps with all this. For my project, I found the max bin magnitude, averaged it with the max from the last cycle, and then divided everything by it. The result is that my visualization is showing something much more like what my ears feel they would see.

Now for the Q's (assuming all signals are normalized to -1 < x < 1):

1. Magnitudes of ~200 +-100, in the bins near the signal wave freq

2. Same, but magnitudes of ~300 +-50. The larger buffer limits cases of more or less high amplitude chunking (sin(x) at [0 to Pi/4] vs [Pi/4 to Pi/2])

3. Magnitudes of ~150 +-100, in the bins near the signal wave freq. The sweeping spreads out the bins a little, but the small buffer size (relative to sweep speed) helps.

4. Magnitudes of ~50 +-100, in the bins touched by the signal wave freq. If the sweep is larger or faster expect the magnitudes to be even more spread out. Note: the more bins touched, the lower the magnitudes overall.

5. Magnitudes of ~5 +-5, in all bins which happen to be strongly represented in this small buffer chunk

6. Magnitudes of ~1 +-1 in all bins.

So, it seems there is probably some form of log function which would help make the perceived levels of all these sounds not so distant when graphed by magnitude, but, overall, using normalization has worked in my project to pick out the most noteworthy note tones. (btw notes and tones are anagrams)

P.S. If any of this is wrong, please do make an answer covering and correcting these topics. I will not be marking this as the accepted answer for at least a month because I'm not experienced enough to be an authority on this subject. That being said, this is what I think may be true based off the results I've been getting, and it's better than nothing for a newcommer.