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The unfortunate use and confusion of the term Frequency Resolution

OWC

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Very recently I bumped into another conversation/discussion again around the term "frequency resolution".

This term seems to be quite confusing, or mostly the usage of it can be.
I am not going into full detail about everything here, because that means I basically have to write an entire book.
So I will skip though this very quickly and assume certain knowledge is already available, but I will provide enough links to articles to read further if people would like to. (If you have additional articles to share, please feel free)

A good read in general is "Digital Signal Filtering, Analysis and Restoration" (but there are many other books as well)
I will stick to hyperlinks, since they are a bit easier to refer to compared to pages in books and such.



For everyone who has a background in electronics and/or data acquisition, in general this term is being used to calculated the frequency steps (called bins) for digital samples in a frequency response.
This frequency resolution can be calculated by dividing the sample rate by the FFT length.
So for example, a sample rate of 48kHz with a FFT length of 4k gives you 48k /4000 = 12Hz resolution.

See here for some more examples and explanations;

What this also means, is that if a certain frequency falls in between two of those two bins, the energy of this frequency is divided between the two.
This could potentially result in a high Q peak that will show up as a wider and lower peak (= lower Q)
Luckily these days there is plenty of CPU power, so very large FFT lengths are possible (depending on the test signal sequence length)
Resolutions of something like 0.5-2Hz are easily achievable or better when the sample rate can be reduced as well.
(tip when area of interest has limited bandwidth and low frequency ;) )

So where does the confusion starts?
This is were we need to introduce gated measurements or time windows.
The reason we would like to use these, is to get rid of any kind of reflections in our measurements.
This is possible because reflections take longer to reach the microphone compared to the direct signal coming from the sound source.
By applying a "gate" or "window" these reflections will be filtered out (this is an extremely over simplified explanation)
The downside of this, is that it also means that we only have a limited bandwidth, since lower frequencies also will be cut off.
(the waves are longer than the window itself)

Since it's not important for this topic, I am not going into the technical details of those time window functions and all, that would literally require many many pages of explanation as well as some high level math.

So I would highly suggest reading along here;

Anyway, this brings us to the confusion part.

For in the field of audio and loudspeakers, I personally think this all started* with a chapter in @Floyd Toole 's book Sound Reproduction (must read if you haven't! ).

Toole_189.png
Chapter 18, page 382 (fig 18.9) to be more exact. (see graph).
(*although this has been already known for many years by people who are involved in data acquisition and such, bud that's a different story)

In this chapter he describes the issue that can happen when window functions are applied with gated frequency measurements.

"From the listening experience one could deduce that a very frequency-specific,
high-Q resonance was the problem. A quick pitch match using an oscillator
revealed that the problem was around 280 Hz. The time gating used for the
measurement was 10 ms, chosen to eliminate room reflections from the data.
This window yields data with 100 Hz frequency resolution (resolution = 1/
window duration), so it was clear that any high-Q events around 280 Hz would
simply not be visible."


So in short, there is some "smoothing" effect going on when windows are being used. (see references below)
When windows functions are being used, the resolution of the frequency will be limited by the length of the window.

An extremely good read about this;
"On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform"

A more practical a visual explanation;

To sum up
So we eventually end up with two "definitions" for "frequency resolution".
1: df = samplerate / FFT length
2: df = 1 / window size

Through the years I have seen some people using "bin resolution" instead for the first one, but not very often.
Just in general and at most universities (that I am aware of) they will just call (1) frequency resolution and is also often the only one people are aware of.
We can basically use (1) when very long window sizes are being used (which some programs use)



Why this is important?
Well hopefully it is clear by now, that it depends what kind of measurements we are looking at and talking about.
We have to be VERY careful with what we actually mean and what is being done.

- When the measurements are not gated, the frequency resolution is determined by (1)
Which means that the resolution can be actually very high (as Floyd Toole shows us as well in fig 18.9).
Think about close range and near field measurements and such as well as in-room measurements (room modes) or true anechoic measurements.


- When we talk about gated measurements, the resolution is determined by (2), meaning that the resolution is dependent on its window size.
Typically done as an alternative for anechoic measurements
(often combined with the stitching method, were (1) can be used again for the close range measurements)

Hope this cleared some dust about the subject!
Cheers!



Some side notes;

- So a solution that wasn't mentioned in Floyd's book and I was very surprised about, is to just perform one (or multiple) close range measurement(s) with no time window at all for example.
Which in the field of general acoustics is actually standard practice. (But there are many other ways.)
Those high-Q resonances will perfectly show up in those graphs.
In general it's good advice to use more than just far-field anechoic measurements as well as cross-correlate those with some theoretical expected values
(when possible)



- Very often I see people using the rectangular type window.
As can be seen in the reference mentioned earlier, rectangular windows are only suitable for two types of signals.
1 - continues signals, 2 - (acoustics) transients

In our case measuring an loudspeaker is neither of those and in fact rectangular windows don't perform so well.
The first one is quite obvious, since we don't have a continues signal.
But since we have to deal with reflections, which have quite some energy left (relatively speaking), the signal also definitely isn't a transient.
Therefor one can (and will) see quite some bleed through from these reflections.

A much better type is a Hann (ideally Hann 25%) or something like a Cosine (which is the same as Hann).
Those are available in ARTA as well as VituixCAD and will give a better result. (Also recommended by VituixCAD)
 
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