What is FFT Frequency Resolution?

[Spimon]

Hi everyone,

Please excuse the somewhat basic question. I'm a mechanical engineering student a little out of my realm!
I'm doing some work with signal processing and have been given some data (a series of vibration measurements, out of interest).

This includes the following:

Note the following:
1. Recording Time Length (s): 5 seconds
2. FFT Frequency Resolution (1/s): 0.2 s
3. Sampling Rate (Hz): 100 Hz
4. Max frequency output from the FFT (Hz) (0.5*Sampling Rate): 50 Hz

I'm a little confused about the difference between #2 and #4 mean.
My understanding was that the max frequency output from the Fast Fourier Transform is a resolution and determines the smallest change in frequency that can be detected.

I'm quite unsure what FFT Frequency Resolution is...

Would someone kindly explain these terms in very simple words (mathematical derivations not required)?
Thanks :)

 

[Floid]

The FFT divides the signal up by frequency, but it does so in a discrete manner. So you can think of the output as a bar chart where every bar represents the signal level of some frequency range. We call that range (or width of the bar or bin) the frequency resolution.

The max frequency output is determined by the sample rate. According to the Nyquist rate, the max frequency you can detect by sampling a signal is your sample rate/2.

So from the situation described above you would have a bar chart from 0 to 50Hz, where every bar in the chart was 5Hz wide.

 

[Spimon]

Ah, makes a little more sense! Thanks so much :D

 

[AlephZero]

Somebody made a typo in #2. It should be 0.2 Hz, not 0.2 s. If the "somebody" wasn't you, the wrong units might be part of the confusion.

You measured 100 data points/second for 5 seconds, i.e. 500 data pionts in total.

The FFT analysis assumes those 500 points repeat continuously (even though that is very unlikely to be true in real life). Therefore the lowest non-zero frequency you can get from the FFT is one cycle of a sine or cosine wave that takes the whole of the sample time. The sample time was 5 seconds, so that frequency is 1/5 = 0.2 Hz.
All the other frequencies in the FFT are multiples of the lowest frequency, i.e. 0, 0.2, 0.4, 0.6. 0.8, 1.0 Hz, etc. That's what #2 means.

The highest frequency you can measure from the sampled data depends on the sampling rate. The highest frequency signal consists of alteranating data ponits of of +a, -a, +a -a, etc for some amplitude a. The period of the sine/cosine wave going through those points is 2 sample intervals. Your sampling rate was 100 Hz, so the highest frequency you can measure (#4) is 100/2 = 50 Hz.

It's not a coincidence that you have 250 different frequencies of 0, 0.2, 0.4, ... 49.8, 50 Hz, each with a sine and cosine component (which you can describe by an amplitude and phase), so your FFT contains 500 items of data, whch is the same number as the number of points you measured. (OK, I've skipped over the fact that there are actually 251 frequecies in the list above, but there are no "sine" components for 0 and 50 Hz - if you want to get to the bottom of that, look up the math!). The FFT doesn't create any new information, or throw any information away - it just converts the 500 data values in the time domain into 500 data values in the frequency domain.

 

[Spimon]

Thanks once again. It's much clearer now. There was a typo (s instead of Hz. Maybe it was meant to be s^-1) in the notes I was given, but that was the least of my problems! All fixed up now.

I also found quite a good video to show how it all comes together (well some of it).
Being a mechanical engineer, I'm more interested (well examined on) the interpretation of the FFT and what is all means, as opposed to the mathematics of performing the transform.

For others' reference, here's the video.